Question about the Kelvin–Planck statement

Question

It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy, by work, to its surroundings, while receiving energy by heat transfer from a single thermal reservoir.

The Kelvin–Planck statement does not rule out the possibility of a system developing a net amount of work from a heat transfer drawn from a single reservoir. It only denies this possibility if the system undergoes a thermodynamic cycle.

According to the Kelvin–Planck statement, a system undergoing a cycle while communicating thermally with a single reservoir cannot deliver a net amount of work to its surroundings. That is, the net work of the cycle cannot be positive. However, the Kelvin–Planck statement does not rule out the possibility that there is a net work transfer of energy to the system during the cycle or that the net work is zero. Thus, the analytical form of the Kelvin–Planck statement is: $W_\text{cycle} \le 0$. We associate the "less than" and "equal to" signs with the presence and absence of internal irreversibilities, respectively.

These are the statements mentioned in the book: FUNDAMENTALS OF ENGINNERING THERMODYNAMICS by Michael J. Moran and Howard N. Shapiro.

I have drawn these conclusions and have the following doubts:

CONCLUSIONS:

1. It is possible for a system to develop a net amount of work due to heat transfer from a single reservoir, just that the system should not undergo a thermodynamic cycle.

2. If the system undergoes a thermodynamic cycle, and exchanges heat only with a single reservoir, then the only condition is that the system undergoing a thermodynamic cycle cannot deliver net amount of work (positive) to the surroundings. This arrangement can have a negative net work (work received, hence negative) or zero net work.

Are these conclusions correct?

DOUBTS:

Does the Kelvin-Planck statement not comment on a system, which is not undergoing a thermodynamic cycle, which is exchanging heat only with a single reservoir, and about the conversion efficiency of heat to work? (Example: gas contained in a piston cylinder arrangement, which is insulated). (Please explain using this example as well.)

If the second conclusion that I have drawn is correct, what systems interact with a single reservoir, either by absorbing some work, or with no net work?

Answer 1

Before addressing your conclusions it's important to understand the conditions required to satisfy the equation $W(cycle)\le 0$ in the third statement by Moran.

An example of condition $W(cycle)=0$ is a reversible isothermal expansion (as done in the Carnot cycle) immediately followed by a reversible isothermal compression along the same path back to the initial state. See FIG 1 below. The positive work is the area under the curve moving from state 1 to state 2. The negative work is the area under the curve moving from state 2 to state 1. Since the expansion and compression follow the same path, the areas are equal and the net work is zero. The amount of heat in from the single reservoir during the expansion equals the amount of heat returned to the the same reservoir during the compression.

For the condition $W(cycle)\lt 0$, we take an irreversible isothermal* return path as shown in FIG 2. Instead of gradually increasing the external pressure as done in the reversible compression of FIG 1, we increase the external pressure suddenly. The increase in pressure happens so quickly that there is not enough time for heat transfer to occur.Then we allow the gas to rapidly (irreversibly) compress under constant external pressure transferring heat back to the reservoir and returning the system to its original state.

In FIG 2 the negative work is the area under the irreversible path from state 2 to 1. This area is larger than the positive work equal to the area under the reversible expansion from state 1 to 2. Thus the net work is negative.

From the above, with regard to your conclusions:

Conclusion 1 is correct as applied to only the reversible isothermal expansion process of FIG 1, where all the heat taken from the single reservoir is converted to work. It doesn't violate the second law because it is not a cycle.

Conclusion 2 applies to the irreversible cycle of FIG 2. It is important to realize you can only get net negative work done if the cycle is irreversible, as noted in the third statement of Moran.

Hope this helps.

*(Note: we call the irreversible compression isothermal only because the temperature at the boundary with the surroundings is held constant equal to the reservoir temperature, not because the gas temperature is constant. It varies spatially due to the rapid irreversible compression)