Easy Tank Model? Outflow of tank proportional to volume of tank.

[takbq2]

1. Suppose we have a tank partially filled with water. There is a pipe
feeding water to the tank as a variable
ow rate and there is also a drain
pipe with a computer controlled variable valve hooked to a sensor in
the tank. The valve opens exactly enough to let water drain from the
tank at a rate proportional to the volume of the tank. The program
allows for us to set one number: the constant of proportionality. Write
a model for this physical problem. Be sure to dene all the variables
in your model. (b) Suppose the in
ow rate is constant. How should
the proportionality constant in the control mechanism be set to keep
the tank near a constant desired volume? (c) Suppose the in flow rate
is periodic. To be denite let's say the flow rate is sinusoidal and
known exactly, how should the constant of proportionality be set for
the controller to best keep the tank at a constant desired volume.

Homework Equations

flow in = flow out (if desired in this case)

The Attempt at a Solution

I call f₀ the flow out and f_i the flow in.
f_i varies with, say, t.

f₀ is proportional to V, the volume of the tank. The volume of the tank is: V = the volume initially in the tank, V_i, + f_i(t) - f₀.

f₀ is proportional to V by c., but in my statement about the V, f₀ is on that side so it can't really be in the model. If I could get help figuring out the model, I could answer parts (b) and (c) pretty easily it seems.

My first proportion was f₀=Vc thus,
f₀ = (V_i+f_i(t))c

But I know this can't be right because in answering part b, f_i would need to be as close as possible to f₀, but any amount for c would mean that the amount out was equal t the entire amount in the tank.

help?

 

[uart]

Use the fact that the "inflow minus outflow" is equal to the rate of change in volume to write a simple DE (differential equation) for the system. The DE is the system model.

 

[takbq2]

dV/dT = c(f_i(t)-f_o) ?

On second thought,

dV/dt = f_i(t) - c*f_o(t) ??

seems better, can someone verify this or otherwise please? Thanks!

 

[rude man]

What does "constant of proportionality" mean? Is it h, the target height of water?

 

[uart]

dV/dT = c(f_i(t)-f_o) ?

On second thought,

dV/dt = f_i(t) - c*f_o(t) ??

seems better, can someone verify this or otherwise please? Thanks!

That's on the right track, but use the fact, in the problem statement, that the outflow is proportional to V so as to write your DE with just one input variable (f_i) and one state variable (V). The state variable V also happens to be the output variable in this case, which is nice.

 

[takbq2]

That's on the right track, but use the fact, in the problem statement, that the outflow is proportional to V so as to write your DE with just one input variable (f_i) and one state variable (V). The state variable V also happens to be the output variable in this case, which is nice.

I'm sorry, I'm confused. They are proportional so f_out = kV, thus,

dV/dt = f_in(t) - k*V(t)

which is not right? >=\

 

[takbq2]

Well I know it's not right. I'm just coming to the same answers over and over again because I've done it so much I can't think outside my current train of thought :S

It must be

q_in(t) = dV/dt + kV ... one final check on this please??

 

[uart]

I'm sorry, I'm confused. They are proportional so f_out = kV, thus,

dV/dt = f_in(t) - k*V(t)

which is not right? >=\

No that's the correct DE for the system.

Now for part b) you can take F_in as a constant and look at what value of "k" you require to keep V constant, say V_desired. Note that V = const means dV/dt = 0.

 

[takbq2]

thanks a lot for your help, uart. I got part b, K would = Qin/V. Working on last part.. if it is a sinusoid nothing changes, you still want amount into equal amount out, so just sinusoid(t)_in/V = k for part (c) I would think