An Energy Balance

Although a full energy balance should take into account all the possible forms of energy (e.g. thermal, kinetic, potential, etc), in most cases, for process plant problems, energy balances are concerned solely with thermal energy. The reason for this is that thermal energy changes in a typical process are larger by one or more orders of magnitude than changes in the other energy forms. Omitting changes in the non-thermal energies is a simplification, but usually doesn’t significantly affect the results obtained.

In this exercise, we’ll be doing an energy balance on a stirred tank heater, where the heat is provided by an electrical heater which delivers heat at a fixed rate of Q (W). Cool liquid flows into the tank at temperature T_in. We’ll make the following simplifying assumptions:

Perfect mixing
Perfect level control
Constant and equal densities
Constant liquid specific heat capacity

To form the balance we first need to form the accumulation term. Remember that this is simply the first derivative with respect to time of the amount of ‘stuff’ held up in the capacity. In this case, the ‘stuff’ is energy and the capacity is the thermal capacity of the liquid held in the tank. Have a go at forming the accumulation term yourself

We need to define enthalpies in relation to some reference temperature - this is constant, and its first derivative will be zero.

Now try forming the complete balance. Energy will flow into and out of the system in the liquid flows, but will also be flowing into the system through the electrical heater.

The balance can be made a little more realistic by modelling heat transfer from the heating coil - usually tanks are heated by steam rather than electricity. In steam heating systems, it is usual to use saturated steam and a steam trap (a special device on the exit of the coil which only allows condensate to pass through). This means that the internal temperature of the coil will be at a close to constant temperature ( the condensation temperature) which will be dependant on the pressure inside the coil. Heat transfer can then be modelled using a standard overall heat transfer relationship. Have a go at modifying the balance to model a steam heater.

With heat transfer equations you need to be careful that you get the signs in the correct directions - net heat only flows from hot stuff to cold stuff!