Process dynamics modelling for heated tank, differential equations

[maistral]

I can't seem to model this properly. This isn't an assignment, I'm just curious how this will go, lol.
So I have this tank with an incoming feed stream with temperature Ti, and an output stream T. It has a jacket where q would be modified depending on the desired output stream T.

So I assembled this:

ρ*cp*vdT/dt = q + m*cp*(T-Ti)


I can't seem to assemble the differential equation required for q.

Thanks!

 

[Chestermiller]

I can't seem to model this properly. This isn't an assignment, I'm just curious how this will go, lol.
So I have this tank with an incoming feed stream with temperature Ti, and an output stream T. It has a jacket where q would be modified depending on the desired output stream T.

So I assembled this:

ρ*cp*vdT/dt = q + m*cp*(T-Ti)


I can't seem to assemble the differential equation required for q.

Thanks!

Hey maistral. Let me understand what you are trying to do. You are looking as a well-mixed CST. The inlet temperature may be a function of time, and you are trying to control the outlet temperature. You measure the outlet temperature as a function of time, and compare it to the desired set point. The question is, based on these measurements, what strategy do you use to control q to minimize the deviation from the desired set point. Correct?

Chet

 

[maistral]

Yes. That's what I'm after, and I can't seem to set the differential equations for it :|

 

[Chestermiller]

Yes. That's what I'm after, and I can't seem to set the differential equations for it :|

You already have the differential equation. The only thing you are missing is how the heat flux is controlled to try to maintain the set point. For example, if it is a proportional controller, then q = k(T_s-T), where k is a constant of proportionality and T_s is the set point temperature.

 

[blue_raver22]

I understand your frustration with trying to model this heated tank system. It is a complex process with many variables and it can be challenging to accurately represent it in a mathematical model. However, with careful consideration and analysis, it is possible to develop a differential equation that accurately describes the behavior of the system.

One approach to modeling this type of system is to use the laws of thermodynamics, specifically the first law which states that energy cannot be created or destroyed, only transferred between different forms. In this case, the energy is being transferred through heating and cooling processes.

To develop a differential equation for the heating process, we can use the following equation:

q = U*A*(Tj - T)

Where q is the heat transfer rate, U is the overall heat transfer coefficient, A is the surface area of the jacket, Tj is the temperature of the jacket, and T is the temperature of the tank. This equation takes into account the heat transfer rate through the jacket, which is dependent on the temperature difference between the jacket and the tank.

Next, we can combine this equation with the energy balance equation you have already assembled:

ρ*cp*vdT/dt = q + m*cp*(T-Ti)

By substituting the value of q from the first equation into the second equation, we can develop a differential equation that includes the heating process:

ρ*cp*vdT/dt = U*A*(Tj - T) + m*cp*(T-Ti)

From here, we can continue to refine and modify the equation to accurately represent the behavior of the system. This may involve incorporating additional variables or terms, depending on the specific details and complexities of the system.

In conclusion, while it may be challenging to model the heated tank system accurately, with careful consideration and application of thermodynamic principles, it is possible to develop a differential equation that can effectively describe the dynamics of the system. I hope this helps you in your modeling process. Good luck!