Question

A hot plate is placed in contact with a cold plate of a different thermal conductivity as shown in the figure. The initial temperature (at time $t = 0$) of the hot plate and cold plate are $T_h$ and $T_c$, respectively. Assume perfect contact between the plates. Which one of the following is an appropriate boundary condition at the surface $S$ for solving the unsteady state, one-dimensional heat conduction equations for the hot plate and cold plate for $t>0$?

 

• A. Temperature at S is same for both the plates
• B. Gradient of temperature at S is same for both the plates
• C. Gradient of temperature vanishes at S
• D. Temperature at S is the average of $T_h$ and $T_c$

Hint:

In heat transfer problems involving contact between two materials, ensure the continuity of temperature at the interface unless otherwise specified by additional resistive elements or discontinuities.

Answer 1

The Correct Option is A

Step 1: Analyze Heat Transfer Dynamics.
The key to solving this problem lies in understanding the heat transfer dynamics at the interface of two materials. Perfect contact between the plates implies that there is no thermal resistance at the interface, allowing heat to transfer freely. 
Step 2: Evaluate the Boundary Conditions.
Option (A) states that the temperature at the surface $S$ is the same for both plates. This condition aligns with the principle of continuity of temperature at the interface in heat conduction, making it the most appropriate choice.
Option (B) implies equal temperature gradients at the interface. While heat flux (related to the product of gradient and thermal conductivity) must be continuous across the interface, the gradients themselves may differ if the thermal conductivities are different.
Option (C) suggests that the temperature gradient vanishes at the surface, which would incorrectly imply no heat transfer.
Option (D) proposes that the temperature at the interface is the average of $T_h$ and $T_c$. This is not a necessary condition and does not generally hold unless derived from specific system conditions or symmetries. 
Step 3: Conclusion.
Continuity of temperature across the interface is essential for correct modeling of heat transfer in contact scenarios. Therefore, Option (A) provides the correct boundary condition ensuring that the temperatures are the same at the surface $S$ of both the hot and cold plates.