A nearly transparent medium with extinction coefficient βλ is between two diffuse parallel walls D apart at temperatures T1 and T2. What total heat flux is being transferred between the walls in the absence of heat conduction and convection?

Question

A nearly transparent medium with extinction coefficient [latex]β_λ[/latex] is between two diffuse parallel walls D apart at temperatures [latex]T_1[/latex] and [latex]T_2[/latex]. What total heat flux is being transferred between the walls in the absence of heat conduction and convection?

Accepted Answer

For the nearly transparent approximation, [latex]q_{rλ}=J_{λ,1}-J_{λ,2}[/latex] Substituting [latex]J_{λ,1}[/latex] and [latex]J_{λ,2}[/latex] from Equation 11.27 yields

[latex]J_{\lambda ,1}=\epsilon _{\lambda ,1}\pi I_{\lambda b,1}+2(1-\epsilon _{\lambda ,1})\left[J_{\lambda ,2}E_3 ( au_{\lambda ,D})+\pi \int_{ au _\lambda ^*=0}^{ au _{\lambda ,D}}{\widehat{I}_\lambda ( au _\lambda ^*) E_2( au _\lambda ^*)}d au _\lambda ^* ight] [/latex] (11.27a)

[latex]J_{\lambda ,2}=\epsilon _{\lambda ,2}\pi I_{\lambda b,2}+2(1-\epsilon _{\lambda ,2})\left[J_{\lambda ,1}E_3 ( au_{\lambda ,D})+\pi \int_{ au _\lambda ^*=0}^{ au _{\lambda ,D}}{\widehat{I}_\lambda ( au _\lambda ^*) E_2( au _{\lambda ,D}- au _\lambda ^*)}d au _\lambda ^* ight] [/latex] (11.27b)

[latex]q_{r\lambda }d\lambda =\frac{\epsilon_{\lambda ,1}E_{\lambda b,1}+\epsilon_{\lambda ,2}E_{\lambda b,2}(1-\epsilon _{\lambda ,1})-\epsilon _{\lambda ,2}E_{\lambda b,2} +\epsilon_{\lambda ,1}E_{\lambda b,1}(1-\epsilon _{\lambda ,2})}{1-(1-\epsilon _{\lambda ,1})(1-\epsilon _{\lambda ,2})} d\lambda [/latex]

Simplifying and integrating with respect to λ gives the required result,

[latex]q_{r }= \int_{\lambda =0}^{\infty }{\frac{E_{\lambda b,1}-E_{\lambda b,2}}{1/\epsilon _{\lambda ,1}+1/\epsilon _{\lambda ,2}-1} }d\lambda [/latex] (12.13)

This is the same as Equation 6.6;

[latex]q_{1}=-q_2=\int_{\lambda =0}^{\infty }{q_{\lambda ,1}d\lambda } =\int_{\lambda =0}^{\infty }{\frac{E_{\lambda b,1}(T_1)-E_{\lambda b,2}(T_2)}{\frac{1}{\epsilon _{\lambda ,1}(T_1)}+\frac{1}{\epsilon _{\lambda ,2}(T_2)} -1 } } d\lambda[/latex] (6.6)

the radiant energy flux transferred is uninfluenced by the nearly transparent medium between the boundaries. The medium temperature is obtained in the next example.

Question 12.2: A nearly transparent medium with extinction coefficient βλ i...

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