Fourier law and thermal conductivity

Fourier's law: the rate of heat conduction in a direction is proportional to the area normal to that direction and to the temperature gradient in it:

$$Q_x = -kA\frac{dT}{dx}\qquad q'' = -k\frac{dT}{dx}$$

The negative sign makes $Q$ positive in the direction of decreasing temperature (second law of thermodynamics).

Thermal conductivity

It is the heat flow per unit time through a unit area of a material of unit thickness when the temperature difference across it is 1 K: $k = \dfrac{Q\,L}{A\,\Delta T}$, in W/m K. It is a property of the material and measures how well it conducts heat.

Variation with temperature

• Pure metals: $k$ is high (free electrons) and decreases slightly as temperature rises, because lattice vibration scatters the electrons. Alloys and impurities reduce $k$ strongly; for alloys $k$ may rise with temperature.
• Gases: $k$ rises with temperature (about as $\sqrt{T}$), since molecular speed and collision energy exchange increase. Pressure has little effect.
• Liquids: $k$ generally falls with temperature (water is the exception, rising to about 130 °C).
• Insulating solids: $k$ increases with temperature and with moisture or density, because pore conduction and radiation across the voids increase.

Typical values (about room temperature)