# Is thermal conductivity constant in Fourier's law?

In Fourier's law we take thermal conductivity as constant in a particular direction, but if there is different temperature at two ends then intuitively there will be a temperature gradient, which means different temperature at different point throughout the solid. Also thermal conductivity is dependent on temperature, means it is different at different point, so why do we determine temperature gradient using Fourier's law where thermal conductivity is constant?

If the thermal conductivity varies significantly between the two end temperatures, Fouriers law still applies locally at each point within the solid. To get the heat flow however, we need to integrate the temperature gradient with respect to position: $$q=-k(T)\frac{dT}{dx}$$so$$qL=\int_{T_C}^{T_H}{k(T)dT}$$where $T_H$ is the temeperature at x = 0 and $T_C$ is the temperature at x = L.