(Solved): The shield of a nuclear reactor can be idealized as a slab with a face area of 6ft2 and a thermal ...

The shield of a nuclear reactor can be idealized as a slab with a face area of $$6{\mathrm{ft}}^2$$ and a thermal conductivity $$k=2\mathrm{Btu}/{\mathrm{hr}}^{?}{\mathrm{ft}}^{?}{}^{?}\mathrm{F}$$. Heat is generated in the shield at a steady rate $$S=S_0\exp (?bx)\left[\mathrm{Btu}/{\mathrm{hr}}^{?}{\mathrm{ft}}^3\right]$$ where $$x=0$$ denotes the inner surface close to the nuclear reactor and the outer surface is at $$x=L$$. The temperature is maintained at $$300^{?}\mathrm{F}$$ at the inner surface $$(x=0)$$ and $$80^{?}\mathrm{F}$$ at the outer surface $$x=L$$. Take the thickness of the slab $$L=0.83\mathrm{ft},S_0=17,280\mathrm{Btu}/{\mathrm{hr}}^{\mathrm{ft}}{}^3,b=5.5{\mathrm{ft}}^{?1}$$ and $$k=2.0\mathrm{Btu}/{\mathrm{hr}}^{?}{\mathrm{ft}}^{?}=\mathrm{F}$$ a. Carry out a steady-state shell balance of energy to derive a differential equation for the heat flux variation in the slab. b. Apply Fourier's law of conduction to derive a differential equation for the temperature distribution $$T(x)$$ in the slab. Apply the given boundary conditions to solve for $$T(x)$$. c. Determine the heat flow rate $$Q_H$$ at the outer surface of the slab (answer should be in units of Btu).