tisdag 25 oktober 2011

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

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
  • $IE = \int h^2U_{xt}^2\, dxdt$
  • $F = R + IE$
  • 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 $R$.

15 kommentarer:

  1. "...two models with deep historical roots"

    Your insane ideas about one way energy flow have no deep historical roots at all. Your notion that somehow an inanimate object knows where all the other objects in the universe are, knows their temperature, and therefore knows whether or not it can absorb energy reaching it from them, is just laughable.

  2. Maybe insane and laughable to you. The truth is that two bodies in radiative contact feel the temperature of the other by the spectrum of radiative contact and this determines the direction of the heat transfer from hot to cold. Nothing to laugh about really.

  3. Yes, the net transfer of energy is from hot to cold. But how does the cold body know to emit energy only in directions where there is not a warmer body? And which directions are those, anyway? Does the Earth not emit any radiation along any line of sight which encounters a star?

  4. Well, the star is so far away that the intensity of light is so small that it is overpowered by the radiation from the Earth.

  5. So radiation is being emitted by the Earth in the direction of the star?

  6. Yes, the presence of the star does not change the low background temperature (0 or 3 K) for the radiation from the Earth.

  7. This is becoming ridiculous. How does a body know where its emitted photons are going to end up? Or/and how does a body know where incoming photons were emitted? And what happens to the photons if a body decides not to absorb them?

  8. It is not so strange: The radiative contact carries the frequency information required. And there are no IR-photons.

  9. The EM wave acts within two coexistent electric and magnetic fields that obey the rule of the univocity of the field lines and so of the univocity of the field intensity. Thus we have also the univocity of the effect produced by them at any point of the space.

  10. This discussion is leading to a repeat of the discussion of this post


  11. Ha ha, no IR photons? But there are visible light photons? Or are you anti-photons generally? Anyway, nice that you have conceded that energy travels from colder bodies to hotter ones, though you seem to think that the surface temperature of a star depends on its distance from Earth.

  12. Yes it is a repetition.

    The frequency of the light from a star does not change frequency, but the amplitude is so small that the heating effect on the Earth is nil.

  13. Small != 0. Are you anti photons generally or just photons with certain wavelengths?

  14. I am skeptical to IR-photons in particular. Maybe there are no photons at all but that is another story, of no relevance to climate.

  15. Claes says:" a body can radiatively transfer heat energy to a body of lower temperature with lower cut-off, but not the other way around."
    But two black bodies with small temp differences, eg earth and atmosphere, have overlapping spectra so they can both emit and absorb IR of the same frequencies.
    But as the radiation from earth is larger the colder atmosphere will be warmed and this will lower the radiation from earthm acc to the SB law, which indirectly will warm it as the input from sun is constant.