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MOMENTUM TRASNFERINTRODUCTIONTRANSPORT PHENOMENABYBIRD, STEWART AND LIGHTFOOTINTRODUCTIONQ.What are Transport Phenomena ?Ans.A combination of three closely related topicsQ.Why these transfer processes be studied together ?The basic molecular mechanism of the three transfer processes is very similar. The same molecules transfer momentum, energy and mass, through viscosity, thermal conductivity and diffusivityThey very frequently occur simultaneously in natureThe basic equations describing these transfer processes are very closely relatedThe close similarity of these equations lead to analogiesMathematics required to solve the three transfer processes equations is also very similar

Momentum Transfer/TransportEnergy Transfer/TransportChemical Species Mass Transfer/TransportFluid DynamicsHeat TransferMass TransferINTRODUCTION, contd. - 2THREE LEVELS OF STUDY OF TRANSPORT PHENOMENAMacroscopic LevelMicroscopic LevelMolecular LevelMacroscopic LevelNo attempt to understand the details of what is going on within the control volumeMainly used for the global assessment of the problemIntegral AnalysisA Macroscopic Balance ofMassMomentumEnergyDue to various inputs & outputs from our control volumeINTRODUCTION, contd. - 3Microscopic LevelAn attempt to understand the details of what is going on within the control volumeMainly used to get information ofVelocity profilesTemperature profilesConcentration profilesA Microscopic Balance ofMassMomentumEnergyDue to various inputs & outputs from our control volumeDifferential AnalysisTo understand the process and optimize itINTRODUCTION, contd. - 4Molecular LevelTo seek the fundamental understanding of the process ofMass transferMomentum transferEnergy transferIn terms of molecular structure & intermolecular forcesA job primarily for Theoretical PhysicistsPhysical ChemistSome times Engineers/applied scientists do get involved in cases ofComplex moleculesExtreme temperatures/pressuresChemical Reacting FlowsINTRODUCTION, contd. - 5Each of these levels involve typical length scalesMolecular Level1 to 1000 nanometersMacroscopic LevelOrder of cm or mMicroscopic LevelMicron to cm rangeRequirements For Good Understanding Of This SubjectPhysical Interpretation of key mathematical resultsGet into the habit of relating physical ideas to equationsComparison of intuition and results obtainedMATHEMATICS, Differential Equations, Vectors, CalculusUnderstanding of dimensional analysisVISCOSITY, MOMENTUM TRANSFER MECHANISMCONCEPT OF VISCOSITYFriction is felt only when you move either slower or faster than the other spectators.Extent of friction depends on the type of clothes they are wearing.It is this type of clothes that gives rise to the concept of viscosity.

Cricket Stadium GateViscosity and Newtons Law of ViscosityExample of two parallel plates

Shear force acting on the second molecular layer of fluid is due to the difference in the velocities of the two adjacent layersTop layer stationary, Bottom layer moves with constant velocity VA fluid is filled between the platesNo slip condition between fluid and plates at both the plate surfacesViscosity & Newtons Law of Viscosity, contd. -2Common sense suggests the following.A constant force F is required to maintain the motion of lower plateThis force is directly proportional toArea of platesVelocity of lower plateThis force is inversely proportional toDistance between the platesMathematical Interpretation Of Common Sense

V/Y is the gradient or slope

The force applied, F is the shear forceYt < 0xyFluid initially at restxyt = 0VLower plate set in motionxysmall tVvx(y, t)Velocity buildup in unsteady flowxylarge tVvx(y)Final velocity distribution in steady flowNote, directions of V & yViscosity & Newtons Law of Viscosity, contd. -3The shear stress exerted in the x-direction on a fluid surface of constant y by the fluid in the region of lesser y is designated as yxFluid surface of constant y, Shear force on unit area perpendicular to the y-directionx-directionShear StressThe shear stress is moving in the direction of y because the bottom layer of fluid exerts a shear stress on the next layer which then exerts a shear stress on subsequent layerShear stress is induced by the motion of the plate. Shear stress can be induced by a pressure gradient or a gravity force.Pressure force is a force acting on a surface while the gravity force is the force acting on a fluid volumeViscosity and Newtons Law of Viscosity, contd. -4The shear stress is a function of Velocity gradientProperties of the fluidIf this functional dependence is linear: fluids are called Newtonian Fluids

Where, vx = fluid velocity in the x-direction = fluid viscosity, a property of the fluid, not the physical systemThe viscosity of Newtonian fluids is constantThe temperature dependence is between T0.6 and T. Some theories are available.Behaviour Of Gases At Moderate PressuresViscosity Is Independent Of PressureIncreases With TemperatureViscosity and Newtons Law of Viscosity, contd. -5Behaviour Of LiquidsViscosity Is Independent Of PressureDecreases With TemperatureUnits of viscosity is g/cm/sec (poise) or Pa-s.Magnitudes:Air @ 20C0.00018 g/cm/secLiquid water @ 20 C0.001 Pa-s, 0.01 g/cm/sNon-Newtonian FluidsFor non-Newtonian fluidsThe functional dependence between The shear stress and the velocity gradient is more complex. We can write in the most general format:

non-Newtoniannon-NewtonianNewtonianNon-Newtonian Fluids, contd. -2Mathematically one can write for Non-Newtonian Fluids in the form

Where, = Apparent Fluid Viscosity, a function of either yx / dvx/dy / bothApparent fluid viscosity is dependent on the current state of fluidIf apparent viscosity decreases with increasing rate of shear (-dvx/dy)The behaviour is termed pseudoplasticIf apparent viscosity increases with increasing rate of shear (-dvx/dy)The behaviour is termed dilatantEffects of Non-Newtonian Fluids, contd. -3Models of Non-Newtonian fluidsBingham model: applicable for fine suspensions and pastesBinghamOstwald-de Waele model: e.g. used for CMC in waterOstwaldEyring model: derived from the Eyring kinetic theoryEyringEllis model: CMC in waterEllisReiner-Philippoff modelReinerNon-Newtonian Fluids, contd. -4These model equations should not be used outside their range of validity.Graphical representation of two-parameter model

Things To RememberThese models are empirical, that is the parameters of the models are obtained by curve fittingThe parameters are function of T, P and compositionPrediction of Viscosity of Gases & Liquids

END OF CHAPTER ONE

SHELL BALANCESHELL BALANCEA mathematical expression showing the balance of rate of change of momentum and forces acting on the control volume.(NEWTONS SECOND LAW OF MOTION)Balance is made on a small shell of dimensions x, y, zAll quantities are written in terms of fluxesSolution gives velocity distribution leading to maximum velocity, average velocity, flow rates and stresses at surfaces etcThis procedure of analysis is called analysis through first principleGenerally can be applied to simple geometries and idealized flow situationsA combination of these simple analysis lead to complex geometries and flow systemsSimple system analysis help in understanding complex systemsSHELL BALANCE BY 1st PRINCIPLEPROCEDURE OF TRANSPORT PHENOMENA ANALYSIS

Draw a physical diagram.Identify all transport mechanismsSet a frame of coordinates and draw the direction of all transport processes identified in step 2.Draw a shell, whether it be one, two or three dimensional depending on the number of transport direction, such that its surfaces are perpendicular to the transport direction.Carry out the momentum shell balance as below:

This should give a first-order ODE in terms of shear stress

SHELL BALANCE BY 1st PRINCIPLE, contd. - 2Procedure Of Transport Phenomena Analysis, contd.If the fluid is Newtonian, apply the Newton law. However, if the fluid is non-Newtonian, apply any appropriate non-Newtonian law empirical equation.This should give a second order ODE in terms of velocity.Impose physical constraint on the boundary of the physical system.This gives rise to boundary conditions.Note that the number of boundary conditions must match the order of the differential equation.Solve the equation for the velocity distribution. Then obtain the mean velocity, flow rate and the shear force.NO-SLIP AT THE WALLAlso called boundary condition of the first kind (Dirichlet BC)At solid-fluid interface, the fluid velocity equals to the velocity of the solid surface.

BOUNDARY CONDITIONSCommon Boundary Conditions in Fluid MechanicsSYMMETRYAt the plane of symmetry in flows the velocity field is the same on either side of the plane of symmetry, the velocity must go through a minimum or a maximum at the plane of symmetry. Thus, the boundary condition to use is that the first derivative of the velocity is zero at the plane of symmetry

STRESS CONTINUITYWhen a fluid forms one of the boundaries of the flow, the stress is continuous from one fluid to another, there are two possibilitiesFor a viscous fluid in contact with an inviscid (zero or very low viscosity fluid)At the boundary, the stress in the viscous fluid is the same as the stress in the inviscid fluid. Since the inviscid fluid can support no shear stress (zero viscosity) this means that the stress is zero at this interface.The boundary condition between a fluid such as a polymer and air, for example, would be that the shear stress in the polymer at the interface would be zero.

This is also called Boundary Condition of Second Kind OR Newmann BCBOUNDARY CONDITIONS, contd. - 2

STRESS CONTINUITY, contd.Alternatively if two viscous fluids meet and form a flow boundary, This same boundary condition would require that the stress in one fluid equal the stress in the other at the boundary.

BOUNDARY CONDITIONS, contd. - 3

VELOCITY CONTINUITYWhen a fluid forms one of the boundaries of the flow then along with stress at the boundary, the velocity is also continuous from one fluid to another.

This is also called Boundary Condition of Fourth Kind

SHELL BALANCE, EXAMPLE-1Step 1: Draw the physical diagramExample 1: Flow on flat plateLWxx+xDirection of TransportShellxzyExample 1: Flow on flat plate, contd. - 2Step 4: Draw a shellThe shell is one dimensional as there is only one transport direction.The shell is drawn such that the surfaces are at x and x+x, which are perpendicular to the direction of transport.Step 2: Possible transport mechanism Transport of shear across the thickness of the film.Step 3: Frame of coordinates and direction of transport Choose a frame of coordinate (x,z) with x across the film and z along the film. The direction of transport is in the x-direction.Example 1: Flow on flat plate, contd. - 3Step 5: Momentum balanceRate of z-momentum in at the surface at x = x

surface areashear (force/area) at the surface xRate of z-momentum out at surface x = x + x

surface areashear (force/area) at the surface x+ x Gravity force in the z-direction acting on the volume of the shell =

Volume of Shellgravity force per unit volume in z-directionExample 1: Flow on flat plate, contd. - 4Momentum balance equation

Divide the equation by the volume of the shell and make the shell as thin as possible:The momentum balance equation is finally a differential equation (valid at any point in the fluid film)

LinearUpon deriving this equation nothing has been said about the fluid behavior. Hence, it is applicable to Newtonian as well as non-Newtonian fluids.Definition of derivative:

Step 6: Apply the Newtonian lawExample 1: Flow on flat plate, contd. - 5From this step onward, we have to decide that what kind of fluid we are dealing with

second-order ODE in terms of velocityStep 7: Impose physical constraintExample 1: Flow on flat plate, contd. - 6

Step 8: Solve ODE for velocity distribution

x = x = 0 is zero at the gas-liquid interfaceAt solid-fluid interface the velocity is zero

Example 1: Flow on flat plate, contd. - 6Is the velocity distributionof any use to us? Sort ofThis is the velocity distribution in the film. It is parabolic in shape (only for Newtonian fluids)Step 9: Useful quantities

Maximum velocityAverage velocityvolumetric flow rateShear force acting on solid surfaceMaximum velocity

Average velocity (defined as the mean velocity when multiplied by the cross section area will give the volumetric flow rate)

Volumetric flow rate

Example 1: Flow on flat plate, contd. - 6Shear force acting on solid surface

Example 1: Flow on flat plate, contd. - 6What if the fluid is non-NewtonianStep 6: Apply the Appropriate Non-Newtonian modelLet us try the Ostwald-de Waele model for non-Newtonian fluids

Momentum Equation

Step 7: physical constraintA Nonlinear ODE

Example 1: Flow on flat plate, contd. - 6Step 8: Solving for Velocity Distribution

Maximum velocity

Average velocity

Volumetric flow rate

Example 1: Flow on flat plate, contd. - 6For 3% CMC in water, n= 0.566Compare with Newtonian Fluids

Stronger than parabolic dependenceImplication of this example:This study of thin film is useful in the analysis of:1. wetted wall tower2. evaporation and gas absorption3. coating to paper4. drainage from large tanks/plates

EX-2: Flow Of Incompressible Fluids IN A Circular TubeDifferences in Problem Definition as compared to Flat PlateFlat plateCircular tubeLaminar flowLaminar flowNo end effectsNo end effectsRectangularCylindricalGravity as the driving forceGravity force &Pressure forceThe procedure of solving the flat plate problem is used here tosolve this problem.Example-2: Flow Through A Circular Tube, contd. -2Step 1: Draw the physical diagram0zrrr+rLR Direction of transport Shellzz+zExample-2: Flow Through A Circular Tube, contd. -3Step 2: Transport mechanismTransport of rate of momentum in the r-direction.Transport of rate of momentum in the z-direction.Step 3: Frame of coordinatesChosen as shown in the diagramTypical for any system having cylindrical geometry Step 4: Draw a shellShell in this case is an annulus having surfaces perpendicular to the direction of momentum transport. Shell is at the position (r,z) and has thicknesses r and z in the r and z coordinates, respectively.Example-2: Flow Through A Circular Tube, contd. -4Step 5: Momentum shell balanceRate of z-momentum in the shell across the cylindrical surface at r =

Surface contact areashear (force/area) at the surface rRate of z-momentum out across the cylindrical surface at r+ r =

Surface contact areashear (force/area) at the surface r + rExample-2: Flow Through A Circular Tube, contd. -5Rate of z-momentum in across the annular surface at z =

Cross-sectional areaVolumetric flow rateMass flow rateRate of z-momentum out across the annular surface at z+z =

Example-2: Flow Through A Circular Tube, contd. -6Pressure force acting on the z surface:

Cross-sectional areaPressure along zPressure force acting on the z+z surface:

Gravity force acting on the shell volume along z direction:

Volume of the shellWeight force of the fluid in shellEx-2: Flow Through A Circular Tube, contd. -7Add all the termsDivide through out by the volume of shellTake the limits by making the shell as thin as possibleMomentum balance equation after the shell is made as thin as possible

equal to zero as vz is independent of zthe fluid is incompressibleThe tube area is constantIndependent of rFirst-order ODE in terms of shearstressEx-2: Flow Through A Circular Tube, contd. -8Final momentum balance equation

C1 is zero because the shear stress must be finite at r=0applicable to any fluidsshear stress is a linear function of r

Ex-2: Flow Through A Circular Tube, contd. -9Step 6: Newtonian law

Step 7: Physical constraints

Ex-2: Flow IN A Circular Tube, contd. 9.1

Parabolic velocity distribution

Linear momentum flux distribution

Step 8: Velocity distributionEx-2: Flow IN A Circular Tube, contd. -10

Parabolic profileVolumetric flow rateAveragevelocityMaximumvelocityForce of fluidon wetted wallStep 9: Useful quantities

Ex-2: Flow IN A Circular Tube, contd. -11Maximum velocity, at r = 0

Volumetric flow rateIntegrate the differential volumetric flow ratedifferential volumetric flow rate

Can be measured easilyCalculatedMeasuredEx-2: Flow IN A Circular Tube, contd. -12

Famous Hagen-Poiseuille equationOften used to determine fluid viscosityAverage velocity

Force exerted by the flowing fluid on the walls of tube

HPEEx-2: Flow IN A Circular Tube, contd. -13

Inner Surface area of tubeShear Stress at the wallResults obtained are for Newtonian fluids. For non-Newtonian fluids, substitute the appropriate model equation after step-5.Step 6: Apply the Appropriate Non-Newtonian modelLet us try the Ostwald-de Waele model for non-Newtonian fluidsFOR NON-NEWTONIAN FLUIDSEx-2: Flow through a circular tube, contd. -14

For n = 1

Compare with

Ex-2: Flow Through A Circular Tube, contd. -15Try the Bingham model for non-Newtonian fluids

The shear is zero at center of the tube and maximum at surface of the tube; thus, one expects a plug flow region in the central part of the tube.

Max finite shear at wall r=R zero shear at r=0 shear stressprofilePhysical Constraints of Bingham fluidsVelocityprofile

zero slip at wallEx-2: Flow Through A Circular Tube, contd. -16Velocity distribution of a Bingham fluid flow in a circular tube

Critical radius ro and o where dvz/dr=0

FOR, r ro

FOR, r ro

Ex-2: Flow Through A Circular Tube, contd. -17 For Volumetric flow rate

For practice do examples of Annular flowFlow of two immiscible liquidsEND OF CHAP-2HAGEN-POISEUILLE EQUATION LIMITATIONSa. The flow is laminar; that is, Re < 2100.b. The density is constant (incompressible flow).c. The flow is steady (i.e., it does not change with time).d. The fluid is Newtonian e. End effects are neglected. Entrance Length, of the order of Le = 0.035D Re, is needed for the buildup to the parabolic profile. If the section of pipe of interest includes the entrance region, a correction must be applied.f. The fluid behaves as a continuumthis assumption is valid, except for very dilute gases or very narrow capillary tubes, in which the molecular mean free path is comparable to the tube diameter (the slip flow region) or much greater than the tube diameter (the Knudsen flow or free molecule flow regime).g. There is no slip at the wall, so that B.C. 2 is valid