In the modelling section, one of the models I derived for the stirred tank heater was:
We'll now solve this for step changes in Tin and Ts. You might think that looking at changes in Ts is a bit pointless, but this isn't the case. In most steam heated systems, condensing steam is used as the heating medium. The pressure of the steam in the jacket or shell-side is regulated by the control valve - since the steam is condensing the pressure effectively sets the temperature.
Assuming the following nominal conditions for the heater:
You should be able to show that the heat balance becomes:
By inspection we can see that this is a first-order differential equation with a time constant of 7.5 minutes, a steady-state gain on inlet temperature changes of 0.375, and a gain on steam temperature changes of 0.628. This means that changes in the tank temperature due to inlet or steam temperature changes will take around 37.5 minutes to complete (5 time constants). For every degree the inlet steps by, the steady-state tank temperature will change by 0.375 degrees, and for every degree the steam temperature changes, the tank temperature will change by 0.628 degrees when the system reaches steady state.
The numerical solution can be found here. In this case, because the equations are linear when solving the problem we specified (they wouldn't be if, for example, F was allowed to change), the analytical solution will be exact and will match the numerical results.