Chapter 3, Problem 3.141P

An experimental arrangement for measuring the thermal conductivity of solid materials involves the use of two long rods that are equivalent in every respect, except that one is fabricated from a standard material of known thermal conductivity $k_A$ while the other is fabricated from the material whose thermal conductivity $k_B$ is desired. Both rods are attached at one end to a heat source of fixed temperature $T_b,$ are exposed to a fluid of temperature $T_{\infty },$ and are instrumented with thermocouples to measure the temperature at a fixed distance $x_1$ from the heat source. If the standard material is aluminum, with $k_A=200\text{W/m}\cdot \text{K,}$ and measurements reveal values of $T_A=75°\text{C}$ and $T_B=60°\text{C}$ at for $T_b=100°\text{C}$ and $T_{\infty }=25°\text{C,}$ what is the thermal conductivity $k_B$ of the test material?