Any Errors in My Differential Equation for a Leaking and Refilling Container?

[SidVicious]

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?!
Thanks!

 

[torquil]

Seems fine to me. As long as the formula for v is correct.

Torquil

 

[SidVicious]

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)
x=t
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)
etc..

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!

 

[torquil]

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.

Torquil