Question

Why is there a negative sign in Fourier's Law?

Answer 1

The flow of heat via conduction through a brick wall or through a pane of glass is an example of heat flow in one dimension. We can let the x-axis denote the direction of heat flow and consider the flow of heat between the faces of a flat plate of thickness Δx and face area A (Figure 23.1). Let ΔT denote the temperature difference that is maintained across the thickness Δx. This temperature difference is what gives rise to the heat flow. Let q denote the rate of heat flow across the slab (q is measured in watts). We might expect that the thicker the slab, the smaller q. For small
Δ
x
. experiment bears out this expectation and shows that


Figure 23.1. Heat flow by conduction. For small ΔT and Δx the rate of heat flow is proportional to A ΔT/Δx. The rates of heat flow across the two faces are equal provided that the temperatures T and T + ΔT are maintained constant. For heat flow in the direction shown, ΔT is negative.
q
∝
1
Δ
x
Likewise, we would expect q to increase if the temperature difference increases. Again, experiment shows that our expectation is correct. For small ΔT
q
∝
Δ
T
Finally, experiment confirms the expectation that the rate of heat flow should be directly proportional to the face area:
q
∝
A
Thus, for small ΔT and small Δx, the rate of heat flow is proportional to A ΔT/Δx:
q
∝
A
Δ
T
Δ
x
This relation can be converted into an equation by introducing a proportionality factor, k, called the thermal conductivity.
(23.1)
q
=
−
k
Δ
T
Δ
x
Equation 23.1 is known as Fourier's law. *The minus sign accounts for the fact that heat flows from a higher temperature to a lower temperature. Thus, for example, in Figure 23.1, heat flows in the positive x-direction. For this to happen the temperature must decrease as x increases. Thus, ΔT is negative in Figure 23.1. The units of k are W/m · C°. The value of the thermal conductivity depends primarily on the physical composition of the material. A material with a large value of k would be described as a good thermal conductor. Metals are among the best conductors. A material with a small value of k would be classified as a poor thermal conductor, or a good thermal insulator. Gases and various porous plastics such as styrofoam are among the best insulators. Table 23.1 presents values of the thermal conductivities of many different materials.
Table 23.1. Thermal conductivity *
Material k(W/m · C°) Material k(W/m · C°))
Gases Solids
Air 0.0237 Glass 0.760
Carbon dioxide 0.0145 Ice 2.21
Oxygen 0.0246 Steel 16.3
Hydrogen 0.167 Iron 72.7
Helium 0.142 Aluminum 228
Methane 0.0305 Copper 386
Silver 417
Liquid
Water 0.566
Solids
Corrugated
cardboard 0.064
Asbestos 0.150
Wood 0.190
*
Values refer to a temperature of 0°C and a pressure of 1 atm.)