Does it take some time for the hot combustion products from a flame to reach local thermodynamical equilibrium (i.e. for the energy state populations to follow the Boltzmann distribution)? If so, on what type of time scales might this departure from local thermodynamical equilibrium occur?


I am trying to model infrared emission spectra in the mid-infrared region to reproduce emission spectra of hot exhausts from the flame of an industrial flare stack measured with an FTIR-spectrometer.

The emitted spectral radiance is calculated using the Planck function and the absorption crossections of different molecular species. The absorption crossections are calculated for different species using the HITRAN and HITEMP line databases.

This seems to work fine for some species and some regimes. However, I seem to have problems especially when the exhaust is really hot. For instance, in the region 900-1000 cm$^{-1}$ I can qualitativly see that I have strong water emission lines, but with only varying the temperature parameter I cannot get the relative intensities of the different lines right.

I keep thinking that one possible problem might be that the exhaust product from the combustion are not in local thermodynamical equilibrium. In that case the state populations for the species would not follow the Boltzmann distribution which is assumed for calculating the line intensities and their temperature dependence.

  • $\begingroup$ Does your model take into account that combustion takes time - in other words, that different parts of the flame have different degree of "completeness of combustion"? $\endgroup$ – Floris Jun 8 '14 at 21:45
  • $\begingroup$ Of course it takes time for the flame (ionized gas, plasma) distribution function to become Maxwellian, that's the collision time for electrons and ions. In fact, electrons will thermalize among themselves, and ions among themselves on a shorter time scale than the two species thermalize between each other, so the system will relax to a two-temperature system for some time. However, this is not an initial value problem here, looks like there are source terms that drive the system away from Maxwellian. Anyway, reading a basic plasma physics book (e.g. F. Chen) should be your starting point. $\endgroup$ – Maxim Umansky Jun 8 '14 at 22:24
  • $\begingroup$ @MaximUmansky Could you post that as an answer? $\endgroup$ – rob Jun 9 '14 at 14:04
  • $\begingroup$ Am I correct in thinking that we must be careful to distinguish between the thermalization of the free electrons, and the thermalization of the molecular energy levels? For instance, I could imagine a situation in which the free electrons rapidly come to a Maxwellian distribution, but the population of the water molecule energy levels are not what one would calculate based on Boltzmann factors. This is often the case in astrophysics, for example, and there is a huge literature concerning radiative non-LTE effects. I don't know if this is analogous to the combustion problem posed here, though. $\endgroup$ – kleingordon Jun 10 '14 at 6:06

I'm no specialist on this, so, take this answer as thoughts from someone that also wants to know the answer to your problem.

This is a very interesting problem, and one that I have interested myself. If you try modelling this process using some kind of kinectic theory (e.g. Boltzmann Equation) approach, naturally apears several time scales from the diferent relaxation processes that can appear. If you try modeling as Navier-Stokes, and afterwards fitting the data with the thermal equilibrium, you resolve the first moment on the Knudsen number, but you ignore any kind of relaxation time, this could be one source of the problem.

As @Floris said, it can happen that you have different stages of the chemical reaction occuring in diferent parts of the flame, that you come from a finite relaxation time for the chemical reactions.

Other problem that might be closer to reality, and way simpler to solve, is that you may have huge temperature gradients, and you are ignoring spatial gradients when doing the calculation, and this results in some kind of smearing of your spectral emmision. If so, it might be interesting to either simulate this system, to get some kind of information of the temperature distribution of the flame, or to try measuring it directly on a smaller flame, but with a spectometer with a smaller apperture.

One last possibility would be the change of the emission function. If you have (really) out of equilibrium fenomena, you might have to calculate corrections on the distribution function, in the case of light from the Bose-Einstein distribution, to include corrections that come from either heat fluxes, viscous tensors and the like. I haven't seen this used very much, but I did seen in a very diferent context (Relativistic Heavy Ion physics).

  • $\begingroup$ I really appreciate this input, but unfortunately it will probably not bring me much closer to a clear answer to my question. The kinetic theory modelling would probably be a bit out of my league. Temperature gradients and other limitations of my geometric assumptions are definitely error sources that I have not ruled out. The idea of a non-Boltzmann distribution, is one of several explanations I'm considering. Unfortunately, I will not have the time to solve my problem. For the time being, I will have to leave this on the ever-growing heap of "things I wish I understood better". $\endgroup$ – jkej Jun 16 '14 at 22:15

The kinetic equation for the distribution function $f(x,v,t)$ is

$\partial_t f + v \partial_x f + F \partial_v f = S$

This equation is basically the conservation law for the density in the phase space (x,v); F is the force acting on particles; the term on the right-hand side is the so-called collision integral, it is what drives the distribution function toward the Maxwellian, other terms can drive it away from Maxwellian. In these low-temperature plasmas that we are considering here there are numerous processes that can selectively populate or depopulate some energy levels, so it is not uncommon to see deviation from the Maxwellian.

If we consider an initial value problem when the distribution function for the flame (better call it partially ionized gas, or low-temperature plasma) deviates from Maxwellian initially then it takes time for the distribution function to become Maxwellian, that's the collision time for electrons and ions. In fact, electrons will thermalize among themselves, and ions among themselves, on a shorter time scale than the two species thermalize between each other, so the system will relax to a two-temperature distribution for some time.

However, it looks like this is not an initial value problem here, seems to be a steady state situation where there are some source terms present that drive the system away from the Maxwellian. Some subtle factors like whether the emitted radiation is trapped or escapes freely can make a large difference. More information on this would be needed to carry out a detailed analysis. Reading a basic plasma physics book (e.g. Introduction to Plasma Physics: Francis F. Chen) should be your starting point.

  • $\begingroup$ I realize that the actual flame might be a partially ionized gas, but I am trying to measure the radiation from the exhaust slightly downwind from the flame so I am assuming that electrons and ions have recombined. Of course I cannot be totally sure of this, but I guess emission signatures of ions and free electrons should not be expected to be remotely similar to that of, for instance, water molecules. For this reason, I don't believe that this is what I'm seeing. Or are you suggesting that the two-temperature distribution of the plasma would somehow carry over to the recombined molecules? $\endgroup$ – jkej Jun 13 '14 at 14:20
  • $\begingroup$ The two-temperature distribution will not apply when ions and electrons recombine into atoms. But it is not clear what fraction recombined and what is still ionized. Also, a question is what radiation you are seeing - can you identify particular spectral lines? Do you have any measurements of the temperature and density of the gas? If not, let's at least make some assumptions and use the Saha equation to estimate the ionization degree en.wikipedia.org/wiki/Saha_ionization_equation. $\endgroup$ – Maxim Umansky Jun 13 '14 at 23:19

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