Two long rods A and B have same diameter of 10 cm The ends o

Two long rods A and B have same diameter of 10 cm. The ends of both rods are attached to a surface maintained at 200 degree C, while the surfaces of the rods are exposed to ambient air at 25 degree C and convection coefficient of 12 W/m^2. K. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions x_A = 0.13 m and x_B = 0.085 m, where x is measured from the base surface and subscripts A and B refer to rod A and rod B. If the thermal conductivity of rod A is k_A = 45 W/m. K, determine the thermal conductivity of rod B.

Solution

(Txa - Tamb) / (Tbase - Tamb) = e^-m_ax_a and

(Txb - Tamb) / (Tbase - Tamb) = e^-m_bx_b

Since Txa = Txb, we get

e^-m_ax_a = e^-m_bx_b

or, m_ax_a = m_bx_b

m_a / m_b = x_b / x_a

= 0.085 / 0.13

= 0.6538

m = (hP / kA)^0.5

Since both rods have equal diameters, their peremetrs P and cross-section areas A will be equal.

Moreover, since both are exposed to same ambient, h is same for both the rods.

Hence, m_a / m_b = (k_b / k_a)^0.5

0.6538 = (k_b / 45)^0.5

k_b = 19.24 W/m-K