Uniform energy generation per unit volume at $=2\mathrm{.}4{10}^6\frac{W}{m^3}$ is occurring within a spherical nuclear fuel element of $20$ cm diameter. Under steady conditions the temperature distribution is given by

$T=900-10,000r^2$

where $r$ is in meters and $T$ is in degrees Celsius.

a$)$ Determine the thermal conductivity of the nuclear fuel element.

b$)$ What is the average heat transfer coefficient at the surface of the sphere if the ambient temperature is $35C?$