darcy weisbach equation

Darcy–Weisbach equation

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In fluid dynamics, the Darcy–

relates the head loss — or pressure

average velocity of the fluid flow. The equation is named after

Weisbach.

The Darcy–Weisbach equation contains a

friction factor. This is also called the

factor. The Darcy friction factor is four times the

should not be confused.[1]

Head loss form

Head loss can be calculated with

where

• hf is the head loss due to friction (SI units: m);

• L is the length of the pipe (m);

• D is the hydraulic diameter

internal diameter of the pipe) (m);

• V is the average velocity of the fluid flow, equal t

cross-sectional wetted area

• g is the local acceleration due to

• fD is a dimensionless coefficient called the

found from a Moody diagram

Do not confuse this with the Fanning Friction factor, f.

Weisbach equation

From Wikipedia, the free encyclopedia

–Weisbach equation is a phenomenological equation, which

pressure loss — due to friction along a given length of pipe to the

average velocity of the fluid flow. The equation is named after Henry Darcy

Weisbach equation contains a dimensionless friction factor, known as the

. This is also called the Darcy–Weisbach friction factor or

. The Darcy friction factor is four times the Fanning friction factor, with which it

can be calculated with

is the head loss due to friction (SI units: m);

L is the length of the pipe (m);

hydraulic diameter of the pipe (for a pipe of circular section, this equals the

internal diameter of the pipe) (m);

is the average velocity of the fluid flow, equal to the volumetric flow rate

wetted area (m/s);

the local acceleration due to gravity (m/s2);

is a dimensionless coefficient called the Darcy friction factor.[citation needed

Moody diagram or more precisely by solving the Colebrook equation

this with the Fanning Friction factor, f.

equation, which

along a given length of pipe to the

Henry Darcy and Julius

friction factor, known as the Darcy

or Moody friction

, with which it

of the pipe (for a pipe of circular section, this equals the

volumetric flow rate per unit

citation needed] It can be

Colebrook equation.

Pressure loss form

Given that the head loss hf expresses the

where ρ is the density of the fluid, the Darcy

of pressure loss:[2]

where the pressure loss due to friction

• the ratio of the length to diameter of the pipe,

• the density of the fluid,

• the mean velocity of the flow,

• Darcy Friction Factor; a (dimensionless) coef

Since the pressure loss equation can be derived from the head loss equation by multiplying

each side by ρ and g.

Darcy friction factor

See also Darcy friction factor formulae

The friction factor fD or flow coefficient

the pipe and the velocity of the fluid flow, but it is known to high accuracy within certain

flow regimes. It may be evaluated for given conditions by the use of various empirical or

theoretical relations, or it may be obtained from published charts. These charts are often

referred to as Moody diagrams

called the Moody friction factor

the approximate formula he proposed.

For laminar (slow) flows, it is a consequence of

the Reynolds number calculated substituting for the characteristic length the hydraulic

diameter of the pipe, which equals the inside diameter for circular pipe geometries.

expresses the pressure loss ∆p as the height of a column of fluid,

where ρ is the density of the fluid, the Darcy–Weisbach equation can also be written in terms

e the pressure loss due to friction ∆p (Pa) is a function of:

the ratio of the length to diameter of the pipe, L/D;

the density of the fluid, ρ (kg/m3);

the mean velocity of the flow, V (m/s), as defined above;

Darcy Friction Factor; a (dimensionless) coefficient of laminar, or turbulent flow


Darcy friction factor

Darcy friction factor formulae

or flow coefficient λ is not a constant and depends on the parameters of


. It may be evaluated for given conditions by the use of various empirical or


Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes

Moody friction factor. It is also sometimes called the Blasius friction factor, after

the approximate formula he proposed.

For laminar (slow) flows, it is a consequence of Poiseuille's law that λ = 64/

calculated substituting for the characteristic length the hydraulic


as the height of a column of fluid,

Weisbach equation can also be written in terms

turbulent flow, fD.


is not a constant and depends on the parameters of


. It may be evaluated for given conditions by the use of various empirical or


, after L. F. Moody, and hence the factor itself is sometimes

friction factor, after

64/Re, where Re is

calculated substituting for the characteristic length the hydraulic


For turbulent flow, methods for finding the friction factor

the Moody chart; or solving equations such as the

Swamee–Jain equation. While the diagram and Colebrook

methods, the Swamee–Jain equation allows

pipe.

Confusion with the Fanning friction factor

The Darcy–Weisbach friction factor,

attention must be paid to note which one of these is meant in any "friction factor" chart or

equation being used. Of the two, the Darcy

civil and mechanical engineers, and the Fanning factor,

should be taken to identify the correct factor regardless of the source of the chart or formula.

Note that

Most charts or tables indicate the type of friction factor, or at least provide the formula for the

friction factor with laminar flow. If the formula for lami

factor, f, and if the formula for laminar flow is

Which friction factor is plotted in a Moody diagram may be determined by

publisher did not include the formula described above:

1. Observe the value of the friction factor for laminar flow at a Reynolds number of

1000.

2. If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in

the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the

formula for the laminar Darcy friction factor:

3. If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in

the Moody diagram. Note th

formula for the laminar Fanning friction factor:

The procedure above is similar for any available Reynolds number that is an integral power

of ten. It is not necessary to remember the value

integral power of ten is of interest for this purpose.

urbulent flow, methods for finding the friction factor f include using a diagram such as

; or solving equations such as the Colebrook–White equation

. While the diagram and Colebrook–White equation are iterative

Jain equation allows f to be found directly for full flow in a circular

Confusion with the Fanning friction factor

Weisbach friction factor, fD is 4 times larger than the Fanning friction factor


equation being used. Of the two, the Darcy–Weisbach factor, fD is more commonly used by

civil and mechanical engineers, and the Fanning factor, f, by chemical engineers, but care



friction factor with laminar flow. If the formula for laminar flow is f = 16/Re

, and if the formula for laminar flow is fD = 64/Re, it's the Darcy–Weisbach factor,

Which friction factor is plotted in a Moody diagram may be determined by

publisher did not include the formula described above:

Observe the value of the friction factor for laminar flow at a Reynolds number of

If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in

Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the

formula for the laminar Darcy friction factor: fD = 64/Re.

If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in

the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the

formula for the laminar Fanning friction factor: f = 16/Re.


of ten. It is not necessary to remember the value 1000 for this procedure – only that an

l power of ten is of interest for this purpose.

include using a diagram such as

White equation, or the

e equation are iterative

to be found directly for full flow in a circular

Fanning friction factor, f, so


is more commonly used by

, by chemical engineers, but care



Re, it's the Fanning

Weisbach factor, fD.

Which friction factor is plotted in a Moody diagram may be determined by inspection if the

Observe the value of the friction factor for laminar flow at a Reynolds number of

If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in

Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the

If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in

at the nonzero digits in 0.016 are the numerator in the


only that an

History

Historically this equation arose as a variant on the

by Henry Darcy of France, and further refined into the form used today by

of Saxony in 1845. Initially, data

Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many

cases. In later years it was eschewed in many special

empirical equations valid only for certain flow regimes, notably the

equation or the Manning equation

calculations. However, since the advent of the

major issue, and so the Darcy–

Derivation

The Darcy–Weisbach equation is a phenomenological formula obtainable by

analysis.

Away from the ends of the pipe, the characteristics of the flow are independent of the

position along the pipe. The key quantities are then the pressure drop along the pipe per unit

length, ∆p/L, and the volumetric flow rate. The flow rate can b

velocity V by dividing by the wetted area

the pipe if the pipe is full of fluid).

Pressure has dimensions of energy per unit volume. Therefore, the pressure drop between two

points must be proportional to

below) the expression for the kinetic energy per unit volume. We also know that pressure

must be proportional to the length of the pipe between the two points

per unit length is a constant. To turn the relationship into a proportionality coefficient of

dimensionless quantity we can divide by the hydraulic diameter of the pipe,

constant along the pipe. Therefore,

The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow

coefficient". This dimensionless coefficient will be a combination of

as π, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe

(the ratio of the roughness height

ally this equation arose as a variant on the Prony equation; this variant was developed

of France, and further refined into the form used today by

in 1845. Initially, data on the variation of f with velocity was lacking, so the

Weisbach equation was outperformed at first by the empirical Prony equation in many

cases. In later years it was eschewed in many special-case situations in favor of a variety of

valid only for certain flow regimes, notably the Hazen–

Manning equation, most of which were significantly easier to use in

calculations. However, since the advent of the calculator, ease of calculation is no longer a

–Weisbach equation's generality has made it the preferred one.

Weisbach equation is a phenomenological formula obtainable by



, and the volumetric flow rate. The flow rate can be converted to an average

wetted area of the flow (which equals the cross

the pipe if the pipe is full of fluid).


t be proportional to (1/2)ρV2, which has the same dimensions as it resembles (see


must be proportional to the length of the pipe between the two points L as the pressu


dimensionless quantity we can divide by the hydraulic diameter of the pipe,

constant along the pipe. Therefore,


coefficient". This dimensionless coefficient will be a combination of geometric factors such

, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe

roughness height to the hydraulic diameter).

; this variant was developed

Julius Weisbach

with velocity was lacking, so the

Weisbach equation was outperformed at first by the empirical Prony equation in many

case situations in favor of a variety of

–Williams

, most of which were significantly easier to use in

, ease of calculation is no longer a

Weisbach equation's generality has made it the preferred one.

Weisbach equation is a phenomenological formula obtainable by dimensional



e converted to an average

cross-sectional area of


, which has the same dimensions as it resembles (see


as the pressure drop


dimensionless quantity we can divide by the hydraulic diameter of the pipe, D, which is also


geometric factors such

, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe

Note that (1/2)ρV2 is not the kinetic energy of the fluid per unit volume, for the following

reasons. Even in the case of laminar flow

the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and

the velocity in the center of the pipe must therefore be larger than the average velocity

obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy

then involves the mean-square velocity, which always exceeds the square of the mean

velocity. In the case of turbulent flow

directions, including perpendicular to the length of the pipe, and thus turbulence contributes

to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.

Practical applications

In hydraulic engineering applications, it is often desirable to express the head loss in terms of

volumetric flow rate in the pipe. For this, it is necessary to substitute the following into the

original head loss form of the Darcy

where

• V is, as above, the average velocity of the fluid flow, equal to the

per unit cross-sectional

• Q is the volumetric flow rat

• Aw is the cross-sectional

is not the kinetic energy of the fluid per unit volume, for the following

laminar flow, where all the flow lines are parallel to the length of



metric flow rate by the wet area. The average kinetic energy

square velocity, which always exceeds the square of the mean

ent flow, the fluid acquires random velocity components in all



Practical applications

applications, it is often desirable to express the head loss in terms of


original head loss form of the Darcy–Weisbach equation

is, as above, the average velocity of the fluid flow, equal to the volumetric flow r

sectional wetted area (m/s);

volumetric flow rate (m3/s);

sectional wetted area (m2).

is not the kinetic energy of the fluid per unit volume, for the following

are parallel to the length of



metric flow rate by the wet area. The average kinetic energy

square velocity, which always exceeds the square of the mean

, the fluid acquires random velocity components in all



applications, it is often desirable to express the head loss in terms of


volumetric flow rate

For the general case of an arbitrarily

known, being an implicit function of pipe slope, cross

variables. If, however, the pipe is assumed to be full flowing and of circular cross

is common in practical scenarios, then

where D is the diameter of the pipe

Substituting these results into the original formulation yields the final equati

in terms of volumetric flow rate

where all symbols are defined as above.

For the general case of an arbitrarily-full pipe, the value of Aw will not be immediately

function of pipe slope, cross-sectional shape, flow rate and other

variables. If, however, the pipe is assumed to be full flowing and of circular cross

is common in practical scenarios, then

of the pipe

Substituting these results into the original formulation yields the final equati

volumetric flow rate in a full-flowing circular pipe

where all symbols are defined as above.

will not be immediately

sectional shape, flow rate and other

variables. If, however, the pipe is assumed to be full flowing and of circular cross-section, as

Substituting these results into the original formulation yields the final equation for head loss

darcy weisbach equation

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