Unsteady state problem - draining from a tank

[_Greg_]

Homework Statement

The flowrate leaving the tank is regulated by a partially open valve, You may assume that the pressure drop in the exit pipe system is across the valve itself.

The cross sectional area is 1m². At steady state, the water level inside the tank is measured to be 1m for a feed stream flowrate of 0.5 m³/min.
Since the water into the tank has a temperature of 20*C, then the temperature out is also 20*C (assuming no heat losses)

Develop a MathCAD solution to find how the level and temperature changes when the flow and temperature of the feed stream changes

Question What equation do I need for this?

I have one for the liquid level but don't know how to combine it with temperature, some sort of energy balance.

Homework Equations

I have this, describing the liquid level in the tank

dh/dt = 1/A (F₁+F₂-k(sqrt h))

The Attempt at a Solution

I can't do it , all I could get is the above equation

 

[KataKoniK]

. Any help would be appreciated.
To solve this problem, we can use the energy balance equation for the tank. This equation takes into account the changes in both liquid level and temperature.

The energy balance equation for the tank is as follows:

dE/dt = m_in * (h_in - h_out) - W + Q

Where:
- dE/dt is the rate of change of energy in the tank
- m_in is the mass flow rate of the feed stream
- h_in and h_out are the enthalpies of the feed stream and the outflow, respectively
- W is the work done by the valve (which is equal to the pressure drop times the flow rate)
- Q is the heat transfer rate (assuming no heat losses, this term will be zero)

We can rearrange this equation to solve for the outflow enthalpy (h_out):

h_out = h_in - (dE/dt + W)/m_in

Now, we can substitute this equation into the continuity equation you mentioned in your post:

dh/dt = 1/A (F1+F2-k(sqrt h))

Where:
- dh/dt is the rate of change of liquid level
- A is the cross sectional area of the tank
- F1 is the inflow rate (equal to m_in)
- F2 is the outflow rate (equal to m_out)
- k is the valve constant (which depends on the valve opening)

By combining these two equations, we can solve for the changes in liquid level and temperature as the feed stream flow rate and temperature change. You can use MathCAD to solve this system of equations and plot the results.

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!