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 k₁  = 200 W/m ⋅ K, but the thermal conductivity of Rod 2, k₂, is not known. To determine the thermal conductivity of Rod 2, both rods are thermally attached at one end 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_∞. 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 x₁  = 40 cm from the base is the same as that of the rod of unknown thermal conductivity (Rod 2) at x₂  = 20 cm from the base. Determine the thermal conductivity k₂ of the second rod (W/m⋅K).

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Tb k₁= 200 W/m-K x₁ = 40 cm x₂ = 20 cm T₂(x) FIGURE P17-112 T₁(x) Rod 1 h, To Rod 2