For Fourier's law \(\displaystyle \vec{J}=-k\nabla{T},\) where

\(\displaystyle \vec{J}=\) conductive heat flux in \(\displaystyle \frac{W}{m^2}\)

k=thermal conductivity in W/m-K

T=temperature,

\(\displaystyle \frac{1}{k}\vec{J}\) certainly appears to be a conservative field with temperature as potential and for any path from position a to position b, \(\displaystyle \int\limits_{path}\vec{J}\cdot d\vec{r}=-k(T_b-T_a)\), i.e., conductive heat flows from an equipotential surface at \(\displaystyle T_a\) to an equipotential surface at [lower] \(\displaystyle T_b\)

This is analogous to:

\(\displaystyle \vec{F}_{grav.}=-\nabla{\mbox{[potential energy]}}\) where \(\displaystyle \int\limits_{path}\vec{F}_{grav}\cdot d\vec{r}=-(P.E._b-P.E._a)\) and mass flows (falls) from a to b.

Note: The path from position a to position b is a path in three dimensional space with coordinates x,y,z, not a path in a thermodynamic phase space, such as p-V diagram, in which heat is always path-dependent.

But this path-independent heat seems to only be useful when heat is considered to be an idealized conserved quantity, i.e.,

1. heat is a perfect fluid - no meaningful internal or thermal energies, no chemical interaction, nuclear interactions, particle collisions, particle velocity distributions, etc.

2. heat is conserved - no unwanted heat losses.

Which is a clearly limited model.

On the other hand, isn't it true that, whenever we do any relatively-straightforward engineering analysis involving conductive heat flow, such as the very-common one-dimensional application of Fourier's law, we are treating heat as an ideal conserved quantity?

But on the other hand, heat transfer requires \(\displaystyle \Delta T\Rightarrow\) heat transfer never occurs in thermodynamic equilibrium \(\displaystyle \Rightarrow\) heat transfer is always irreversible \(\displaystyle \Rightarrow\) heat transfer always involves unwanted heat losses \(\displaystyle \Rightarrow\) heat transfer can never involve a perfectly idealized conserved quantity.

So I think the conclusion is:

We do pretend that heat is an idealized conserved quantity when we do relatively-straightforward conductive heat transport analysis, even though the 2nd law \(\displaystyle \Rightarrow\) there are always unwanted heat losses.

Does this seem reasonable, or am I missing something important?

\(\displaystyle \vec{J}=\) conductive heat flux in \(\displaystyle \frac{W}{m^2}\)

k=thermal conductivity in W/m-K

T=temperature,

\(\displaystyle \frac{1}{k}\vec{J}\) certainly appears to be a conservative field with temperature as potential and for any path from position a to position b, \(\displaystyle \int\limits_{path}\vec{J}\cdot d\vec{r}=-k(T_b-T_a)\), i.e., conductive heat flows from an equipotential surface at \(\displaystyle T_a\) to an equipotential surface at [lower] \(\displaystyle T_b\)

This is analogous to:

\(\displaystyle \vec{F}_{grav.}=-\nabla{\mbox{[potential energy]}}\) where \(\displaystyle \int\limits_{path}\vec{F}_{grav}\cdot d\vec{r}=-(P.E._b-P.E._a)\) and mass flows (falls) from a to b.

Note: The path from position a to position b is a path in three dimensional space with coordinates x,y,z, not a path in a thermodynamic phase space, such as p-V diagram, in which heat is always path-dependent.

But this path-independent heat seems to only be useful when heat is considered to be an idealized conserved quantity, i.e.,

1. heat is a perfect fluid - no meaningful internal or thermal energies, no chemical interaction, nuclear interactions, particle collisions, particle velocity distributions, etc.

2. heat is conserved - no unwanted heat losses.

Which is a clearly limited model.

On the other hand, isn't it true that, whenever we do any relatively-straightforward engineering analysis involving conductive heat flow, such as the very-common one-dimensional application of Fourier's law, we are treating heat as an ideal conserved quantity?

But on the other hand, heat transfer requires \(\displaystyle \Delta T\Rightarrow\) heat transfer never occurs in thermodynamic equilibrium \(\displaystyle \Rightarrow\) heat transfer is always irreversible \(\displaystyle \Rightarrow\) heat transfer always involves unwanted heat losses \(\displaystyle \Rightarrow\) heat transfer can never involve a perfectly idealized conserved quantity.

So I think the conclusion is:

We do pretend that heat is an idealized conserved quantity when we do relatively-straightforward conductive heat transport analysis, even though the 2nd law \(\displaystyle \Rightarrow\) there are always unwanted heat losses.

Does this seem reasonable, or am I missing something important?

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