Question

A quartz window of thickness L serves as a viewing port in a furnace used for annealing steel. The inner surface (x=0) of the window is irradiated with a uniform heat flux q''o due to emission from hot gases in the furnace. A fraction, β, of this radiation may be assumed to be absorbed at the inner surface, while the remaining radiation is partially absorbed as it passes through the quartz. The volumetric heat generation due to this absorption may be described by an expression of the form q˙(x)=(1−β)q''oα^e−αx where α is the absorption coefficient of the quartz. Convection heat transfer occurs from the outer surface (x=L) of the window to ambient air at T[infinity] and is characterized by the convection coefficient h. Convection and radiation emission from the inner surface may be neglected, along with radiation emission from the outer surface.

Required:
Determine the temperature distribution in the quartz, expressing your result in terms of the foregoing parameters.

Answer 1

Assuming steady state condition the temperature distribution is calculated as expressed in the attached solution below

Step-by-step explanation:

Given data :

thickness : L , inner surface (x) : 0, uniform flux : q"o

fraction : β

volumetric heat generation : q˙(x)=(1−β)q''oα^e−αx

determine the temperature distribution in the quartz

attached below is the detailed solution