A Note on the Stefan-Boltzmann Equation
According to American climatologist, Judith Curry, there are plenty of things to be skeptical about when it comes to Anthropogenic Global Warming, but the basic physics of gaseous infrared radiative transfer is not one of them.
Dr Curry, is the chair of the School of Earth and Atmospheric Sciences at the Georgia Institute of Technology and the co-author of ‘Thermodynamics of Atmospheres and Oceans’ (1999), and was speaking out against claims in a book ‘Slaying the Sky Dragon: Death of the Greenhouse Gas Theory’.
There has been some discussion of radiative transfer theory at this blog mostly via notes I have posted from Alan Siddons, Holden, Massachusetts, who is a chapter author in the book and is very skeptical about the basic physics of radiative transfer as applied by mainstream climate scientists and in the IPCC reports.
I was recently alerted to a blog post Dr Curry has started, ostensibly to discuss this physics and its application to global warming theory. It seems to have stirred up interest again in this issue.
An occasional commentator at this blog, known as Cementafriend, is also skeptical, in particular about how the Stefan-Boltzmann equation is applied by the climate scientists.
Following is a note from Cementafriend to resurrect the issue here:
Is the Stefan-Boltzmann Equation used correctly
D Kern in the text book “Process Heat Transfer” states “Radiant energy is believed to originate within molecules of the radiating body, atoms of such molecules vibrating in simple harmonic motion as linear oscillators. The emission of radiant energy is believed to represent a decrease in the amplitudes of the vibrations within the molecules, while an absorption of energy represents an increase.”
But from where does the Stefan-Boltzmann equation come?
It appears that Josef Stefan determined around 1879 from experimental results that the radiative flux (W/m2) was proportional to the fourth power of the absolute temperature. His student Ludwig Boltzmann around 1884 derived a confirming equation and proportionality constant from statistical thermodynamics. This became known as the Stefan-Boltzman law.
Most laws have some boundary conditions i.e. they do not apply universally. Einstein demonstrated that with his theory of relativity.
It was quickly found that the Stefan-Boltzmann law/equation only applies to black bodies.
There is a question mark about the constant as it seems that it was derived from a flat surface inside a hemisphere. That is OK when considering heat transfer been two surfaces but may be inaccurate when considering a radiating sphere.
A different derivation of the Stefan-Boltzmann equation is give by Miles Mathis who the radiant heat flux to electromagnetic fields. This is interesting as one of the criticisms of Boltzman’s work is that he did not consider gravity in his thermodynamics but this is outside the scope of this short note.
Boltzmann proposed a distribution of energy flux which was later developed by Max Planck into Planck’s Law. The Stefan-Boltzmann equation can be derived from Planck’s law with some assumptions. (It seems that Boltzmann has a lot to answer for –one of his students was Arrhenius)
Engineers work differently to scientists. They have to consider best practice, what actually works to give a solution to a problem. They have to take account of boundary conditions and assumptions, which have been applied to give a workable solution for similar problems.
The first modification to the Stefan-Boltzmann equation is to apply an emissivity factor because a black body is a theoretical consideration which does not apply in reality.
The emissivity is the ratio of the actual power to the power of a black body integrated over the whole flux density Planck distribution at a particular temperature.
Engineering researchers have determined the emissivities for many substances over a range of temperatures. Some emissivities are shown in figure 5-12 and table 5-6 of Perry’s Chemical Engineering Handbook (Perry’s CEH). It will be noted that the emissivity varies with temperature. For some materials the emissivity decreases with temperature and for others it increases with temperature.
Perry’s CEH states “According to Kirchhoff’s law, the emissivity and absorptivity of a surface in its surroundings at its own temperature are the same for both monochromatic and total radiation.” However, the emissivity and absorptivity needs to be considered in relation to the temperatures of the source and receiver.
For gases, researchers have determined a wavelength spectrum of emission/absorption.
It can be found that pure 100% CO2, at one atmosphere pressure, absorbs energy in only narrow wavelengths. At 4.2 micron there is approximately 100% absorption over less than 0.1 micron range and in the wavelengths range 14.5 to 15.5 micron there is approximately 60% absorption. Now the sun has less than 1% of its total spectrum in wavelengths greater than 3.9 micron and less than 0.1% greater than 9 micron so the absorptivity/emissivity of CO2 for incoming radiation from the very high temperature sun is practically zero. A black body at 288K has about 0.1% of its spectrum at wavelengths less than 3.9 micron and roughly 8% in the range 14.5 to 15.5 micron, giving an absorptivity/emissivity of radiation from such a body for CO2 of less than 0.05.
While the change in absorptivity in the case of CO2 is small, it will be recognised that the emissivity/absorptivity is temperature dependent.
So far only a single remote radiating source has been considered. From engineering measurements it was found that the temperature of the receiver affected the amount of energy transferred. In other words there was a net heat transfer which requires modification of the Stefan-Boltzmann equation.
Some may call this back radiation. Engineers normally do not care about the detailed mechanism while the equations can give a useful result. However, the concept of back radiation would be in conflict with the second law of thermodynamics. Infra-red radiation is part of the continuum of electromagnetic waves. It is possible that a reduced series of waves set out only in one direction or that incoming and outgoing waves cancel. Professor Claes Johnson, an author of ‘Slaying the Sky Dragon’ gives a plausible explanation with the concept of threshold energy levels.
In heat exchangers the area of surfaces are usually not equal and they may not be parallel.
From measurements and geometry calculations Engineers have introduced View factors.
This is outside the scope of this short note. However, view factors must be a consideration in the Urban Heat Island (UHI) effect. The heat loss/gain of a bitumen road in a open rural area will be different from the surrounding due to different emissivity but it will also be different to a road in a city where there are adjacent buildings of various heights due to view factor corrections. Anyone saying UHI does not exist is either not telling the truth or has no understanding of heat transfer.
A gas is not a surface and requires a different treatment to determine heat transfer. Engineering researchers have over many years made measurements on combustion products in heat exchangers and furnaces. Hoyt Hottel, Professor of Chemical Engineering MIT in the late 1950’s developed an equation which allows the absorptivity/emissivity of a combustion gas to be calculated from the partial pressures of CO2 and H2O vapour, the beam length, the total pressure and individual emissivity factors from the black body radiation curves. The equation is given in Perry’s CEH as 5-145 and an example 6 “Calculation of Gas Emissivity and Absorptivity”. Text books such as Process Heat Transfer by D Kern have graphs which allow quick selection for temperatures down to 100F (310K).
There are computer programs now to ease calculations (but one must understand what the program achieves and its limitations). My calculations making assumptions such as 8km beam length, average pressure 62kPa, average temperature 262K, surface temperature 320K and CO2 partial pressure 2.4 Pa show the absorptivity of CO2 insignificant (less than 0.01). For water vapour in a clear sky (ie no clouds) I calculated an absorptivity of about 0.4. Clouds have varying emissivities and need separate determination.
In climate discussions the term albedo seems to be frequently used. This is just (1-emissivity). To me as an engineer the term albedo is gobbledegook and shows to a large extent the users do not understand heat transfer.
Beside the modification of the Stefan-Boltzmann equation to fit reality there are other complications. Heat transfer by radiation is really only applicable to a vacuum. Calculations, using appropriate emissivity/absorptivity factors and view factors, work when there is a large temperature difference because the heat transfer relates to the difference of the fourth power of temperatures. However, calculations have large errors because the emissivity is not well known and temperatures may be incorrect especially if measured by instruments which assume black body radiation. For example the sun is not a perfect black body. Measurements have found surface temperature spikes. No one truly knows the surface temperature at any point in time. Similarly no one knows the exact diameter. Calculations of insolation at some point of the earth’s atmosphere need lots of assumptions and there is no way it can be reported to two decimal places.
If there is a fluid between surfaces or above a surface convective heat transfer (natural and forced) and phase change (evaporation & condensation) can be more important with small temperature differences (less than 50K) than radiation. Such heat transfer will change surface temperatures over time. Conduction below surfaces may then become a controlling factor. This needs elaboration at some other time.
In summary there is nothing settled about the Stefan-Boltzmann equation. It cannot be used on its own to determine atmospheric temperatures.
Judith Curry’s blog