Our aim in this chapter is to study engineering models for unsteady heating and cooling processes. These models are used to estimate rates of temperature change, heat fluxes, and transient temperature distributions.
In the steady conduction problems examined in the previous chapter, thermal resistance or thermal conductivity was the physical variable of greatest importance. In unsteady heat transfer processes, the heat capacitance of the objects involved must also be considered. Modeling unsteady heat conduction hinges on recognizing the heat capacitances and thermal resistances present.
In some situations, a thermal resistance external to a solid object is much larger than the internal conduction resistance of the object. In such cases, it is possible to neatly separate the main thermal resistance from the main thermal capacitance; such lumped capacity situations are characterized by a small value of the Biot number.
In other circumstances, the conduction resistance within a body may be as large or larger than that outside the body (Biot number of order 1 or greater). In these cases, the thermal resistance of a body is distributed throughout, as is the thermal capacitance, and it is usually necessary to solve a partial differential equation to determine the precise temperature distribution within the body. Even so, ballpark estimates of the temperature response can be obtained by using an appropriate approximation of the body's internal resistance (see Section 3.2).
In this chapter, we first examine simple resistance–capacitance models for unsteady thermal response. These estimates extend the usual lumped capacity model. Then, we consider adaptations of analytical conduction solutions for the response of bodies at large time, using series and chart results. Short time models, for semi-infinite behavior, are discussed next. Scaling analysis of conduction problems is introduced in the following section; such analyses are used here mainly as a tool for simplifying complex situations. These ideas are explored further in the end-of-chapter problems.
Physical Properties in Unsteady Conduction
The thermal energy storage of an object, per unit weight and per unit temperature change, is measured using its specific heat capacity, J/kg·K, when no phase change occurs.