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Archived Max flow through a pipe

  1. Jul 30, 2014 #1
    1. The problem statement, all variables and given/known data

    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".


    2. Relevant equations
    bernoulli's principle


    3. 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?
     
  2. jcsd
  3. Feb 5, 2016 #2

    haruspex

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    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.
     
  4. Feb 5, 2016 #3
    You cant 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 cant 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.
     
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