Tank Change in Flow Process Modeling

[PaxFinnica96]

Hi All,
I'm really struggling with the below question - I'm not sure if I've taken the correct approach to determining the model as there may be an easier way to do this?

Any help in much appreciated, please let me know if I have submitted this template correctly as this if my first time!
Cheers

1. Homework Statement

a. FIGURE 2 shows two cylindrical tanks interconnected with a pipe which has a valve that creates a constant resistance to flow of Rf when fully open. The height of liquid (of density ρ) in the first tank is hin and the second tank hout. The cross-sectional area of the first tank is Ain m2 and the second tank Aout m2.

Produce a mathematical model of the process to determine the change in height of fluid in the second tank when the valve is open.

b. Determine the time constant for the system.

Homework Equations

The flow rate of liquid through the valve is given by:[/B]

Q = 1/Rf (Pin - Pout)

Where; Q = flow rate in m^3 s^-1

Pin = pressure due to height of liquid in first tank (Pa)

Pout = pressure due to height of liquid in second tank (Pa)

The Attempt at a Solution

a[/B]

b

Giving a time constant of 63.2% for the system.

 

[Chestermiller]

You don't need the final differential equation to determine $h_{out}$. You have two algebraic equations in two unknowns:
$$A_{in}h_{in}+A_{out}h_{out}=A_{in}h_{in,0}+A_{out}h_{out,0}$$
and $$h_{in}-h_{out}=(h_{in,0}-h_{out,0})e^{-t/\tau}$$

 

[PaxFinnica96]

So instead of differentiating at the end I should just leave the equations as they are like below?

 

[Chestermiller]

So instead of differentiating at the end I should just leave the equations as they are like below?

View attachment 239115

No. You solve the two equations in two unknowns for the change in height in the second tank.

 

[PaxFinnica96]

No. You solve the two equations in two unknowns for the change in height in the second tank.

Ok, that's great I think I've managed to solve the two equations and demonstrate the proof but I'm still struggling with how best to determine the time constant - would it just be a simple matter of rearranging the model equation (the one I've just solved from the two above equations) for tau?

 

[Chestermiller]

Ok, that's great I think I've managed to solve the two equations and demonstrate the proof but I'm still struggling with how best to determine the time constant - would it just be a simple matter of rearranging the model equation (the one I've just solved from the two above equations) for tau?

You already provided an equation for tau.