Liquid nitrogen \((T=77 \mathrm{~K})\) is stored in a thin-walled, spherical container that is covered with a uniformly thick insulation layer of thermal conductivity \(k=0.15\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\). The outer surface temperature of the insulation is at \(T_{s, 2}=20^{\circ} \mathrm{C}\). Due to space constraints, the outer radius of the insulation is fixed at \(r_{2}=0.5 \mathrm{~m}\). Determine the radius of the thin-walled, spherical container that will yield the minimum heat transfer rate per unit nitrogen volume. Also calculate the minimum heat transfer rate per unit nitrogen volume.