Maximizing Flow Rate in a Water Pipe: Accounting for Pipe Losses

[soronemus]

Homework Statement

What is the maximum flow rate that can be seen in a water pipe where city water is supplied to a building. The pressure behind the water is 50psi, and the inner diameter of the pipe is 2".

Homework Equations

bernoulli's principle

The Attempt at a Solution

I am more over verifying that my process is correct in solving this. I assumed no pipe losses. I used the bernoulli equation. The first point of the bernoulli equation I estimated as the surface of a pond, ocean, or some infinite water source where v=0 (the water elevation does not change) to cancel out the velocity term for point 1. The pressure on top of the surface is 50psi.

The elevation for both points I assumed to be equal, which cancels out all terms for point 1 other than P/density.

For point 2 I used the exit of the pipe where the fed water is first exposed to the atmosphere (p=0). Since the elevation is the same this leaves only the following equation:

Pressure_1/density=V_2^2/2
V=Q/A
therefore
Pressure/density=(Q/A)^2/2
The only unknown in this equation is Q, so it can be solved for.

My question is: is this a valid solution if I am ignoring pipe losses?

 

[haruspex]

In a pipe of constant cross section, a pressure difference between the ends implies a force difference, yet the flow rate must be constant. This proves the question is all about viscous drag and Bernoulli does not help.

 

[OrangeDog]

You can't use Bernoulli, and I will show you why:

Create a balance between the ends of the pipes, 1 designating the front of the pipe and 2 designating the end.

You will have:
P1,T1,V1,rho1,Z1 and P2,T2,V2,rho2,Z2

You know that the temperature, density, and elevation are constant, so T2=T1, rho1=rho2 and Z2=Z1.

You can now apply the Bernoulli equation.

(P2-P1) - rho/2 (V2^2 - V1^2) + hl = 0

Immediately we have a problem. While we do know P2 - P1, we don't know V2 or V1. Additionally, if we wish to apply the Darcy friction/loss formulas (see: https://en.wikipedia.org/wiki/Darcy–Weisbach_equation) we do not know the length of the pipe or the operating Reynolds number, so we can't use these equations.

My recommendation:
Assume the flow is laminar. You know there is a certain velocity profile from laminar flow. From here you can calculate the flow rate using Q = V*A where V is the average velocity and A is the cross sectional area. You can do this if the flow is turbulent too, but you have to make some additional assumptions.