The shield of a nuclear reactor is idealized by a large 25 cm thick flat plate having a thermal conductivity of 3.5 W/(m K). Radiation from the interior of the reactor penetrates the shield and there produces heat generation that decreases exponentially from a value of 187.6 kW/ m3. at the inner surface to a value of 18.76 kW/m3 at a distance of 12.5 cm from the interior surface. If the exterior surface is kept at 38°C by forced convection, determine the temperature at the inner surface of the shield. Hint: First set up the differential equation for a system in which the heat generation rate varies according to (0)e–Cx
IVEN
Large flat plate with non-uniform internal heat generation Thickness (L) = 25 cm=0.25 m Thermal conductivity (k) = 3.5 W/(m K) Exterior surface temperature (To) = 38°C Heat generation is exponential with values of
? 187.6 kW/m3 at the inner surface
? 18.76 kW/m3 at 12.5 cm from the inner surface
FIND
The inner surface temperature (Ti)
ASSUMPTIONS
One dimensional, steady state conduction The thermal conductivity is constant No heat transfer at the inner surface of the shield
SKETCH
From the hint, the internal heat generation is
Solving for the constant c using the fact that q(x) = 18.76 kW/m3 at x = 12.5 cm = 0.125 m
The one dimensional conduction equation is given
The boundary conditions are
Integrating the conduction equation
The constant C1 can be evaluated by applying the first boundary condition
Integrating again
The constant C2 can be evaluated by applying the second boundary condition
Therefore, the temperature distribution is
Solving for the temperature at the inside surface (x = 0)
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