Two very long, slender rods of the same diameter and length are given. One rod
(Rod 1) is made of aluminum and has a thermal conductivity $k_{1}=200
\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, but the thermal conductivity of
Rod \(2, k_{2}\), is not known. To determine the thermal conductivity of Rod 2 ,
both rods at one end are thermally attached to a metal surface which is
maintained at a constant temperature \(T_{b}\). Both rods are losing heat by
convection, with a convection heat transfer coefficient \(h\) into the ambient
air at \(T_{\infty}\). The surface temperature of each rod is measured at
various distances from the hot base surface. The measurements reveal that the
temperature of the aluminum rod (Rod 1) at \(x_{1}=40 \mathrm{~cm}\) from the
base is the same as that of the rod of unknown thermal conductivity (Rod 2) at
\(x_{2}=20 \mathrm{~cm}\) from the base. Determine the thermal conductivity
\(k_{2}\) of the second rod \((\mathrm{W} / \mathrm{m} \cdot \mathrm{K})\).