The hardest thing for me about studying thermodynamics, is connecting the formal theory with actual problem instances. Here is an example:
In Callen’s Thermodynamics and an Introduction to Thermostatics (second edition), Chapter 4, Example 1, the following scenario is described:
A particular system is constrained to constant mole number and volume, so that no work can be done on or by the system. [...]. Two such systems, with equal heat capacities, have initial temperatures $T_{10}$ and $T_{20}$, with $T_{10} < T_{20}$. An engine is to be designed to lift an elevator (i.e., to deliver work to a purely mechanical system), drawing energy from the two thermodynamic systems. What is the maximum work that can be so delivered?
What I don't understand here is: Why does the question mention two such systems? More generally, I don't understand how I'm supposed to envision this scenario.
I understand that the given system cannot do mechanical work, so it can only transfer (or receive) heat to another system, which I believe is the engine. Therefore, the engine needs to be able to do the mechanical work. What I imagine is: the given system is in contact with an engine. The engine is simply (for example) a gas in a cylinder with a movable piston. The gas in the engine is heated by the system, and therefore expands, and this moves a piston which gives us the required mechanical motion. I understand that this process can continue until the temperature of the given system equals the temperature of the gas in the engine.
So where does the second system enter? Why do we need it? Am I imagining the scenario correctly?
EDIT
Here is a synthesis of all the answers I received, in words that are easier for me to understand. I hope I got it right.
We are supposed to think about the engine as a rather limited entity, in at least these aspects:
It has a small heat capacity. This means that, when it comes into contact with the high temperature system (which I'll call system $A$), a small amount of heat transferred to the engine is enough to raise its temperature to the level of system $A$. At this point, no more energy (heat) can be transferred to the engine from system $A$.
The engine's ability to do work might be mechanically bounded. For example, if the work is performed by expanding gas in a piston, the piston might have a maximal expansion length, after which it cannot do anymore work (because the piston cannot mechanically move anymore).
It follows that, when running the process, the engine will "quickly" get from its initial state to a final state in which it cannot do anymore work - either because it cannot take in anymore heat from system $A$ (this heat is what drives the work to be done), or because it is mechanically "stuck".
However, more work can be done if the engine is brought back to its initial state, and then the entire process is repeated. But to bring the engine back to its initial state, the engine must dump heat to another system - a low temperature system (which I'll call system $B$) - so that the engine's temperature can return to the initial value. System $B$ allows the engine to work in cycles, and hence to perform more and more work.
As these cycles are repeated many times, system $A$'s temperature is slowly reduced, and system $B$'s temperature is slowly increased, until both temperature become equal. At this point, no more work can be done.
If the given scenario did not include system $B$, then we would need to consider more carefully the details of the engine, e.g its temperature and heat capacity. In this case, the engine would take heat from system $A$ until the temperatures are equal, while performing work. And we don't have repeating cycles.