How to Calculate the Proportionality Constant for a Two Tank System

[Merk]

Greetings! We are designing a very basic two tank system with water flowing in one tank, and out the other. As illustrated here:

We have come up with two equations, which should be pretty self-explanatory. W is waterflow, A is area, h is the water level and so forth.

As you can see, we assume the waterflow out is proportional with the water level. The problem is, we have no idea how to calculate this value k. Any suggestions?

Thanks!

 

[Jmf]

I did something very similar recently for an experiment in control engineering.

The system is actually nonlinear, and gives rise to equations like:

A_1 \dot{h_1} = Q_{in} - Q_1
A_2 \dot{h_2} = Q_1 - Q_{out}

where:

Q_1 = \sigma_1 A_1 \sqrt{2g(h_1 - h_a)}
Q_{out} = \sigma_2 A_2 \sqrt{2g(h_2 - h_b)}

the Q's are the flowrates, g is the gravitational constant, A the areas and h_a and h_b the height of the orifices at the exits of the first and second tank, respectively. Sigma 1 and 2 are termed the 'orifice constants' and are generally known, or found experimentally.

Since water cannot flow back from tank 2 to tank 1 etc. it must also be the case that:

Q_{in} \geq 0
Q_1 \geq 0
Q_{out} \geq 0

Which are also nonlinearities. The equations need to be linearized about an equilibrium point (for some value of Q_in) - simply assuming that the flow out of each tank is proportional to the height of water in the tank, like you have, will give a very poor model of the system.

Instead you need to define something like:

z_1 = h_1 - h^0_1
z_2 = h_2 - h^0_2

where the second terms are the (steady-state) equilibrium values of h for a given Q_in.

You should then find a system that is linear in the purtubations z_1 and z_2.

Hope this helps.