Fluid Mechanics: Friction Losses

Pipe Pressure Loss 
Wall drag and changes in height lead to pressure drops in pipe fluid flow.

To calculate the pressure drop and flowrates in a section of uniform pipe running from Point A to Point B, enter the parameters below. The pipe is assumed to be relatively straight (no sharp bends), such that changes in pressure are due mostly to elevation changes and wall friction. (The default calculation is for a smooth horizontal pipe carrying water, with answers rounded to 3 significant figures.)

Note that a positive Dz means that B is higher than A, whereas a negative Dz means that B is lower than A.

Equations used in the Calculation
Changes to inviscid, incompressible flow moving from Point A to Point B along a pipe are described by Bernoulli's equation,
where p is the pressure, V is the average fluid velocity, r is the fluid density, z is the pipe elevation above some datum, and g is the gravity acceleration constant.

Bernoulli's equation states that the total head h along a streamline (parameterized by x) remains constant. This means that velocity head can be converted into gravity head and/or pressure head (or vice-versa), such that the total head h stays constant. No energy is lost in such a flow.

For real viscous fluids, mechanical energy is converted into heat (in the viscous boundary layer along the pipe walls) and is lost from the flow. Therefore one cannot use Bernoulli's principle of conserved head (or energy) to calculate flow parameters. Still, one can keep track of this lost head by introducing another term (called viscous head) into Bernoulli's equation to get,

where D is the pipe diameter. As the flow moves down the pipe, viscous head slowly accumulates taking available head away from the pressure, gravity, and velocity heads. Still, the total head h (or energy) remains constant.

For pipe flow, we assume that the pipe diameter D stays constant. By continuity, we then know that the fluid velocity V stays constant along the pipe. With D and V constant we can integrate the viscous head equation and solve for the pressure at Point B,

where L is the pipe length between points A and B, and Dz is the change in pipe elevation (zB - zA). Note that Dz will be negative if the pipe at B is lower than at A.

The viscous head term is scaled by the pipe friction factor f. In general, f depends on the Reynolds Number R of the pipe flow, and the relative roughness e/D of the pipe wall,

The roughness measure e is the average size of the bumps on the pipe wall. The relative roughness e/D is therefore the size of the bumps compared to the diameter of the pipe. For commercial pipes this is usually a very small number. Note that perfectly smooth pipes would have a roughness of zero.

For laminar flow (R < 2000 in pipes), f can be deduced analytically. The answer is,

For turbulent flow (R > 3000 in pipes), f is determined from experimental curve fits. One such fit is provided by Colebrook,
The solutions to this equation plotted versus R make up the popular Moody Chart for pipe flow,
The calculator above first computes the Reynolds Number for the flow. It then computes the friction factor f by direct substitution (if laminar; the calculator uses the condition that R < 3000 for this determination) or by iteration using Newton-Raphson (if turbulent). The pressure drop is then calculated using the viscous head equation above. Note that the uncertainties behind the experimental curve fits place at least a 10% uncertainty on the deduced pressure drops. The engineer should be aware of this when making calculations.
Absolute Pipe Roughness
Included here is a sampling of absolute pipe roughness e data taken from Binder (1973). These values are for new pipes; aged pipes typically exhibit in rise in apparent roughness. In some cases this rise can be very significant.
Pipe Material Absolute Roughness, e
x 10-6 feet micron
(unless noted)
drawn brass 5 1.5
drawn copper 5 1.5
commercial steel 150 45
wrought iron 150 45
asphalted cast iron 400 120
galvanized iron 500 150
cast iron 850 260
wood stave 600 to 3000 0.2 to 0.9 mm
concrete 1000 to 10,000 0.3 to 3 mm
riveted steel 3000 to 30,000 0.9 to 9 mm
Relative pipe roughness is computed by dividing the absolute roughness e by the pipe diameter D,

 

 

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