To calculate flowrate in a pipe the calculation generally requires that you know the values of the "primary variables, inlet pressure, outlet pressure, the inside pipe diameter and the length of pipe, Q, Pi, Po, D, L, respectively, along with several other "minor variables", the pipe's roughness, viscosity of the fluid, etc. Leaving the minor variables aside, if you know 4 of the 5 primary variables, you can calculate the unknown variable. You can generalize the equation for a pipe by using dP = (Po-Pi) or the equivalent in terms of head, H, dH = Ho-Hi.
Calculating flows in a network requires similar knowledge of each pipe. For each pipe in the network you must know 4 of the Q, Pi, Po, D, L, or, 3 of the 4 variables, Q, dP, D, L, or Q, dH, D, L. If a pump is included, you must use an equivalent expression for the pump relating the pressure (or head) increase with the flowrate, instead of the pressure (or head) decrease for a pipe.
To calculate flowrates in your network, generally you would have to know the inlet pressure (or head) at the pump inlet.
The pump, where the equation is for dH of the pump relating to Q of the pump, the equation of the pump curve. And a flow loss equation for each pipe in the network, also an equation where dH proportional to Q for each pipe. Therefore you must also know either the outflow of each pipe out of the network, or alternatively the outlet pressure (or head) at each pipe outlet from the network.
Do you have an inlet pressure at the pump suction and do you know the outlet pressures at each pipe's discharge point from the network?
If you have those values, then you can use an iterative procedure to find the flowrate in each pipe, where the pump suction pressure + pump added pressure - pressure loss along the pipes to any outlet must equal the discharge pressure. There are two other equations you must employ for networks. At all junctions, the pressures must be equal at all connecting pipes and the flows into the junction must equal the flow out of the junction.
