# Reduction of Heat Conduction Equation to get Diffusion Time

There is a question that asks me to find the diffusion time assuming an initial temperature and given thermal conductivity $$k$$, specific heat capacity $$C$$ and density $$\rho$$. From the heat conduction equation (assuming diffusion in only one direction):

$$\frac{\rho C}{k}\frac{\partial T}{\partial t} = \frac{\partial^2T}{\partial x^2}$$

Then there is somehow a conclusion that given the length of the diffusion wall $$L$$, the diffusion time is $$t_D = L^2/\alpha$$. How does this come about? What additional assumptions must be made?

$$\frac{\rho C}{k}\frac{\partial T}{\partial t} = \frac{\partial^2T}{\partial x^2};$$
$$\frac{\rho C}{k}\frac{\Delta T}{\Delta t}\sim\frac{\Delta T}{(\Delta x)^2}.$$
(Note that $$\Delta T$$ doesn't get squared on the right because the operator is $$\frac{\partial^2}{\partial x^2}$$.)
From this we obtain $$\Delta t=\frac{\rho C(\Delta x)^2}{k}=\frac{(\Delta x)^2}{\alpha}$$, where $$\alpha\equiv\frac{k}{\rho C}$$ is the thermal diffusivity.
This relation isn't exact by any means; it just tells us that we can expect the diffusion process over distance $$x$$ to have progressed fairly far along by time $$\frac{x^2}{\alpha}$$. (For some configurations, we're at $$1-e^{-1}=63\%$$ of the asymptotic final solution; two time constants would give us $$86\%$$, and three $$95\%$$.) A better solution may require solution of the original differential equation.