# How fast is heat transferred by conduction?

How fast is heat transferred by conduction? Is there some simple, but quantitative way that starts from some properties of the material (e.g. its thermal conductivity) and makes rough predictions, for example about how much time is needed for temperature to change at one end of the body when it is placed, at the other end, in thermal contact with another?

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The conduction coefficient for slow temperature gradients can be calculated in many materials using a Kubo type formula, but I am not sure if the result is reliable in metals with a large thermal gradient, in the case that electrons are delocalized over a region which is big enough for the temperature gradient to be non-infinitesimal. – Ron Maimon Apr 5 '12 at 19:01

## 2 Answers

The heat equation for these kind of problems (assume 1D), reads

$$\frac{\partial T}{\partial t} = a \frac{\partial^2 T}{\partial x^2}$$

Where of course $T$ is temperature, $t$ is time, $x$ is position and $a$ the thermal diffusivity: $a=\frac{\lambda}{\rho c_p}$ (respectively thermal conductivity, density and heat capacity. This equation can be derived from Fourier's law.

Suppose you have a block at temperature $T_0$ which you put in contact with a block of temperature $T_1$ at $x=0$. Then you have a set of boundary conditions

$$T(x,0) = T_0 \\ T(0,t) = T_1 \\ T(x\to\infty,t)=T_0$$

It is not easy, but it has been derived that the solution to this equation is

$$\frac{T-T_0}{T_1-T_0}=1-\frac{2}{\sqrt{\pi}}\displaystyle\int_0^{\frac{x}{2\sqrt{at}}} e^{-s^2}ds$$ Where the solution to this integral is referred to as error function

This solution describes the transient and spatial profile of the heated piece of material.

For short times, only a certain amount of the material is heated. Using the error function, one can define the penetration depth, which is $x_p=\sqrt{\pi a t}$, which is obviously a measure for how far the increased temperature ranges into the material.

Suppose you domain has a finite length and is isolated at the other end. The heat transfer coefficient from the wall at $x=0$ is nearly constant, and the average temperature of the block will converge with an exponential decay to the boundary temperature (see Vladimir's answer). This can be derived from the heat equation by assuming that $\frac{T-T_0}{T_1-T_0}=1 - f(t) g(x/L)$

Note: This book was used as a reference for some of the equations.

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One can easily estimate the time of reaching a steady state. If you have a layer of a certain thickness and you know the thermal properties of the material, the problem is solved in any textbook on heat conduction. The temperature at the other end varies with time and the final stage is very simple (called a regular regime): $T(t)\approx T(\infty)+Ae^{-\lambda_0 t}$. The lowest eigenvalue $\lambda_0$ is determined with the layer thickness $L$, its heat conductivity $\kappa$, specific heat capacity $c$ and the material density $\rho$.

$$\lambda_0 = \frac{\pi^2 \cdot\kappa}{4\rho \cdot c \cdot L^2}$$

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