Radiative Heat Transfer: Kirchhoff's Law in New Light

Kirchhoff's Law of Thermal Radiation: 150 Years starts off by:

• Kirchhoff’s law is one of the simplest and most misunderstood in thermodynamics.

Let us see what we can say about Kirchhoff's Radiation Law stating that the emissivity and absorptivity of a radiating body are equal, in the setting of the wave model with damping presented in Computational Blackbody Radiation and Mathematical Physics of Blackbody Radiation:

(1) $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f $

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$ models a vibrating material string with $U$ displacement
2. $- \gamma U_{ttt}$ is a dissipative term modeling outgoing radiation
3. $- \delta^2U_{xxt}$ is a dissipative term modeling internal heating by friction
4. $f$ is the amplitude of the incoming forcing,
5. $T$ is temperature with $T^2=\int\frac{1}{2}(\vert U_t^2+U_x^2)dxdt$,
6. (1) expresses a balance of forces,

where $\gamma$ and $\delta^2$ are certain small damping coefficients defined by spectral decomposition as follows in a model case:

• $\gamma = 0$ if the frequency $\nu >\frac{1}{\delta}$
• $\delta = 0 $ if the frequency $\nu < \frac{1}{\delta}$,

where $\delta = \frac{h}{T}$ represents a "smallest coordination length" depending on temperature $T$ and $h$ is a fixed smallest mesh size (representing some atomic dimension).

This represents a switch from outgoing radiation to internal heating as the frequency $\nu$ passes the threshold $\frac{T}{h}$, with the threshold increasing linearly with $T$.

The idea is that a hotter vibrating string is capable of radiating higher frequencies as coherent outgoing radiation. The switch acts as a band filter with frequencies outside the band being stored as internal heat instead of being radiated: The radiator is then muted and heats up internally instead of delivering outgoing radiating.

A spectral analysis, assuming that all frequencies share a common temperature, shows an energy balance between incoming forcing $f$ measured as

• $F = \int f^2(x,t)\, dxdt$

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation $R$ measured by

• $R = \int \gamma U_{tt}^2\, dxdt$,

and (rate of) internal energy measured by

• $IE = \int \delta^2U_{xt}^2\, dxdt$,

together with the oscillator energy

• $OE =T^2 = \frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt$

with the energy balance in stationary state with $OE$ constant taking the form

• $F = \kappa (R + IE)$

with $\kappa\lessapprox 1$ is a constant independent of $T$, $\gamma$, $\delta$ and $\nu$. In other words,

• incoming energy = $\kappa\times$ outgoing radiation energy for $\nu <\frac{1}{\delta}$
• incoming energy = $\kappa\times$ stored internal energy for $\nu > \frac{1}{\delta}$,

which can be viewed as an expression of Kirchhoffs' law that emissivity equals absorptivity.

The equality results from the independence of the coefficient $\kappa$ of the damping coefficients $\gamma$ and $\delta^2$, and frequency.

Summary: The energy of damping from outgoing radiation or internal heating is the same even if the damping terms represent different physics (emission and absorption) and have different coefficients ($\gamma$ and $\delta^2$).

PS: Note that internal heat energy accumulating under (high-frequency) forcing above cut-off eventually will be transformed into low-frequency outgoing radiation, but this transformation is not part of the above model.