A cylindrical nuclear fuel rod under steady conditions in a cooling bath has a 2-d temperature distribution of: T(r,z)=Ar^2+Bz+C, where A,B and C are constants with the appropriate units and r and z are the radial and vertical position components respecively. Note the rod's z axis values are chosen to range from 0.5 to 1.5 meters. The thermal conductivity varies linearly with distance according to the expression k(z)=Dz.
Calculate an experession in terms of the constants A,B,C and D for the thermal energy conversion as a function of r and z within the rod.
Suppose that the heat generation suddenly ceases. At that instant, calculate the rate of the rod's temperature change at z=1.0 meter, assuming the rod's density and specific heat are 2500 [kg/m^3] and 2000 [kJ/kg*K], respectively and the values of the constants are: A=-5e4 [K/m^2], B=200 [K/m], C=600 [K], D=50 [W/m^2*K].