Characteristic time for heat diffusion

Question

I often see that one can write the characteristic time (or time scale) of a diffusive process as:

$\tau = \frac{L^2}{d}$

where $L$ is the characteristic length and $d$ is the diffusion coefficient.

1) How to prove that statement starting from the heat equation?

2) Suppose I have a 1D system of length $L$ with fixed temperatures at both boundaries (T1 $\ne$ T2) and a uniform initial temperature distribution. Can I say that $\tau = \frac{L^2}{d}$ (with $d$ the thermal diffusivity) is the time necessary for the system to reach its steady state? Or am I oversimplifying?

Answer 1

Since the heat equation for 1D time-dependent conduction is $$k\frac{\partial^2 T}{\partial x^2}=\rho c\frac{\partial T}{\partial t},$$

the hand-wavy way to derive the characteristic time is to replace partial differentials with finite differences and note that the thermal diffusivity $D=k/\rho c$:

$$D\frac{\Delta T}{(\Delta x)^2}\approx\frac{\Delta T}{\Delta t}$$ $$\Delta t\approx\frac{(\Delta x)^2}{D}$$

But note that $L^2/D$ is not an exact time for anything to occur—it's just an estimate. In your example, the bar never reaches its steady state; it only asymptotically approaches it. (In practice, random fluctuations would soon hide any difference.) For a given geometry and material, the progress in reaching this steady state (e.g., the middle of the object is now halfway to its steady-state temperature) will scale with $L^2/D$.

You can often assume that the process of reaching equilibrium is well on its way after time $L^2/D$ has passed and that the process is nearly complete after several time constants have passed. However, I emphasize again that this is just an approximation, one that depends on the nature of the boundary conditions (e.g., constant temperature vs. constant flux), for example.

Answer 2

The characteristic time in mathematical physics ,also called Time constant and relaxation time in dynamical physical or chemical phenomena analogously reverberation time in audio rooms is used to describe the speed with which the system goes exponentially in time towards equilibrium .Simplifying the phenomena-time dependence as F(t)=F(0)*(1-EXP(-t/T)) where T is the time constant ilucidates that the the system march 63% of its way towards equilibrium in time T and 86% after 2T and so on .As t goes to infinity the system goes to its final time independent steady state.How ever this is only explantory over simplified mathematical picture. For the case of heat equation in 1 D Boundary value problem described it can be shown that the characteristic time T is L**2/d .
The question it self is composed of two parts the first one is completely solved while this answer may clarify the second .