THERMAL SYSTEMS
Thermal systems are encountered in chemical processes, heating, cooling and air conditioning systems, power plants, etc. Thermal systems have two basic components: thermal resistance and thermal capacitance. Thermal resistance is similar to the resistance in electrical circuits. Similarly, thermal capacitance is similar to the capacitance in electrical circuits. The across variable, which is measured across an element, is the temperature, and the through variable is the heat flow rate. In thermal systems there is no concept of inductance or inertance. Also, the product of the across variable and the through variable is not equal to power. The mathematical modelling of thermal systems is usually complex because of the complex distribution of the temperature. Simple approximate models can, however, be derived for the systems commonly used in practice.
Thermal resistance, R, is the resistance offered to the heat flow, and is defined as:
where T1 and T2 are the temperatures, and q is the heat flow rate.
Thermal capacitance is a measure of the energy storage in a thermal system. If q1 is the heat flowing into a body and q2 is the heat flowing out then the difference q1 − q2 is stored by the body, and we can write
An example thermal system model is given below.
Example 2.14
Figure 2.25 shows a room heated with an electric heater. The inside of the room is at temperature Tr and the walls are assumed to be at temperature Tw . If the outside temperature is To, develop a model of the system to show the relationship between the supplied heat q and the room temperature Tr .
Solution
The heat flow from inside the room to the walls is given by
Figure 2.26 shows a heated stirred tank thermal system. Liquid enters the tank at the temperature Ti with a flow rate of W . The water is heated inside the tank to temperature T . The temperature leaves the tank at the same flow rate of W . Derive a mathematical model for the system, assuming that there is no heat loss from the tank.
Solution
The following equation can be written for the conservation of energy:
where C is the thermal capacity, i.e. C = ρ VC p and V is the volume of the tank. Substituting (2.114)–(2.116) into (2.113) gives
