A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature [latex]T_0[/latex]. It has attenuation coefficient β, density ρ, and specific heat [latex]c_υ.[/latex] At time t = 0 the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation.

From the results of Example 12.4 , the instantaneous transient heat flux emerging from both boundaries of the layer is [latex]q(t=)4\beta \sigma T^4(t)D[/latex].The energy equation for the layer then becomes [latex]ρc_υdT /dt=-4\beta \sigma T^4(t),[/latex]or, in dimensionless form,[latex]d\vartheta /\overline{dt} =-4\vartheta ^4(\overline{t} )[/latex] where [latex]ϑ=T/T_0[/latex] and [latex]\overline{t}=(\beta \sigma T_0^3/\rho c_v)t. [/latex] Integrating with the condition that ϑ = 1 at [latex]\overline{t}=0 [/latex] gives the transient uniform temperature throughout the layer for the emission approximation as [latex]\vartheta (\overline{t} )=\frac{1}{(1+2\overline{t})^{1/3}} [/latex]

Question 12.5: plane layer of gray medium with thickness D and constant pro...

Thermal Radiation Heat Transfer

A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature $T_0$ . It has attenuation coefficient β, density ρ, and specific heat $c_υ.$ At time t = 0 the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation.

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From the results of Example 12.4, the instantaneous transient heat flux emerging from both boundaries of the layer is $q(t=)4\beta \sigma T^4(t)D$ .The energy equation for the layer then becomes $ρc_υdT /dt=-4\beta \sigma T^4(t),$ or, in dimensionless form, $d\vartheta /\overline{dt} =-4\vartheta ^4(\overline{t} )$ where $ϑ=T/T_0$ and $\overline{t}=(\beta \sigma T_0^3/\rho c_v)t.$ Integrating with the condition that ϑ = 1 at $\overline{t}=0$ gives the transient uniform temperature throughout the layer for the emission approximation as

$\vartheta (\overline{t} )=\frac{1}{(1+2\overline{t})^{1/3}}$