$(4$ Points$)$ A hollow spherical shell $\{r_{\text{i}\text{n}\text{n}\text{e}\text{r}}=0\mathrm{.}5(m),r_{\text{o}\text{u}\text{t}\text{e}\text{r}}=0\mathrm{.}7(m),k=40(1+0\mathrm{.}001(T))\}$

stores a liquid at $250C,$ while the outside surface temperature of the sphere is $100C.$

The ambient surrounding air is at $30C.$ Find the heat flux through the spherical shell

in W and the outside heat transfer coefficient in $\frac{W}{m^2}K.$ Note that the thermal

conductivity k is temperature dependent $($ T is in Celsius$)$ and the surface area of a

sphere is given by $4r^2.$