Is Mean Temp in 2 phase Heat Exchangers Higher Than Logarithmic Mean?

[Devin-M]

I made this to illustrate.

After 2, same weight, compressed samples of C02 at constant pressure transfer the same amount of energy to sub-zero C environments in separate experiments (one at 5MPa and the other at 2.5MPa, both starting at 125C), at the end of the experiments, the 5MPa CO2 sample is 25C warmer than the 2.5MPa CO2 sample... the 2.5MPa sample follows the log mean formula so wouldn't the 5MPa sample be warmer than the log mean formula implies, on average?

 

[Chestermiller]

The 5 MPa would not be higher than its own log-mean formula (which would correspond to a straight line drawn from tip to toe of the green line).

 

[Devin-M]

The 5 MPa would not be higher than its own log-mean formula (which would correspond to a straight line drawn from tip to toe of the green line).

If the hot side CO2 refrigerant of a heat pump is 2.5MPa from 125C to 0C, I thought the average temperature of the hot heat exchanger can be calculated by the log mean formula, and its phase stays a gas.

But if we have 1kg samples of CO2 at fixed pressure 2.5MPa and 5MPa starting at 125C, and we extract the same 133kJ energy from the 5MPa and 2.5MPa samples to a cold sub-zero C environment at constant CO2 pressure, the 5MPa temperature is 25C warmer at the end than the 0C 2.5MPa temperature.

 

[Chestermiller]

If the hot side CO2 refrigerant of a heat pump is 2.5MPa from 125C to 0C, I thought the average temperature of the hot heat exchanger can be calculated by the log mean formula, and its phase stays a gas.

But if we have 1kg samples of CO2 at fixed pressure 2.5MPa and 5MPa starting at 125C, and we extract the same 133kJ energy from the 5MPa and 2.5MPa samples to a cold sub-zero C environment at constant CO2 pressure, the 5MPa temperature is 25C warmer at the end than the 0C 2.5MPa temperature.

All this is telling us is that, for the same amount of heat transferred, the 5 MPa case requires less heat transfer area.

 

[Devin-M]

The heat capacities are different along the curves for the two different pressure cases.

So the Lorenz COP of the heat pump with ~125C (126.85C) to ~0C (1.85C) on the hot side is 3.95... That's assuming -48.15C coming out of the expansion valve heated to 1.85C by the ocean water.

Compared to this case:
"If the hot side CO2 refrigerant of a heat pump is 2.5MPa from 125C to 0C, I thought the average temperature of the hot heat exchanger can be calculated by the log mean formula, and its phase stays a gas."

...but with the same mass flow rate, since the 5MPa CO2 has higher heat capacity per kg than the 2.5MPa CO2, couldn't we hook one stirling up to the part of the 5MPa hot side of the heat exchanger that goes from 125C to 25C, and get the same amount of energy out or slightly more because of slightly higher efficiency because the same amount of energy comes out of the the coolant from 125C to 25C at 5MPa as with 125C to 0C at 2.5MPa, but the avg temp of 125C and 25C is hotter than 125C and 0C, so the higher average temp would make that stirling more efficient, because the hot side avg temperature is hotter from 125C to 25C than 125C to 0C, then add a second stirling to the rest of the 5MPa hot side heat exchanger from 25C to 0C (cold side stirlings connected to cold side heat pump)... wouldn't that stirling be less efficient because the the average temp on its hot side is lower, but it doesn't matter because the 125C to 25C hot side stirling was putting out as much if not more power as the single 125C to 0C 2.5MPA hot side stirling on account of the higher heat capacity and higher average hot temp?

 

[Chestermiller]

Did you read what I said about the different heat transfer areas required for the same heat load?

 

[Chestermiller]

So the Lorenz COP of the heat pump with ~125C (126.85C) to ~0C (1.85C) on the hot side is 3.95... That's assuming -48.15C coming out of the expansion valve heated to 1.85C by the ocean water.

Compared to this case:
"If the hot side CO2 refrigerant of a heat pump is 2.5MPa from 125C to 0C, I thought the average temperature of the hot heat exchanger can be calculated by the log mean formula, and its phase stays a gas."

...but with the same mass flow rate, since the 5MPa CO2 has higher heat capacity per kg than the 2.5MPa CO2, couldn't we hook one stirling up to the part of the 5MPa hot side of the heat exchanger that goes from 125C to 25C, and get the same amount of energy out or slightly more because of slightly higher efficiency because the same amount of energy comes out of the the coolant from 125C to 25C at 5MPa as with 125C to 0C at 2.5MPa, but the avg temp of 125C and 25C is hotter than 125C and 0C, so the higher average temp would make that stirling more efficient, because the hot side avg temperature is hotter from 125C to 25C than 125C to 0C, then add a second stirling to the rest of the 5MPa hot side heat exchanger from 25C to 0C (cold side stirlings connected to cold side heat pump)... wouldn't that stirling be less efficient because the the average temp on its hot side is lower, but it doesn't matter because the 125C to 25C hot side stirling was putting out as much if not more power as the single 125C to 0C 2.5MPA hot side stirling on account of the higher heat capacity and higher average hot temp?

View attachment 286339View attachment 286340

Sorry, this is too complicated. Too much for me to deal with.

 

[Devin-M]

Sorry, this is too complicated. Too much for me to deal with.

This is what I meant:

2.5MPa 125C (400K) to 0C (275K) hot side (all gas) - avg temp of hot side is the logarithmic mean temp:

5MPa 125C to 0C hot side (gas to liquid):

For the same mass flow rate gas loses the same amount of heat energy from 125C (400K) to 25C (298K) as the 1st case (from 125C to 0C) through the upper stirling, but the average temp between 125C and 25C is higher than the 1st case of 125C to 0C, which makes the upper stirling more efficient while transferring the same amount of heat as the single stirling in the 1st case, so it should make as much power if not more as compared to the 1st case. The lower stirling then generates additional power from the remaining 25C (298K) to 0C (275K) on the hot side.

 

[Chestermiller]

Let's take a step back to post #36. For the same amount of heat transferred, the temperature decrease for the 2.5 MPa is larger than for the 5 MPa, and the average temperature for the 2.5 MPa is less than the 5 MPa. This means that there is a larger temperature driving force for the 5 MPa than for the 2.5 MPa. For the same heat transferred, mass flow rate, and overall heat transfer coefficient, this tells us that more heat transfer area will be required for the 2.5 MPa cased than for the 5 MPa.

Regarding the comparisons between the LMTD and the actual effective temperature driving force, the LMTD will be about equal to effective temperature driving force for the 2.5 MPa case; but, for the 5 MPa case, the LMTD (based on the initial and final temperature for this case) will be less than the effective temperature driving force for this case.

 

[Chestermiller]

ok so:

1kg, 7MPa, 127C to 0C goes from 560kJ/kg to 200kJ/kg at constant pressure so loses 360kJ to the -10C environment.

1kg, 2.5MPa, 127C to 0C goes from 590kJ/kg to 450kJ/kg at constant pressure so loses 140kJ to the -10C environment.

Since the environment acquires 220kJ more with the 7MPa than the 2.5MPa for the same temperature change, doesn't that make the 7MPa, warmer on average, and wouldn't it mean the 2.5MPa loses it's temperature according to the logarithmic formula since it has no phase change (stays as a gas) while the 7MPa converts from a gas to a liquid?
View attachment 286330

For the same mass flow rate, more heat is transferred in the 7 MPa case than in the 2.5 MPa case, even though the average temperature on the CO2 side of the heat transfer surface is lower for the 7MPa case than the 2.5 MPa case. This is an indication that more heat transfer area will be required for the 7 MPa case both because of the lower heat transfer driving force and because of the larger amount of heat transferred.

 

[Devin-M]

Let's take a step back to post #36. For the same amount of heat transferred, the temperature decrease for the 2.5 MPa is larger than for the 5 MPa, and the average temperature for the 2.5 MPa is less than the 5 MPa. This means that there is a larger temperature driving force for the 5 MPa than for the 2.5 MPa. For the same heat transferred, mass flow rate, and overall heat transfer coefficient, this tells us that more heat transfer area will be required for the 2.5 MPa cased than for the 5 MPa.

Regarding the comparisons between the LMTD and the actual effective temperature driving force, the LMTD will be about equal to effective temperature driving force for the 2.5 MPa case; but, for the 5 MPa case, the LMTD (based on the initial and final temperature for this case) will be less than the effective temperature driving force for this case.

In this picture (5MPa), when the C02 cools from 125C (400K) to 25C (298K), it’s still a gas, but has released the same energy as the 2.5MPa case from 125C all the way to 0C (275k).

Since the same energy has transferred from 125C to 25C & no phase change occurs, if I take the logarithmic mean temperature of 125C and 25C, that’s higher than the logarithmic mean of 125C and 0C.

A higher logarithmic mean means higher stirling efficiency, but the same heat energy transferred for the same mass flow rate, so the upper stirling should produce more mechanical power than the only stirling in in the 2.5MPa case. But the gas has only cooled to 25C, and we can go all the way down to ~0C with the setup, so I added a second stirling.

The 2.5MPa setup already gets 100% mechanical power out as mechanical power in if you multiply the Lorenz COP by the stirling efficiency for the logarithmic mean temps of both sides of that setup.

 

[Chestermiller]

I'm somewhat confused. Can you please explain to me how this device operates, including how the Stirling Engine comes in, and what its working fluid is? It looks to me like the CO2 is going through a standard vapor compression refrigeration cycle.

 

[Devin-M]

I'm somewhat confused. Can you please explain to me how this device operates, including how the Stirling Engine comes in, and what its working fluid is? It looks to me like the CO2 is going through a standard vapor compression refrigeration cycle.

I used equations 2, 3, and 4 in this paper to calculate the heat pump lorenz COP:

https://backend.orbit.dtu.dk/ws/portalfiles/portal/149827036/Contribution_1380_final.pdf

If you calculate the log mean temps of both sides, and use equation 1 to calculate the stirling efficiency, when you multiply the stirling efficiency by the lorenz cop, it’s always exactly 100% as far as I can tell.

My understanding is that describes the 2.5MPa setup with no phase changing heat exchangers.

 

[Chestermiller]

I used equations 2, 3, and 4 in this paper to calculate the heat pump lorenz COP:

https://backend.orbit.dtu.dk/ws/portalfiles/portal/149827036/Contribution_1380_final.pdf

If you calculate the log mean temps of both sides, and use equation 1 to calculate the stirling efficiency, when you multiply the stirling efficiency by the lorenz cop, it’s always exactly 100% as far as I can tell.

My understanding is that describes the 2.5MPa setup with no phase changing heat exchangers.

I need something much more basic than this. I need you to describe in words the operation of the exact system the you are using here.

 

[russ_watters]

I think there's a design issue in the diagram and still a conceptual issue with the entire exercise:

The Carnot efficiency is based on the reservoir temperatures. Lorentz efficiency is based on the working fluid temperatures and is necessarily worse than the Carnot efficiency. Always keep that in mind, because if you are getting a heat pump and Stirling engine efficiency that are exact inverses via the Lorenz efficiency, something is wrong: the Stirling efficiency should be worse. But if you're going through all this effort to define the cycles, it seems to me like you might as well calculate a more "real" efficiency using the heat transfer. If you have the starting and end states you can just calculate the heat transfer.

The cycle you drew is missing the heat rejection from the Stirling engine. CO2 liquid comes in at 225C -- so then what temperature does it leave at? I didn't go back through and look for where you did the calc, but the fact that it doesn't appear in the diagram implies to me that you used 225C as the cold side reservoir, which is the wrong temperature (too cold), giving you way too high of an efficiency on the Stirling engine.

One of the other problems with just using the temperatures is that it can hide another flaw in the cycle: if the heat transfers don't match then you also have a problem with the flow rates not matching. One of the problems with Stirling engines is because of the low delta-t's, you need huge flow rates and heat exchangers to capture a useful amount of heat.

[edit]
Er...wait a minute, why do we even have the water in the cycle? There's a heat leak here. The heats won't sum to zero.

 

[Devin-M]

Energy stored in a flywheel powers the compressor. The compressor heats the 275K gas via the compression to 400K. Heat exits the C02 via the hot side of the stirling engine towards the cold side of the C02 loop. After the expansion valve, the temperature of the CO2 drops from 275K to 225K. The stirling engine heats this side from the heat from the hot side. The water ensures the C02 reaches 275K before it is recompressed.

If we assume the C02 is gas only on both sides, only sensible heat is involved. I used the logarithmic mean temp of the hot side and the logarithmic mean temp of the cold
side to calculate the stirling efficiency. The lorenz COP multiplied by the stirling efficiency gives the same mechanical power out from the stirling as goes into the compressor from the flywheel.

But the part I don’t understand is if we have phase changing in the heat exchangers, on the cold side where the C02 heats from a liquid to gas (absorbs heat from ocean water to 275k), the water is adding latent heat in addition to sensible heat. On the hot side, where the C02 changes phase from gas to liquid, it is discarding latent heat in addition to sensible heat.

What I’m trying to understand is how the latent heat in addition to the sensible heat affects the calculated efficiencies of ideal stirling engines, compared to the case where only sensible heat is involved.

 

[russ_watters]

The basic exercise here should start with the most basic system: A heat pump driving a heat engine, with nothing else in the system -- no external reservoirs. Using Carnot efficiencies you don't even need to define the cycles, you just note that your "reservoir" temperatures are your working fluid temperatures (since you don't have separate reservoirs), plug and chug the math and that shows the efficiencies are inverses (multiply to 1).

Anything you do to add complexity to the system or calculations will decrease the efficiency (they'll multiply to something less than 1) unless there's an error or another source of input not accounted for.

Here you seem to have added an external reservoir for an additional input of heat. That's a 2nd Law violation and worse it's the opposite of what your problem is. The problem with a non-ideal system is that it needs to reject more heat, not that it can provide excess cooling.

 

[russ_watters]

Energy stored in a flywheel powers the compressor.

No, get rid of the flywheel. It is an unnecessary complication and is probably a separate, additional wrong perpetual motion belief you hold. Flywheels are not continuous sources of power, so this hides an additional energy leak in your system.

After the expansion valve, the temperature of the CO2 drops from 275K to 225K. The stirling engine heats this side from the heat from the hot side.

To what temperature does the Stirling engine heat the CO2 and what temperature are you using in your efficiency calc for the Stirling engine cold side?

The water ensures the C02 reaches 275K before it is recompressed.

How do you know you even need a heat addition there? (hint: you don't)

What I’m trying to understand is how the latent heat in addition to the sensible heat affects the calculated efficiencies of ideal stirling engines, compared to the case where only sensible heat is involved.

To be honest, while the exercise @Chestermiller took you through is a nice learning experience, the intricacies of heat transfer should be learned after one has mastered the thermodynamics. I don't think you're ready yet for that level of complexity and it may not even be needed to begin with.

 

[Devin-M]

No, get rid of the flywheel. It is an unnecessary complication and is probably a separate, additional wrong perpetual motion belief you hold. Flywheels are not continuous sources of power, so this hides an additional energy leak in your system.

Constant 1kW mechanical (or 1MW for that matter) in from external power.

To what temperature does the Stirling engine heat the CO2 and what temperature are you using in your efficiency calc for the Stirling engine cold side?

The stirling engine cools the C02 on the hot side of the heat pump. As the hot heat exchanger passes the hot side of the stirling engine, the C02 cools from 400k to 275K. The hot stirling temperature is calculated from the logarithmic mean between 400k and 275k.

On the cold side, the stirling cold temperature is the logarithmic mean between 225K and 275K. 225k is the temperature as the C02 exits the expansion valve. It heats to 275K from the stirling and certainly, the water.

 

[russ_watters]

On the cold side, the stirling cold temperature is the logarithmic mean between 225K and 275K.

Great, then we can get rid of the water bath. And did you calculate and then use that temperature in the efficiency calc?

I don't think you've found the main energy leak yet. It's in the first point I made in the first post. The difference between the logarithmic means at the heat exchangers and the compressor temperatures is where the loss is.

 

[Devin-M]

And did you calculate and then use that temperature in the efficiency calc?

I did.

When you multiply the 3.95… lorenz cop by the .253… stirling efficiency it’s 1 exactly (at the very bottom).

 

[russ_watters]

I did.View attachment 286370

Yeah, so you're using the wrong temperatures to calculate the efficiency for the heat pump. They should be 400 and 225K.

Additional potential problem: these are the temperatures of the heat pump working fluid, not the temperatures of the Sterling engine working fluid (though perhaps we can ignore that and assume it can be made equal with a 100% effective heat exchanger).

 

[Devin-M]

Yeah, so you're using the wrong temperatures to calculate the efficiency for the heat pump. They should be 400 and 225K.

https://backend.orbit.dtu.dk/ws/portalfiles/portal/149827036/Contribution_1380_final.pdf

I used equations 2, 3 and 4 on this paper.

 

[russ_watters]

https://backend.orbit.dtu.dk/ws/portalfiles/portal/149827036/Contribution_1380_final.pdf

I used equations 2, 3 and 4 on this paper.

Those equations don't apply to what you are doing. The source and sink for your heat pump are the temperatures of the Stirling engine working fluid (which you aren't even looking at). You're applying the equation to calculate the mean temperature of the heat pump working fluid from its own inlet and outlet temps. That's just not what those equations are for.

That paper is still using ideal heat exchangers.

Note: I'm not sure I buy what the paper is trying to say; it says you can do better than Carnot efficiency by putting heat pumps in series. That doesn't seem right, but I don't have time to dig in right now. This problem may reduce to a question of whether you can apply the Lorenz efficiency to a heat pump while simultaneously applying a Carnot efficiency to the heat engine in order to violate conservation of energy.

Either way, in order to dig into this further I'd want to dispense with all the unnecessary complications and just use ideal cycles, constant temperatures.

 

[russ_watters]

Note: I'm not sure I buy what the paper is trying to say; it says you can do better than Carnot efficiency by putting heat pumps in series. That doesn't seem right, but I don't have time to dig in right now. This problem may reduce to a question of whether you can apply the Lorenz efficiency to a heat pump while simultaneously applying a Carnot efficiency to the heat engine in order to violate conservation of energy.

Either way, in order to dig into this further I'd want to dispense with all the unnecessary complications and just use ideal cycles, constant temperatures.

Ahh, ok, I've got it: the paper is talking about progressive cooling/heating of a source/sink. For example, if you want to heat your house from 10C to 20C if you first supply it 11C air, then 12C air, then 13C air, etc. you get higher efficiency than if you'd provided 20C air from the start. That's true, but that isn't what we are doing with your system. Your system is operating at steady state.

 

[Devin-M]

Ahh, ok, I've got it: the paper is talking about progressive cooling/heating of a source/sink. For example, if you want to heat your house from 10C to 20C if you first supply it 11C air, then 12C air, then 13C air, etc. you get higher efficiency than if you'd provided 20C air from the start. That's true, but that isn't what we are doing with your system. Your system is operating at steady state.

If that’s true, and you remove the part of the pipe that goes through the water (so the heat pump is a closed loop entirely enclosed in vacuum with no contact with the water), and the stirling outputs less than than the compressor mechanical input (when the system reaches steady operation), where is the energy going that’s not coming out of the stirling engine?

 

[russ_watters]

If that’s true, and you remove the part of the pipe that goes through the water (so the heat pump is a closed loop entirely enclosed in vacuum with no contact with the water), and the stirling outputs less than than the compressor mechanical input (when the system reaches steady operation), where is the energy going that’s not coming out of the stirling engine?

The Stirling engine has two outputs:
1. Heat (rejecting too much)
2. Mechanical power (not enough)

The loss can be in either or both.

Or if it isn't absorbing enough heat in the input, the heat pump loop just gets hotter and hotter.

 

[Devin-M]

Does this snippet have any applicability:

“ Transcritical CO2 refrigeration cycle integrated with mechanical subcooling (MS) cycle operating with zeotropic mixture is proposed in this study, based on the concept of Lorenz cycle. An energetic model is developed and analyses are conducted in detail. A maximum overall coefficient of performance (COP) is achieved at the optimum discharge pressure and optimum subcooling degree. The maximum overall COP, optimum subcooling degree and discharge pressure are closely related to the temperature glide of the mixtures. The energy efficiency of the transcritical CO2 refrigeration cycle can be efficiently improved, and the high pressure can be reduced when mixtures with proper temperature glide are used as MS cycle refrigerant. ”
https://www.researchgate.net/publication/323326860_Energetic_performance_of_transcritical_CO2_refrigeration_cycles_with_mechanical_subcooling_using_zeotropic_mixture_as_refrigerant

 

[russ_watters]

Does this snippet have any applicability:

Kind of. It's an article about picking the optimal temperatures/pressures for your refrigerant to maximize efficiency.

It's an interesting article. The broader issue they are investigating is alternative refrigerants to replace ozone depleting and global warming causing refrigerants. It's a problem in the industry right now because the refrigerants we have work really well and the "environmentally friendly" ones don't. It's saying CO2 is a leading candidate but its operating temperatures/pressures aren't good for normal HVACR needs.

 

[Devin-M]

It says “based on the concept of Lorenz cycle”

“The maximum overall COP, optimum subcooling degree and discharge pressure are closely related to the temperature glide of the mixtures.”

From my readings I had gathered that “temperature glide” means when the temperature in the C02 changes in the condenser. If the C02 was losing heat entirely during the gas to liquid transition (vapor % is changing), it’s temperature would remain constant, so temperature glide is when the temperature doesn’t remain constant across the condenser (ie releasing heat while CO2 is entirely liquid or gas).

From my reading I had gathered that the “lorenz COP” is the COP you get when the C02 does have temperature glide, and the “carnot cop” is the COP you get when the CO2 doesn’t have temperature glide.

For example with 400k to 275k on the hot side and 225k to 275k on the cold side, it’s more efficient to lift the temperature from 275k to 400k (lorenz cop) than it is to lift the temperature from 225k to 400k when there is no “temperature glide” (carnot cop).

 

[Devin-M]

https://www.researchgate.net/figure/Comparison-of-Carnot-and-Lorenz-cycles-respective-to-their-adapted-application-cases_fig2_37413723

 

[Devin-M]

For example with 400k to 275k on the hot side and 225k to 275k on the cold side, it’s more efficient to lift the temperature from 275k to 400k (lorenz cop) than it is to lift the temperature from 225k to 400k when there is no “temperature glide” (carnot cop).

From this paper:
https://backend.orbit.dtu.dk/ws/portalfiles/portal/149827036/Contribution_1380_final.pdf

If I use the same temperatures as circled in the table from the paper, I calculate the same lorenz cop and carnot cop as shown in the paper's table (see above and below)...

... but if the Cold in and Cold Out are both essentially the same temperature (15C - no temperature glide in the evaporator) and the hot in and hot out are the same temperature (90C - no temperature glide in the condenser), then the lorenz cop is the same as the carnot cop, and the carnot cop is the same carnot cop as shown in the paper's table (see below):

 

[russ_watters]

It says “based on the concept of Lorenz cycle”

“The maximum overall COP, optimum subcooling degree and discharge pressure are closely related to the temperature glide of the mixtures.”

From my readings I had gathered that “temperature glide” means when the temperature in the C02 changes in the condenser. If the C02 was losing heat entirely during the gas to liquid transition (vapor % is changing), it’s temperature would remain constant, so temperature glide is when the temperature doesn’t remain constant across the condenser (ie releasing heat while CO2 is entirely liquid or gas).

Almost. For the Lorenz cycle, there isn't just a temperature change in the condenser, there is more than one heat pump/condenser. To put the issue another way: ordinarily you want a constant temperature with a phase transition in each heat exchanger. Due to the poor characteristics of CO2 as a refrigerant, they are mixing in with it a more traditional refrigerant, which results in a varying temperature instead of a constant temperature. This is a bad thing, not a good thing. One way to mitigate this problem is to add additional heat pumps, moving energy at different delta-Ts. You don't have this.

From my reading I had gathered that the “lorenz COP” is the COP you get when the C02 does have temperature glide, and the “carnot cop” is the COP you get when the CO2 doesn’t have temperature glide.

No. The Lorenz COP comes from combining/using multiple refrigerant cycles. It helps reduce the downside of the temperature glide by having separate cycles for the different temperatures. The paper describes a system with two heat pumps working together.

None of this has anything to do with your cycle, though. Your system only has one heat pump. You're taking bits and pieces of papers about cycles that aren't the same as yours and mis-applying them. You really need to go back to the basic principles and analyze your cycle for what it is, according to those principles:

• What is the temperature at the outlet of your compressor? If that temperature needs to be higher, does that help or hurt your efficiency? In other words, does it increase or decrease the compressor input power?
• What is the temperature of the Stirling engine working fluid? If this temperature is lowered, does that help or hurt your efficiency? In other words, does it increase or decrease the power output of the Stirling engine?

The answers to these questions and implications for your idea should be obvious. The problem you are having - not knowing what temperatures to use in the equations - is part of the reason I prefer using energy to calculate efficiency. It's obvious what the input and output states are at the compressor. There's no way to accidentally use the wrong state.

But still; you're attempting to use an equation meant to be applied to an infinite series of heat pumps. Clearly, you don't have that.

P.S. I still have my college thermodynamics book. It doesn't mention Lorenz cycles. I hade to google that. To me, what Lorenz did is more of a mathematical curiosity than a useful/real-world concept. It seems that it is being resurrected - sort of - as a way to help mitigate performance/efficiency problems with new refrigerants.

 

[Devin-M]

The Lorenz COP comes from combining/using multiple refrigerant cycles.

The lorenz cop is the same as the carnot cop when there is no temperature glide...

See below COP Lor and COP Car...

 

[russ_watters]

The lorenz cop is the same as the carnot cop when there is no temperature glide...

Yes...
[note, there were some late edits to my post]

I really think you need to take a step back and define and evaluate the problem you are trying to solve for what it is. I think you are getting lost in the weeds.