# Heat Equation Equalities

While studying the heat equation, I ran into a few equalities that I cannot understand.

For example, Fourier's law of heat conduction claims that

$$\varphi(x,t)=-K_0\frac{\partial u}{\partial x},$$

where $\varphi$ is the heat flux (the amount of thermal energy per unit time flowing to the right per unit surface area), $K_0$ is the thermal conductivity, and $u$ is the temperature.

How did Fourier end up with this?

Another equality that I cannot understand is the relationship between thermal energy and temperature:

$$e(x,t)=c(x)\rho(x)u(x,t),$$

where $e$ is the thermal energy density, $c$ is the specific heat (the heat energy that must be supplied to a unit mass of a substance to raise its temperature one unit), and $\rho$ is the mass density (mass per unit volume).

Why is this true? I cannot seem to make the physical connection.

## migrated from math.stackexchange.comSep 6 '12 at 4:23

This question came from our site for people studying math at any level and professionals in related fields.

• Perhaps this would be better suited for physics.SE? – Alex Becker Sep 6 '12 at 4:01
• I believe you're right. Is there a way to move this question from here to there without having to start a new one? – Josué Molina Sep 6 '12 at 4:02
• I voted to migrate it. I'll flag it so the mods can do so unilaterally. – Alex Becker Sep 6 '12 at 4:05

In a sense, Fourrier's law is the definition of $K_0$. We want heat to flow from a high temperature to a low one and we believe in locality-things depend only on what is happening at a point, not somewhere far away. $\frac{\partial u}{\partial x}$ is how fast the temperature changes at the point, which is how hard we are pushing the heat. $K_0$ says how hard it is to move the heat.

Added: this is the simplest equation one can write for heat flow. Having written it, one can do experiments to see if it is satisfied. It works well for isotropic solids, and terribly if you have convection or localized boiling.

For the second, it helps to look at the units. $e(x,t)=c(x)\rho(x)u(x,t)$ has $e$ in joules/m^3, $c$ in joules/(kg-K), $\rho$ in kg/m^3, and $u$ in K. So $u$ is the temperature. If you want to raise it, you need to supply heat. If the density doubles, there is twice as much matter, so it takes twice as much heat to change the temperature 1K. If the heat capacity doubles, again you need twice as much heat to change the temperature. Water has a high heat capacity, which is why it takes a long time to boil on the stove.

I may have something to add to the first part of your question...

The Heat Equation is just another of many diffusion equations, which all follow Fick's laws. As the wikipedia article explains, you can arrive at those results from two different approaches:

• A phenomenological approach, starting from the laws, calculating their consequences, and comparing that to experimental results. This is probably what Fourier did.
• An atomistic approach, where you start by considering random walks of molecules, and compute how certain parameters vary with time and location. This is sort of a statistical mechanics derivation from first principles, and may serve the purpose of providing a physical intuition of why flux is proportional to the gradient. The deduction for the one-dimensional case of Fick's law is relatively easy to follow.

Fourier did derive his law from Newton's law of cooling http://en.wikipedia.org/wiki/Newton%27s_law_of_cooling

But, it is as Ross Millikan said a very natural law. If between two points there is a temperature difference then the restoring process is proportional to it.

For the second pqrt of your question, it is just the local version of the thermodynamical relation:

DQ=m*C*dT