Pressure Drop, Friction, and the Darcy-Weisbach Equation

We have covered the significance of pipe roughness, both absolute and relative, and how this relates to friction factor most commonly within a Moody-type diagram. Let us finish things off by looking at a worked example where we pull all the aspects together to determine the pressure drop in a pipe due to pipe wall friction.

Example:

A pipe of 75 mm internal diameter with a friction factor of 5.5 x 10-3 contains liquid flowing at an average velocity of 1.9 m/s. The fluid has a density of 1,200 kg/m³. If the pipe is 150 m long, what is the pressure loss due to friction per metre length, and the overall pressure drop?

The Darcy-Weisbach equation is:

increment P subscript բ equals 4 phi L over d rho u ²

For the first part of the question, we are asked to determine the pressure drop per metre length, hence L = 1 m initially:

increment P subscript բ equals 4 phi L over d rho u squared equals 4 cross times 5.5 cross times 10 to the power of negative 3 end exponent cross times fraction numerator 1 over denominator 0.075 end fraction cross times 1 comma 200 cross times 1.9 squared

increment P subscript բ equals 1270.72 space P a

Therefore, the pressure drop per metre is 1270.72 Pa/m.

For a 150 m length of this pipe, we can either redo this calculation with L = 150 m or more simply multiply the pressure drop per metre value by 150, thus:

increment P subscript բ equals 1270.72 cross times 150 equals 190.6 space k P a

This would be a good time to summarise the key points of the lesson so far.

The Darcy-Weisbach equation can be used to determine the pressure drop due to friction between the inner pipe wall and the fluid:

increment P բ equals 4 phi space L over d space rho u ²

For laminar flow, the friction factor φ is a function of Reynolds number only and given by:

phi equals fraction numerator 16 over denominator R e end fraction

For turbulent flow, the friction factor φ is a function of Reynolds number and relative roughness ε, of the pipe wall. Moody diagrams are most easily used to determine φ. ε is the ratio of absolute roughness e to the pipe diameter d:

epsilon equals e over d

Values of absolute roughness e are readily available in design guides, from pipe suppliers or within literature.

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