A solid material that has thermal conductivity K K in kilowatts per meter-kelvin and temperature given at each point by w ( x , y

A solid material that has thermal conductivity $K$ in kilowatts per meter-kelvin and temperature given at each point by $w(x,y,z)$ has heat flow given by the vector field $F=-K\nabla w$ and rate of heat flow across a surface $S$ within the solid given by $-K{\iint }_S\nabla wdS$.

An insulated cylinder of solid gold \((K=310 \mathrm{~kW} / \mathrm{m}-\mathrm{k})\) of radius \(\sqrt{2} \mathrm{~m}\) and height \(5 \mathrm{~m}\) is heated at one end until the temperature at each point in the cylinder is given by \(w(x, y, z)=\left(30-z^{2}ight)\left(2-\left(x^{2}+y^{2}ight)ight)\). Determine the rate of heat flow across each horizontal disk given by \(z=1, z=2\), and \(z=3\), identifying which has the greatest rate of heat flow across it.