Extremely strange consequence of the speed of heat value

[fluidistic]

By chance, I have read a paper that left me in shambles. I would like you to help me figure out if it makes sense or not. I tried to follow Bridgman's logic, without a complete success. The extraordinary claim is that, if you start with a system consisting of a 1cm^3 copper cube where 2 faces are initially at different temperatures in a steady state, and then the system is left alone (so no more Dirichlet boundary conditions are applied to the 2 surfaces), then the cube's final temperature will be higher than the average temperature of the 2 initial surface temperatures, by an amount that depends on the speed of heat. By speed of heat, I do not mean the common misconception of how fast Tmax travels in a material, I am actually talking about the real speed of heat, i.e. the speed at which the heat disturbance travels through the material, measured in units of distance over time.

Here is the passage of the paper:

Imagine a centimeter cube of copper between the opposite faces of which there
is a temperature difference of 100°. The thermal flux is approximately 100
cal ./sec. The volume density of energy corresponding to this flux is such that
its product into the velocity of flux is equal to 100. For the velocity we may
take, in accordance with the Debye picture of thermal conduction, the veloc-
ity of sound, which for copper is about $3.5 x10^6$ cm/sec. The space density
of energy is therefore 100/3.5 X106 = 3 X10- 5 gm cal./cm3. The heat capacity
of 1 cm3 of copper is about 0.8 gm cal. This means, therefore, that if a copper
cube in which a thermal current of 100 cal ./sec is flowing is suddenly isolated
from the source and sink of heat flow, its final equilibrium temperature will
be about $4\times 10^{-5}$ °C higher than its average temperature during the flow.

This, of course, would be very difficult to detect. It is interesting, however,
that if a different velocity were assumed, as for example a velocity of the order
of a few cm per sec, which is the order of the apparent velocity with which the
maxima or minima of ordinary periodic thermal disturbances sink into the
metal, a temperature effect of the order of many degrees would have been
found. This affords rather direct confirmation of the correctness of the Debye
point of view. The experiment might be worth making to find how far the
velocity limit could be pushed.

Here is my attempt to understand his argument. So the 2 faces have an initial ΔT=100K. Then he mentions a heat flux but gives units of heat, which confuses me a bit. Let's try to be more rigorous than him and say the thermal flux is $\vec J_Q=100 cal /scm^2$. He claims that we can rewrite $\vec J_Q$, the heat flux, as the product of a speed with an energy density (this makes sense to me, if the speed is the actually speed of whatever constitute the heat, which may be phonons/electrons and other quasiparticles), and he says the product must be worth 100 (but in reality, the units should be J/m^2 in SI, i.e. it should also be a heat flux). Mathematically, $uv=100 cal/scm^2$ where $u$ is the internal energy divided by the volume of the cube ($=U/V$). From there, he gets that the energy density $u=|\vec J_Q|/v=3.5\times 10^6 g cal/cm^3$. Then he mentions that the heat capacity of 1cm^3 of Cu is worth 0.8 g/cal. Then I think he uses the relation $Q=mc\Delta T$ but the exact step he does is a bit mysterious and I am not quite sure how he got his answer. As if he had done $\Delta T =3\times 10^{-5}/0.8 K$...

 

[hutchphd]

read a paper that left me in shambles

Please reference the whole paper.

 

[fluidistic]

Please reference the whole paper.

Alright, I'll do it when I get the time, as soon as I can. Possibly tomorrow, but I am not 100 percent sure. It's by Percy Williams Bridgman.

 

[Frabjous]

Bridgman was a serious physicist. Maybe @Chestermiller can help?

What was the paper about? I have a copy of his collected papers in the office and I can look up the full ref.

 

[Chestermiller]

I have no idea what any of this is about. Sorry.

 

[fluidistic]

Bridgman was a serious physicist. Maybe @Chestermiller can help?

What was the paper about? I have a copy of his collected papers in the office and I can look up the full ref.

I think it was something along.the lines of ''relations in thermoelectricity''. The part I quoted is the very last part of the paper, so it should be quick to find the paper, I'll try to do it today.

 

[fluidistic]

The ref. Is '' a new kind of e.m.f. and other effects thermodynamically connected with the four transverse effects'' by Bridgman.

 

[hutchphd]

I think he is saying something quite simple (or maybe that is a description of me!)
If we have a fluid (say a gas) flowing in pipe, we don't include the flow velocity when we reckon the temperature of the flowing gas. The gas flow is not included in that reckoning.
Bridgman seems to be arguing that thermal energy moves not at the diffusion rate but rather at the speed of sound in a solid and so the concerted motion producing that flux does not produce a temperature rise. When the door is slammed shut ("suddenly isolated") that residual energy slightly raises the T.
All I could access was the ABSTRACT

 

[fluidistic]

I think he is saying something quite simple (or maybe that is a description of me!)
If we have a fluid (say a gas) flowing in pipe, we don't include the flow velocity when we reckon the temperature of the flowing gas. The gas flow is not included in that reckoning.
Bridgman seems to be arguing that thermal energy moves not at the diffusion rate but rather at the speed of sound in a solid and so the concerted motion producing that flux does not produce a temperature rise. When the door is slammed shut ("suddenly isolated") that residual energy slightly raises the T.
All I could access was the ABSTRACT

Hmm, I'm not sure I get it. To me, it's not so much about the amount of the energy flux, but about its speed. I.e. for a given energy flux, if the velocity is "low", then the temperature rise will be high. And vice-versa, if the velocity is high, then the temperature rise will be low. And that's assuming the same energy flux magnitude for both cases. That's really strange, to me...

 

[DaveE]

OK, I didn't really read the paper*, just skimmed it super fast. But here's my, probably incorrect, analogy (analogies are always incorrect, somehow):

Imagine a water pipe between two really big lakes at different altitudes. There is a constant pressure difference between the inlet and outlet and a constant flow of water downhill in the pipe. Then you instantaneously seal both ends. The final pressure in the pipe is increased above average because of the energy in the initial water flow.

So, the initial conditions aren't just constant temperature at the surfaces, but also include a constant heat flow due to the external maintenance of those temperatures before the transient. Or alternatively, the ICs aren't just constant temperature, but also no surface temperature change at t=0. Immediately after the transient, there is heat flowing to the cold surface that now has nowhere to go.

*[Footnote redacted by the Mentors for copyright reasons]

 

[berkeman]

The ref. Is '' a new kind of e.m.f. and other effects thermodynamically connected with the four transverse effects'' by Bridgman.

All I could access was the ABSTRACT

That paper is from 1932?!

 

[hutchphd]

That paper is from 1932?!

He got Nobel in 1946 (not for this). I haven't found the idea or this paper much cited so maybe I don't really get the argument. Would read it if I could. My sketchy understanding is that if 1 cm³ of fast moving stuff carries a the same heat flux as 1 cm³ of slow moving stuff then the slow moving stuff has more internal energy per volume.

 

[Vanadium 50]

Perhaps the Mentors should close this thread. It's been seven months, and we still don't have access to this paper.

Honestly, "Help me understand this 90 year old paper that I don't have and anyway is behind a paywall" is a very, very heavy ask.

If the paper shows up, then we can reopen and discuss it.

 

[DaveE]

Perhaps the Mentors should close this thread. It's been seven months, and we still don't have access to this paper.

Honestly, "Help me understand this 90 year old paper that I don't have and anyway is behind a paywall" is a very, very heavy ask.

If the paper shows up, then we can reopen and discuss it.

a) The paper is accessible to some.

b) I don't see how the age of the paper, or problem statement is relevant. It's newer than the double slit experiment or relativity which we talk about all the time.

c) The problem is pretty well defined by the OP anyway, so I'm not sure reading the paper is necessary.

d) It was the OP that resurrected the thread with an on topic comment. So it's not a completely ignored thread.

e) Posts in this thread are basically on topic and not obnoxious.

f) Since PF doesn't have a regular policy of closing inactive threads anyway (maybe they should, IDK), I don't see what the basis is for closing this one compared to many others.

What problem would be solved by preventing others from posting here?

 

[Vanadium 50]

The paper is accessible to some.

But not so many.

I don't see how the age of the paper, or problem statement is relevant

Langauge changes. Styles change. For example, expecting to see modern error analyses in old papers (Milikan, Michaelson) is pretty hopeless. Figuring out what is going on is that much harder.

PF has a long, long, long history of papers not saying what people saying they do. I think it's reasonable for someone who wants help in understanding what a paper says to point us to the paper.

But since I can't see it, you all go right ahead.

 

[berkeman]

Not my area of expertise, but is there a way to tell how many other papers have cited this paper and the work? Or did the work discussed in the paper die out? 90 years seems like a pretty long time for nothing fruitful to have happened if it was a valid line of exploration...

 

[DaveE]

But since I can't see it, you all go right ahead.

Which is a better response than suggesting no one else should comment either. I am constantly ignoring threads myself for a huge variety of reasons.

 

[256bits]

He got Nobel in 1946 (not for this). I haven't found the idea or this paper much cited so maybe I don't really get the argument. Would read it if I could. My sketchy understanding is that if 1 cm³ of fast moving stuff carries a the same heat flux as 1 cm³ of slow moving stuff then the slow moving stuff has more internal energy per volume.

It is the energy, or momentum of a 'gas' of phonons.
They do not have an actual momentum as they have no mass, so one should call it a pseudo-momentum.
Thermodynamically, the guy is arguing that that the phonons and their wavelike nature gives an extra energy to the one dimensional heat flow moving through the cm cubed amount of material.
How that correlates with the speed of sound in the material - ie lessor energy with a greater speed of sound, and vice versa, is beyond my knowledge of phonons.
Any thermodynamic phonon expert available, or one who can dot the i's and cross the t's.

the article mentions the the Debye picture of thermal conduction, which expresses the phonon nature of heat capacity.

there was this post below,
https://www.physicsforums.com/threa...erls-approaches-of-the-heat-transfer.1008839/
which received no responses though.

 

[fluidistic]

I agree with DaveE. I think the passage I quoted is self consistent in that, in principles, we don't need the rest of the paper to work through it. However I also recognize that the rest of the paper would be relevant and possibly help, too. The paper has apparently prohibitive copyrights, so that we cannot display it here.

I like PF for the fact that we can discuss topics (something Physics Stack Exchenge lacks). I would prefer not to close.this thread, if possible, and I might post in the future if I make any progress. I'm thinking to possibly test if I can experimentally measure.the ''speed of heat'' in Cu, but I'm not quite sure of how to interprete the result I would get.

 

[hutchphd]

They do not have an actual momentum as they have no mass

That is not really correct. Their momentum is conserved but only to within a vector of the reciprocal lattice. The momentum of the whole crystal is rigorously conserved, and the phonon can indeed transfer momentun via scattering.
Also there are optical phonons that have a different dispersion relation. I suggest Kittel for this.