There's one situation relating to heat engines where there's one equation I never understood completely. The situation is something like: there's a heat engine operating between two systems $A$ and $B$ with temperatures $T_A<T_B$.
As we know there's one heat input from system $B$ into the working substance and one work output. What I'm in doubt is one equation for the heat input.
Suppose, for simplicity, that $B$ has constant heat capacity $C$, in that case one has
$$Q = C(T_B-T_A).$$
This equation always puzzled me. The main issue is that this is the heat transfer in a process where system $B$ is taken from initial state $\xi_1$ with temperature $T_B$ to the final state $\xi_2$ with temperature $T_A.$
Now as I always understood in a heat engine the working substance changes its state, while the reservoirs do not.
Furthermore, if the hot reservoir cools in one cycle, in the next one the efficiency would be something different, and everything would have changed.
My question is: why in this case $Q$ is given by this formula? How can we derive it considering that we are dealing with reservoirs in a heat engine? And how can we understand these changes in temperature of the reservoirs in a heat engine? This really doesn't make much sense to me.
EDIT: The whole point here is that the result follows trivially if we know the following facts:
- The system $B$ is in thermal contact just with the working substance;
- In a cycle the system $B$ goes from a state with temperature $T_B$ to a state with temperature $T_A$.
If these two things are true, we know that the heat for system $B$ will be $Q_B = C(T_A-T_B)$ and consequently by conservation of energy if system $B$ has only thermal contact with the working substance, the heat input is $Q = C(T_B-T_A)$.
The first fact is obvious. The problem is the second fact: I can't see why system $B$ goes from a state with temperature $T_B$ to a state with temperature $T_A$. Furthermore, it seems to me that this would affect the heat engine after the first cycle, since temperature changed.
So, why system $B$ goes from a state with temperature $T_B$ to another with temperature $T_A$ in one cycle?