Why is heat flux sometimes assumed to be proportional to surface temperature? Suppose a fluid flows over a heated surface. In "Newtonian heating", it's assumed that the heat flux through the surface, $q$, is proportional to the surface temperature, $T_\text{surface}.$
Written as an equation, it's$$
q ~=~ h \, T_{\text{surface}}
\,,$$where $h$ is heat transfer coefficient.
Questions:


*

*What is the physical significance of assuming that heat flux is proportional to the surface temperature?

*Why might someone make this assumption?
 A: The answer to the question "Why is convection (Newtonian) heat transfer linear?" is that, in general, it's not! We simply treat it as linear in many cases because


*

*It's a good way to deal with convective heat transfer in a pedagogical way.

*It's a decent approximation to many real-world scenarios where the temperature difference between objects isn't too big.


In some sense, the Newtonian law for convective cooling is really just a disguised, less applicable version of Fourier's heat conduction law (written below in 1-D):
$$q = k\frac{dT}{dx}$$
This usually works well for solids since everything is standing still, but not as much for fluids because fluids move around in response to temperature gradients (or are in motion already).
You may be wondering why I make this comparison when Fourier's law involves $\frac{dT}{dx}$ and Newton's law involves $\Delta T$. That's because the fluid naturally creates a thermal boundary layer of some specific length $\delta$ through which the thermal gradient is non-zero, and we usually disregard its existence because it's very difficult to model. Hence, we can approximate convection heat transfer between two objects using this fact as:
$$q_{conv} = k_{eff}\frac{dT}{dx} \approx k_{eff}\frac{\Delta T}{\delta} = h\Delta T$$
where $k_{eff}$ is some "effective" thermal conductivity. In this sense, convective heat transfer is approximately linear for the same reasons it is in the conductive case: because temperature gradients are usually low and the non-linear terms "fall off" in the same way they do for a Taylor expansion near the expansion point.
Note that the convective heat transfer coefficient $h$ is almost always an empirically or experimentally derived quantity: you'll be hard-pressed to find a derivation for $h$ from first principles for any systems other than the very simplest. This is partially because of the complexity of these thermal boundary layers and because they're very very often not static.
For a look into thermal boundary layers, take a look at this book by Hermann Schlichting. Hope this helps!
A: I guess you wanted to say
$$
q ~=~ h \, \left(T_\text{fluid} - T_\text{surface}\right)
\,,$$
right? So this means that the heat flux density $\left(\mathrm{W} \cdot {\mathrm{m}^{-2}}\right)$ is proportional to the temperature difference between the fluid and the wall. It seems reasonable since when $T_\text{fluid} = T_\text{surface},$
$$
\require{cancel}
q
~=~ h \, \cancelto{0}{\left(T_\text{fluid} - T_\text{surface}\right)}
~=~ 0
\,,$$you don't have any heat flux anymore.
A: tl;dr-  The described relationship, $q=hT_\text{surface},$ is probably a mistake.  However, there are a few cases in which it might be used, and I've actually seen it before, so here're cases in which it might pop up either as an approximation or reframing.

The common heat-transfer coefficient equation is$$
q
~~=~~h \left(T_\text{fluid}-T_\text{surface}\right)
~~=~~h\,\Delta T
\,,$$where:


*

*$q$ is the heat flux;

*$h$ is a proportionality constant called the "heat-transfer coefficient";

*$T_\text{fluid}$ is the temperature of the fluid;

*$T_\text{surface}$ is the temperature of the surface; and

*$\Delta T \equiv T_\text{fluid} - T_\text{surface} \,.$
In principle, the correlation that you're talking about could be approximately correct when $T_{\text{surface}} \gg T_{\text{fluid}}$ and we flip the sign of the heat-transfer coefficient such that$$
h_{\text{normal definition}}
~~ \longrightarrow ~~ -h_{\text{new definition}}
\,.$$Then,$$
\require{cancel}
q
~~=~~-h \left(T_\text{fluid}-T_\text{surface}\right)
~~\approx~~-h\left(-T_\text{surface}\right)
~~=~~h \, T_\text{surface}
\,,$$recovering the description that you're asking about.
Seems like this could work in four situations:


*

*We select a temperature scale such that $T_\text{fluid} \approx 0 \,.$Seems like the potential confusion that could arise from this trick would more than counterbalance and mild convenience that might be derived from it in most cases.

*The temperature $T_{\text{surface}}$ is far greater than $T_\text{fluid}.$Should pretty much never happen in practice because this correlation doesn't tend to work well for large temperature gradients, plus if you're working with huge temperature scales, that almost always implies that it's serious business – better models should be used.

*We're considering a set of cases in which $\Delta T$ is roughly constant.In this case, we basically stop talking about $h$ and instead start talking about a new constant, $h \Delta T,$ which we might start to call "$h$" again simply for convenience.  Then it becomes$$
  q
  ~~\approx~~ h_{\text{pseudo}} \, T_{\text{surface}}
  \,,$$where $h_{\text{pseudo}} \equiv -h_{\text{normal definition}} \left. \frac{\Delta T}{T_{\text{surface}}}\right|_{\text{reference}} \, .$  The weird thing about this approximation is that I think that I've actually seen people use it before, though I generally wouldn't recommend it.

*We're discussing separable heat flows.This is probably the most legitimate case, though perhaps not too common.  Here, we discuss heat flows as a combination of component heat flows,$$
  q
  ~~=~~\sum_{\forall \text{heat flows}~i} {q_i}
  \,,$$in which case we might rewrite the heat-transfer coefficient equation as$$
  \begin{alignat}{7}
  q &~~=~~ q_{\text{fluid}} &&+ q_{\text{surface}} \\[5px]
    &~~=~~ h \, T_\text{fluid} &&+ h \, T_{\text{surface}} \,,
  \end{alignat}
  $$in which case we can say that $q_{\text{surface}}=h \, T_{\text{surface}} \,,$ so long as we don't confuse $q_{\text{surface}}$ with the total heat flux through the surface, $q.$
So there are cases in which one might conceivably encounter $q=hT_\text{surface},$ though they tend to be niche approximations or special contexts.  In general, I wouldn't recommend worrying too much about this particular correlation.
