Pipe Flow Calculations 3: The Friction Factor & Frictional Head Loss

Pipe flow under pressure is used for a lot of purposes. Energy input to the gas or liquid is needed to make it flow through the pipe or conduit. This energy input is needed because there is frictional energy loss (also called frictional head loss or frictional pressure drop) due to the friction between the fluid and the pipe wall and internal friction within the fluid. The Darcy Weisbach Equation, which will be discussed in this article, is commonly used for a variety of calculations involving frictional head loss, pipe diameter, flow rate or velocity, and several other parameters. The friction factor, which is used in the Darcy Weisbach equation, depends upon the Reynolds number and the pipe roughness.

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Most pipe and conduit flow of gases and liquids with a viscosity similar to water will be turbulent flow. If the total pipe length is large compared to the entrance length, then the entrance effects are negligible and the total pipe length is used for calculations.

The Darcy Weisbach Equation provides an empirical relationship among several pipe flow variables as shown here:

The equation is: h_L = f (L/D)(V²/2g), where

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h_L = frictional head loss, ft-lb/lb

L = pipe length, ft

D = pipe diameter, ft

V = average flow velocity of fluid (= Q/A), ft/sec

g = acceleration due to gravity = 32.2 ft/sec²

f = friction factor, a dimensionless empirical factor that is a function of Reynolds Number (Re = DVρ/μ) and/or ε/D, where

ε = an empirical pipe roughness, ft

The table at the right shows some typical values for pipe roughness for common pipe materials.

The energy loss in pipe flow due to friction can be expressed as a pressure drop instead of as a head loss. Chemical and mechanical engineers often work with pressure drop, whereas civil engineers usually work with head loss. The relationship between frictional head loss and frictional pressure drop is simply:

ΔP_f = ρgh_L = γh_L, where:

ΔP_f = frictional pressure drop, lb/ft²,

h_L = frictional head loss, ft-lb/lb (as noted above).

ρ = fluid density, slugs/ft³,

g = acceleration due to gravity, ft/sec²,

γ = specific weight, lb/ft³.

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There are equations available for friction factor for each of the four regions of the chart identified above as follows.

For laminar flow (Re < 2100): f = 64/Re

For the completely turbulent region: f = [1.14 + 2 log₁₀(D/ε)]^-2

For smooth pipe turbulent flow: f = 0.316/Re^1/4

For the transition region: f = {-2 log₁₀[(ε/D)/3.7 + 2.51/Re(f^1/2)]}^-2

Note that the last equation requires an iterative solution to find f for given values of ε/D and Re, or "solver" can be used in the Excel spreadsheet.

The Darcy Weisbach Equation relates the variable, h_L, D, L, V, ε, ρ and μ. It's typical use is to calculate h_L, D, L, or V, when all of the other parameters are known. Some of these require iterative calculations.

For Excel spreadsheet templates that can be downloaded to make pipe flow/friction factor calculations, see the article, "Pipe Flow/Head Loss/Friction Factor Calculations Using Excel Spreadsheet Templates."

For further information:

1. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

2. Darcy Weisbach equation history - http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm

3. Source for pipe roughness values - http://www.efunda.com/formulae/fluids/roughness.cfm