Equations
In an isotropic medium the thermal conductivity is the parameter k in the Fourier expression for the heat flux
where is the heat flux (amount of heat flowing per second and per unit area) and the temperature gradient. The sign in the expression is chosen so that always k > 0 as heat always flows from a high temperature to a low temperature. This is a direct consequence of the second law of thermodynamics.
In the one-dimensional case q = H/A with H the amount of heat flowing per second through a surface with area A and the temperature gradient is dT/dx so
In case of a thermally-insulated bar (except at the ends) in the steady state H is constant. If A is constant as well the expression can be integrated with the result
where TH and TL are the temperatures at the hot end and the cold end respectively, and L is the length of the bar. It is convenient to introduce the thermal-conductivity integral
The heat flow rate is then given by
If the temperature difference is small k can be taken as constant. In that case
Read more about this topic: Thermal Conductivity