# Determining the headloss according to Darcy-Weisbach with pipes in parallel and series

Given the following situation:

Pipe A and B have a known and identical length and diameter, and are in parallel. I'm assuming the flow trough pipes A and B are also identical, as is the roughness of the pipes. Pipe C also has a known length and diameter (which is different from the diameter of pipe A/B).

I'm interested in the total head loss at the end of Pipe C. I am aware of the following Darcy-Weisbach equations that are relevant in this case (parallel and series):

Here the total headloss is the sum of the headloss per pipe, placed in series.

Here the total headloss over set of 2 parallel pipes is equal to the headloss of one of them.

But, how do I combine these two equations to retrieve the total headloss over pipes A, B and C?

Of course, the total headloss is equal to the pressure difference that is driving the flow. For example, if the tank is filled with a liquid to depth H, and if the end of the pipe C is opened to the atmosphere, the pressure driving the flow due to the elevation difference is known. In other words, you usually know the total headloss because you know the pressure difference, and the calculation is to find the flow rate. You can determine that from the first equation because you are assuming that $$Q_A=0.5Q_C$$.