# What is temperature as a function of time in Fourier's Law?

Can we find temperature as a function of time using Fourier's Law of thermal conduction?

For example:

If we have two square parallel plates of a given length, width, and distance apart with one plate hotter than the other, how do find the temperature of the two plates as a function of time?

Fourier's Law in 1D says:

$$q_x = -k\frac{dT}{dx}$$

I'm not sure how to integrate/manipulate this equation to get temperature with a time dependence. Any advice would be appreciated!

## 2 Answers

Fourier's law is just expressing the heat flux as a function of the gradient of temperature in a medium characterized by the heat conductivity $$k$$. What's the effect of such heat flux on the temperature of the material depends on something else which are not present in the Fourier's law: the effect of a heat transfer on the temperature rate of variation. Under the hypothesis that all the heat entering in a small volume of material goes into internal energy, one can obtain the Fourier equation of heat ( https://en.wikipedia.org/wiki/Heat_equation ), which in one dimension, and for a uniform sample can be written as $$\frac{\partial{T}}{\partial{t}}= \frac{k}{c_p \rho}\frac{\partial^2{T}}{\partial{x}^2}$$ where $$c_p$$ is the constant pressure specific heat and $$\rho$$ the mass density. This is a partial differential equations and methods are known to solve it.

However, from your question I am wondering if you have in mind a picture like this (coming from youtube: In the context of Fourier's law, the two "plates" have not to be intended as two physical objects, but are just two geometric surfaces of the same material which fills the whole space in between, used to evaluate the integral of the heat flux density.

The physical problem of finding the temperature field between two $$physical$$ plates at different temperature, one in front of the other, may require more than the Fourier law (for example, if between the two surfaces there would be a gas, convective motion could play an important role).

In order to obtain the temperature time- and space-dependences $$T(x,t)$$, you have to solve the heat conduction equation with some initial conditions $$T(x,0)=f(x)$$ and with some boundary conditions. The Fourier law is not sufficient for that.