Refilled Leaking container ODE

[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

 

[blue_raver22]

Your equation looks correct for a cylindrical container with a hole at the bottom being refilled at a constant rate. However, there are a few things to consider when setting up a differential equation for this scenario:

1. The equation assumes that the height of the water (h) is changing continuously and smoothly, which may not be the case in real life. The water level may jump or fluctuate due to disturbances or turbulence.

2. The equation also assumes that the hole at the bottom is small enough that the water flow can be approximated as a point source. In reality, the size and shape of the hole can affect the rate of water flow.

3. The equation does not take into account the effects of air pressure on the water flow. If the container is sealed, the air pressure inside will increase as the water level rises, which can affect the rate of water flow.

Overall, your equation is a good starting point for modeling the scenario, but it may need to be adjusted or refined depending on the specific conditions and assumptions of the experiment.