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I was reading Feynman-Leighton-Sands 's physics lecture note in Volume 2 chapter 2 when deducing the equation $$h = -k\ \text{grad}T$$

It seems we use the assumption that heat flow orthogonal to the isothermal surface.

Which I find is not mentioned in the book,so I find a post here,I'm not sure whether this point corresponds to the sentence "but for many metals and other substances that conduct heat is quite accurate"

But I find a proof for this fact here

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.

Hence we can deduce that if we assume that isothermal surface has constant thermal energy,we must has heat flow orthogonal to isothermal surface correct?

I mean isothermal surface may not have constant thermal energy in general correct?

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  • $\begingroup$ An isothermal surface is one on which the temperature is the same on the entire surface. Heat is energy transfer due solely to temperature difference. So clearly you can not have a component of energy transfer in the form of heat along an isothermal surface. $\endgroup$
    – Bob D
    Commented Mar 30, 2021 at 13:38
  • $\begingroup$ @Bob D thanks Why physics.stackexchange.com/a/67783/264677 this post state it does not hold in general then? $\endgroup$
    – user264677
    Commented Mar 30, 2021 at 13:43
  • $\begingroup$ It seems in the book we first define heat flow as the change of thermal energy through the surface unit per unit time(which does not correspond to the temperature),then deduce the relation between heat flow and change of temperature?So we need to prove the statement "Heat is energy transfer due solely to temperature difference" ?How to prove it? $\endgroup$
    – user264677
    Commented Mar 30, 2021 at 13:46
  • $\begingroup$ In general, the heat flux vector does not have to be parallel with the the temperature gradient For crystalline solids the conductivity $\kappa$ is a tensor that can be non-diagonal and thus there can be tangential component to the heat flux vector. $\endgroup$
    – hyportnex
    Commented Mar 30, 2021 at 13:52
  • $\begingroup$ @hyportnex what does tensor here means a (0,0) tensor?isothermal surface may not have constant thermal energy in general correct? $\endgroup$
    – user264677
    Commented Mar 30, 2021 at 13:56

1 Answer 1

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The heat flux vector being proportional to the temperature gradient vector is an empirical observation applicable exclusively to isotropic materials (materials whose physical properties at a given location are independent of direction). For anisotropic materials (such as crystalline materials or composites), the heat flux vector can have a component parallel to the local isotherms.

Your inclusion of the terms thermal energy and thermal energy change in this discussion make no sense to me, and are irrelevant as far as I am concerned.

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  • $\begingroup$ thanks,does the definition of heat flux as $h = \frac{\Delta J}{\Delta a} e_f$ where $\Delta J$ as change of thermal energy per unit of time and $\Delta a$ is area of surface where $e_f$ is direction of flow correct? If so " heat flux vector can have a component parallel to the local isotherms." implies can have thermal energy change on isothermal surface correct? $\endgroup$
    – user264677
    Commented Mar 30, 2021 at 14:22
  • $\begingroup$ @ChetMiller isn't heat a transfer of energy due to a difference in temperature. If there is heat flux parallel to the isotherm then how is it heat since it is not driven by a difference in temperature. Is this using a different definition of heat? $\endgroup$
    – Dale
    Commented Mar 30, 2021 at 14:29
  • $\begingroup$ @Dale I didn't say that the heat flux is parallel to the isotherm. I said that the heat flux vector has a component parallel to the isotherm. Suppose that the composite has highly conductive rods oriented at an angle to the isotherms. Then the average of the rods plus matrix will have a component of heat flux parallel to the isotherms. $\endgroup$
    – Chet Miller
    Commented Mar 30, 2021 at 14:34
  • $\begingroup$ @Chet Miller Does isotherm has different thermal energy on it? $\endgroup$
    – user264677
    Commented Mar 30, 2021 at 14:38
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    $\begingroup$ See Transport Phenomena by Bird, Stewart, and Lightfoot $\endgroup$
    – Chet Miller
    Commented Mar 30, 2021 at 15:47

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