(adsbygoogle = window.adsbygoogle || []).push({}); 1. Suppose we have a tank partially filled 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 dene 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 denite 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.

2. Relevant equations

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

3. The attempt at a solution

I call f_{0}the flow out and f_{i}the flow in.

f_{i}varies with, say, t.

f_{0}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) - f_{0}.

f_{0}is proportional to V by c., but in my statement about the V, f_{0}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 f_{0}=Vc thus,

f_{0}= (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 f_{0}, but any amount for c would mean that the amount out was equal t the entire amount in the tank.

help?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Easy Tank Model? Outflow of tank proportional to volume of tank.

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