The derivation for Heat Equation I am reading starts by stating

Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total heat generated inside $[x,x+\Delta x]$ and writes the conservation equation

$$\textit{Total Heat Inside} [x,x+\Delta x]= cpA \int _{ x}^{x+\Delta x}u(s,t) ds $$

and later equates net flux across the boundary with

$u_x(x+\Delta x,t) - u_x(x,t)$,

so the heat flow is proportional to the spatial gradient of temperature at both ends. What I dont understand is why is net flux proportional to the spatial gradient, why cant we use the temparature variable $u$ itself to measure the difference? If the temparature difference at both ends is high, then we would expect higher flow from high temparature to low, right? What would be the physical intuition behind this? Was this model arrived at by experimentation?

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    $\begingroup$ Suppose that the temperature at each end, $u(x)$ and $u(x+\Delta x)$ is equal to $T_0$, while the temperature in the center is $T_1<T_0$. This means there is no temperature difference between the ends, would you expect there to be no heat flux into the region? $\endgroup$ – user2963 Jan 25 '12 at 22:38
  • $\begingroup$ @zephyr So you mean that spatial gradient differences of both ends would capture the $T_1<T_0$ case? But doesn't a spatial gradient infinitesimally mean the temp difference of two particles next to eachother? So we are still roughly speaking about the end points, far from the middle. $\endgroup$ – BBAerodyn Jan 26 '12 at 10:58
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    $\begingroup$ You could imagine a case where the temperature is symmetric about $x+\Delta x/2$ (but lower in the middle). In this case, the temperature at both ends would always be equal, however heat would still be flowing into the region until it reached equilibrium. $\endgroup$ – user2963 Jan 26 '12 at 13:51
  • $\begingroup$ Let' say there are 9 physical points or particles between $x$ and $x + \Delta x$, therefore 9 temparature measurements. The values are 8,8,2,2,0,2,2,10,10. Spatial gradient at both ends are 0, the difference of zeros is also zero. Intuitively there should be flow of course, but I cant see how spatial gradient can capture this. The model uses spatial gradient diff proportion before $\Delta x \to 0$ by the way. That's why I am confused. $u_x$ is already infinitesimally small which (I assume) is seperate from $\Delta x$ business. $\endgroup$ – BBAerodyn Jan 26 '12 at 16:13
  • $\begingroup$ I think you are getting too hung up on the distinction between $\endgroup$ – user2963 Jan 26 '12 at 16:38

You have to be precise with the definition here. You mention a temperature difference at both sides, but this is a 1D problem. You have different temperatures at both sides, and a temperature difference across the rod.

The heat flux is proportional with the temperature gradient, this is referred to a Fourier's law (see e.g. http://en.wikipedia.org/wiki/Fourier's_law#Fourier.27s_law )

Now I understand your question as: why is $q=-k \frac{du}{dx}$ (u is temperature in your case), and not just proportional to $u$. Now, suppose the flux would be proportional to the temperature. Then you get some problems. First, suppose you have a rod of uniform temperature. The heat flux is constant across the rod everywhere, but in what direction? And what is the new equilibrium than?

Just the notion that the energy should flow from high to low temperature, is given by the gradient, making the energy always traveling down the gradient, trying to reach a more uniform state. To make it work out, some constant of proportionality was introduced, which is most often a system (or material) property.

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