Problem $2.$ Solid, short, cylindrical metal rods $($length$-$to$-$diameter ratio $=3)$ are used as heat transfer promoters on the exterior of a hot surface with a surface temperature of $700C.$ The ambient air flowing around the rod promoters has a temperature of $T_A=30C.$ The metal conductivity $(k)$ can be assumed to be $0\mathrm{.}247ca\frac{l}{scmK}.$ The heat transfer coefficient $(h)$ around the surface of the promoter is constant at $3\mathrm{.}6$Kca$\frac{l}{m^2(h)C}.$

a$)$ Analyze a single rod of $4mm$ in diameter, and show that the steady$-$state differential balance yields the following for the case where the metal temperature changes only in the axial $x-$direction, and rod radius $R$ and $2R=$ diameter, $D.$

$\frac{d^{2}T}{d{x}^2}-(\frac{2h}{Rk})(T-T_A)=0$

b$)$ Use the following boundary conditions and dimensionless variables below and render the entire system $($ODE and BCs dimensionless$)$:

Boundary Conditions:

Dimensionless variables $($and Biot number$)$:

$=\frac{T-T_A}{T_H-T_A},$ and $,x^{**}=\frac{x}{L},$ and $,Bi=\frac{hD}{k}$

c$)$ Compute the Biot number $($Bi$)$ based on the given problem parameters. Using Maple, analytically solve and plot the dimensionless temperature $()$ versus dimensionless axial distance $(x^{**})$ for different values of Biot number, including the Bi value for this system. Comment on any physical significance of what you observe as: What happens when $Bi$ is small, for example, compared to when it is large?