Question $1(25$ points$)$ : Consider a $2$ mm diameter solid cylindrical rod, $10$ cm long $($ L $),$ connecting two plane walls as shown in the figure below. Each end of the rod is fixed at a temperature of $T_b=100C.$ The rod is surrounded by a fluid at $T_{}=20C$ with a convection heat transfer coefficient of $20\frac{W}{m^2}.K.$ The thermal conductivity of the rod is $15\frac{W}{\text{I}}II.N.$

a$.$ Determine the temperature of the rod halfway between the walls $(5$ cm from each wall$).$

b$.$ Determine the rate of heat transfer from the rod to the fluid.