What so special about Q(heat) and not special for "X(any other energy transfer)" in non-heat engines?

Question

I understand Bolzmann definition of entropy well enough, but I keep struggling to understand entropy $S$ and it's relation to heat and efficiency in classical (non statistical) thermodynamics.

As Carnot said, heat engine efficiency depends only on temperature difference of it's hot and cold sinks. That is: $$η = 1 - T_c/T_h$$

"Well, what's a big deal" — I always asked my self. "*There are lots of differentials in other engines like:

1. height differential in waterwheel
2. pressure differential in .... some engines
3. wind speed "differential" in windmills (Betz Law)

I mean, why $S$ wasn't invented for a waterwheel at first? Why thermodynamics lead the way for $S$ discovery?

*"But recently I understood that

4. For many engines there is no known theoretical efficiency limit, so no one knows on what this possible limit depends: design or physical constrain.
5. There must be engines efficiecy of which depend on design and not on some physical constrain. Examples?
6. Even if for some engines there is a theoretical efficiency limit independent of it's design, still you can't derive entropy from this.

So this lead to conclusion that "there is something special about heat(energy transported through temperature differential)"... I know it's silly to figure this out when it's written in every book, but I'm quiet dumb in these matters.

Now the question: For Carnot case it is true that:

If $Q$ drops from $T_h$ to $T_c$ only $W=Q(1-T_c/T_h)$ amount can be extracted as work. Why it's not the case for any other engine (but without heat transfer) and their differential (pressure, velociry, voltage ect)?

Example: Let $X$ Joules of energy "drops" from high level $L_h$ to low level $L_l$ passing through a waterwheel. Then as far as I know, there is no such thing (and any other kind) as $W=X(1-L_l/L_h)$. Why? Is there a way for the blind (like me) to see what's SO SPECIAL about Q and nothing special about X(in waterwheel example).

Answer 1

why S wasn't invented for a waterwheel at first?

Your question goes into the heart of thermodynamics and has to do with microscopic irreversibility. Classical mechanics is reversible, and large macroscopic mechanical systems whose performance is not immediately or appreciably affected by the molecular state, such as a waterwheel for example, are practically reversible, which is to say their entropy is zero: you do $X$ amount of work to rotate the wheel by some amount, it returns that work to you when you reverse the motion.

When the microscopic state is affected by the process, it is not possible to reverse it. When steam runs through a reversible Carnot cycle, starting from state $A$ and ending at the same state later, the macroscopic state (i.e., pressure and temperature) has been restored, but the internal state has not. In fact, it cannot be restored, because this would require us to bring every molecule to its original position and velocity. This microscopic irreversibility is the origin of entropy and heat.

To get more insights on entropy one needs to get deeper into statistical thermodynamics and away from classical thermodynamics and cycles.