5.1. DERIVATION OF BASIC EQUATION

5.1A. Introduction

In Chapter 4 we considered various heat-transfer systems in which the temperature at any given point and the heat flux were always constant over time, that is, in steady state. In the present chapter we will study processes in which the temperature at any given point in the system changes with time, that is, heat transfer is unsteady state or transient.

Before steady-state conditions can be reached in a process, some time must elapse after the heat-transfer process is initiated to allow the unsteady-state conditions to disappear. For example, in Section 4.2A we determined the heat flux through a wall at steady state. We did not consider the period during which one side of the wall was being heated up and the temperatures were increasing.

Unsteady-state heat transfer is important because of the large number of heating and cooling problems occurring industrially. In metallurgical processes it is necessary to predict cooling and heating rates for various geometries of metals in order to predict the time required to reach certain temperatures. In food processing, for example, the canning industry, perishable canned foods are heated by immersion in steam baths or chilled by immersion in cold water. In the paper industry wood logs are immersed in steam baths before processing. In most of these processes the material is suddenly immersed in a fluid of higher or lower temperature.

5.1B. Derivation of Unsteady-State Conduction Equation

To derive the equation for unsteady-state conduction in one direction in a solid, we refer to Fig. 5.1-1. Heat is being conducted in the x direction in the cube Δx, Δy, Δz in size. For conduction in the x direction, we write

Equation 5.1-1

The term ∂T/∂x means the partial or derivative of T with respect to x, with the other variables, y, z, and time t, being held constant. Next, making a heat balance on the cube, we can write

Equation 5.1-2

The rate of heat input to the cube is

Equation 5.1-3

Also,

Equation 5.1-4

The rate of accumulation of heat in the volume Δx Δy Δz in time ∂t is

Equation 5.1-5

The rate of heat generation in volume Δx Δy Δz is

Equation 5.1-6

Substituting Eqs. (5.1-3)–(5.1-6) into (5.1-2) and dividing by Δx Δy Δz,

Equation 5.1-7

Letting Δx approach zero, we have the second partial of T with respect to x or ∂²T/∂x² on the left side. Then, rearranging,

Equation 5.1-8

where α is k/pc_p, thermal diffusivity. This derivation assumes constant k, p, and c_p. In SI units, α = m²/s, T = K, t = s, k = W/m · K, p = kg/m³, = W/m³, and c_p = J/kg · K. In English units, α = ft²/h, T = °F, t = h, k = btu/h · ft·°F, ρ = lb_m/ft³, = btu/h · ft³, and c_p = btu/lb_m · °F.

For conduction in three dimensions, a similar derivation gives

Equation 5.1-9

In many cases, unsteady-state heat conduction is occurring but the rate of heat generation is zero. Then Eqs. (5.1-8) and (5.1-9) become

Equation 5.1-10

Equation 5.1-11

Equations (5.1-10) and (5.1-11) relate the temperature T with position x, y, and z and time t. The solutions of Eqs. (5.1-10) and (5.1-11) for certain specific cases as well as for the more general cases are considered in much of the remainder of this chapter.