It is the roughness k-not f-that is a property of the pipe. While f behaves well, the roughness can take on negative values in the parallel pipe case. Therefore, only solutions for D will be developed.

The other problem with the Darcy-Weisbach equation is that D and f are not uniquely related and depend on the Reynolds number, which is a function of velocity. So the question that must be first answered is, Which value of f should be used in the equations? This is especially tricky when the individual pipes have different values of k. First, a velocity of 1 m/s will be used as a reference velocity to calculate Reynolds number for the individual pipes. Second, an iterative solution must be used to solve for D.

That is

1. Pick a D and k based on the dominant pipe.
2. Calculate f for the resultant pipe using Swamee-Jain formula.
3. Use that f for fr in the equations below.
4. Check if Dr is close enough to D used to calculate f.
5. Repeat until convergence.

The Swamee-Jain equation is:

where

v must be selected so that the units cancel. Typical values are 1.00e-6 m²/s or 1.088e-5 ft.²/s.

Series Pipes

Parallel Pipes