Confusion on Callen's Maximum Work Theorem

[EE18]

This question was, effectively, asked here (please refer to that question for additional context); however, I don't think the given answer is correct (or at least complete) despite my having added a bounty and having had a productive discussion with the answerer there. In particular, I don't believe the usage of reversible there is what Callen has in mind; Callen is referring to a process which is reversible in the sense of the entire composite system (i.e. isentropic), not with respect to a particular subsystem. At any rate, it doesn't get to the crux of my question, which is why the essential logic below is wrong. I have since reread the section of Callen and still cannot convince myself of what's going on.

In Chapter 4.5 of his famous textbook, Callen gives his "Maximum Work Theorem"; effectively, it says that for a composite system (and process defined thereon) composed of three subsystems: (1) a primary subsystem which is taken from some initial state to some final state (2) a "reversible heat source" (RHS) which can only exchange heat and (3) a "reversible work source" (RWS) which can only exchange work. The theorem then states that

"for all processes leading from the specified initial state to the specified final state of the primary system, the delivery of work is maximum (and the delivery of heat is minimum) for a reversible process. Furthermore the delivery of work (and of heat) is identical for every reversible process."

The "reversible" caveats above are perhaps a bit of a misnomer; they are just intended to indicate that every process is quasistatic with respect to said subsystem.

As shown in Callen, one can show that in an infinitesimal process, the maximum work delivered to the RWS is given by
$$dW_{RWS} = \left(1 - \frac{T_{RHS}}{T}\right)(-dQ) - dW$$
where unsubscripted quantities refer to the primary subsystem. My question is why the following logic is wrong:
By the definitions of the two "reversible sources", any heat lost by the primary subsystem must be exchanged with RHS, and any work must be exchanged with the RWS. Therefore, $dQ_{RHS} = -dQ$ and $dW_{RWS} = -dW$.

This clearly contradicts $dW_{RWS} = \left(1 - \frac{T_{RHS}}{T}\right)(-dQ) - dW$. But what is wrong with the logic above? Certainly, we must have $-dW - dQ = -dU = dQ_{RHS} + dW_{RWS}$ (indeed this is part of Callen's derivation), but clearly I am wrong and it's not true that $dQ_{RHS} = -dQ$ and $dW_{RWS} = -dW$.

To be clear, I am not disputing that the derivation given by Callen is wrong; I follow it through completely. I am asking why my particular logic above fails, since I think any answer to this "paradox" may offer some insight (to me anyway) into how heat is converted to work.

My question here is different than the linked one precisely because that answer does not (it seems to me) explain why my logic above is wrong directly. I am not asking for an explanation of Callen's derivation, but why my (wrong) derivation is wrong.

 

[TSny]

My question is why the following logic is wrong:
By the definitions of the two "reversible sources", any heat lost by the primary subsystem must be exchanged with RHS,

Callen does not assume that any heat lost by the primary subsystem must be exchanged with the RHS. Callen allows for the possibility that some of the heat lost by the primary subsystem could be converted to work $\Delta W'$ (using a heat engine for example). This work would be delivered to the RWS and the remaining heat $\Delta Q^c$ (the exhaust of the engine) would then be delivered to the RHS. See the second answer posted in your link. The total work delivered to the RWS would be the work done by the primary subsystem plus the work $\Delta W'$ that came from converting some of the heat lost by the subsystem to work. Only part of the heat $-\Delta Q$ lost by the primary subsystem is delivered to the RHS.

In the first edition of Callen's text, he included the following diagram that I don't see in the second edition.

 

[EE18]

Callen does not assume that any heat lost by the primary subsystem must be exchanged with the RHS. Callen allows for the possibility that some of the heat lost by the primary subsystem could be converted to work $\Delta W'$ (using a heat engine for example). This work would be delivered to the RWS and the remaining heat $\Delta Q^c$ (the exhaust of the engine) would then be delivered to the RHS. See the second answer posted in your link. The total work delivered to the RWS would be the work done by the primary subsystem plus the work $\Delta W'$ that came from converting some of the heat lost by the subsystem to work. Only part of the heat $-\Delta Q$ lost by the primary subsystem is delivered to the RHS.

In the first edition of Callen's text, he included the following diagram that I don't see in the second edition.

View attachment 327974

I think I see, thanks so much for your answer. In short, would it be fair to characterize your answer as agreeing with the second answer in my link -- namely that Callen tacitly allows for some auxiliary system to "transmit" heat to work as long as its state is not changed by the end of the given process?

 

[TSny]

I think I see, thanks so much for your answer. In short, would it be fair to characterize your answer as agreeing with the second answer in my link -- namely that Callen tacitly allows for some auxiliary system to "transmit" heat to work as long as its state is not changed by the end of the given process?

Yes.

 

[TSny]

…as long as its state is not changed by the end of the given process?

That’s a good point to emphasize. The auxiliary system needs to return to its initial state in the overall process.

 

[EE18]

That’s a good point to emphasize. The auxiliary system needs to return to its initial state in the overall process.

Do you have any sense as to why Callen would have omitted mention of this in his Chapter 4.5 (indeed, he explicitly seems to say at the start of Chapter 4.5 that the only components of the composite system are the primary subsystem, the RHS, and the RWS -- with no mention of auxiliary systems)? It seems like a crucial bit and he misses it (or at least decides not to mention it). He does mention it in Chapter 4.7 in the particular case of a reversible process (with respect to the composite system) of a Carnot cycle, but I'm not sure whether I'm correct to interpret a Carnot cycle as a particular example of the reversible processes described in Chapter 4.5 unless we take that Callen was wrong in Chapter 4.5 not to include mention of the auxiliary subsystems.

 

[TSny]

Do you have any sense as to why Callen would have omitted mention of this in his Chapter 4.5 (indeed, he explicitly seems to say at the start of Chapter 4.5 that the only components of the composite system are the primary subsystem, the RHS, and the RWS -- with no mention of auxiliary systems)? It seems like a crucial bit and he misses it (or at least decides not to mention it). He does mention it in Chapter 4.7 in the particular case of a reversible process (with respect to the composite system) of a Carnot cycle, but I'm not sure whether I'm correct to interpret a Carnot cycle as a particular example of the reversible processes described in Chapter 4.5 unless we take that Callen was wrong in Chapter 4.5 not to include mention of the auxiliary subsystems.

I believe Callen does allude to “auxiliary systems” or “engines” in the introductory section 4.1. Here, he considers the example of an engineer who wishes to use some of the internal energy in a furnace to lift an elevator. The “engine” in this case corresponds to “the various pistons, levers, and cams” used to somehow link the furnace to the elevator. I think that Callen is pointing out that the first and second laws of thermodynamics (energy is conserved, total entropy can’t decrease) limit the amount of work that can be delivered to the elevator. It doesn’t matter how clever the engineer is in designing an engine or auxiliary system to perform the task. The laws of thermodynamics can be used to find the maximum work that can be delivered to the elevator for a given amount of energy taken from the furnace.

I think Callen purposely avoids discussing any details of auxiliary systems in the first few sections of chapter 4 since he wants to emphasize the generality of the laws of thermodynamics. This is just my opinion. For example, consider Callen’s Example 1 starting at the bottom of page 91 in the 2nd edition:

The solution given in the text arrives at the answer by just using the general principles of energy conservation and the requirement that total entropy cannot decrease. Callen does not indicate how one would design an “engine” or auxiliary system to carry out the steps that deliver work to the elevator from the two subsystems “1” and “2” described in the problem statement.

Since systems “1” and “2” are specified to have fixed volumes, these systems cannot deliver work directly to the elevator. Nevertheless, the maximum work that can be delivered to the elevator using energy of the systems “1” and “2” can be found. The beauty is that we don’t need to imagine how energy from “1” and “2” is engineered to be delivered as work to the elevator.

We can relate this example to the schematic diagram used later in the chapter:

We take systems “1” and “2” together as the “System”. We can let the elevator represent the reversible work source.

In Example 1 we do not need to make use of a reversible heat source, so it is not shown.

The choice of the auxiliary system (shown by the “?” in the diagram), which converts the energy extracted from the System to the work delivered to the elevator, is not required to be known in order to solve the problem in Example 1.

After studying heat engines in the later sections of the chapter, you can see that the auxiliary system could be a Carnot engine using the hotter system “1” as the heat source and the cooler system “2” as the heat sink for the engine.

Each cycle of the engine takes a little heat $dQ_1$ from "1”, delivers a little heat $dQ_2$ to “2”, and delivers some work $dW$ to the elevator. The temperature of “1” drops while the temperature of “2” increases until “1” and “2” reach the same final temperature $T_f$. A calculation using the efficiency of a Carnot engine and the heat capacity $C$ of systems “1” and “2” leads to a total work delivered to the elevator that matches the result obtained in Example 1.

 

[EE18]

I believe Callen does allude to “auxiliary systems” or “engines” in the introductory section 4.1. Here, he considers the example of an engineer who wishes to use some of the internal energy in a furnace to lift an elevator. The “engine” in this case corresponds to “the various pistons, levers, and cams” used to somehow link the furnace to the elevator. I think that Callen is pointing out that the first and second laws of thermodynamics (energy is conserved, total entropy can’t decrease) limit the amount of work that can be delivered to the elevator. It doesn’t matter how clever the engineer is in designing an engine or auxiliary system to perform the task. The laws of thermodynamics can be used to find the maximum work that can be delivered to the elevator for a given amount of energy taken from the furnace.

I think Callen purposely avoids discussing any details of auxiliary systems in the first few sections of chapter 4 since he wants to emphasize the generality of the laws of thermodynamics. This is just my opinion. For example, consider Callen’s Example 1 starting at the bottom of page 91 in the 2nd edition:

View attachment 328024
The solution given in the text arrives at the answer by just using the general principles of energy conservation and the requirement that total entropy cannot decrease. Callen does not indicate how one would design an “engine” or auxiliary system to carry out the steps that deliver work to the elevator from the two subsystems “1” and “2” described in the problem statement.

Since systems “1” and “2” are specified to have fixed volumes, these systems cannot deliver work directly to the elevator. Nevertheless, the maximum work that can be delivered to the elevator using energy of the systems “1” and “2” can be found. The beauty is that we don’t need to imagine how energy from “1” and “2” is engineered to be delivered as work to the elevator.

We can relate this example to the schematic diagram used later in the chapter:

View attachment 328025

We take systems “1” and “2” together as the “System”. We can let the elevator represent the reversible work source.

View attachment 328026

In Example 1 we do not need to make use of a reversible heat source, so it is not shown.

The choice of the auxiliary system (shown by the “?” in the diagram), which converts the energy extracted from the System to the work delivered to the elevator, is not required to be known in order to solve the problem in Example 1.

After studying heat engines in the later sections of the chapter, you can see that the auxiliary system could be a Carnot engine using the hotter system “1” as the heat source and the cooler system “2” as the heat sink for the engine.

View attachment 328027

Each cycle of the engine takes a little heat $dQ_1$ from "1”, delivers a little heat $dQ_2$ to “2”, and delivers some work $dW$ to the elevator. The temperature of “1” drops while the temperature of “2” increases until “1” and “2” reach the same final temperature $T_f$. A calculation using the efficiency of a Carnot engine and the heat capacity $C$ of systems “1” and “2” leads to a total work delivered to the elevator that matches the result obtained in Example 1.

Beautifully explained, thank you!