Parallel plane nongray walls are D = 2.5 cm apart and are at uniform temperatures T1 = 1100 K and T2 = 550 K. Pure CO2 gas at 10-atm pressure and Tg = 550 K is between the walls. The

Question

Parallel plane nongray walls are D = 2.5 cm apart and are at uniform temperatures [latex]T_1 = 1100[/latex] K and [latex]T_2 = 550 K [/latex]. Pure [latex]CO_2[/latex] gas at 10-atm pressure and [latex]T_g = 550 K[/latex] is between the walls. The hemispherical spectral emissivity for both walls is approximated as a function of wave number by the following table:

[latex]\eta (cm^{-1})[/latex]	[latex]\epsilon _\eta [/latex]	[latex]\eta (cm^{-1})[/latex]	[latex]\epsilon _\eta [/latex]
0 to 500	0.37	1150 to 2200	0.45
500 to 750	0.26	2200 to 2500	0.65
750 to 850	0.32	2500 to 3600	0.61
850 to 1000	0.37	3600 to 3750	0.69
1000 to 1150	0.46	3750 to ∞	0.73

Assume that only the 15-, 10.4-, 9.4-, 4.3-, and [latex]2.7-μm CO_2[/latex] bands participate in the radiative transfer. Compute the total heat flux being added to wall 2.

Accepted Answer

In Example 10.3 the spectral exchange was found for radiation between infinite parallel plates with a gas between them. The total energy added to plate 2 is found by integrating Equation 10.77b over all wave numbers:

[latex]q_{\lambda ,2}=\frac{1}{1-(1-\epsilon _{\lambda ,1})(1-\epsilon _{\lambda ,2})\overline{t_\lambda ^2} } \left\{\epsilon _{\lambda ,1}\epsilon _{\lambda ,2}\overline{t}_\lambda(E_{\lambda b,2}-E_{{\lambda b,1}})+\epsilon _{\lambda ,2}(1-\overline{t}_\lambda ) imes [1+(1-\epsilon _{\lambda ,1})\overline{t}_\lambda](E_{\lambda b,2}-E_{{\lambda b,g}}) ight\} [/latex] (10.77b)

[latex]q_{2}=\int_{\eta =0}^{\infty }{\frac{\epsilon _{\eta ,1}\epsilon _{\eta ,2}\overline{t}_\eta (E_{\eta b,2}-E_{\eta b,1})+\epsilon _{\eta ,2}(1-\overline{t}_\eta )[1+(1-\epsilon _{\eta ,1})\overline{t}_\eta ](E_{\eta b,g}) }{1-(1-\epsilon _{\eta ,1})(1-\epsilon _{\eta ,2})\overline{t^2_\eta } } }d\eta [/latex]

In this example [latex]\epsilon_{η,1} = \epsilon_{η,2}[/latex] and [latex]T_g = T_2[/latex], so [latex]q_2[/latex] simplifies to

[latex]q_{2}=-\int_{0}^{\infty }{\frac{\epsilon _{\eta ,1}\overline{t}_\eta (E_{\eta b,1}-E_{\eta b,2}) }{1-(1-\epsilon _{\eta ,1})^2\overline{t^2_\eta } } }d\eta [/latex]

The integration is expressed as a sum over wave number bands. For the lth band let [latex]\epsilon _{\eta ,1}= \epsilon _{I}[/latex] and [latex]\overline{t}_\eta =\overline{t} _l [/latex]

[latex]q_2=-\sum\limits_{I}{\frac{\epsilon _I^2\overline{t}_I[E_b(T_1)-E_b(T_2)]_I\Delta \eta _I }{1-(1-\epsilon _I)^2\overline{t^2_I} } } [/latex]

where [latex](E_b)_I Δη_I[/latex] is the blackbody radiation in the Ith band. From Equation 10.31, [latex]\overline{t}[/latex] is written as

[latex]U_v(v,r)=\frac{1}{c}\int_{\Omega =0}^{4\pi }{I_v}(r,S)d\Omega =\frac{4\pi }{c}\left[\frac{1}{4\pi } \int_{\Omega =0}^{4\pi }{I_v}(r,S)d\Omega ight] =\frac{4\pi }{c}\overline{I}_v(r) [/latex] (10.31)

[latex]\overline{t}_I=1-\overline{\alpha }_I=1-\frac{\overline{A}_I }{\Delta \eta _I} [/latex]

where [latex]\overline{A}_I [/latex] is the integrated effective bandwidth that includes the integrated path-length variation for a parallel-plate geometry. The [latex]q_2 [/latex] now becomes

[latex]q_2=-\sum\limits_{I}{\frac{\epsilon _I^2(1-\overline{A}_I/\Delta \eta _I )[E_b(T_1)-E_b(T_2)]_I\Delta \eta _I }{1-(1-\epsilon _I)^2(1-\overline{A}_I/\Delta \eta _I )^2} } [/latex]

Some of the calculated quantities and the flux results are in the tables that follow. Values of [latex]\overline{A}_I [/latex] for the bands were computed from the exponential wide-band correlations using the basic quantities in Table 9.2. The mean beam length from Table 10.2, [latex]L_e = 1.8D = 1.8 × 0.025 = 0.045[/latex] m, was used as the effective path length for the radiant intensities in the layer. The gas density is [latex]ρ = 9749 g/m^3[/latex] which gives a mass path length of [latex] ρL_e = 438.7 g/m^2[/latex] .
The procedure in Example 9.1 was followed for calculating values for each band. The band wave number spans [latex]Δη_l[/latex] were computed from Table 10.3 with the [latex]\bar{A}_I [/latex] obtained using the values from Table 9.2. Since the calculation for [latex]\bar{A}_I [/latex] depends on the band limits, an iteration may be required to obtain the correct band limits. The band energies [latex][E_b (T_1 ) -E_b (T_2 )]_I Δη_I[/latex] were

TABLE 10.2
Mean Beam Lengths for Radiation from Entire Medium Volume
Geometry of Radiating System	Characterizing Dimension	Mean Beam Length for Optical Thickness κλLe → 0, Le,	Mean Beam Length Corrected for Finite Optical Thickness,a Le	C = Le/Le,0
Hemisphere radiating to element at center of base	Radius R	R	R	1
Sphere radiating to its surface	Diameter D	2	0.65D	0.97
Circular cylinder of infinite height radiating to concave bounding surface Circular cylinder of semi-infinite height radiating to:	Diameter D	3 D	0.95D	0.95
Element at center of base	Diameter D	0.77D	0.90D	0.9
Entire base Circular cylinder of height equal to diameter radiating to:	Diameter D	2qqq	0.65D	0.8
Element at center of base	Diameter D		0.71D	0.92
Entire surface Circular cylinder of height equal to two diameters radiating to:	Diameter D	0.77D	0.60D	0.9
Plane end	Diameter D	0.73D	0.60D	0.82
Concave surface	Diameter D	0.82D	0.76D	0.93
Entire surface Circular cylinder of height equal to one-half the diameter radiating to:	Diameter D	0.80D	0.73D	0.91
Plane end	Diameter D	0.48D	0.43D	0.9
Concave surface	Diameter D	0.52D	0.46D	0.88
Entire surface	Diameter D	0.50D	0.45D	0.9
Cylinder of infinite height and semicircular cross section radiating to element at center of plane rectangular face Infinite slab of medium radiating to:	Radius R		1.26R	0.9
Element on one face	Slab thickness D	2D	1.8D	0.9
Both bounding planes	Slab thickness D	2D	1.8D	0.9
Cube radiating to a face Rectangular parallelepipeds 1 × 1 × 4 radiating to:	Edge X	qqqqq	0.6X	0.9
1 × 4 face	Shortest edge X	0.90X	0.82X	0.91
1 × 1 face	Shortest edge X	0.86X	0.71X	0.83
All faces 1 × 2 × 6 radiating to:	Shortest edge X	0.89X	0.81X	0.91
2 × 6 face	Shortest edge X	1.18X
1 × 6 face	Shortest edge X	1.24X
1 × 2 face	Shortest edge X	1.18X
All faces	Shortest edge X	1.20X
Medium between infinitely long parallel concentric cylinders	Radius of outer cylinder R and of inner cylinder r	2(R–r)	See Anderson and Handvig (1989)
Medium volume in the space between the outside of the tubes in an infinite tube bundle and radiating to a single tube:
Equilateral triangular array:	Tube diameter
S = 2D	D, and spacing	3.4(S–D)	3.0(S–D)	0.88
S = 3D	between tube	4.45(S–D)	3.8(S–D)
Square array:	centers, S
S = 2D		4.1(S–D)	3.5(S–D)	0.85
a Corrections are those suggested by Hottel (1954), Hottel and Sarofim (1967) or Eckert and Drake (1959). Corrections were chosen to provide maximum Le where these references disagree.

TABLE 10.3 Approximate Band Limits for Parallel-Plate Geometry
			Band limits [latex]η (cm^{−1}) [/latex]ª
Gas	Band λ (μm)	Band center [latex]η (cm^{−1}) [/latex]	Lower	Upper
[latex]CO_2[/latex]	15	667	[latex]667- (\overline{A}_{15} / 1 .78)[/latex]	[latex]667 + (\overline{A}_{15} / 1 .78)[/latex]
	10.4	960	849	1013
	9.4	1060	1013	1141
	4.3	2350	[latex]2350 – (\overline{A}_{4.3} / 1 .78)[/latex]	2430
	2.7	3715	[latex]3715 –(\overline{A}_{2.7} / 1 .76)[/latex]	3750
[latex]H_2O[/latex]	6.3	1600	[latex]1600 –(\overline{A}_{6.3} / 1 .6)[/latex]	[latex]1600 +(\overline{A}_{6.3} / 1 .6)[/latex]
	2.7	3750	[latex]3750 –(\overline{A}_{2.7} / 1 .4)[/latex]	[latex]3750 + (\overline{A}_{2.7} / 1 .4)[/latex]
	1.87	5350	4620	6200
	1.38	7250	6200	8100
ª [latex]\overline{A}[/latex] are found for various bands as in Example 9.1. Terms such as [latex]\overline{A}_{15} /1.78[/latex] are [latex]\overline{A} /2(1− τ_g ) [/latex]from Table 1 and 2 and Equation 17 of Edwards and Nelson (1962). Source: Edwards, D. K., and Nelson, K. E.: Rapid Calculation of Radiant Energy Transfer between Nongray Walls and Isothermal [latex]H_2O[/latex] or [latex]CO_2[/latex] Gas, JHT, vol. 84, no. 4, pp. 273–278, 1962; Edwards, D. K.: Radiant Interchange in a Nongray Enclosure Containing an Isothermal Carbon Dioxide–Nitrogen Gas Mixture, JHT, vol. 84, no. 1, pp. 1–11, 1962; Edwards, D. K., Glassen, L. K., Hauser, W. C., and Tuchscher, J. S.: Radiation Heat Transfer in Nonisothermal Nongray Gases, JHT, vol. 89, no. 3, pp. 219–229, 1967.

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