32.1 Introduction

The same kind of equation can be used to describe the fact that heat flows down a temperature gradient (see Figure 32.1),

q=–KdTdx

dT/dx is the temperature gradient in K m^− 1, q is the heat flux (the heat flowing down the temperature gradient per second per unit area) in J m^− 2 s^− 1, and K is the thermal conductivity in W m^− 1 K^− 1 (W = J s^− 1).

We have already looked at one application of thermal conductivity in Section 30.3—the conduction of heat away from the sliding surfaces between snow and skis (or snow and sledge runners). We saw another example in Section 17.4, where we used thermal insulation materials (polymer foam and mineral wool) to slow down the rate of heat transfer from the environment to liquid methane at − 162°C. K is a critical parameter in a vast range of applications, from thermal insulation in buildings to heat shields for spacecraft.

Equation (32.2) describes the one-dimensional flow of heat through a solid under steady-state conditions—the temperature at any given point within the solid does not change with time (as in Section 17.4, for example). Under nonsteady-state conditions, the temperature at any given point within the solid does change with time (see Figure 32.2), and the heat flow equation becomes

∂T∂t=λ∂2T∂x2

where t is time, and λ = K/ρC is the thermal diffusivity of the material. It has dimensions of m² s^− 1. ρ is the density in kg m^− 3, and C is the specific heat in J kg^− 1 K^− 1.

Worked Example 1

We can derive Equation (32.3) quite simply, by looking at Figure 32.2. We take a volume element of 1 × δx (the “1” is because the cross section of the conducting bar has unit area). In time δt, an amount of heat q_in δt enters the volume element from the left-hand side, and an amount of heat q_out δt leaves the volume element from the right-hand side. q_in δt > q_out δt, because the temperature gradient entering the element is greater than leaving the element. The difference in heat, q_in δt − q_out δt goes into warming up the volume element by amount δT, in other words some of the heat passing through the solid is “diverted” into heating it up. The equation for this is

qin–qoutδt=ρCδTδx

From Equation (32.2),

qin–qout=–KdTdx–dTdx–d2Tdx2δx=Kd2Tdx2δx

Substituting for (q_in – q_out) leads directly to Equation (32.3).

32.2 Thermal Conductivities and Specific Heats

Approximate experimental values for K, C, and λ are shown in Table 32.1. These are generally measured at, or fairly close to, 300 K. Since K, C, and λ can change with temperature, in any thermal design problem it may be necessary to obtain experimental data for the particular temperature range under consideration.