Fourier’s Law of Heat Conduction

The basic relation for conductive heat transport is Fourier’s law, which states that the heat flux q, or the flow of heat per unit area and per unit time, at a point in a medium is directly proportional to the emperature gradient at the point. In one dimension, Fourier’s law takes the form q = −kdT dy (4.1) where k is the coefficient of thermal conductivity and y is the coordinate in the direction of the temperature variation. The minus sign appears in Equation (4–1) since heat flows in the direction of decreasing temperature. With dT/dy > 0, T increases in the positive y direction, so that heat must flow in the negative y direction. Figure 4–1 is a simple example of how Fourier’s law can be used to give the heat flux through a slab of material of thickness l across which a temperature 4.2 Fourier’s Law of Heat Conduction 239

Heat flux and the local slope of the temperature profile when T(y) has nonzero curvature. difference ∆T is maintained. In this case, the temperature gradient is dT dy = −∆T l , (4.2) and the heat flux, from Fourier’s law, is q = k∆T l . (4.3) Fourier’s law applies even when the temperature distribution is not linear, as sketched in Figure 4–2. In this case, the local slope of the temperature profile must be used in Fourier’s law, and for constant k the heat flux is a function of y, q = q(y). We will see that curvature in a emperature profile implies either the occurrence of sources or sinks of heat or time dependence.