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Before the question: I am working on numerical calculation of three dimension parabolic equation that based on Fourier's Law of which I am a little confused.

Here comes the law in modern mathematics language.

"The local heat flux is proportional to temperature gradient" $$ \vec{q}=-k\nabla T, $$ where $k$ is the material's conductivity.

How extremely concise it is, but how to understand the Law? I read the book written by Fourier in 1822 but I know neither of the law in modern mathematics language nor in Fourier's language. I found that every statement or formula related to proofing the Law are not done with rigour enough. Here is some statement from a book by YUNUSA.CENGEL on its page 65 chapter 2.

To obtain a general relation for Fourier's law of heat conduction, consider a medium in which the temperature distribution is three-dimensional. The figure below shows an isothermal surface in that medium. The heat flux vector at a point $P$ on this surface must be perpendicular to the surface, and it must point in the direction of decreasing temperature. If $n$ is the normal of the isothermal surface at point $P$, the rate of heat conduction at that point can be expressed by Fourier's law as $$ \dot{Q_n} = -kA\cfrac{\partial T}{\partial n} $$

enter image description here

My questions toward the issue I have mentioned are

  • How could heat flux be a vector?
  • What's the meaning of the direction of the heat flux?
  • Why heat flux at a point is normal to the isothermal surface?
  • What's the definition of heat flux vector, not heat flux which is defined as the quantity per second per area?

You might say it is true just because of the Second law of thermodynamics.

The heat always flows spontaneously from regions of higher temperature to regions of lower temperature, and never the reverse, unless external work is performed on the system.

It is lower but not fastest decrease, isn't?

If the direction is not through the line in the tangent plane of the isothermal surface it would transfer to a colder palce, isn't? So why choice the normal line to be the heat flux direction since there is infinity line to the colder place. Maybe it is that project works when considering the other line! However, it is human beings not the nature define the direction of heat flux for convenient. Am I right?

It may be related with Fick's Law. I am not sure about the proof of three dimension situation.

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In accordance with the second law of thermodynamics, the heat flux acts to reduce the temperature gradient. –  Dilaton Jun 4 '13 at 14:45
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Possible duplicate: physics.stackexchange.com/q/66550 –  joshphysics Jun 4 '13 at 19:16
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Brooks, stop editing your question. –  David Z Jun 10 '13 at 19:01
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3 Answers

Heat flux is a vector beceause it has a magnitude and a direction. Furthermore it has these properties in every point in space, which makes it a vector field. You can think of an analogy with the mass flux in a medium with inhomogenius density; diffusion will tend to equalize the denisty everywhere, so there will be specific motion of mass at every point determined by its imediate surroundings.

The direction of the heat flux specifies for each point the direction of the fastest drop in temperature.

Finally, heat flux is normal to a isothermal surface, because if it wasn't it would have a tangential component along the isothermal surface at that point. That would in turn mean that there would be a non-zero temperature gradient (difference) along the surface, which would mean that it isn't a isothermal surface.

Further resources:

http://www.et.byu.edu/~vps/ME340/ME340.htm

http://www.amazon.com/books/dp/0470501960

http://freevideolectures.com/Course/3005/Heat-Transfer/1

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The direction of the heat flux specifies for each point the direction of the fastest drop in temperature. Is it a law? –  Brooks Jun 9 '13 at 9:04
    
Yes, since it is related to the gradient in the definition. Gradient always points in the direction of the greatest increase of the scalar field (en.wikipedia.org/wiki/Gradient). Therefore the additional negative sign points it in the opposite direction - the direction of the greatest decrease of temperature. –  mgphys Jun 9 '13 at 9:10
    
Thanks. Why the fastest direction? I am good at gradient but not heat! –  Brooks Jun 9 '13 at 9:19
    
How can I get a ebook of the book Introduction to Heat Transfer since I couldn't buy it on amazon? –  Brooks Jun 9 '13 at 9:20
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I do not agree with the statement that the heat flux has to be normal to the isothermal surface because it would create a transverse gradient otherwise. For example, one could imagine that the vector is not normal to the surface because it did not have time to generate the transverse gradient. The true explanation, in my opinion is a symmetry reason: imagine an infinite isothermal plane. The heat flux has to be normal to it because all contributions will be statistically distributed symmetrically. –  fffred Jun 10 '13 at 6:11
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EDIT: To answer the new formulated questions, it's a basic principle of thermodynamics that heat flows from hot to cold bodies. The direction of the heat flux vector is precisely that. Therefore, it should be obvious why this vector is orthogonal to isothermal surfaces once we accept that principle. Fourier's law is just a refined statement of that principle which also tells us the relation between the magnitudes of temperature gradient and heat flux.

Suppose you have a box over which there is a temperature gradient from the left side of the box to the right side, the left side being the warmer side. The amount of heat that flows through the box per time unit is proportional to the temperature gradient and the surface area of the side of the box. That's Fourier's law.

$$\frac{\Delta Q}{\Delta t} = -k A \frac{\Delta T}{\Delta x}$$

In a more complex situation, heat could flow in various directions. The heat flow $dQ/dt$ described in the above formula is to heat flux $\vec{q}$ as current $I$ is to current density $\vec{j}$ in electromagnetism. The temperature gradient then becomes $\nabla T$ in its most general vector form, where $\nabla$ is the gradient operator. We then have

$$\vec{q}=-k\nabla T$$

The temperature gradient is the analog of potential difference in electromagnetism. And Fourier's law is the analog of Ohm's law.

Here's a clarification of the definition of flux

$$\vec{q}=\frac{d (dQ/dt)}{d\vec{A}}$$

or alternatively

$$\frac{dQ}{dt}={\int \!\!\! \int}_{A} \vec{q}\cdot d\vec{A}$$

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See my edit, it addresses this issue. –  Raskolnikov Jun 4 '13 at 10:24
    
It is not the original form of the law. But while vector calculus didn't exist in Fourier's time, Fourier was well aware that heat fluxes could flow through any side of my hypothetical box I was talking earlier. He would have had a $dQ/dt$ for each side, hence a vector. –  Raskolnikov Jun 4 '13 at 10:59
    
Your link doesn't seem to work. –  Raskolnikov Jun 4 '13 at 11:00
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I'm sorry, but what you say doesn't make any sense. I suggest you follow a course on vector calculus before attempting to read about thermodynamics in this book. You are very confused. There are some good Schaum series books on vector calculus. Look for them, they are quite cheap. –  Raskolnikov Jun 5 '13 at 21:12
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It is the lower temperature but not the lowest temperature, isn't? –  Brooks Jun 9 '13 at 10:44
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Actually, this is not even correct. The temperature gradient is normal to isothermal surface, which is a simple mathematical consequence of the local Taylor expansion $T({r}_0+\delta{{r}})=T(r_0)+(\partial{T}/\partial{r}) \delta r$. However, in general the heat flux is not local (i.e., the heat flux at a given point is not defined only by the local temperature and its gradient); but even if it is local, the heat flux is not in general collinear with the temperature gradient due to transport anisotropy, so the right relation is $\vec{q}=-\hat{\kappa} \nabla(T)$ where $\kappa$ is the heat conduction tensor. For example, in magnetized plasma the anisotropy of heat transport can be many orders of magnitude, and in a magnetically confined plasma the heat flux is usually not orthogonal to the isothermal surface but almost exactly along the surface (along the magnetic field line, to be exact).

However, if we assume isotropic transport (as the question seems to be implying) then the standard type of argument used for a diffusive process as in, e.g., the Wikipedia article https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion explains why the flux is down the temperature gradient.

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thanks. Your answer is pretty good. Do we must go to Fick's law when we trying to explain Fourier's law? –  Brooks Jun 12 '13 at 8:30
    
There is a proof about one dimension situation, but I am not sure about three dimension. –  Brooks Jun 12 '13 at 9:25
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