Closed Conduit Hydraulics – Friction and Minor Losses

Friction and minor losses are glossed over in the darcy-weisbach article but I want to throw in some extra notes here. Friction occurs over every bit of length of a close-conduit system and is usually a surprisingly high amount of energy loss. Friction depends on the material of the pipe and the velocity of flow. The formula for frictional head loss IS the Darcy-Weisbach equation.

$$ h_f = f \frac{L}{D}\frac{V^2}{2g} $$

Minor Losses

Minor losses are solved similarly to friction but represent a different thing. Minor losses are incurred in systems whenever there is a bend/elbow, angle connector, faucet, tee intersection, pipe contraction or enlargement, valves, and other things. All of these minor losses have an associated K value that you must look up in a table. Use this K value, in addition to the velocity of flow, to find the head loss.

$$ h_f = K \frac{V^2}{2g} $$

Notice how similar the equation is to friction. A speed tip is to just sum all of the K values in the system that share the same velocity:

$$ (K_1 + K_2 + K_3 + \text{etc} + f\frac{L}{D})(\frac{V^2}{2g}) $$

This speeds up calculation but remember you can only group the K values for minor losses that share the same velocity. If the pipe diameter changes, or the velocity in the pipe changes for any other reason, the K values should be added up and multiplied separately in those areas. Also not that the friction loss can be included as well to further simplify head loss calculation in a segment.

Equivalent Length

Some problems will ask for an equivalent length of the minor losses in a given pipe. The equivalent length is a “fake” section of pipe where the friction head loss due to friction is equivalent to the head loss from the minor losses. You can find an equivalent length by equating the sum of the minor head losses to the Darcy-Weisbach (you need to know the friction factor of the pipe for this):

$$ K\frac{V^2}{2g} = f\frac{L_e}{D}\frac{V^2}{2g} $$
$$ L_e = \frac{KD}{f} $$

I suppose you could also use the head-loss form of the Hazen-Williams for this too, using C instead of the friction factor, just remember that Hazen-Williams is only used for Turbulent flow. It would look ugly however as the \(\frac{V^2}{2g}\) portion would not cancel out. Not recommended!

While studying through material for this I realized I have forgotten the conversion between gallons per minute and cubic feet per second. I need to look this up and am considering crafting up my own table or chart of conversion factors.

Note:

1 cfs = 448.8 gpm

For those of you that also don’t know… the inside cover of the CERM has an excellent conversion table. I have not pored over it yet but was able to find this easily and no longer really have a need to write up my own conversions table after finding it.