Do Carnot and Stirling Engines Have the Same Efficiency?

[FranzDiCoccio]

Hi,
I have some doubts and questions about the above thermodynamic cycles. These questions arise from some statements I find in a couple of textbooks: "Physics, 10th edition" by Cutnell et al [A], and the other is "Fundamentals of Physics" by Halliday and Resnick . Actually, I have the versions of these books for Italian high schools. The quoted sentences are my translations of what I read.

In [A] I read that "a Carnot engine is a reversible thermal engine that operates between two heat reservoirs only".
Also, as a part of Carnot's theorem, "all reversible engines that operate between the same two temperatures have the same efficiency".

Then I thought about the (idealized) Stirling cycle (isothermal, isochoric, isothermal, isochoric) and started wondering what's the difference.
I mean, at first it seemed to me that the (idealized) Stirling engine operates between two heat reservoirs only, like the Carnot engine.
Therefore, if I get the second statement above right, a Stirling engine and a Carnot engine operating between the same two reservoirs should have the same efficiency.

Then I took a look into , which maintains that, in order to be reversible, the isochoric processes needs a variable-temperature reservoir that gradually changes the temperature at constant volume.
Thus one cannot say that there are only two reservoirs.
Also, says that from this it follows that "the efficiency of an idealized Stirling engine is necessarily lower than that of a Carnot engine operating between the same two temperatures".
While I agree with the need of more than two reservoirs, I fail to see the implication that the efficiency is lower.

The statement about the reversibility and the variable-temperature reservoir seemed kind of strange to me, especially after seeing that the "cartoon" representation of the Stirling engine in this wikipedia page does not seem to involve a variable temperature reservoir. The substance is put into thermal contact with one reservoir (e.g. the hottest) and after a while reaches thermal equilibrium at the temperature of the reservoir.Somewhere else I read that "operates between two reservoirs only" does not really mean what it seems to mean, i.e. that there are only two heat reservoirs. It rather means that the heat exchange happens at two temperatures only. Since this means that there are two isothermal and two adiabatic processes, it necessarily implies a Carnot engine. Hence a Stirling neither of the above Stirling engines (reversible and irreversible) operates between two reservoirs (or better between two temperatures).

Can I say the following?

• the reversible version clearly needs more than two reservoirs/temperatures, and hence (?) it has a lower efficiency than the Carnot engine.
• the irreversible version may (or may not) operate between two temperatures only, but it has a lower efficiency than the Carnot engine, simply because it is irreversible.

Sorry if my thoughts are a bit confused. It's a bit late and my English is fading out.
Thanks a lot for your help.
Franz

 

[stockzahn]

Can I say the following?

• the reversible version clearly needs more than two reservoirs/temperatures, and hence (?) it has a lower efficiency than the Carnot engine.
• the irreversible version may (or may not) operate between two temperatures only, but it has a lower efficiency than the Carnot engine, simply because it is irreversible.

The Carnot cycle and the Stirling cycle should have the same efficiency. Let's assume the two processes numbered from 1 to 4, where 1 represents the point at low temperature and low entropy. Since for an ideal gas with an isochoric change of state the entropy difference can be calculated with $\Delta s = c_v ln\frac{T_2}{T_1}$ the heat transferred from 1 → 2 and 3 → 4 are identical. This is the heat stored and released by the regenerator periodically during operation. What's left is the heat supply from 2 → 3 and the heat discharge from 4 → 1, which occur isothermally and therefore can be treated as in the Carnot cycle. The efficiency of the Carnot and the Stirling cycle therefore are the same.

Regarding your second statement, I'd agree. Irreversible processes have lower efficiency than reversible ones.

 

[Chestermiller]

The Carnot cycle and the Stirling cycle should have the same efficiency. Let's assume the two processes numbered from 1 to 4, where 1 represents the point at low temperature and low entropy. Since for an ideal gas with an isochoric change of state the entropy difference can be calculated with $\Delta s = c_v ln\frac{T_2}{T_1}$ the heat transferred from 1 → 2 and 3 → 4 are identical. This is the heat stored and released by the regenerator periodically during operation. What's left is the heat supply from 2 → 3 and the heat discharge from 4 → 1, which occur isothermally and therefore can be treated as in the Carnot cycle. The efficiency of the Carnot and the Stirling cycle therefore are the same.

Regarding your second statement, I'd agree. Irreversible processes have lower efficiency than reversible ones.

I have a little different perspective on this. If the efficiency is defined as work done divided by total amount of heat added (during all heat addition portions of the cycle), then since, heat is being added from 1 to 2, the efficiency on this basis is less than that of the Carnot cycle.

 

[stockzahn]

I have a little different perspective on this. If the efficiency is defined as work done divided by total amount of heat added (during all heat addition portions of the cycle), then since, heat is being added from 1 to 2, the efficiency on this basis is less than that of the Carnot cycle.

So you mean $\eta = \frac{q_{12}+q_{23}-\left(q_{34}+q_{41}\right)}{q_{12}+q_{23}} = 1-\frac{q_{34}+q_{41}}{q_{12}+q_{23}}$ compared to $\eta = \frac{q_{23}-q_{41}}{q_{23}} = 1-\frac{q_{41}}{q_{23}}$ ?

 

[Chestermiller]

So you mean $\eta = \frac{q_{12}+q_{23}-\left(q_{34}+q_{41}\right)}{q_{12}+q_{23}} = 1-\frac{q_{34}+q_{41}}{q_{12}+q_{23}}$ compared to $\eta = \frac{q_{23}-q_{41}}{q_{23}} = 1-\frac{q_{41}}{q_{23}}$ ?

Yes. But I was thinking of it more like $$\eta=\frac{q_{23}-q_{41}}{q_{23}+q_{12}}$$

 

[stockzahn]

Yes. But I was thinking of it more like $$\eta=\frac{q_{23}-q_{41}}{q_{23}+q_{12}}$$

Of course, a much more elegant expression. But they should be identical, since in an ideal gas $q_{12} = q_{34}$. However, so the difference between the two definitions for the efficiency is whether the heat transferred from and to the regenerator respectively is taken into account in the energy balance. Isn't that then a question of the definition of the system's boundary? For a system defined as Stirling machine, the regenerator heat stays inside. If only the processed gas is defined as system, the regenerator heat enters and exits periodically. Is it valid to say: The efficiency of an ideal Stirling process is lower than the efficiency of an ideal Stirling machine?

 

[Chestermiller]

Of course, a much more elegant expression. But they should be identical, since in an ideal gas $q_{12} = q_{34}$. However, so the difference between the two definitions for the efficiency is whether the heat transferred from and to the regenerator respectively is taken into account in the energy balance. Isn't that then a question of the definition of the system's boundary? For a system defined as Stirling machine, the regenerator heat stays inside. If only the processed gas is defined as system, the regenerator heat enters and exits periodically. Is it valid to say: The efficiency of an ideal Stirling process is lower than the efficiency of an ideal Stirling machine?

As an engineer, the exact definition of the efficiency for a specific process is not too important to me as long as I know which one is being used. Maybe to a physicist, the distinction is more relevant (for whatever reason).

 

[gary350]

There are several types of engines and designs. Alpha engine has 2 separate cylinders, 2 pistons, 2 rods, 1 crank shaft. Beta engine has 2 pistons inside 1 cylinder with 2 rods and 1 crankshaft. Some engine have a connecting rod from the crank shaft to both pistons while other engines have 1 rod connected to 1 piston while the other piston floats with no rod connecting it to the crankshaft. There is a whole assortment of different designs. Some have re generators and some don't. Some re generators work better than others. One thing all engines have in common is, if the hot & cold end swap places the engine runs in the opposite direction. An engine that runs on heat can also run on ice. If air is replaced with different types of gas the engines run different with more or less power. Engine can be powered by the top of the piston or the bottom of the piston. If you put any of these engines inside of an air compressor tank then increase the air pressure inside the tank by 10 atmospheres the power of that engine increases 10 times. If you add more heat the engine produces more power and runs faster. There is a video that shows an Alpha engine with 2000 psi crank case pressure that produced 65 hp they put it in a small car it has enough power to burn the rubber off the tires. I am an engineer I have my own machine shop at home I can build just about anything. Here are youtube videos of 4 of my beta engines. Beta engines all have 2 pistons inside 1 cylinder. The power piston does not have rings like a car it has small groves that create a pressure drop at each groove that act like friction less piston rings. 30 years ago I knew the name of these rings grooves someone on this forum probably knows what they are called, I forgot?

 

[Philip Koeck]

Hi,
In [A] I read that "a Carnot engine is a reversible thermal engine that operates between two heat reservoirs only".
Also, as a part of Carnot's theorem, "all reversible engines that operate between the same two temperatures have the same efficiency".

Then I took a look into , which maintains that, in order to be reversible, the isochoric processes needs a variable-temperature reservoir that gradually changes the temperature at constant volume.
Thus one cannot say that there are only two reservoirs.
Also, says that from this it follows that "the efficiency of an idealized Stirling engine is necessarily lower than that of a Carnot engine operating between the same two temperatures".
While I agree with the need of more than two reservoirs, I fail to see the implication that the efficiency is lower.

Can I say the following?

• the reversible version clearly needs more than two reservoirs/temperatures, and hence (?) it has a lower efficiency than the Carnot engine.

You are definitely onto something here. The (bachelors level) textbook version of a Stirling engine doesn't usually include a regenerator so that means that all heat exchange is with the surroundings, even during the isochoric processes. That means that you have to include the heat going into the system during the isochoric temperature (and pressure) increase in the calculation of the efficiency. So efficiency becomes (work / (isothermal heat in + isochoric heat in)).
This efficiency (even for the reversible Stirling) will always be lower than the corresponding Carnot efficiency. It will also depend on the ratio of the volumes during the two isochoric processes. Try to derive the expression for the Stirling efficiency in terms of the isothermal temperatures and the isochoric volumes and put in some reasonable numbers. If the ratio of volumes goes to infinite you should get the Carnot efficiency (which depends only on the temperatures). All these calculations are for reversible engines. Shout if you need help!

Regenerators have been mentioned a few times in this thread. A regenerator is simply a lump of metal that absorbs the heat leaving the working substance (the gas in the cylinder) during the isochoric cooling and gives it back to the working substance during the isochoric temperature increase. This amount of heat is the same (as has been pointed out). So an ideal Stirling engine with a regenerator only receives heat from the surroundings during the isothermal expansion and gives off heat to the surroundings during the isothermal compression. It therefore has the same efficiency as a Carnot engine.
This is not surprising since in a way a Stirling engine with a regenerator is just another version of a Carnot engine. You can see that if you simply include the regenerator into the system. Then the two isochors become adiabats (sort of) since the system (the gas and regenerator together) doesn't give off heat during these two processes. This is what stockzahn talks about in his latest post.

In summary: Without regenerator I agree with Chestermiller, with regenerator I agree with stockzahn