FIGURE P$2-80$

$2-81$ In a food processing facility, a spherical container of inner radius $r_1=40cm,$ outer radius $r_2=41cm,$ and thermal conductivity $k=1\mathrm{.}5\frac{W}{m}*K$ is used to store hot water and to keep it at $100C$ at all times. To accomplish this, the outer surface of the container is wrapped with a $500-$W electric strip heater and then insulated. The temperature of the inner surface of the container is observed to be nearly $100C$ at all times. Assuming $10$ percent of the heat generated in the heater is lost through the insulation, $($a$)$ express the differential equation and the boundary conditions for steady one$-$dimensional heat conduction through the container, $($b$)$ obtain a relation for the variation of temperature in the container material by solving the differential equation, and $($c$)$ evaluate the outer surface temperature of the container. Also determine how much water at $100C$ this tank can supply steadily if the cold water enters at $20C.$ d$)$ find the rate at which water can be heated from $20\backslash$deg C to $100\backslash$deg C in units of kg$/$hr$.$