Entropy transfer between finite heat capacity systems

Question

Consider two systems $A$ and $B$ in an insulated vessel with constant finite heat capacities $C_A$ and $C_B$, which are initially at different temperatures $T_{A1} \neq T_{B1}$ for the initial state $1$.

Then the combined system runs down to thermal equilibrium, so that the final temperatures of $A$ and $B$ are equal for state 2, $T_{A2} = T_{B2}$.

How can you calculate the entropy transferred to each system?

Take system $A$ for example. By definition, the entropy transferred to $A$ from states $1$ to $2$ is

$\int_1^2 dS = \int_1^2 \frac{\delta Q_{12}}{T_{bA}} $

where the boundary temperature of $A$ is $T_{bA}$. Is this correct? Do we need to know the boundary temperature as a function of time?

Answer 1

Yes. You need to know the boundary temperature (and heat flux) as a function of time and location. This can be done by solving the transient heat conduction equation for each of the bodies, with matching of both the local temperature and local heat flux at the boundary. After solving the heat conduction problem, you would then integrate over the area of the boundary and over time to get the net transfer of entropy between the two bodies.

A simple version of this problem is if both bodies are cubes made out of the same material, and contacted at one face of each cube. The boundary temperature would then be constant, and equal to the arithmetic average of the initial temperatures of the two cubes. So the exchanged entropy would just be equal to the heat exchanged divided by the constant boundary temperature.