Fourier - Heat Conduction

*PRLog*-

*Feb. 21, 2013*-*YOUNGWOOD, Pa.*--**Background**

Since the 19th century, many efforts have been made to determine the velocity of heat propagation that underlies the one-step parabolic differential equation of a solid known as Fourier’s law. See thumbnail. T is temperature, t time, x distance, rho density, C heat capacity, K thermal conductivity, and Q heat generation.

Although verified in an uncountable number of heat transfer simulations, the Fourier law itself remains a paradox in that the temperatures T through the gradient of dT/dx at all points in the solid are instantaneously related to a thermal disturbance dT/dt at some other point. Since the thermal disturbance may be distantly removed, Fourier’s law suggests action at a distance or that the disturbance moves at an infinite velocity both of which are in violation of Einstein’s theory of relativity that precludes motion of anything faster than the speed of light.

Indeed, the literature is replete with proposals [1, 2] of modifying the Fourier law to allow the thermal disturbance to move at finite velocity while still preserving the validity of the Fourier law. One such modification described by Marin [3] is based on the Cattaneo and Vernotte (C-V) model that assumes a delay time for the thermal disturbance to build-up to the temperature at a distant point for the Fourier equation to proceed. But then the question is:

How is the delay time

*a priori*known in terms of the geometry and properties of the solid?

Modifying the Fourier law to avoid infinite velocity is difficult to implement, e.g., in the C-V model the delay time for every point in the solid relative to the location of the disturbance would be required, and although implementable is cumbersome.

Perhaps, it is better reconsider what the incredible success of the Fourier law in explaining thermal conduction is telling us - the heat carrier does indeed move at a very high velocity. Instead of modifying the Fourier equation by mathematical trickery to avoid infinite velocity, we should instead be looking for a heat carrier that moves at a very high, although not infinite velocity.

**Problem**

Based on theories of Einstein and Debye, the heat carrier in the thermal conduction of non-metals is phonons, but the phonon is too slow to be the heat carrier in thermal conduction. Similarly, the heat carrier in metals might include the electron. But like the phonon, the electron is too slow to be the heat carrier. Therefore, the elevant question is:

What heat carrier in thermal conduction is consistent with the Fourier law?

**Proposed Heat Carrier**

The BB radiation present in all solids – metals and non-metals alike – can only be the heat carrier in thermal conduction at the macroscale. BB stands for blackbody. Planck’s QM allows the BB radiation heat carrier to move at the speed of light at least consistent with the Fourier law that requires the heat carrier to move at infinite velocity. QM stands for quantum mechanics.

**Discussion**

Depending on the temperature of a thermal disturbance, BB radiation is propagated throughout the solid depending on its absorption spectrum, but if transparent the BB radiation escapes to the surroundings. Transit times are very short because the BB radiation moves at the speed ~ c/n, where c is the velocity of light in the vacuum and n the refractive index of the solid. Therefore, the Fourier equation that implies the thermal disturbance propagates at infinite speed is reasonably approximated with BB radiation heat carriers that move at the speed of light, but certainly not phonons and electrons. No other proof of BB radiation as the heat carrier in solids is necessary as the argument is self-evident.

**Extensions to the Nanoscale**

Traditional heat transfer based on phonons is generally thought to not follow the Fourier law at the nanoscale. Indeed, the BB radiation heat carrier is not applicable at the nanoscale as the heat capacity of the atom vanishes thereby precluding any temperature changes necessary to implement the Fourier lawn.

At the nanoscale, heat may only be transferred by QED induced EM radiation. QED stands for quantum electrodynamics and EM for electromagnetic. By this theory, absorbed EM energy in a nanostructure cannot be conserved by an increase in temperature because QM requires the atom to have vanishing heat capacity under its own TIR confinement. TIR stands for total internal reflection. Instead, conservation proceeds by the creation of non-thermal EM radiation at the TIR resonance of the nanostructure. Depending on photoelectric properties, the EM radiation creates excitons that charge the nanostructure or is emitted to the surroundings. See QED radiation at http://www.nanoqed.org , 2009-2013.

In contrast, the Fourier law at the nanoscale based on phonons assumes the atom has the same heat capacity as the macroscale. Marin [3] describes the thermal time constant at the macroscale proportional to ~ R

^{2}/alpha, where R is the particle radius and alpha = K/rho/C. For condensed matter, alpha ~ 10

^{-6}m

^{2}/s which if applied to nanostructures upper bound by a 1 micron spherical particle having R = 0.5 microns, the time constant < 0.25 microseconds. However, QM requires the heat capacity C to vanish at the nanoscale, and therefore alpha diverges and the response time vanishes. Similarly, QED radiation for n ~ 1.5 gives the response time ~ R/c/n ~ 1 fs. Clearly, the delay time in the C-V model is not applicable as divergence occurs at femtosecond time scales.

**Conclusions**

1. At the macroscale, only BB radiation heat carriers moving at the speed of light are consistent with Fourier’s law that requires thermal disturbances to propagate at infinite velocity. Phonons and electrons are simply too slow to be considered the heat carriers in thermal conduction.

2. At the macroscale, the validity of using Fourier’s law need only require a statement that heat is transferred at the speed of light in the solid by BB radiation. No other justification for the validity of the Fourier equation is required. Hyperbolic wave equation simulations of BB radiation at the speed of light may be used as an alternative to the Fourier law, but the complication of obtaining analytical solutions is not justified as the Fourier law upper bounds the thermal response.

3. Fourier’s law is no longer valid at the nanoscale. Instead, the heat transfer may be determined by QED induced radiation

**References**

[1] D. D. Joseph and L. Preziosi, “Heat waves,” Rev. of Modern Phys., vol. 61, pp. 41 – 73, 1989.

[2] D. D. Joseph and L. Preziosi, Addendum to the paper "Heat waves" Rev. Mod. Phys. 61, 41(1989) Rev. of Modern Phys., vol. 62, pp. 375 – 391, 1990.

[3] E. Marin, “Does Fourier’s Law of Heat Conduction Contradict the Theory of Relativity?” Lat. Am. J. Phys. Educ., vol. 5, pp. 402-407. June 2011