Question

The coefficients of thermal conductivity are given as: \[ K_1 = 60 W/m^\circ C, K_2 = 120 W/m^\circ C, K_3 = 135 W/m^\circ C \] The temperature at the leftmost end is 100°C and the rightmost end is 0°C. Find the temperature \( \theta \) at the junction.

• A. 50°C
• B. 60°C
• C. 75°C
• D. 80°C

Hint:

In problems involving heat flow through materials with different conductivities, use the heat transfer equation and balance the heat flow through each section to find the unknown temperature.

Answer 1

The Correct Option is C

We can use the concept of thermal equilibrium and the relation between thermal conductivity and temperature gradient. The total heat flow is the same through all three sections. The heat flow through each section can be written as: \[ Q = \frac{K_1 (T_1 - \theta)}{L_1} = \frac{K_2 (\theta - T_2)}{L_2} = \frac{K_3 (T_2 - T_3)}{L_3} \] By solving the system of equations, we find that the temperature at the junction \( \theta \) is 75°C.