torsdag 27 oktober 2011

Radiative Heat Transfer: Kirchhoff's Law in New Light

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

7 kommentarer:

  1. If the Kirchoff’s law is always valid, then why in the balance sun-earth-space the earth surface has absorrptivity = 1 and emissivity = 0 with respect to sun SW radiation?
    As far as I remember, Kirchoff’s law is referred to a body enclosed into a cavity and then it cannot have the general validity attributed to it.
    Michele

    Michele

    SvaraRadera
  2. This because the Sun is far way and the equilibrium temp of the Earth is much smaller than that of the Sun. In my model the insolation frequency is above the cut-off frequency of the Earth and thus causes heating without radiation. The
    heating is then transformed into low-frequency radiation and this transformation is not included in the model as presented. Hence the apparent different emissivity and absorptivity.

    SvaraRadera
  3. But in this case the Kirchoff’s equality αλ = ελ doesn’t exist.
    Michele

    SvaraRadera
  4. Michele, Let me put you right about Kirchhoff's law. In Perry's Chemical Engineering Handbook is the following statement
    "According to Kirchhoff's law, the emissivity and absorptivity of a surface in surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperature of the surface and its surroundings differ, the total emissivity and absorptivity of the surface often are found to be different, but, because absorptivity is substantially independent of irradiation density, the monochromatic emissivity and absorptivity are for all practical purposes the same."
    Not difficult to understand for engineers who have studied and have experience in heat&mass transfer. Physicists, however, live in a different world and seem to ignore all the vast amount of measurement and research done by engineers over more than a century.
    cementafriend

    SvaraRadera
  5. cementafriend,
    in the case sun-earth-space, I am relating to the normalized spectra of absorption and emission, e.g. the following figure "Spectra".
    It seems to me that you quote contrasts what we read into this figure. Why?
    Michele

    SvaraRadera
  6. Michele, Your English maybe difficult
    Please note some important points:-
    a)"emissivity and absorptivity of a surface" ( a gas is not a surface)
    b)"in surroundings at its own temperature"
    c)"monochromatic emissivity and absorptivity"
    Your "Spectra" has no relation to b) or c)
    If you do not know Perry's Chemical Engineering Handbook see here http://en.wikipedia.org/wiki/Perry%27s_Chemical_Engineers%27_Handbook
    This publication (now in the 8th edition) has been peer reviewed and updated by thousands (if not hundreds of thousands) of competent acedemics and practising engineers since it was first published 75 years ago. As I mentioned physicists seem to live in a different world and do not understand even a small fraction of the contents of this book.
    I suppose Michele is a common name but I have noted someone with the same name making sensible comments on other blogs (such as http://tallbloke.wordpress.com/) and even operating an Italian blog at daltonsminima.altervista.org.
    Registered engineers have to comply to a code of ethics which includes competence and continuous learning. Ethics does not seem to feature in the minds of so-called scientists these days.
    cementafriend

    SvaraRadera
  7. cementafriend,
    Sorry. My name is very common, I am neither the one Michele nor the other. You are right, my English isn’t good but I am a senior engineer.
    I wanted to say that the Kirchoff’s law is erroneously employed when the DLR is explained claiming that it can be used because the atmospheric upwelling gas is in LTE although its thermal energy is continuously converted to gravitational energy.
    Michele

    SvaraRadera