Question 10.5: Parallel plane nongray walls are D = 2.5 cm apart and are at...

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:

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 )\times [1+(1-\epsilon _{\lambda ,1})\overline{t}_\lambda](E_{\lambda b,2}-E_{{\lambda b,g}})\right\}             (10.77b)

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

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

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

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

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} } }

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

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 \right] =\frac{4\pi }{c}\overline{I}_v(r)                 (10.31)

\overline{t}_I=1-\overline{\alpha }_I=1-\frac{\overline{A}_I }{\Delta \eta _I}

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

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} }

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