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Q. 11.6

Develop an analysis for the onset of transient heating by a unidirectional flux $q_i$ incident at angle θ on a semi-infinite gray solid, as in Figure 11.14a. The absorption coefficient of the solid is κ and there is no scattering. Include partial reflection at the layer surface.

Verified Solution

A fraction $\widehat{t_0}(\theta )$ of the incident energy is transmitted through the interface, and within the material the energy is refracted into direction χ (an optically smooth interface has been assumed). The fraction of entering radiation that reaches depth x is $e^{-kx/\cos \chi }$ ,and the amount $(κ/cos χ)e^{-kx/\cos \chi }$ is absorbed at x per unit volume. The solid is assumed to be in its initial state of heating so that temperatures are low, and emission within the solid and heat loss from its surface can be neglected. The energy equation, initial condition, and boundary conditions are then

$\rho c\frac{\partial T}{\partial t} =k\frac{\partial^2T}{\partial x^2} +q_i\cos \theta \widehat{t_0}(\theta )\frac{k}{\cos \chi }e^{-kx/\cos \chi }$          (11.112)

$T(x,0) =T_0$                  (initial condition)

$\frac{\partial T}{\partial x} =(0,t);\underset{x\longrightarrow \infty }{\lim} T(x,t)=T_0$

The solution is in Carslaw and Jaeger (1959) and can be placed in the form

$\frac{[T(x,t)-T_0]\sqrt{k\rho c} }{q_i\cos \theta \widehat{t_0}(\theta )\sqrt{t} } =\frac{2}{\sqrt{\pi } }e^{-x^2}-2X \ erfc \ X -\frac{1}{A}e^{-2AX} +\frac{1}{2A}[e^{A(A-2X)}erfc(A-X)+e^{A(A+2X)}erfc(A+X)]$            (11.113)

where $X=x/2\sqrt{\alpha t}$,and $A=(κ/cosχ)\sqrt{\sigma t}$,and α = k/ρc.(Note that t = time,$\widehat{t_0}$=surface transmissivity.)Some results are in Figure 11.14b. Since the parameter on the curves depends on the absorption coefficient κ, an increase of κ increases the surface temperature reached at any time. A small κ increases the penetration of the temperature distribution into the solid. As $(κ/cos χ)\sqrt{\sigma t} \longrightarrow \infty$ the temperature distribution approaches that for an opaque solid.