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Heat flux and heat flux density is the same thing, while electric flux density and electric flux is not the same thing? It makes me confused since we compare Fourier's law with Ohm's law. Here is a statement from Wikipedia.

To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small.

Is heat flux defined at a point or on a surface? I have never been found any defintion of heat flux or heat flux density.

As a mathematical concept, flux is represented by the surface integral of a vector field, $$\Phi_F = \iint_A \mathbf{F}\cdot \mathrm{d}\mathbf{A}$$ where $\mathbf{F}$ is a vector field, and $\mathrm{d}\mathbf{A}$ is the vector area of the surface $A$, directed as the surface normal. Heat is often denoted $\vec{\phi_q}$ and we integrated the heat flux density $\vec{\phi_q}$ over the surface of the system to have the heat rate but we integrated the $\mathbf{E}$-filed to get the electric flux?

Thanks.

Brooks
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2 Answers2

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If we take the definition of heat flux given here seriously, then heat flux is defined as a vector field $\vec\phi$ with units of energy per unit time, per unit area. At every point $\vec x$ in space, the vetor $\vec\phi(\vec x)$ tells you the direction and magnitude of heat flow in a neighborhood of that point. In particular, if we consider some two-dimensional surface $d\vec A$ containing $\vec x$, then $$ \vec\phi(\vec x) \cdot d\vec A $$ will tell us the amount of energy per unit time flowing through that surface. In particular, notice that here flux is being using to describe a vector field, not a scalar as in electric flux in EM. Perhaps this is rather bad terminology for this reason.

joshphysics
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  • Do we have infinity many choice of two-dimensional surface $\mathrm{d}\vec{A}$ containing $\vec{x}$. – Brooks Jun 06 '13 at 19:45
  • Would any different choice leads a different result? – Brooks Jun 06 '13 at 20:00
  • @Brooks Yes you can choose any $d\vec A$ you'd like, and the discussion still applies. Different choices would indeed lead to different results in general. For example, if the direction of the heat flow were parallel to a given choice of $d\vec A$, then one would have $\vec\phi(\vec x)\cdot d\vec A = 0$. – joshphysics Jun 06 '13 at 21:41
  • Would you mind to help me with these two probelm?http://physics.stackexchange.com/questions/67033/how-to-understand-fouriers-law-or-more-specifically-heat-flux . http://physics.stackexchange.com/questions/67307/how-could-flux-can-be-a-vector-and-a-scalar – Brooks Jun 06 '13 at 22:04
  • @joshphysics After reading your answer, I still don't get the difference between heat flux and heat flux density. I thought that in general, flux density is the flux per unit area. However, heat flux seems to already be defined per unit area. Are "heat flux" and "heat flux density" two terms for the same thing? – cdwilson Sep 14 '14 at 00:38
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    @joshphysics sorry, I didn't really understand your "notice that here flux is being using to describe a vector field, not a scalar as in electric flux in EM" statement before I posted my comment. Is it correct to say that "flux" is defined per unit area for heat transfer, but "flux" is defined as a surface integral for E&M? I.e. the definition of "flux" changes depending on the context it's used in? And therefore, in heat transfer context, "heat flux" is similar to "flux density" in E&M? In that case, does the term "heat flux density" make sense? – cdwilson Sep 14 '14 at 00:56
  • @cdwilson Yes I think it's fair to say that the term's definition is context-dependent, and also, possibly, dependent on who's using it, although I'm no expert on this subject, so I'm completely unaware of relevant, prevailing terminological conventions. – joshphysics Sep 14 '14 at 01:00
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As I understand it:

Electromagnetism and mathematics defines "flux" as the current or flow rate (dimensions of quantity/time), which is equal to the surface integral of a "flux density" (dimensions of quantity/time/area).

In the study of transport phenomena (including thermal transport and mass diffusion), "flux" refers to integrand (quantity/time/area), and the integrated result (quantity/time) is called a flow or transport rate.

However, I have found the field of heat transport to have the most inconsistency. As OP noted, often the same symbol will be given to the two quantities (flow rate and flux by Transport convention, or flux and flux density, by E&M/math convention). To further complicate the matter, in heat transport one may see both conventions used, so "flux" could refer to either quantity/time/area or quantity/time. "Flux density" and "flow rate" are unambiguous, but are based on different conventions. Personally I prefer the Transport convention for heat transport, defining flux as the area derivative of flow rate.

electronpusher
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