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takbq2
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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.
flow in = flow out (if desired in this case)
I call f0 the flow out and fi the flow in.
fi 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, Vi, + fi(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 = (Vi+fi(t))c
But I know this can't be right because in answering part b, fi 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?
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.
Homework Equations
flow in = flow out (if desired in this case)
The Attempt at a Solution
I call f0 the flow out and fi the flow in.
fi 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, Vi, + fi(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 = (Vi+fi(t))c
But I know this can't be right because in answering part b, fi 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?