$18\mathrm{.}22$ In the canning process, sealed cans of food are sterilized with pressurized steam in order to kill any microorganisms initially present in the food and thereby prolong the shelf life of the food. A cylindrical can of food has a diameter of $4$ cm and length of $10$ cm $.$ The food material has a heat capacity of $\backslash (4000\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ),$ a density of $\backslash (1500\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ),$ and thermal conductivity of $\backslash (0\mathrm{.}6\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash$cdot $\backslash$mathrm$\{$K$\}\backslash ).$ In the present process, steam, at $120$ C $,$ is used to sterilize the can. The convective heat$-$transfer coefficient is $\backslash (60\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$ Initially, the can and its contents are at a uniform temperature of $20$ C $.$

a$.$ If the heat$-$transfer resistance offered by the can itself is neglected, and the can ends are thermally insulated, how long will it take for the center of the can to reach a temperature of $100$ C $,$ which is sufficient to kill all microorganisms?

b$.$ What is the required time if the ends of the can are not sealed?

c$.$ What will be the center temperature of the can and contents after $2500$ s have elapsed?