The Path Length Approximation

As anyone who has spent a wee bit of time reviewing the theoretical method of determining how much radiation is absorbed/re-emitted by CO2 can tell you, the method is spectacularly complicated. It is not feasible for daily calculations. In most fields, an approximation of the solution is used. The Path Length Method is the method 0f choice for engineers, in particular combustion engineers. I’ll not get into the development of this method. It dates back a long way (at least 1950, possibly earlier).

Anyway, if I’ve done this right, this should link to the paper.

November 9, 2014. I have updated the original PDF. Minor grammar corrections (two).



About John Eggert

A minerals processing engineer in Canada. A cynic by nature, but open to being convinced!
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14 Responses to The Path Length Approximation

  1. plazaeme says:

    Sorry (and hello), the link is not working.

  2. Potentially very helpful.

    There is an obvious error in the figure number and title associated with the second figure.

    It takes time to digest which curves are relative to each other. There are two pairs of curves, for q and for delta-F, within each pair the IPCC and Leckner curves are located relatively close to each other.
    Comprehension would be speeded by adding “leckner” to the annotation of the thick purple line. (If I have comprehended correctly. 😉
    The phrase “Figure 2 shows the two curves relative to each other…” appears to be referring to the delta-F curves since that paragraph says the IPCC curve goes through zero at 278 ppm.
    “They are nearly identical in shape…” is misleading, there is a fundamental difference at the top end – the point of your article, as I understand it. (Perhaps you could say “The curves appear close, but there is a fundamental difference at high CO2 concentrations – the Leckner curve becomes asymptotic to a limit value, the IPCC’s increases without limit.”
    The significance of the statement that the IPCC delta-F curve is artificially offset to be 0 at 278 appears to be from your best understanding of what IPCC’s means by forcing, in which you say it “seems to be the difference obtained by subtracting Heat Absorption at 278 ppm…”
    I don’t grasp the significance of “The IPCC curve closely models the curve generated using standard engineering methods…”. Are you saying the IPCC uses some very basic formula that does not include atmospheric factors you use in the Leckner curves?
    I don’t understand where the 100% number comes from – I understand that the atmosphere could hold far higher concentrations of CO2 (from reports of the distant past), so perhaps you mean to say “The IPCC curve does not reach a maximum – rather it continues to grow without limit.” (You only plot to 800 ppm.)
    (“Leckner” may not be the right label for the non-IPCC curves, as I understand they are your calculation applying Leckner’s generic values to the particular case of the atmosphere.)

    Also, the paper is headed Part I, is there a Part 2?

    – Keith, sometimes aka “editor” 😉

    • John Eggert says:


      Thanks for your input. The second curve is indeed incorrectly edited. There are also some other issues with the paper beyond your comments that I will not correct. I am writing something new that includes this, but also includes further refinements. For instance, it deals with the “atmosphere is a blanket” meme that even Willis repeats. (Short answer, this should be seen at all levels of CO2. There is nothing magical about the saturation point). It also clarifies that looking at emissivity is NOT looking at what gets absorbed. Absorbtivity is related to, but not numerically equal to emissivity. In addition, I look at the answer to the question: How can a grey body subject to a constant heat source be hotter than a black body subject to an identical heat source? it is complicated and I’m not done yet. I have a real job that takes about 60 hours a week (and often involves heat balance calculations). Part 2 is a layman’s guide and Part 3 is a summary for policy makers. I dropped editing of these.

  3. cementafriend says:

    Hottel presents an equation for absorptivity of H20 and CO2 in Perry’s Chemical Engineering Handbook -chapter 5 Heat and Mass transfer. I may made some calculations of absorptivity which could be in error but get about 0.42 for H2O in the atmosphere and about 0.007 for CO2. I have worked on measurements in furnaces of flames with different fuels. I found that the emissivity of a natural gas flame is about 0.45 which mainly comes from the H2O (the emissivity can be increased by internal flame recirculation to crack the CH4 to elemental carbon which has an emissivity of 1-a burner I designed using supersonic velocity at the multiple jet exit, arranged in a hollow circle, got over 0.65). I believe in the atmosphere the radiation absorption of CO2 is insignificant because the H2O swamps the CO2 in quantity and overlap of wavelengths.

    • John Eggert says:

      Thanks for your input. Yes, the effect of water is substantially larger. That being said, if there is a change in absorbance of CO2 that is independant of water, then that will have an impact independent of water. That is, it doesn’t mater that the effect of water is much larger. What matters is that, as CO2 increases, it’s effect increases, hence the gross effect increases. I am in the process of writing a more detailed paper than my “Path Length” paper. The absorptivity of CO2 is not trivially small relative to water. Small, but not trivially so. My day job is going extremely well however which means I’m working 12 hours a day for the next number of days. Doesn’t leave much room for this stuff.

  4. cementafriend says:

    What Hottel found ( in experiments and from theory) that the emissivity/absorptivity of CO2 and H2O overlap and that a correction factor needs to be applied so that the total absorption or emission is less than that which would occur separately. For interest (which is what I did) one can calculate under the assumption the other does not exist to find the the relative amount of absorption. I found that basically water vapor swamps the CO2 absorbing all the radiation in its wavelength range so it does not matter if the CO2 increases. Others have found that the absorption of CO2 wavelengths occurs at very low CO2 concentrations (or partial pressures) so that it is saturated (or all the wavelengths absorbed) with less than 200ppm. That leads to the same conclusion that increasing CO2 does not lead to further absorption.
    This post was an attempt to put some of the discussion on radiative heat transfer on track.
    Wider consideration of clouds has just begun. Hottel in various texts makes mention of clouds of particles (particularly in relation to flames) but clouds of liquid water or solid ice particles could be treated similarly because the emissivity of water and ice in the infrared wavelength range say 2 to 40 micron is close to 1.0. This will also overwhelm any contribution of increased CO2.
    I have just thought that so-called “climate science” is like political science -all politics and no science.

  5. Frank says:

    John: The absorption and emission in the center of the CO2 bands is saturated. You are correct believing that doubling CO2 will not reduce the energy flux at these wavelengths. However, every absorption band has shoulders and the radiative forcing associated with 2X CO2 occurs at these shoulders. In fact the maximum radiative forcing for 2XCO2 occurs at the wavelengths where CO2 absorbs only 50% of outward surface LWR. Doubling CO2 decreases transmission at those wavelengths from 50% to 25%. See Figure 16 from“greenhouse”-effect-–-part-nine/ The importance of absorption shoulder can also be seen at

    Engineers interested in radiation transfer near the surface of the earth may not be interested in what happens at the extreme edges of the CO2 absorption bands. They may not be interested in how the line shape of the individual lines changes with temperature and pressure. So your source of information may not take into account what is happening on the extreme edges of absorption bands. Science of Doom has a long series of articles on this subject that are certainly worth reading. The database of absorption lines used by MODTRAN and HITRAN was initially begun for aeronautical and aerospace engineers needing to predict heat build-up and radiative cooling at various altitudes in the atmosphere.“greenhouse”-effect-–-part-nine/

    • John Eggert says:


      I’ve read all of the stuff at scienceofdoom. It is a link on my side bar. As I say, that is where you should go if you want a good resource. Regarding the extreme edges of CO2 absorption, combustion engineering and furnace design deal with CO2 path lengths far greater than seen in the atmosphere. I would say it is the atmospheric study that doesn’t test the bounds of the extreme. The derivations and examples at scienceofdoom are relevant to all gaseous absorption of radiant energy. The absorption for water certainly follows the logarithmic model very closely. The absorption for CO2 does not. This does not contradict what is written there. I’ve discussed this at scienceofdoom. His only critique was that the path length approximation does not involve temperatures below 0 celsius. At temperatures above 0 celsius, it can be said that CO2 is, practically speaking, at saturation at a path length of 500 (about 335 ppm) Beyond this, the increase in absorption is so small that it can be ignored. If we displaced 100% of the nitrogen and oxygen in our atmosphere with CO2, the increase in absorption would be relatively trivial. This is why there was life in the Jurassic when CO2 levels were over 1,000 ppm or more. This is why life began to florish in the Ordovician when CO2 levels were somewhere between 3,000 ppm to 6,000 ppm. The climate alarmist community asserts that a doubling of CO2 will increase temperature by 3°C. If this is the case, going from 300 ppm to 600 ppm is 3°, continuing from 600 ppm to 1200 ppm is a total of 6°C, continuing from 1200 ppm to 2400 ppm is a total of 12°C and continuing from 2400 ppm to 4800 ppm (path length of 7,200 is a total of 24°C. What they are saying is that the earth’s temperature was somewhere around 24 degrees hotter in the Ordovician than today. That is, the global average temperature was 40 celsius. Given that the geological record confirms the presence of glaciers in the Ordovician, I’m . . . skeptical. By the way, there is a third database called HITEMP. At 1,000 K, for the 4.3 micron CO2 band alone HITEMP includes close to 200,000 lines. As I said, combustion tests the extremes.

  6. I have read many posts at scienceofdoom (SOD) and have seen many of his comments at other sites where he has taken on well qualified people who have presented empirical data opposed to the AGW concepts. To me SOD presents clear evidence that he does not understand the engineering subjects of thermodynamics, heat & mass transfer, or fluid dynamics. Many of his thought bubbles are pure nonsense. He particularly has no idea about convective heat transfer (natural and forced-by winds), and phase change heat transfer (evaporation and condensation). I suppose he could be forgiven about cloud formation and the significance of clouds in climate assessment because there is no one in the world that fully understands that. However, to kept on about the significance of CO2 when a) one can show from empirical data such as that from Hottel that CO2 in the atmosphere has a theoretical insignificant effect b) there is no measured evidence that CO2 is linked to temperature c) that measurements by respected scientists of atmospheric conditions (such as temperature, pressure, solar radiation, CO2 level, precipitation etc) show that solar radiation leads temperature which in turn leads CO2 on a daily, seasonal and many year cyclical basis, and d) that ice core data show temperature leading CO2 over tens of thousand year cycles, shows that SOD has a closed political mindset.
    I personally do not agree with your linking to a website that presents misleading information.

    • John Eggert says:

      I understand your point, but I don’t have the time to put together most of the stuff he has there. Most of what is there is accurate, though incomplete. My only problem with his site is he doesn’t make clear that what he is describing is the general features of absorption by intervening gasses. The specifics of CO2 are important. Most important is the fact that the slope of the relation between absorption and concentration is nearly flat for CO2. If it weren’t, there never would have been life on this planet.



  7. Billy Liar says:

    Thanks for an interesting exposition.

    A small point – please edit the two occurrences of the word ‘affect’ in your conclusion to ‘effect’.

    (I’m surprised editor ‘Keith’ above didn’t notice!)

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