$(30$ points$)$ see Figure $2,$ side view $-$ The eight exhaust headers from a Formula I race car engine heat the engine block through radiation heat transfer. The engine block is at $450$ K and has an emissivity of $0\mathrm{.}9.$ The exhaust headers are at $1200$ K and are radiating as blackbodies. The engine compartment $($surroundings$)$ is at $400$ K $.$ The exhaust headers are $40$ mm in diameter and are spaced $70$ mm center$-$to$-$center. The eight exhaust headers radiate a rectangular area of the engine block measuring $560mm500mm,$ and the exhaust headers are $500$ mm long.

a$.$ Draw the radiation circuit between the exhaust headers $($surface $1),$ the engine block $($surface $2),$ and the engine compartment or surroundings $($surface $3).$ Ignore the non$-$radiated $($back$)$ side of the engine block.

b$.$ Give values for all the necessary view factors modeling the geometry as that of an infinite plane and row of cylinders $($see Table $13\mathrm{.}1).$ Assume $F_{11}=0\mathrm{.}1.$

c$.$ What is the net radiation $[$W$]$ to the engine block? Hint: first solve the radiation circuit for the radiosity J of the engine block using an energy balance at that node.

d$.$ What is the heat input to the exhaust headers $[$kW$]?$

e$.$ If the engine produces $740$ kW $,$ what fraction is spent heating the exhaust headers?