Fourier's Law of Heat Transfer: Conceptual Explanation

[E12-1]

TL;DR

Why is heat flux inversely proportional to the length of the conductor?

Hello,

I'm looking over some old school notes and re-learning some basic heat transfer. I have known Fourier's Law (1D: $q = k\frac{dT}{dx}$) for a long time, but when I look at it now I find it strange that heat flux is inversely proportional to the length of the conductor. I would think that the amount of heat being transferred is the same if the temperatures at each end are the same (and the area, but flux normalized by area so no need to worry about that).

Why is it that a conductor of length L will conduct twice as much power as a conductor of length 2L, given t1 and t2 are the same? Why does the distance the heat energy travels affect the amount of energy which is transferred per unit time?

I feel like the answer is obvious I just need someone to help me unblock my brain.

 

[erobz]

TL;DR Summary: Why is heat flux inversely proportional to the length of the conductor?

Hello,

I'm looking over some old school notes and re-learning some basic heat transfer. I have known Fourier's Law (1D: $q = k\frac{dT}{dx}$) for a long time, but when I look at it now I find it strange that heat flux is inversely proportional to the length of the conductor. I would think that the amount of heat being transferred is the same if the temperatures at each end are the same (and the area, but flux normalized by area so no need to worry about that).

Why is it that a conductor of length L will conduct twice as much power as a conductor of length 2L, given t1 and t2 are the same? Why does the distance the heat energy travels affect the amount of energy which is transferred per unit time?

I feel like the answer is obvious I just need someone to help me unblock my brain.

My Heat Transfer text says Fourier's Law is phenomenological; It is developed from observation rather than first principles.

 

[E12-1]

Interesting. May I ask which book you are reading? I use Lienhard's A Heat Transfer Textbook which generally gives pretty good explanations.

 

[erobz]

Fundamentals of Heat and Mass Transfer ( Incropera, DeWitt, Bergman, Lavine)

 

[Dullard]

Unblock attempt (in the form of a question):

Why is there more water flow in a hose of length 'L' (vs '2L), when the pressure difference across the hoses is the same?

 

[DaveE]

Perhaps if you consider the question in reverse (sort of) it will be easier to see. For a given, constant, heat flow, what is the temperature difference in a path of length L vs. 2L (with everything else equal). What is the temperature gradient in any small section of each path? What happens if you add all of those sections together in series?

 

[E12-1]

Unblock attempt (in the form of a question):

Why is there more water flow in a hose of length 'L' (vs '2L), when the pressure difference across the hoses is the same?

Rude! Now I don't know 2 things.

Just kidding, I think the point you and DaveE are trying to make is that it is the magnitude of the gradient which determines the flow rate, and "stretching" the distance effectively makes this gradient more gradual.

This isn't a totally satisfying answer, and maybe that's not the point you are even trying to make. If that is the point you are making then my next question is still "why?". Why does a steeper pressure gradient cause a faster flow? I'm trying to formulate an analogy using traffic or some other macro concept. (Maybe a ramp? Using potential energy as an analogy surely we all know why we slide faster when the ramp is steeper. Still brainstorming these...).

Thanks for the unblocking clues.

 

[DaveE]

Rude! Now I don't know 2 things.

LOL. That's kind of how learning works. Isn't it?

then my next question is still "why?"

A universal problem in physics. Why questions never ultimately seem to have satisfying answers. There's a great Feynman video about this. Why? That's how it works, that's why. OTOH, there's a lot more to learn about how; via statistical mechanics, thermodynamics and such.

One thing to consider relates to the general structure of the problem. Let's say in a 1m bar conducting heat in a steady state*, what is the relationship between the solution (temperature, temp. gradient, heat flow, etc) at the 3cm distance and the 3.01cm distance, what about the effect of the solutions at 10cm or 70cm on that 3cm solution. The point here is that the problem can be deconstructed into a collection of smaller problems, which will lead you to the linear nature of the solutions. If you know the solution of the bar of length L, isn't a bar of length 2L just two of your shorter bars put end to end? This business of breaking up a problem and then adding up their solutions, often called superposition, is really powerful when it works (i.e. in linear systems). You'll see it in many places.

* This bit is important. If things are varying in time you get a different more complex solution.

This video won't answer your why questions, but I thought it was good:


PS: A key feature of these steady state heat flow problems is the concept that the heat flow only depends on the temperature difference (or gradient) so the heat flow through a bar with 30C and 20C temperatures at the ends is the same as when that bar has 50C and 40C temperatures, and is twice the flow as when the temperatures are 30C and 25C, or 50C and 45C. That is the key thing that Fourier is saying in that equation. Heat flow is proportional to the temperature gradient regardless of what the temperature value is. Of course there are examples in the real world where this doesn't apply, like the difference between ice and water.

 

[256bits]

TL;DR Summary: Why is heat flux inversely proportional to the length of the conductor?

Why is it that a conductor of length L will conduct twice as much power as a conductor of length 2L

Heat flow is a measure of power in watts as you have mentioned.
(1D: q=k dT / dx) , which really should be dq / dt = -k A dT / dx , where the unit area is implied in what you wrote, but with a negative sign, as the heat flow is along the negative temperature gradient.

Why the power should be a linear function of temperature difference is anyone's guess, I suppose, if one is good at guessing.

The other forms of power equations, electricity, fluids, mechanical involve a squared term of one of the more familiar units that we use.
ie electrical power
P = E² / R
or
P = E² σ, where σ is the conductivity of the material ( in line with k, thermal conductivity )
One can see that increasing the voltage between the end points of a resistor results in a power factor increase in wattage, in the case of doubling the voltage the wattage increase is quadruple.
For the thermal conduction case, as mentioned above, doubling the temperature difference doubles the heat flow.

Good question.
Not one I really had thought about, because to me it did seem sensible that a longer thermal conductor would be more of a hindrance to heat flow than a shorter one.

 

[Dullard]

"The other forms of power equations, electricity, fluids, mechanical involve a squared term of one of the more familiar units that we use."

I think that 'Power' is a bit of a red herring here. We're talking about flow - the fact that heat flow is also 'power' (simply by virtue of the fact that heat is energy) doesn't suggest that power consumption and heat transport should be regarded as similar.

I = V/R seems very similar to the heat transport case. Current is the proper analog for heat flow. It is also possible that I just don't understand the question.

 

[E12-1]

which really should be dq / dt = -k A dT / dx

Thanks for correcting my negative sign. I have seen 'q' to be defined as both W (Incorpera) and W/m2 (Lienhard). You are probably right that most sources define 'q' as W.

It is also possible that I just don't understand the question.

The basic question is why is heat flux inversely proportional to the length of the conductor. Since Fourier's Law isn't derived from first principles (as erobz pointed out) I was hoping to get a more intuitive explanation as to why the flow of heat is slowed by 2x when you double the length of the conductor, when all other things are held constant. It's also possible (read: highly likely) that I just don't understand the underlying physics.

Why is it that a conductor of length L will conduct twice as much power as a conductor of length 2L

Based on this logic, it seems that the reason that power is inversely proportional to length has to do with the random nature of heat transfer (via lattice vibrations, electron flow). To me this would explain why a conductor of length 2L would be twice as cool, if one side was hot and the temperature on the other side was determined by convection / radiation (such as a pot handle: longer = cooler at the cool end). But when the temperature is controlled on both sides I thought you are effectively "forcing" the flow of energy to be constant too. I think the assumption in my last sentence is probably wrong, but I don't know why! I haven't had much time to research this since posting but I appreciate all of your input!