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Refilled Leaking container ODE

  1. Feb 6, 2010 #1
    I have to set up a differential equation for a leaking cylindrical container that is being refilled at a constant rate. Its leaking from the bottom, refilled from the top, and starts empty.
    Wondered if someone could check if what i have done so far is right..?
    h=height of water
    A=cylinder cross section
    a=hole cross section
    v=speed of water out = Sqrt(2gh) using Torricelli’s Law
    b=rate of water going in

    Rate of change of volume dV/dt= -av+b = A*dh/dt
    Sub in v = Sqrt(2gh):
    A*dh/dt=-a*sqrt(2gh)+b -----> dh/dt = (-a*Sqrt(2gh)+b)/A

    Any errors?!
  2. jcsd
  3. Feb 7, 2010 #2
    Seems fine to me. As long as the formula for v is correct.

  4. Feb 7, 2010 #3
    Thanks Torquil.
    I have to solve this using Eulers method:
    yn+1 = yn + hf(xn,yn)
    xn+1 = xn + h
    where h is a step in x, to be chosen.
    y=height of water(original h)
    f(x,y) = dh/dt

    As there is no t on the rhs of the differential equation will this formula work?

    n=o : yn=0(t=o)=0
    n=1 : yn=1 = (yn=0) + h((-a*Sqrt(2g(yn=0))+b)/A)
    n=2 : yn=2 = (yn=0) + h((-a*Sqrt(2g(yn=0))+b)/A) + h((-a*Sqrt(2g(yn=1))+b)/A)

    I want to make a plot of y against x (height water vs time) at the end, can i do that using this method as there is no t on rhs...?!

    Thanks again!
  5. Feb 7, 2010 #4
    Yes that seems correct. To plot the height vs time, plot the points y_n vs h*x_n, for all values of n. Try doing everything for various small values of h to determine if the result has converged. Maybe you can even perform the integral analytically and get an exact result to compare with your computer simulation.

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