Thermal conductivity not strictly a material property? Fourier's law breaks down? 3

[NotReallyKelvin]

A few vague references out there say you can't just state a thermal conductivity for a bulk material, for example insulation used in building construction, because it depends on thickness. Anybody know what this is getting at? Trivially, packing a flexible insulation into a smaller volume spoils its resistivity, and if manufacturing methods change for different weight insulations their intrinsic properties may change, and making stacks of sheet insulation may give different effective bulk conductivities for the stack because all the contact conductances between layers need accounting for, and if thermal radiation penetrates the sample then thermal conductivity will poorly model the system behavior. But references make it sound like it's the fault of Fourier's law itself.

Another example concerns layers of silicon in integrated circuit manufacture. There is a new issue, Kapitza resistance, involving resistance at atomically perfect bonds because the mechanism of conduction changes or because phonons scatter at the interface due to lattice discontinuities. For example alternating layers of bismuth and diamond create great resistance because the conduction mechanism alternates between the electron gas and phonon conduction. Maybe this reference is just failing to consider these interfaces separately from the bulks, but it doesn't sound like it.

So -- Fourier's law might be repealed, or what?

Thanks!!

 

[IRstuff]

No, you are confusing R-value and thermal conductivity (rho). R-value is the thermal conductance of a fixed thickness of insulation. Thermal conductivity is ALWAYS in terms of W/m-K, which thermal conductance is rho/thickness, resulting in W/m^2 of thermal conductance.

Thermal conductivity of insulation, particularly fiberglass is heavily dependent on the density of the fiberglass, and the AIR, since the thermal conductivity of typical fiberglass insulation is comparable to the same thickness of air. Only when the fiberglass is compressed, and the proportion of air is reduced would the thermal conductivity change, by up to a couple orders of magnitude.

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[NotReallyKelvin]

>No, you are confusing R-value and thermal conductivity (rho). R-value is the thermal conductance of a fixed thickness of insulation. Thermal conductivity is ALWAYS in terms of W/m-K, which thermal conductance is rho/thickness, resulting in W/m^2 of thermal conductance.

Thanks, but that isn't it. The one reference I saw said that building insulation manufacturers should not just say that their insulation has a certain thermal conductivity in the bulk, or say that there will be a certain R value per inch of thickness. The reason, they say, is that the thermal conductivity is not a constant independent of thickness. I believe that fixed thermal conductivity or fixed "R value per inch of thickness" should work unless there is something funny going on.

The other reference I saw showed silicon layers of different thickness in an integrated circuit and claimed that their thermal conductivity (not specific conductance) approached a constant value for thick layers but went to zero in the limit as thickness went to zero.

I got questioned about these in a small meeting and only had content cut and pasted from the respective sources. I have to find out the sources to get the whole context. When I can I'll post links.

This definitely isn't the dimensional confusion between "per layer" and "per thickness", though.

 

[IRstuff]

Sure it is. Lay people, and even technical, have a terrrible time using terminology correctly. Thermal conductivity is EXTREMELY well-defined as a term; just open ANY heat transfer text: rho = W/m-K. Thermal conductance is therefore rho *area/thickness. There is no ambiguity, there is no arbitrarily variation.

Moreover, your specific example of silicon can be readily explained through sloppy modeling. If you ignore contact and interface resistances, then you'll get stupid answers. Unless, you are talking about atomic scale thicknesses, in which case, there may well be other things going on.

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[dvd]

Glass fiber contact increases as insulation is compressed. The actual conductivity part of glass wool insulation is due to the conductivity of the air trapped in the glass fibers PLUS the conductivity of the glass itself. Fibrous insulation binder binds fibers into location and provides a predictable product density with a minimum of compression applied. When insulation is compressed the less-conductive air is displaced and the more-conductive glass fiber interfaces increase.

 

[Dougt115]

Taking the limit as insulation goes to zero or infinity is a bad idea.

Building insulation at zero thickness will equal 100% glass fibers, at infinity it approaches air the thermal conductivity of the actual insulation will be between these two numbers. The constant that the manufactures provide is if you install it per instructions. Change the installation and you will impact the R number.

Same is true with your silicon. A thick layer will have a constant equal to only the silicon. Infinite layers will have to add an infinite number of resistance layers which will drive it to zero. But try to make an infinitive number of layers.

 

[moltenmetal]

Thermal conductivity of true solids is well understood, but that doesn't mean it's easy to measure.

Porous insulating materials, and materials in composite, are notoriously hard to thermally model because contact resistance is a very tricky parameter to quantify with any accuracy. Sometimes it can be eliminated from consideration as not mattering all that much, but sometimes it is a dominant resistance.

Radiation through an insulating material is inherently compensated for in the k value given for the insulating material versus temperature- it is impractical to isolate the effects of radiation and conduction through the material at high temperatures.

 

[NotReallyKelvin]

OK, thank you moltenmetal, we have a couple encouraging leads here!

>Porous insulating materials, and materials in composite, are notoriously hard to thermally model because contact resistance is a very tricky parameter to quantify with any accuracy.

We actually had a discussion about this point here in our organization, and we thought it would matter very little for porous insulating materials that have a conductivity near that of air. For one thing, the void spaces at the contact interface would behave nearly the same way if we could magically fill them with insulation, because k wouldn't change much in the void. For another thing, I figure the most significant void spaces in k measurement in general tend to occur where a high spot or contaminant particle holds regions of the two surfaces separated like a tent pole holds the cloth off the ground, but in insulation materials a tent pole just kind of penetrates and doesn't create any clearance.

>Radiation through an insulating material is inherently compensated for in the k value given for the insulating material versus temperature....

Doesn't radiation contribute flux that is less strongly dependent on thickness than conductive flux is? For sheet insulation that is very long and wide compared to its thickness, if it's completely transparent to the radiation, this part of the flux would be totally independent of thickness, so if we called it a conductivity, its value would appear to be proportional to thickness.

Any thoughts on these two things???

 

[IRstuff]

> For an insulating material, the contact resistance is not that significant, particularly for a voidy material like fiberglass. Contact resistance is really only an issue with highly conductive layers/materials. You might effectively consider the interface layer to be 3 or 4 times thicker to account for the poor contact resistance, but that's still a minor fraction of the overall thickness for insulation.

> The radiation being discussed is mostly well above 1000nm, which is not transmitted by any insulation materials. Therefore, it does become thickness dependent. However, that said, for a voidy material, internal radiation is relatively insignificant. Typical insulation radiation shields are for external radiation, to prevent thermal transfer from the insulation to the environment.

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[moltenmetal]

Attached please find the thermal conductivity of copper versus temperature as measured by various researchers over time.

Solid copper.

The plot is log-log.

Obviously, measuring a bulk k value is attempting to fit a simple conduction model to a more complex heat transfer situation- one which exists inside every type of device ever contrived to measure a simple thermal conductivity, despite one's best efforts.

Does this imprecision prevent us from doing useful design? Not at all- but it does give you some sobering advice whenever someone tells you they've measured something precisely and now you'll have to hang your hat- or someone's life- on the result. Take it from hard life experience- measuring anything accurately isn't easy, and relying on measurements as accurate is reckless.

To paraphrase a famous quote, engineering is the art and science of working with bad data and measurements, poor models, imprecise shapes and unknown forces, and doing so in such a way that the public has little reason to suspect the depth of our ignorance.

 

[NotReallyKelvin]

Here are citations that doubt Fourier's law (to oversimplify a bit):

http://www.qats.com/cms/2011/10/21/thermal-conductivity-what-is-it-and-why-you-should-care/

This describes small length scale deviation from Fourier's law, saying "In the electronics industry, the constant push for smaller size and faster speeds has considerably reduced the scale of many components. As this transition now continues from the macro- to micro-scale, it is important to consider the effects on thermal conductivity and not to assume the bulk property is still accurate. Continuum-based Fourier equations cannot predict thermal characteristics at these smaller scales. More complete methods, such as the Boltzmann transport equation and the lattice Boltzmann method, are needed [3].

The effect of thickness on conductivity can be seen in Figure 2. The material characterized is silicon, which is widely used in electronics."

http://www.ecfr.gov/cgi-bin/retriev...L&n=16y1.0.1.4.61&r=PART#16:1.0.1.4.61.0.32.5

See section 460.20 in the above, which says "You can list a range of R-value per inch. If you do, you must say exactly how much the R-value drops with greater thickness. You must also add this statement: “The R-value per inch of this insulation varies with thickness. The thicker the insulation, the lower the R-value per inch.”"

http://en.wikipedia.org/wiki/R-value_(insulation)

This points out that ways of making insulation thicker or thinner may also change its properties, which I think all will agree is obvious.

 

[Dougt115]

Of course the R value will not the same at the Macro level and the micro levels. The macro level R is the Mean of an evenly distributed material well mixed. The micro level cannot assume this as the gap may be either material in pure form, a mixture of unknown ratio, or even a void.

 

[IRstuff]

Again, you are mixing apples and oranges, and possibly some rocks. None of your supposed counter examples are related in any way shape or form.

In the case of the silicon cited, the thickness range is on the order of 1000 nm, which is around the point where silicon is practically visibly transparent. No one can realistically claim that the effective thermal conductivity isn't changing at the scale. BUT, note that they are plotting W/m-K vs. thickness, so for any given thickness, there is a corresponding thermal conductivity. AND, they are futzing with the surface properties, so what they're actually graphing is NOT bulk thermal conductivity to begin with; it includes surface, or contact, effects. No smoking gun, no failure of Fourier's Law. As I alluded to earlier, if the contact interface becomes a sizable portion of the overall resistance, you must be more careful about how that's evaluated.

As for the R-value; note this from tghe Wiki article: "In practice, this linear relationship does not hold for compressible materials such as glass wool batting whose thermal properties change when compressed." And the implication from the Federal code is the same. There are materials for which the R-value is thickness independent, like sheet rock or styrofoam. Fiberglass batting, on the other hand is compressible, and unless great care is taken a 1-in thickness might indeed have a different density than a 2-in thickness. Additionally, this is another case where the interfaces may play a large role in thermal resistance. If you take 2 layers of 1-in thick batting, the R-value WILL double, as expected, therefore, there are other factors at play that are being ignored. Once again, Fourier's Law has not been violated.

Technically, Fourier's Law CANNOT be "violated." It's like Ohm's Law V = I * R. When you think the law is being violated, it just means that your model, which is an abstraction of reality is INCORRECT. When you find the correct model, the relationship holds.

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[moltenmetal]

BTW the plot is from an Ethics in Engineering textbook- and its point has always stuck with me. I learned it the hard way during my Msters' research, losing a month of my life due to repeatability problems in my experiments caused by the position of a thermocouple moving 1/4" within a thermowell.

 

[NotReallyKelvin]

Thanks, everybody.

IRstuff, for the record I'd like to say I'm not actually criticizing Fourier. I've told people the concept of thermal conductivity can be used to predict heat transfer through materials that are thicker than mean free path scale, if they practically opaque to thermal radiation at distances small compared to the material thickness, if there is no bulk flow permeating the material, if we don't assume thermal conductivity stays the same when we alter the material, and if there are either few interfaces that we account for separately, or many interfaces that we treat as part of the substance of our material by averaging. The issue I personally am dealing with is that people are questioning this statement because of things they are finding online.

Moltenmetal, the illustration came from a paper by David R. Lide, Jr. Its title was "Critical Data for Critical Needs" and it appeared in Science volume 212, 19 June 1981, pages 1343 to 1349. You can get the paper, with a clearer image of the graph, at:

http://worldtracker.org/media/libra..._1981-1982/pdf/1981_v212_n4501/p4501_1343.pdf

 

[NotReallyKelvin]

Well, that's great, I just can't get past this one, can I?

"Mainz/Frankfurt. Scientists at the Max Planck Institute for Polymer Research (MPI-P) in Mainz and the National University of Singapore have attested that the thermal conductivity of graphene diverges with the size of the samples. This discovery challenges the fundamental laws of heat conduction for extended materials.
Davide Donadio, head of a Max Planck Research Group at the MPI-P, and his partner from Singapore were able to predict this phenomenon with computer simulations and to verify it in experiments. Their research and their results have now been presented in the scientific journal "Nature Communications".
"We recognized mechanisms of heat transfer that actually contradict Fourier’s law in the micrometer scale. Now all the previous experimental measurements of the thermal conductivity of graphene need to be reinterpreted. The very concept of thermal conductivity as an intrinsic property does not hold for graphene, at least for patches as large as several micrometers", says Davide Donadio."

http://www.mpip-mainz.mpg.de/news/thermal_conductivity

 

[IRstuff]

I don't see anything to "get past."

What do you see in the axes labels for attached graph "d"?
> They're graphing thermal conductivity, and with relatively small error bars. How are they doing that? They're using Fourier's Law, since σ is derived from the ratio of heat flow to delta temperature

What do you see the graph doing as L increases further?
> It looks awfully asymptotic to a constant value at a given temperature. Note that thermal conductivity is NEVER constant over temperature, since the atomic vibrations dictate the ability to propagate increased heat. This is not fundamentally different than the fact that the resistance of a cold light bulb filament NEVER equals wattage/voltage; that relationship only applies when the filament is hot. In this case, resistance increases with temperature, but in the graphene case, thermal resistance decreases with temperature. The variation in thermal conductivity with length in the graph simply means that when you get to the physical scale where you have to be thinking about phonons, you have to be more circumspect in using thermal conductivity.

FYI: the article can be downloaded from Arxiv.org:

http://arxiv.org/ftp/arxiv/papers/1404/1404.5379.pdf

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