# When is path-independent heat a useful idea?

#### trekkiee

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?

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#### Benit13

Math Team
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?
Yes. That's because in most cases we can ignore changes in pressure and volume of the various media which are exchanging heat. In those problems, only the temperature changes with some fixed pressure and volume and the errors associated with path-dependency are minimal. This is okay for most macrosopic objects, such as walls, ovens and pies, but will fail horribly for microscopic objects. It is also often assumed that the conductivity of materials are constant with temperature, which is false, so the level of errors we are dealing with are comparable. Also, for engines and other objects that have important contributions to heat losses, are usually calculated based on a constant efficiency factor in practice.

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?
In short, those heat losses are either considered negligible because changes in pressure and volume are ignored or because the work being done is of no consequence.

Heat is always a conserved quantity given a large enough control volume (after all, it is energy; energy cannot be created or destroyed). What isn't conserved is the heat transfer between two specific objects A and B, because the heat transfer depends on the states of A and B and there is always some sort of waste heat going somewhere else (C). The heat exchange will cause the state variables (PVT) of the system to change. These details become important for some macroscopic cases (phase changes, engines, etc) and almost all microscopic applications.