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Darcy-Weisbach Pressure and Major Head Loss Equation

The Darcy-Weisbach equation can be used to calculate the major pressure or head loss due to friction in ducts, pipes or tubes

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Pressure Loss

The pressure loss (or major loss) in a pipe, tube or duct can be calculated with the Darcy-Weisbach equation

Δpmajor_loss = λ (l / dh) (ρf v2 / 2)                       (1)

where

Δpmajor_loss = major (friction) pressure loss in fluid flow (Pa (N/m2), psf (lb/ft2))

λ = Darcy-Weisbach friction coefficient

l = length of duct or pipe (m, ft)

v = velocity of fluid (m/s, ft/s)

dh = hydraulic diameter (m, ft)

ρf = density of fluid (kg/m3, slugs/ft3)

Note! - be aware that there are two alternative friction coefficients present in the literature. One is 1/4 of the other and (1) must be multiplied with four to achieve the correct result. This is important to verify when selecting friction coefficients from Moody diagrams. The Colebrook friction coefficient calculator corresponds to equation (1). 

The Darcy-Weisbach equation is valid for fully developed, steady state and incompressible flow. The friction factor or coefficient - λ -depends on the flow, if it is laminar, transient or turbulent (the Reynolds Number) - and the roughness of the tube or duct. The friction coefficient can be calculated by the Colebrooke Equation or by using the Moody Diagram.

Example - Pressure Loss in Air Duct

Air flows with velocity 6 m/s in a duct with diameter 315 mm. The density of air is 1.2 kg/m3. The friction coefficient is estimated to 0.019 and the length of the duct is 1 m. The friction loss can be calculated as

Δpmajor_loss = 0.019 ((1 m) / (0.315 m)) ((1.2 kg/m3) (6 m/s)2 / 2)

     = 1.3 Pa   

Note! - in addition to friction loss - there is almost always minor loss in a flow.

Online Pressure Loss Calculator

The calculator below, which is based on formula (1), can be used to calculate the pressure loss in a duct, pipe or tube if the velocity of the fluid is known. The default values are for air flow 20oC, 1.2 kg/m3 and 6 m/s - the same as in the example above. The friction coefficient can be calculated with the Colebrook equation.

This calculator is generic and can be used with SI and Imperial units. Just substitute the values with the values for the actual application.

friction coefficient - λ

length of pipe or duct - l - (m, ft)

 hydraulic diameter - dh - (m, inches)

density of fluid - ρf - (kg/m3, lb/ft3)

velocity of fluid - v - (m/s, ft/min)

SI-units Imperial units


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The calculator below can be used if the volume flow is known

friction coefficient - λ

length of pipe or duct - l - (m, ft)

hydraulic diameter - dh - (m, inches)

density of fluid - ρf - (kg/m3, lb/ft3)

volume flow - q - (m3/s, ft3/min)

SI-units Imperial units


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Head Loss

Alternatively the Darcy-Weisbach equation can express head loss as water column by dividing the pressure loss (1) with the specific weight of water

Δhmajor_loss,w = λ (l / dh) (ρf v2 / 2) / γw      

    = λ (l / dh) (ρf v2 / 2) / ρw g

    = λ (l / dhf / ρw) (v2 / (2 g))                                (2)

where

Δhmajor_loss,w = major head loss (water column) in fluid flow (m H2O, ft H2O)

λ = friction coefficient

l = length of pipe or duct (m, ft)

dh = hydraulic diameter (m, ft)

v = velocity of fluid (m/s, ft/s)

γw = ρw g = specific weight of water (9807 N/m3, 62.4 lbf/ft3)

ρw = density of water (1000 kg/m3, 62.425 lb/ft3)

g = acceleration of gravity (9.81 m/s2, 32.174 ft/s2)

Note! - in the equation above the head is related to water as the reference fluid. Another reference fluid can be used - like Mercury Hg - by replacing the density of water with the density of the reference fluid.

If the density in the fluid flow is the same as the density in the reference fluid - as typical with water flow - eq. (2) can be simplified to 

Δhmajor_loss = λ (l / dh) (v2 / (2 g))                                (2b)

where

Δhmajor_loss = major head loss (column of flowing fluid) (m "fluid", ft "fluid")

For metric units the head loss can alternatively be modified to 

Δhmajor_loss,w (mmH2O) = λ (l / dhf / ρw) (v2 / (2 g)) / 1000          (2c)

where

Δhmajor_loss,w (mmH2O) = head loss (mm H2O)

For imperial units the head loss can alternatively be modified to

Δhmajor_loss,w (inH2O) = 12 λ (l / dhf / ρw) (v2 / (2 g))           (2d)

where

Δhmajor_loss,w (inH2O) = head loss (inches H2O)

The Darcy-Weisbach equation with the Moody diagram are considered to be the most accurate model for estimating frictional head loss in steady pipe flow. Since the approach requires a trial and error iteration process, an alternative less accurate empirical head loss calculation that do not require the trial and error solutions like the Hazen-Williams equation, may be preferred.

Online Head Loss Calculator

The calculator below, which is based on eq. (2), can be used to calculate the head loss in a duct, pipe or tube. The default values used in the calculator are for air flow 20oC, 1.2 kg/m3 and 6 m/s. The default density of water commonly used as reference fluid is 1000 kg/m3. The friction coefficient is calculated with the Colebrook equation.

The calculator is generic and can be used for both SI and Imperial units. Just substitute the values with the values for the actual application. 

friction coefficient - λ

length of pipe or duct - l - (m, ft)

hydraulic diameter - dh - (m, inches)

velocity of fluid - v - (m/s, ft/min)

density actual fluid - ρ - (kg/m3, lb/ft3)

 density reference fluid (typical water) - ρ - (kg/m3, lb/ft3)

SI-units Imperial units


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