Control Theory/flow of licuids in tanks - bernoully and Darcy

[ester_franke]

Homework Statement

Two tanks with water, connected in the bottom by a pipe (known area (a) and length (L)). One of the tanks have known bottom area (A).

a= 0,03m
A= 1 m
L = 40 m

Input: hight of tank1 (h1)
Output: hight of tank2 (h2)

I am asked to:
- Draw block scheme (H1= tank1, H2=pipe, H3=tank2)
- Find the transference function

I am also told to use Darcy's formula (with level difference 0,5 m) to find the "k" of the pipe.

Homework Equations

Formula Darcy (given by professor):
Q = k((0,5*A)/L)

The Attempt at a Solution

First of all the Q at the bottom of the tanks will be:
q=a*srt(2gh) due to Bernoullli
I do a linealization and find my q(s)/h(s) = (-a/2)*srt((2g)/h_0)


I am thinking that the Q trough the pipe is constant. Therefore the two qs should be alike. However, that only states that the levels of the tank will be alike aswell, (which they off course will, but after a while).

However, after this I'm stuck. I've been trying for 1,5 weeks now! This is my first homework in control Theory ever, and it feels like I'm missing something..?!

Thanks a lot for your help.

//L

 

[anathema]

Dear L,

Thank you for your forum post. I would approach this problem by first drawing a block diagram, as suggested. This will help us visualize the system and understand the relationships between the different variables.

I have attached a sketch of the block diagram below. As you can see, the two tanks (H1 and H3) are connected by a pipe (H2). The input to the system is the height of tank 1 (h1) and the output is the height of tank 2 (h2).

To find the transfer function, we can use the fact that the flow rate through the pipe is constant. This means that the flow rate at the bottom of tank 1 (q1) is equal to the flow rate at the bottom of tank 2 (q2). We can express this mathematically as:

q1 = q2

Using the formula given by your professor, we can write:

k((0.5*A)/L) = a*sqrt(2gh2)

Where k is the "k" of the pipe, A is the bottom area of tank 1, L is the length of the pipe, a is the area of the pipe, g is the acceleration due to gravity, and h2 is the height of tank 2.

Solving for h2, we get:

h2 = (k^2*(0.5*A)^2*g)/(a^2*L^2)

Therefore, the transfer function is:

h2/h1 = (k^2*(0.5*A)^2*g)/(a^2*L^2)

To find the "k" of the pipe, we can use Darcy's formula, as you mentioned. This formula relates the flow rate through a pipe to the hydraulic gradient (the pressure difference divided by the length of the pipe). We can write this as:

Q = k*(i*L)

Where Q is the flow rate, k is the "k" of the pipe, i is the hydraulic gradient, and L is the length of the pipe.

In our case, the hydraulic gradient is 0.5 m (given by the level difference between the two tanks). So we can write:

Q = k*(0.5*L)

We know that the flow rate through the pipe is constant, so we can calculate it using the flow rate at the bottom of tank 1 (q1):

Q = q1 = a*