(Solved): Microbes in hay bales release thermal energy causing the temperature within the hay to rise. It is ...

Microbes in hay bales release thermal energy causing the temperature within the hay to rise. It is recommended that the temperature of hay never exceed $T_{\mathrm{c}}=55^{\circ }\mathrm{C}$ to avoid damage. It is found that in fresh damp hay, the volumetric thermal energy generation rate $\mathrm{s}\dot{q}=5\mathrm{W}/{\mathrm{m}}^3$. \&onsider a $D=2\mathrm{m}$ diameter cylindrical bale of fresh damp hay with a thermal conductivity of $K=0.05\mathrm{W}/\mathrm{m}\cdot \mathrm{K}$. If the ambient air provides cooling with a convective coefficient of $h=10\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$, determine the following: (a) The maximum surface temperature of the bale of hay, $T_{\mathrm{s},\mathrm{c}}$, such that the maximum temperature of the hay is less than $T_{\mathrm{c}}$. (b) The maximum ambient air temperature, $T_{\infty ,c}$, such that the maximum temperature of the hay is less than $T_{\mathrm{c}}$.