Hydraulics, flow rate, and tubing size questions Here are a couple assumptions / things I believe to be true:


*

*Pumps create a certain pressure in a piping circuit.

*Pressure is like voltage and constant across the piping circuit, with differences in piping size increasing flow rate by reducing friction loss ie resistance

*Given constant pressure, flow rate is variable as a fuction of pipe radius per the Poiseuille law.
This is where I am confused:
Imagine the case that a 2" pump outlet, runs for 1 foot of 2" piping and then connects in with a 4" pipe for reduced friction.
So if the pressure across both pipes (the system) are the same, the gpm for the larger pipe is something like 16x bigger for the larger pipe than the smaller pipe. The larger pipe is fed by the smaller pipe, so where does this water come from?
Understand what I am getting at?
One of my assumptions seems erroneous.
 A: Two points to make
First - Poiseuille flow is only valid for laminar flow, i.e. very low Reynolds numbers (so usually not true); but let's move past that
Second - When you have two sections of tube of different diameter, and you have an incompressible fluid that goes through both of them, then the flow rate will adjust so that the total pressure difference across the two pieces of tube equals the difference between the pump outlet pressure and the environment. The result is that the majority of the pressure difference will be across the narrower tube (assuming they both have the same length).
This is exactly analogous to the situation of two resistors in series, with a given voltage applied. The current through the resistors must be the same; the voltage drop across the higher resistance will be greater. If you have resistor $R_1$ and $R_2$, then
$$V_1 = V\frac{R_1}{R_1+R_2}$$
Your mistake is the assumption that the two pieces of pipe "see" the same pressure difference. That would be true if they were in parallel - but not if they are in series. By the time the water has passed through the narrower tube, the pressure has dropped.
