Darcy-Weisbach Equation for Pressure and Head Loss

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

Pressure Loss

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

Δp = λ (l / dh) (ρ v2 / 2)         (1)

where

Δp = pressure loss (Pa, N/m2)

λ = Darcy-Weisbach friction coefficient

l = length of duct or pipe (m)

dh = hydraulic diameter (m)

ρ = density (kg/m3)

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.

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 velocity is known. The default values are for air flow 20oC, 1.2 kg/m3 and 6 m/s. The friction coefficient can be calculated with the Colebrook equation.

The 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 - ρ - (kg/m3, lb/ft3)

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

SI-units Imperial units

This calculator 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 - ρ - (kg/m3, lb/ft3)

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

SI-units Imperial units

Head Loss

Alternatively, the Darcy-Weisbach equation (1) can also express the head loss as

Δh = λ (l / dh) (v2 / g 2)         (2)

where

Δh = head loss (m)

λ = friction coefficient

l = length of pipe or duct (m)

dh = hydraulic diameter (m)

v = velocity (m/s)

g = acceleration of gravity (9.81 m/s2)

Note! - the head is related to the actual fluid. If head is wanted expressed related to a reference fluid - example water - use equation (3) below. 

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

Head can be expressed related to a reference fluid by modifying equation (2) to 

Δhr = λ (l / dh) (v2 / g 2) (ρa / ρr)        (3)

ρa = density of the actual fluid (kg/m3)

ρr = density of the reference fluid (kg/m3)

The calculator below, which is based on formula (3), 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 - v - (m/s, ft/min)

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

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

SI-units Imperial units

Related Topics

  • Fluid Mechanics - The study of fluids - liquids and gases. Involves various properties of the fluid, such as velocity, pressure, density and temperature, as functions of space and time.
  • Fluid Flow and Pressure Drop - Pipe lines - fluid flow and pressure loss - water, sewer, steel pipes, pvc pipes, copper tubes and more

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