chapter 1 fluid properties. the engineering science of fluid mechanics has been developed through an...
TRANSCRIPT
Chapter 1Chapter 1
FLUID PROPERTIESFLUID PROPERTIES
The engineering science of fluid mechanics has been developed through an understanding of fluid properties the application of the basic laws of mechanics and thermodynamics and orderly experimentation
The properties of density and viscosity play principal roles in open- and closed-channel flow and in flow around immersed objects (stream river sewer line pipe line groundwater sea water overland flowhellip)
Surface-tension effects are important in the formation of droplets in the flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occur as well as in the formation of capillary waves (phase water air solid phase)
The property of vapor pressure which accounts for changes of phase from liquid to gas becomes important when reduced pressures are encountered
We will use International System of Units (SI) of force mass length and time units
11 DEFINITION OF A FLUID11 DEFINITION OF A FLUID
A fluid is a substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be
Shear stress = Tangential (Shear )Force Area
Normal Stress (pressure) = Normal Force Area
Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point
In figure a substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid
Figure 11 Deformation resulting from application of Figure 11 Deformation resulting from application of constant shear forceconstant shear force
The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)
In the figure 11 the fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate
Experiments show that other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form
Where μ is the proportionality factor and includes the effect of the particular fluid
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The engineering science of fluid mechanics has been developed through an understanding of fluid properties the application of the basic laws of mechanics and thermodynamics and orderly experimentation
The properties of density and viscosity play principal roles in open- and closed-channel flow and in flow around immersed objects (stream river sewer line pipe line groundwater sea water overland flowhellip)
Surface-tension effects are important in the formation of droplets in the flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occur as well as in the formation of capillary waves (phase water air solid phase)
The property of vapor pressure which accounts for changes of phase from liquid to gas becomes important when reduced pressures are encountered
We will use International System of Units (SI) of force mass length and time units
11 DEFINITION OF A FLUID11 DEFINITION OF A FLUID
A fluid is a substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be
Shear stress = Tangential (Shear )Force Area
Normal Stress (pressure) = Normal Force Area
Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point
In figure a substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid
Figure 11 Deformation resulting from application of Figure 11 Deformation resulting from application of constant shear forceconstant shear force
The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)
In the figure 11 the fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate
Experiments show that other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form
Where μ is the proportionality factor and includes the effect of the particular fluid
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
11 DEFINITION OF A FLUID11 DEFINITION OF A FLUID
A fluid is a substance that deforms continuously when subjected to a shear stress no matter how small that shear stress may be
Shear stress = Tangential (Shear )Force Area
Normal Stress (pressure) = Normal Force Area
Shear force is the force component tangent to a surface and this force divided by the area of the surface is the average shear stress over the area Shear stress at a point is the limiting value of shear force to area as the area is reduced to the point
In figure a substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid
Figure 11 Deformation resulting from application of Figure 11 Deformation resulting from application of constant shear forceconstant shear force
The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)
In the figure 11 the fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate
Experiments show that other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form
Where μ is the proportionality factor and includes the effect of the particular fluid
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
In figure a substance is placed between two closely spaced parallel plates so large that conditions at their edges may be neglected The lower plate is fixed and a force F is applied to the upper plate which exerts a shear stress FA on any substance between the plates A is the area of the upper plate When the force F causes the upper plate to move with a steady (nonzero) velocity no matter how small the magnitude of F one may conclude that the substance between the two plates is a fluid
Figure 11 Deformation resulting from application of Figure 11 Deformation resulting from application of constant shear forceconstant shear force
The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)
In the figure 11 the fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate
Experiments show that other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form
Where μ is the proportionality factor and includes the effect of the particular fluid
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The fluid in immediate contact with a solid boundary has the same velocity as the boundary (no slip at the boundary)
In the figure 11 the fluid in the area abcd flows to the new position abcd each fluid particle moving parallel to the plate and the velocity u varying uniformly from zero at the stationary plate to U at the upper plate
Experiments show that other quantities being held constant F is directly proportional to A and to U and is inversely proportional to thickness t In equation form
Where μ is the proportionality factor and includes the effect of the particular fluid
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
If τ = FA for the shear stress
The ratio Ut is angular velocity of line ab or it is the rate of angular deformation of the fluid (rate of decrease of angle bad)
The angular velocity may also be written dudy ndash is more general
The velocity gradient dudy may also be visualized as the rate at which one layer moves relative to an adjacent layer in differential form
- Newtons law of viscosity (111)
is the relation between shear stress and rate of angular deformation for one-dimensional flow of a fluid
The proportionality factor μ is called viscosity of the fluid
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Materials other than fluids cannot satisfy the definition of a
fluid
A plastic substance will deform a certain amount proportional to
the force but not continuously when the stress applied is below
its yield shear stress
A complete vacuum between the plates would cause deformation
at an ever-increasing rate
If sand were placed between the two plates Coulomb friction
would require a finite force to cause a continuous motion Hence
plastics and solids are excluded from the classification of fluids
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Fluids may be classified as Newtonian non-Newtonian
In Newtonian fluid there is a linear relation between the magnitude of applied shear stress and the resulting rate of deformation [μ constant in Eq (111)] as shown in Fig 12
In non-Newtonian fluid there is nonlinear relation between the magnitude of applied shear stress and the rate of angular deformation
An ideal plastic has a definite yield stress and a constant linear relation of τ to dudy
A thixotropic substance such as printers ink has a viscosity that is dependent upon the immediately prior angular deformation of the substance and has a tendency to take a set when at rest
Gases and thin liquids tend to be Newtonian fluids while thick long-chained hydrocarbons may be non-Newtonian
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Figure 12 Rheological diagramFigure 12 Rheological diagram
Rate
of d
efor
mat
ion
dud
y
Idea
l flu
id
New
toni
an
flui
dNon
- N
ewto
nian
fluid Id
eal
plas
tic
Shear stress τ
Thixo
tropic
subs
tance
Yield stress
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
For purposes of analysis the assumption is frequently made that a
fluid is nonviscous
With zero viscosity the shear stress is always zero regardless of th
e motion of the fluid
If the fluid is also considered to be incompressible it is then called
an ideal fluid and plots as the ordinate in Fig 12
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
12 FORCE MASS LENGTH AND TIME 12 FORCE MASS LENGTH AND TIME UNITSUNITS
Consistent units of force mass length and time greatly simplify problem solutions in mechanics derivations may be carried out without reference to any particu
lar consistent system A system of mechanics units is said to be consistent when unit force
causes unit mass to undergo unit acceleration
The International System (SI) newton (N) as unit of force kilogram (kg) as unit of mass metre (m) as unit of length the second (s) as unit of time
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
With the kilogram metre and second as defined units the newton is derived to exactly satisfy Newtons second law of motion
(121)
The force exerted on a body by gravitation is called the force of gravity or the gravity force The mass m of a body does not change with location the force of gravity of a body is determined by the product of the mass and the local acceleration of gravity g
(122) For example where g = 9876 ms2 a body with gravity force of 10 N
has a mass m = 109806 kg At the location where g = 97 ms2 the force of gravity is
Standard gravity is 9806 ms2 Fluid properties are often quoted at standard conditions of 4oC and 760 mm Hg
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Table 11 Selected prefixes for powers of 10 in SI unitsTable 11 Selected prefixes for powers of 10 in SI units
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
13 VISCOSITY13 VISCOSITY
Viscosity requires the greatest consideration in the study of fluid flow
Viscosity is that property of a fluid by virtue of which it offers resistance to shear
Newtons law of viscosity [Eq (111)] states that for a given rate of angular deformation of fluid the shear stress is directly proportional to the viscosity
Molasses and tar are examples of highly viscous liquids water and air have very small viscosities
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The viscosity of a gas increases with temperature but the viscosity of a liquid decreases with temperature ndash it can be explained by examining the causes of viscosity
The resistance of a fluid to shear depends upon its cohesion and upon its rate of transfer of molecular momentum
A liquid with molecules much more closely spaced than a gas has cohesive forces much larger than a gas Cohesion appears to be the predominant cause of viscosity in a liquid and since cohesion decreases with temperature the viscosity does likewise
A gas on the other hand has very small cohesive forces Most of its resistance to shear stress is the result of the transfer of molecular momentum
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
As a rough model of the way in which momentum transfer gives rise to an apparent shear stress considering two idealized railroad cars loaded with sponges and on parallel tracks (see Fig 13)
Assume each car has a water tank and pump so arranged that the water is directed by nozzles at right angles to the track First consider A stationary and B moving to the right with the water from its nozzles striking A and being absorbed by the sponges
Car A will be set in motion owing to the component of the momentum
of the jets which is parallel to the tracks giving rise to an apparent shear stress between A and B Now if A is pumping water back into B at the same rate its action tends to slow down B and equal and opposite apparent shear forces result
When both A and B are stationary or have the same velocity the pumping does not exert an apparent shear stress on either car
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Figure 13Figure 13
Model illustrating Model illustrating transfer of momentumtransfer of momentum
Within fluid there is always a transfer of molecules back and forth across any fictitious surface drawn in it When one layer moves relative to an adjacent layer the molecular transfer of momentum brings momentum from one side to the other so that an apparent shear stress is set up that resists the relative motion and tends to equalize the velocities of adjacent layers in a manner analogous to that of Fig 13
The measure of the motion of one layer relative to an adjacent layer is dudy
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Molecular activity gives rise to an apparent shear stress in gases which is more important than the cohesive forces and since molecular activity increases with temperature the viscosity of a gas also increases with temperature
For ordinary pressures viscosity is independent of pressure and depends upon temperature only For very great pressures gases and most liquids have shown erratic variations of viscosity with pressure
A fluid at rest or in motion so that no layer moves relative to an adjacent layer will not have apparent shear forces set up regardless of the viscosity because dudy is zero throughout the fluid
Hence in the study of fluid statistics no shear forces considered and the only stresses remaining are normal stresses or pressures greatly simplifies the study of fluid statics since any free body of fluid can have only gravity forces and normal surface forces acting on it
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The dimensions of viscosity are determined from Newtons law of viscosity [Eq (111)] Solving for the viscosity μ
and inserting dimensions F L T for force length and time
shows that μ has the dimensions FL-2T
With the force dimension expressed in terms of mass by use of Newtons second law of motion F = MLT-2 the dimensions of viscosity may be expressed as ML-1T-1
The SI unit of viscosity which is the pascal second (symbol Pa middot s) or kilograms per meter-second (kgm middot s) has no name
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Kinematic ViscosityKinematic Viscosity
μ - absolute viscosity or the dynamic viscosity ν - kinematic viscosity (the ratio of viscosity to mass density)
(131)
ν occurs in many applications (eg in the dimensionless Reynolds number for motion of a body through a fluid Vlν in which V is the body velocity and l is a representative linear measure or the body size)
The dimensions of ν are L2T-1 The SI unit of kinematic viscosity is 1 m2s and has no name
Viscosity is practically independent of pressure and depends upon temperature only The kinematic viscosity of liquids and of gases at a given pressure is substantially a function of temperature
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Example 11Example 11
A liquid has a viscosity or 0005 Pa middot s and a density of 850 kgm3 Calculate the kinematic viscosity
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Example 12Example 12In Fig 14 the rod slides inside a concentric sleeve with a reciprocating motion due to the uniform motion of the crank The clearance is δ and the viscosity μ Write a program in BASIC to determine the average energy loss per unit time in the sleeve D = 08 in L = 80 in δ = 0001 in R = 2 ft r = 05 ft μ = 00001 lb sft2 and the rotation speed is 1200 rpm
SolutionSolution The energy loss in the sleeve in one rotation is the product of resisting viscous (shear) force times displacement integrated over the period of the motion The period T is 2πω where ω = dθdt The sleeve force depends upon the velocity The force Fi and position xi are found for 2n equal increments of the period Then by the trapezoidal rule the work done over the half period is found
Using the law of sines to eliminate φ we get
Figure 15 lists the program in which the variable RR represents the crank radius r
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Figure 14 Notation for sleeve motionFigure 14 Notation for sleeve motion
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Figure 15 BASIC program to determine loss in sleeve motionFigure 15 BASIC program to determine loss in sleeve motion
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
14 CONTINUUM14 CONTINUUM
In dealing with fluid-flow relations on a mathematical or analytical basis consider that the actual molecular structure is replaced by a hypothetical continuous medium - continuum
For example velocity at a point in space is indefinite in a molecular medium as it would be zero at all times except when a molecule occupied this exact point and then it would be the velocity of the molecule and not the mean mass velocity of the particles in the neighborhood
This is avoided if consider velocity at a point to be the average or mass velocity of all molecules surrounding the point With n molecules per cubic centimetre the mean distance between molecules is of the order n-13 cm
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Molecular theory however must be used to calculate fluid properties (eg viscosity) which are associated with molecular motions but continuum equations can be employed with the results of molecular calculations
In rarefied gases (the atmosphere at 80 km above sea level) the ratio of the mean free path [the mean free path is the average distance a molecule travels between collisions] of the gas to a characteristic length for a body or conduit is used to distinguish the type of flow
The flow regime is called gas dynamics for very small values of the ratio the next regime is called slip flow and for large values of the ratio it is free molecular flow
In this text only the gas-dynamics regime is studied
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
15 DENSITY SPECIFIC VOLUME UNIT GRAVITY 15 DENSITY SPECIFIC VOLUME UNIT GRAVITY
FORCE RELATIVE DENSITY PRESSUREFORCE RELATIVE DENSITY PRESSURE
The density ρ of a fluid is defined as its mass per unit volume To define density at a point the mass Δm of fluid in a small volume ΔV surrounding the point is divided by ΔV and the limit is taken as ΔV becomes a value ε3 in which ε is still large compared with the mean distance between molecules
(151)
For water at standard pressure (760 mm Hg) and 4oC ρ = 1000 kgm3
(152)
The specific volume vs is the reciprocal of the density ρ that is it is the volume occupied by unit mass of fluid
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The unit gravity force γ - the force of gravity per unit volume It changes with location depending upon gravity
(153) Water γ = 9806 Nm3 at 5oC at sea level
The relative density S of a substance - the ratio of its mass to the mass of an equal volume of water at standard conditions
The average pressure - the normal force pushing against a plane area divided by the area
The pressure at a point is the ratio of normal force to area as the area approaches a small value enclosing the point
If a fluid exerts a pressure against the walls or a container the container will exert a reaction on the fluid which will be compressive
Pressure has the units force per area which is newtons per square metre called pascals (Pa)
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
16 PERFECT GAS16 PERFECT GAS
In this treatment thermodynamic relations and compressible-fluid-flow cases have been limited generally to perfect gases
The perfect gas is defined as substance that satisfies the perfect-gas-law
(161)
and that has constant specific heats
p is the absolute pressure vs is the specific volume R is the gas constant and T is the absolute temperature
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The perfect gas must be carefully distinguished from the ideal fluid An ideal fluid frictionless and incompressible The perfect gas has viscosity and can therefore develop shear stresses and it is compressible according to Eq (161)
Eq (161) is the equation of state for a perfect gas may be written
(162)
The units of R can be determined from the equation when the other units are known
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Real gases below critical pressure and above the critical temperature tend to obey the perfect-gas law As the pressure increases the discrepancy increases and becomes serious near the critical point
The perfect-gas law encompasses both Charles law and Boyles law
Charles law states that for constant pressure the volume of a given mass of gas varies as its absolute temperature
Boyles law (isothermal law) states for constant temperature the density varies directly as the absolute pressure
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The volume V of m mass units of gas is mvs
(163)
With being the volume per mole the perfect-gas law becomes
(164)
If n is the number of moles of the gas in volume V
(165)
The product MR called the universal gas constant has a value depe
nding only upon the units employed It is
(166)
The gas constant R can then be determined from
(167)
knowledge of relative molecular mass leads to the value of R
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The specific heat cv of a gas is the number of units of heat added per unit mass to raise the temperature of the gas one degree when the volume is held constant
The specific heat cp is the number of heat units added per unit mass to raise the temperature one degree when the pressure is held constant
The specific heat ratio k = cpcv The intrinsic energy u (dependent upon P ρ and T) is the energy pe
r unit mass due to molecular spacing and forces
The enthalpy h is important property of a gas given by h = u + Pρ cv and cp have the units joule per kilogram per kelvin (JkgK) 418
7 J of heat added raises the temperature of one kilogram of water one degree Celsius at standard conditions
R is related to cv and cp by
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Example 13Example 13
A gas with relative molecular mass of 44 is at a pressure of 09 MPa and a temperature of 20oC Determine its density
SolutionSolution
From Eq (167)
Then from Eq (162)
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
17 BULK MODULUS OF ELASTICITY17 BULK MODULUS OF ELASTICITY
For most purposes a liquid may be considered as incompressible but for situations involving either sudden or great changes in pressure its compressibility becomes important Liquid (and gas) compressibility also becomes important when temperature changes are involved eg free convection
The compressibility of a liquid is expressed by its bulk modulus of elasticity
If the pressure of a unit volume of liquid is increased by dp it will cause a volume decrease -dV the ratio -dpdV is the bulk modulus of elasticity K
For any volume V of liquid
K expressed in units of p For water at 20oC K = 22 GPa
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Example 14Example 14
A liquid compressed in a cylinder has a volume of 1 liter (L = 1000 cm3) at 1 MNm2 and volume of 995 cm3 at 2 MNm2 What is its bulk modulus of elasticity
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
18 VAPOR PRESSURE18 VAPOR PRESSURE Liquids evaporate because or molecules escaping from the liquid surfa
ce The vapor molecules exert partial pressure in the space known as vapor pressure
If the space above the liquid is confined after a sufficient time the number of vapor molecules striking the liquid surface and condensing is just equal to the number escaping in any interval of time and equilibrium exists
The vapor pressure of a given fluid depends upon temperature and increases with it When the pressure above a liquid equals the vapor pressure of the liquid boiling occurs
At 20oC water vapor pressure - 2447 kPa mercury vapor pressure - 0173 Pa
When very low pressures are produced at certain locations in the system pressures may be equal to or less than the vapor pressure the liquid flashes into vapor This is the phenomenon of cavitation
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
19 SURFACE TENSION19 SURFACE TENSION
CapillarityCapillarity
At the interface between a liquid and a gas or two immiscible liquids a film or special layer seems to form on the liquid apparently owing to attraction of liquid molecules below the surface
The formation or this film may be visualized on the basis of surface energy or work per unit area required to bring the molecules to the surface The surface tension is then the stretching force required to form the film obtained by dividing the surface-energy term by unit length of the film in equilibrium
The surface tension of water varies from about 0074 Nm at 20oC to 0059 Nm at 100oC (Table 12)
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Table 12 Approximate properties of common liquids at Table 12 Approximate properties of common liquids at 2020ooC and standard atmospheric pressure C and standard atmospheric pressure
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
The action of surface tension is to increase the pressure within a droplet of liquid or within a small liquid jet
For a small spherical droplet of radius r the internal pressure p necessary to balance the tensile force due to the surface tension σ calculated in terms of the forces which act on a hemispherical free body
For the cylindrical liquid jet of radius r the pipe-tension equation applies
Both equations show that the pressure becomes large for a very small radius of droplet or cylinder
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Capillary attraction is caused by surface tension and by the relative value of adhesion between liquid and solid to cohesion of the liquid
A liquid that wets the solid has a greater adhesion than cohesion The action of surface tension in this case is to causes the liquid to rise within a small vertical tube that is partially immersed in it
For liquids that do not wet the solid surface tension tends to depress the meniscus in a small vertical tube When the contact angle between liquid and solid is known the capillary rise can be computed for an assumed shape of the meniscus
Figure 14 shows the capillary rise for water and mercury in circular glass tubes in air
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes
Figure 15 Capillarity in circular glass tubesFigure 15 Capillarity in circular glass tubes