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

In summary, for the physical problem of a tank partially filled with water and a pipe feeding water at a variable flow rate and a drain pipe with a computer controlled valve, a model can be written as dV/dt = fin(t) - k*V(t), where fin(t) is the inflow rate and k is the constant of proportionality. To keep the tank at a constant desired volume, the value of k should be Qin/V. If the inflow rate is periodic, such as a sinusoidal function, the constant of proportionality can be set as sinusoid(t)_in/V.
  • #1
takbq2
32
0
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 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?
 
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  • #2
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.
 
  • #3
dV/dT = c(fi(t)-fo) ?

On second thought,

dV/dt = fi(t) - c*fo(t) ??

seems better, can someone verify this or otherwise please? Thanks!
 
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  • #4
What does "constant of proportionality" mean? Is it h, the target height of water?
 
  • #5
takbq2 said:
dV/dT = c(fi(t)-fo) ?

On second thought,

dV/dt = fi(t) - c*fo(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.
 
  • #6
uart said:
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 = fin(t) - k*V(t)

which is not right? >=\
 
  • #7
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??
 
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  • #8
takbq2 said:
I'm sorry, I'm confused. They are proportional so f_out = kV, thus,

dV/dt = fin(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.
 
  • #9
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
 
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What is the Easy Tank Model?

The Easy Tank Model is a mathematical model used to describe the behavior of a liquid within a tank. It is commonly used in engineering and science to analyze and predict the outflow of liquid from a tank.

How does the Easy Tank Model work?

The Easy Tank Model is based on the principle that the outflow of a tank is directly proportional to the volume of liquid within the tank. This means that as the volume of liquid decreases, so does the outflow rate, and vice versa.

What are the assumptions of the Easy Tank Model?

The Easy Tank Model assumes that the tank is completely filled with liquid, there are no external forces acting on the liquid, and the tank has a constant cross-sectional area. It also assumes that the liquid is incompressible and that there are no changes in the density of the liquid over time.

How accurate is the Easy Tank Model?

The accuracy of the Easy Tank Model depends on how closely the assumptions hold true in a given situation. In real-world scenarios, there may be factors that can affect the outflow rate such as changes in atmospheric pressure or temperature. Therefore, the model should be used with caution and should be validated with experimental data.

What are the limitations of the Easy Tank Model?

The Easy Tank Model is a simplified representation of a complex system and does not take into account factors such as turbulence, viscosity, and other external forces. It is also limited to tanks with a constant cross-sectional area and cannot be applied to tanks with irregular shapes. Additionally, it does not consider the effects of fluid dynamics and cannot accurately predict the outflow rate in situations where the liquid is not completely still.

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