A dynamic model for a tank with jacket cooling or heating. The mathematical representation for the fluid in the tank is the same with \(T_{\mathrm{S}}\) replaced by \(T_{\mathrm{C}}\) in Example 2.15 in the text,

\[ho c_{p} V \frac{d \bar{T}}{d t}=\dot{m} c_{p}\left(T_{\mathrm{i}}-T_{\mathrm{e}}\right)+U A\left(T_{\mathrm{c}}-\bar{T}\right)\]

where \(T\) is the average temperature in the tank.

An additional term due to heat generation can be added. This is \((-\Delta H)\left(-R_{\mathrm{A}}\right) V\). However, this makes the model non-linear, and one also needs to set up a mass balance for the reacting species. The concepts and the method of analysis and model formulations are the same, except that more equations are involved and the numerical solution can be a challenging problem as well.

For the coolant we have

\[\left(ho c_{p} V\right)_{\mathrm{j}} \frac{d \bar{T}_{\mathrm{j}}}{d t}=\dot{m}_{\mathrm{j}} c_{p, \mathrm{j}}\left(T_{\mathrm{ji}}-T_{\mathrm{je}}\right)-U A\left(T_{\mathrm{c}}-\bar{T}\right)\]

Note that the sign on the last term is now changed since the jacket is losing heat.

The output variables are the vessel and jacket temperatures. The fluid in the jacket is assumed to be well mixed, which might not be a good assumption.

Set up a case-study problem involving MATLAB simulation and test for some chosen parameter values. Consider both reacting and non-reacting systems.

Example 2.15:

Develop a dynamic model for a heating of a fluid in tank. Heat is supplied from a jacketed vessel with steam heating. The transfer rate across the system may be assumed to be UA (T_s − T ), where U is the overall heat transfer coefficient.
Here T_s is the steam temperature and T is the process fluid temperature in the tank. The tank is assumed to be well mixed in this case.