THE TOTAL EMISSIVITY OF CARBON DIOXIDE AND ITS EFFECT ON THE TROPOSPHERIC TEMPERATURE
Scientific Research Director at Biology Cabinet
May 12, 2010
(Additional editing of this English text by TS)
This didactic article was Peer Reviewed by scientists of the University Regiomontana on June 28, 2010.
Nahle, N. Total Emissivity of Carbon Dioxide and its Effect on the Tropospheric Temperature. 12 May 2010. Biology Cabinet at San Nicolas de los Garza, N. L.
By applying generally accepted algorithms on radiative heat transfer, verified through experimentation by Hottel(1), Leckner(2) and other contemporary scientists and engineers(3)(4)(5), I have found that the carbon dioxide molecules posses a low total emissivity at the current density of CO2 in the atmosphere.
The formula applied by several authors on radiative heat transfer for calculating the atmospheric temperature anomaly caused by carbon dioxide is as follows:
ΔT = α (ln ([ga] ∞ / [ga]st) / 4 (σ) (T)^3 (Formula 1)
Where ΔT is the change of temperature, α is the climate sensitivity of the absorbent gas (5.35 W/m^2 for CO2 at 1000 K of temperature, 1 atm of total pressure and 0 of distance), [[ga] ∞ is for the instantaneous concentration of the absorbent gas, [ga]st is the standard concentration of the absorbent gas, σ is the Stefan-Boltzmann constant (5.6697 x 10^-8 W/m^2 K^4), and T is the standard temperature (290 K).
Although this formula has been widely used, several factors are missing that have a definitive influence on the results. Exempli gratia, it fails to account for the sphericity of the system, the partial pressure of the absorbent gas and its buoyancy.
Another problem with the formula is that it represents homogeneous values for the total emissivity and total absorptivity of the absorbent gas, which is absolutely at odds with what has been observed and verified through experimentation because the total emissivity and total absorptivity of any substance changes in proportion to its partial pressure in any given environment.
The surrounding temperature also imparts a significant influence on the total emissivity and total absorptivity of any substance; the same can be said for the distance which separates the emissive system from the absorbent system. None of these factors are accounted for in the formula derived from the Stefan-Boltzmann principle, which is appropriate when considering blackbodies, but absolutely inappropriate when considering gray bodies at different concentrations in a given medium, such as carbon dioxide in the atmosphere.
The appropriate formula for obtaining the total emissivity of carbon dioxide is as follows:
ECO2 = 1-[(a-1 * 1-PE / a + b – (1 + PE)) * e [-c (Log10 (paL) m / paL)^2]] * (ECO2)0 (Modest. 2003. Pp. 339-346) (4) (F. 2)
T = 35 °C = 308 K
PE = effective pressure of the absorbent gas (p + 0.28 pa) / p0.
p = Total pressure of the mixed gases (1 bar for the atmosphere at sea level).
pa = Partial pressure of the absorbent gas.
(paL) m = Partial pressure of the absorbent gas (a) modified by Planck’s function.
(paL)0 = partial pressure of the absorbent gas (a) at L = 0.
paL = Partial pressure of the absorbent gas (a) at L = 0.1 m.
a, b, c = Proportionality constants.
(ECO2)0 = Emissivity of the CO2 at (L = 0, p = 1 bar, T = 373 K and pCO2 = 0.03 bar cm) = 0.0016, which can be also iterated from the graphs on the total emissivity of the carbon dioxide published in books on Heat Transfer. [Hottel (1954), Leckner (1972), Pitts & Sissom (1998), Modest (2003), Manrique (2002)]
Obtaining the emissivity of CO2, the normal intensity of the radiation, the normal intensity of the radiative heat transfer of CO2, and the change of temperature caused by the CO2 if:
a) The percentage of Carbon Dioxide in the atmosphere is 0.034% (present percentage)
b) The total pressure of the air is 1.01325 bar (real value)
c) The current temperature of the surface is 330 K (instantaneous temperature of the local land surface today at 22 hrs UT)
d) The current temperature of the air is 308 K (instantaneous temperature of the air above the reported surface temperature today at 22 hrs UT)
Obtaining the total emissivity of carbon dioxide
First, let us obtain the total emissivity of carbon dioxide at its current concentration in the atmosphere. To do this, we will use the following formula:
E CO2 / (ECO2)0 = 1-[(a-1 * 1-PE / a + b-(1+ PE)) * e [-c (Log10 (paL) m / paL)^2]] (4) (F. 2)
(ECO2)0 at T = 373 K and p = 1 bar = 0.0016 (2) (See Formula 3)
t = 308 K / 373 K = 0.82
T0 = 373 K
a = 1 + 0.1 / t^1.45 = 1 + 0.1 / 0.82^1.45 = 1.45
b = 0.23
c = 1.47
pCO2 = 0.00034 bar
pabs = 1 bar
p0 (absolute pressure) = 1 bar
PE = pabs + [0.28 (pCO2)] / p0 = 1 bar + [0.28 (0.00034 bar) / 1 = 1.0001 bar
(PCO2L) m / (PCO2L) 0 = 0.225 * t^2 (if t > 0.7) = 0.151
(PCO2L) m = (0.225 * t^2) * (PCO2L) 0 = 0.151 * 0.034 bar cm = 0.005134 bar cm
PCO2L = 0.00034 bar m = 0.034 bar cm
T = 308 K
Emissivity of carbon dioxide at its current concentration in the atmosphere:
ECO2 = 1-[(a-1 * 1-PE / (a + b) - (1+ PE)) * e [-c (Log10 (paL) m / paL)^2]] * (ECO2)0(Formula 2)
ECO2 = 1- [(1.45-1 * 1-1.0001 bar / (1.45 + 0.23) - (1 + 1.0001 bar)) * e [-1.47 (Log10 (0.005134 bar cm /0.034 bar cm))^2]] * 0.0016
ECO2 = 1- [(0.45 * -0.0001 bar / (1.68) - (2.0001)) * e [-1.47 (Log10 (0.005134 bar cm /0.034 bar cm))^2]] * 0.0016
ECO2 = 1- [(-0.000045 bar / -0.3201 bar)) * e [-1.47 (Log10 (0.005134 bar cm /0.034 bar cm)^2)]] * 0.0016
ECO2 = 1 – [(0.00014 * 0.3712)] * 0.0.0016 = (0.999948) (0.0016) = 0.0017
Actually, the total emissivity of the carbon dioxide at its current density in the atmosphere is quite low. The total absorptivity of the carbon dioxide at its current concentration in the atmosphere is 0.0017.
Therefore, for an air temperature of 308 K (35 °C), the carbon dioxide contributes with 0.135 K. The remainder thermal effect of carbon dioxide is exclusively of cooling of the surface and other masses of more efficient absorbent gases.
Another formula for calculating the total emissivity of the carbon dioxide is derived from the formula above mentioned, which applies especially at temperatures below 1000 K, e.g. the temperatures of the Earth’s atmosphere at different altitudes and partial pressures of carbon dioxide. The formula is as follows:
ECO2 = [e ((|√ Log10 (290 K * T∞)| / (- c * 1 K)] * [pCO2 * 100 / 5 (pabs)] (Formula 3)
Where T∞ is for the instantaneous temperature, c is a proportionality constant (1.47), pCO2 is for the instantaneous concentration of the carbon dioxide in the atmosphere, and pabs is for the absolute pressure of the atmosphere.
The proportion of carbon dioxide in the atmosphere is 0.038%, the absolute pressure (pabs) at the sea level is 1 atm, and the instantaneous temperature (T∞) of the air is 32 °C. Calculate the Total Emissivity of the carbon dioxide under these given conditions.
Percentage of CO2 in the atmosphere = 0.038%
T∞ = 32 °C + 273 = 305 K
c = - 1.47
pabs = 1 atm.
Before proceeding on the calculation, we proceed to obtain the partial pressure of the carbon dioxide in the atmosphere as from its present mass fraction; for doing this, we divide the proportion of the carbon dioxide in the atmosphere by 100 and multiply the result by 1 atm:
pCO2 = (0.038%/100) * 1 atm = 0.00038 atm.
ECO2 = [e ((√ |Log10 (290 K * T∞)|) / (- c * 1 K))] * (pCO2 * 100 / 5 (pabs)) (Formula 3)
ECO2 = [e ((√ |Log10 (290 K * 305 K)|) / (- 1.47 * 1 K)] * (0.00038 atm * 100 / 5 (1 atm))
ECO2 = [e (2.24 K / -1.47 K)] * (0.038 atm / 5 atm) = (0.218) (0.0076) = 0.0017
Something valuable to emphasize is that the total emissivity of the carbon dioxide decreases as the density of the gas in the atmosphere increases, as long as the temperature remains constant; as well, when increasing the temperature of the atmosphere, the emissivity of the carbon dioxide decreases logarithmically. This phenomenon is easily observable in the tables on the Total Emissivity of the carbon dioxide obtained by means of experimentation by Hottel (1) and Leckner (2) which have been published in many modern texts on Heat Transfer. The following graph illustrates the mentioned negative feedback: