What fraction of incident solar radiation is absorbed by a thick layer of CO2 at 10 atm and 830 K? Use the approximate absorption bands of Example 10.5.

Question

What fraction of incident solar radiation is absorbed by a thick layer of [latex]CO_2[/latex] at 10 atm and 830 K? Use the approximate absorption bands of Example 10.5.

Accepted Answer

The effective radiating temperature of the sun is [latex]T_s = 5780[/latex] K. The desired result is the fraction of the solar spectrum that lies within the four [latex]CO_2[/latex] bands, as this is the only portion of the incident radiation that will be absorbed. Using the [latex]F_{0→λTs},[/latex] factors obtained using [latex]T_s[/latex] gives

λ (μm)	[latex]λT_s (μm · K)[/latex]	[latex]F_{0→λTs}[/latex]	λ (μm)	[latex]λT_s (μm · K)[/latex]	[latex]F_{0→λT_s}[/latex]
1.8	10,400	0.92195	4.0	23,120	0.99028
2.2	12,720	0.95251	4.6	26,590	0.99340
2.6	15,030	0.96909	9	52,020	0.99902
2.8	16,180	0.97455	19	109,800	0.99989

The fraction absorbed is then

[latex]\alpha =\sum\limits_{absorbing \ bands}{[F_{0\longrightarrow (\lambda T_s)_{upper}}-F_{0\longrightarrow (\lambda T_s)_{lower}}]_{band}} [/latex]
[latex]=(0.95251− 0.92195)+ (0.97455− 0.96909)+ (0.99340 -0.99028 )+(0.99989-0.99902)=0.040[/latex]

Even though the gas layer is thick, only 4.0% of the incident energy is absorbed since the gas transmits well in the λ regions between the absorption bands. For radiative exchange calculations in furnaces, an approximate procedure for determining [latex]α_g[/latex] is in Hottel and Sarofim (1967). The [latex]α_g[/latex] is obtained from the gas total emittance values by using

[latex]α_g= \alpha _{CO_2}+\alpha _{H_2O}-\Delta \alpha [/latex] (10.117)

where

[latex]\alpha _{CO_2}=C _{CO_2}\epsilon _{CO_2} ^+\left\lgroup\frac{T_g}{T_w} \right\rgroup^{0.5} [/latex] (10.118)

[latex]\alpha _{H_2O}=C _{H_2O}\epsilon _{H_2O} ^+\left\lgroup\frac{T_g}{T_w} \right\rgroup^{0.5} [/latex] (10.119)

[latex]Δα = (Δ\epsilon )_{at \ T_w } [/latex] (10.120)

The [latex] \epsilon _{CO_2} ^+[/latex] and [latex] \epsilon _{H_2O} ^+[/latex] are, respectively, [latex]\epsilon _{CO_2}[/latex] and [latex] \epsilon _{H_2O}[/latex] obtained from Equation 9.62

[latex]\epsilon (T,pL_e)=\exp \left\{a_0+\sum\limits_{j=1}^{M}{a_j[\log (pL_e)]^j} \right\} [/latex] (9.62)

evaluated at [latex]T_w[/latex] and at the respective parameters [latex] P_{CO_2}L_e^′[/latex] and [latex] P_{H_2O}L_e^′[/latex] where [latex]L^′_e = L_e T_w / T_g .[/latex] It is pointed out in Edwards and Matavosian (1984) that the exponent 0.5 is now becoming more accepted to replace the values 0.65 and 0.45 originally used in Equations 10.118 and 10.119. At high temperatures and pressures when there is overlapping of absorption lines in the infrared spectrum, [latex]L^{′}_{e} = L_e (T_w / T_g)^{3/2} ,[/latex] which is discussed in Edwards and Matavosian,

Section 10.9.1 considered a uniform isothermal gas in a black enclosure. If the bounding walls are not black and hence are reflecting, radiation can pass through the gas by means of multiple reflections from the boundary. For an enclosure with a single wall, this can be included by integrating Equation 10.74

[latex]Q_g=-Q_1=-A_1\int_{\lambda =0}^{\infty }{q_{\lambda ,1}d\lambda } =A_1\int_{0}^{\infty }{\frac{E_{\lambda b,g}-E_{\lambda b,1}}{\frac{1}{\epsilon _{\lambda ,1}}+\frac{1}{\overline{\alpha }_{\lambda,1-1 } } -1 } }d\lambda [/latex] (10.74)

over all wavelengths. In Edwards and Matavosian, a procedure is proposed to use total emittance values for situations with multiple reflections. The solution of enclosure heat transfer problems with reflecting walls was treated in Section 10.6.4 by integrating spectral relations over the wavelength absorption bands.

An example that considers a black enclosure with walls at differing temperatures is in Example A.2 in the Appendix A at the publishers web site at www.crcpress.com.

Question 10.7: What fraction of incident solar radiation is absorbed by a t...

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