Radiative Heat Transfer: Theory

In a couple of posts I will seek to summarize experience on radiative heat transfer between two bodies of different (or equal) temperature. We will see that there are two models with deep historical roots:

1. One-way net transfer from hot to cold.
2. Two-way transfer between hot and cold with net transfer from hot to cold.

I argue that 1. is physical obeying the 2nd law of thgermodynamics, while 2. is unphysical violating the 2nd law.

2. represents the model underlying CO2 alarmism with DLR/backradiation from a cold atmosphere to a warm Earth.

I will first consider 1. as expressed in the model studied in Computational Blackbody Radiation

and Mathematical Physics of Blackbody Radiation.

This model is based on a frequency cut-off increasing with temperature, which means that a radiating body is only able to absorb and re-radiate frequencies below cut-off, while frequencies above cut-off are absorbed and stored as internal heat energy. The result is that a body can radiatively transfer heat energy to a body of lower temperature with lower cut-off, but not the other way around. In other words, the model satisfies a 2nd law of thermodynamics.

We thus consider the following wave model for radiation as a continuum of oscillators with damping of the form:

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

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

1. $U_{tt} - U_{xx}$ represents the oscillators in a wave model with U displacement
2. $- \gamma U_{ttt}$ is a dissipative term modeling outgoing radiation
3. $- h^2U_{xxt}$ is a dissipative modeling internal heating
4. $f$ is incoming forcing/microwaves,

where $\gamma$ represents the constant in Planck's radiation law and $h$ represents a smallest mesh size, connected to dissipative losses as outgoing radiation and internal heating, respectively. The model establishes 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

• $R = \int \gamma U_{tt}^2\, dxdt$,

oscillator energy $OE$ measured by

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

(rate of) internal energy measured by

• $IE = \int h^2U_{xt}^2\, dxdt$

with the energy balance in stationary state expressed as

• $F = R + IE$
• incoming energy = outgoing radiation energy + stored internal energy.

Assuming that all frequencies have the same temperature, a common temperature $T$ can be defined by $OE$ and $IE$.

The model has temperature dependent frequency cut-off which effectively means that for frequencies $\nu$ below cut-off $IE_{\nu}=0$ and for frequencies above cut-off $R_{\nu}=0$ with $IE_{\nu}$ and $R_{\nu}$ the contribution to $IE$ and $R$ from frequency $\nu$.

The model can be used to describe radiative heat transfer between two bodies in radiative contact established by sharing the forcing $f$, with the first body satisfying (1) and the second body satisfying a similar wave equation with displacement $V$:

(2) $V_{tt} - {V}_{xx} - \gamma {V}_{ttt} - h^2{V}_{xxt} = f$

The model (1) + (2) can now be used to describe the heat transfer between a body 1. with temperature $T_1$ described by (1) and a body 2. with temperature $T_2$ described by (2). Assuming that $T_1>T_2$, we have for frequencies below cut-off for 1. and above cut-off for 2. (leaving out a frequency subindex) in stationary state:

• $F = R_1 = IE_2$

which effectively expresses transfer of heat from the hotter 1. to the colder 2. For frequencies

below the cut-off of 2. (with $R_1=R_2$) and above the cut-off of 1. (with $IE_1=IE_2$) there is no heat transfer.

The above model thus describes one-way radiative heat transfer from hot to cold with the radiation $R_1$ from 1. being transfered into heating $IE_2$ of 2. in the frequency range between the respective cut-offs. The 2nd law of thermodynamics is thus enforced by a temperature-dependent cut-off shifting outgoing radiation to internal heating for frequencies above cut-off.

The above description covers the basic one-way aspect of the heat exchange between the two bodies, assuming oscillator energy to be stationary.

Transfer of $IE$ into $OE$ requires separate modeling. One can think of the internal energy $IE$ as incoherent high frequency energy which can be organized into coherent oscillator motion stored as $OE$, which can be emitted as coherent radiation $R$.