I would really appreciate help on the following:
Let us assume that we have n pumps pumping water through n pipes. The pumps are running at a constant power. It doesn't change.
The flow rates in the pipes are observed to be a function of combination of the pipes which are active due to hydraulic effects. The flow rates in each bore due to the combination are lower than the flow rate when each pipe is activated solely.
Is there a way to model this behaviour and calculate an expected flow rate for a various combination just based on data consisting of pipe active status and observed flow rates as follows without considering the hydraulic effects:
Pipe A|Pipe B|Pipe C|Flow_rate_A|Flow_rate_B|Flow_rate_C
1 | 0 |0 |50 |0 |0
0 | 1 |0 |0 |55 |0
0 | 0 |1 |0 |0 |13
0 | 1 |1 |0 |50 |10
1 | 1 |1 |40 |46 |9
If pumps are running at constant power, then they are running at constant flow: pumps operate based on a characteristic pump curve, and their power usage/efficiency is directly related to their flow rate and discharge pressure. If you know the power (and the measurement is accurate...), you can get the flow rate directly from the pump curve.
If you're asking what I think you're asking, you can't model the system properly without considering hydraulic effects: they literally dictate how pumps/piping systems operate. To model this system, you need to construct a network model of all pumps/pipelines. There are several software tools out there for performing these types of calculations. The overall principle is that you define the inlet/outlet pressures of the pipelines, and assume that the nodal pressures at any connections between multiple pipes ("tees") are equal.
To solve this model, you can then assume a flow rate in each pipe branch, calculate pressure loss in each pipe using an appropriate pressure loss correlation (such as the Darcy friction factor correlation), and calculate the pump discharge pressure from the pump's characteristic curve using your assumed flow rate. You then would iterate using a root-finding algorithm like the Newton method to adjust the flow through each branch until all nodal pressures are equal, and the calculated outlet pressure matches your specified outlet pressure.
I hope that this makes sense and helps!