The Carnot efficiency problem

[Tom Booth]

I will try to describe my contention with so-called "Carnot efficiency" in a nutshell.

Looking at this diagram:

ezgif.com-resize.gif (7.18 KiB) Viewed 20421 times

It appears simple.

Some heat is supplied to the engine, some is converted to work and the remainder passes through to the sink.

Supposing I have an engine that is actually 100% efficient. That is, if I supply 10,000 joules of heat it will convert that to work with no heat left over.

To make things simple, let's call ambient zero degrees. And let's say that the added heat raises the temperature of the hot side 100 degrees.

If we plug that into the efficiency formula

Wmax = Qhigh(1 - Tlow/Thigh
Work = 100(1-0/100)
W = 100(1-0)
W = 100x1
Work output = 100%

We get 100% efficiency, which is factually correct. All the heat supplied was converted to work with no heat left over to transfer to the sink.

When we apply the same formula using the kelvin scale what do we get?

Wmax = Qhigh(1 - Tlow/Thigh)
Work = 400°K(1 - 300°k/400°k)
W = 400(1 - 0.75)
W = 400 x 0.25
Work output = 100

Well, that appears to work out.

Now let's try the Carnot efficiency formula

Efficiency = 1 - Tc/Th
e = 1 - 300°k/400°k
e = 1 - 0.75
e = 0.25
Efficiency = 25%

What happened????

[Tom Booth]

How did we actually come up with 25% "Carnot efficiency"???

Well, if we take the high temperature on the kelvin scale (BTW, 300°k was used as the sink temperature just because it approximates ambient in my temperate climate and it is a good round number making calculations easy)

Which is 400°k (100°k above ambient) the temperature difference (100°k) is 25% of that.

100 goes into 400 four times.
100 is 25% of 400.

Coincidence?

No.

The Carnot limit is supposed to be derived from Carnot's idea that the temperature difference is what determines efficiency, and the ONLY thing that determines efficiency. Though I can find nowhere where Carnot actually said that. This was actually a later interpretation and simplification. If we actually go back to Carnot's book he said something much more sensible.

[Tom Booth]

Here is Carnot's actual statement:

The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the height of its fall, that is to say, the difference of temperature of the bodies between which the exchange of caloric is made.

So Carnot was not unreasonable, the "motive power" or work output, i.e. efficiency depends on both the temperature difference AND the QUANTITY of "caloric".

Just like electrical power depends upon amps and volts water power depends on pressure and volume etc.

So, where is the actual QUANTITY of heat represented in the SO-CALLED Carnot formula?

Well, it isn't. Is it.

How, why, when, where and by whom did this essential element in Carnot's hypothesis get dropped so that we ended up with this over-simplification where only the temperature difference is taken into consideration with no consideration for the actual quantity of heat involved??

Is this a legitimate formula that includes no reference whatsoever to the actual quantity of heat?

Efficiency = 1 - Tc/Th

Suppose we have a "pancake" engine with a large surface area and low ∆T

We compare that with a small high temperature engine with a very small heat exchange area.

The quantity of heat the pancake engine takes in could actually be greater due to the larger surface area, could it not?

So how can efficiency be determined by reference to temperature difference alone?

[VincentG]

Tom, don't we assume a perfect engine for Carnot? So arguing heat transfer comparisons is moot.

Rather than prove Carnot wrong, why not try to prove yourself right?

[Tom Booth]

Tom, don't we assume a perfect engine for Carnot? So arguing heat transfer comparisons is moot.

The carnot formula according to how it is interpreted and applied states that the ratio derived from the temperature difference alone is also applicable to the quantity of heat that can be converted into work.

In the above example that works out to 25%

At best 25% of the heat supplied to the engine can be converted to work the other 75% passes through to the sink.

This is a "perfect engine"?

What is the basis for this? Well, the distance in "height" between 300° and 400° kelvin is 25% of the temperature scale starting at zero all the way up to 400.

And that applies to the supplied number of joules how exactly?

Rather than prove Carnot wrong, why not try to prove yourself right?

Prove myself right about what? That the Carnot limit has no sound basis in reality?

Let's examine a typical textbook example:

Example 4.2

The Carnot Engine
A Carnot engine has an efficiency of 0.60 and the temperature of its cold reservoir is 300 K.

(a) What is the temperature of the hot reservoir?
(b) If the engine does 300 J of work per cycle, how much heat is removed from the high-temperature reservoir per cycle?
(c) How much heat is exhausted to the low-temperature reservoir per cycle?

Strategy
From the temperature dependence of the thermal efficiency of the Carnot engine, we can find the temperature of the hot reservoir. Then, from the definition of the efficiency, we can find the heat removed when the work done by the engine is given. Finally, energy conservation will lead to how much heat must be dumped to the cold reservoir.

Solution

a. From e = 1−Tc/Th we have

0.60 = 1−300K /Th
,
so that the temperature of the hot reservoir is

Th = 300K /(1− 0.60) = 750K
.
b. By definition, the efficiency of the engine is
e = W/Q

So that the heat removed from the high-temperature reservoir per cycle is

Qh = w/e = 300J/0.60 = 500J

c. From the first law, the heat exhausted to the low-temperature reservoir per cycle by the engine is

Qc = Qh - W = 500J -300J = 200J

Significance
A Carnot engine has the maximum possible efficiency of converting heat into work between two reservoirs, but this does not necessarily mean it is 100% efficient. As the difference in temperatures of the hot and cold reservoir increases, the efficiency of a Carnot engine increases.

https://pressbooks.online.ucf.edu/phy20 ... not-cycle/

I've added some emphasis.

In particular "per cycle"

The implication is that the engine cannot complete a single revolution without rejecting heat to the sink.

What is also apparent is that supposedly the minimum heat to be rejected to the sink per cycle can be easily determined by plugging some known values into the Carnot (so-called) formula.

I say so-called "Carnot" limit not to disparage Carnot but to obsolve him from blame. I can find no evidence of Carnot's advocacy of this formula. It apparently did not originate with Carnot at all.

Bottom line is we do have a means of testing, verifying or refuting this formula by simply calculating the "Carnot efficiency" based on known information. The temperature difference and the heat input in joules, and running the engine and taking some actual measurements using thermocouples and whatnot.

According to the application of this formula, as in the above example, given an equivalent input in joules, the less efficient the engine (the lower the temperature difference) the more heat (a greater percentage of the input heat) we should be able to detect being rejected at the sink.

In no event should we be able to detect LESS heat being rejected than whatever is predicted by the formula as supposedly:

"A Carnot engine has the maximum possible efficiency of converting heat into work between two reservoirs"

Therefore whatever engine we use, we should be able to detect, if anything MORE "waste heat" than what is calculated by the formula for the supposedly "perfect" Carnot engine.

In other words, if 85 joules/second enter into the engine from boiling water by a steam generator with a sink at 300k we can easily determine the minimum joules of heat that should be measured arriving at (or being rejected to) the sink.

[stephenz]

Tom, this formula doesn't work with any scale but the Kelvin scale.
These singularities are not unusual in physics and almost always derivated from the math.

[Tom Booth]

Tom, this formula doesn't work with any scale but the Kelvin scale.
These singularities are not unusual in physics and almost always derivated from the math.

I'm not really sure what you mean by "singularities".

With a certain distance on a scale, like a ruler, a measuring stick, we can have a scale model drawn to size. Say for example, an inch on a map represents a mile in the real world.

The Carnot limit seems to have, or in actuality does, apparently have a similar derivation.

If the "distance" on the Kelvin scale between T hot and T cold is n% of the kelvin scale between 0°K and T hot, then the Carnot limit postulate is that for every joule of heat supplied to any engine whatsoever in the real world at that temperature difference, only the same n% of that supplied heat can be transformed into work.

If true, this exact scale ratio relationship between the position of the ∆T on the temperature scale to REAL heat to work conversion capacity seems quite remarkable and must have, or should have been demonstrated to have some actual physical basis.

It is claimed that heat is "different" than other forms of energy. All other forms of energy can be converted one into the other, but for reasons unknown heat conversions are supposedly limited by this mysterious magical limiting scale ratio. Seems a thing like that should be supported by some actual experimental observation or proof, not just declared on the basis of some ratio because somebody 200 years ago thought it was a good idea.

I'm told, "entropy". Or perhaps, the proof is; "no one has yet built a perpetual motion machine, though many have tried, no one has ever succeeded." Or "It would be a violation of the conservation of energy!"

I don't see how these conclusions are reached.

If I boil water by supplying 200,000 joules of heat from an electric stove burner, how is it a violation of conservation of energy, or an example of "perpetual motion" if my LTD Stirling engine manages to convert 41,000 joules of that supplied heat into work, rather than the 40,000 determined allowable by the Carnot limit ratio calculation?

Why do 160,000 joules of my supplied heat that I created by boiling a pot of water need to be "rejected", wasted, to the sink of my Stirling engine to avoid a "violation of conservation of energy"? How is maybe making a timing adjustment, reducing friction or implementing some other improvement that reduces the amount of waste heat to 159,000 joules rather than 160,000 a demonstration of "perpetual motion".

Well, so I am told that no one has ever exceeded the Carnot limit.

How would anyone ever know?

If I say that I have a paper grocery bag that exceeds the Carnot limit how can that be tested?

Well, I have a pot of water boiling at 100°C under the bag and some ice at 0°C on top of the bag.

The Carnot limit of the paper bag is therefore 26.8%

The paper bag therefore has a maximum efficiency of 26.8%

It has been proven mathematically by the Carnot (so-called) formula that the maximum heat conversion efficiency of my paper bag engine is less than the efficiency of a Carnot engine.

Suppose a UFO crashes and we recover the engine that was developed by an advanced alien civilization?

This alien space craft engine operates on the heat of boiling water. How do we determine if it exceeds the Carnot limit or not?

Well, boiling water is 100°c (373.15k) and the ambient temperature is 20°c (293.15k) therefore the Carnot efficiency of this advanced alien technology is 21.44%

How do we know this for a fact?

For every 500,000 joules of heat supplied to the alien space craft engine, it must exhaust or waste 392,800 joules as only 21.44% or the heat from the boiling water can actually be converted into useable work. This must be true because even alien technology cannot violate a "law of the universe".

I feel justified in asking, when, where and by whom was this so-called "Carnot limit" verified experimentally or tested experimentally in any way?

No? It hasn't been tested? No, never?, Not by anybody? Why not?

Well we cannot compare a real engine with the efficiency of a REAL Carnot engine.

Why not?

Because it is impossible to build a real Carnot engine, it is a "perfect engine" it is impossible for any engine to exceed the efficiency of a Carnot engine.

There is no possibility of resolution. The Carnot limit does not rise to the level of Popper's requirement for any scientific theory. It is not "falsifiable". It cannot be subject to test or verification because no "perfect" Carnot engine is available for comparison.

We do however have many imperfect engines and we can apply the formula and calculate the minimum joules that must be transfered to the sink on the cold side of our imperfect engine which must be MORE not less than a Carnot engine, and see how our imperfect real engine compares with our calculations for the "perfect" Carnot engine.

Since the waste heat from our imperfect engine must be more than for a "perfect" Carnot engine, this excessive waste heat should be rather easy to detect and quantify.

We cannot compare this with a so-called Carnot engine, but we do have the numbers which our "imperfect" Stirling engine allegedly, should not be able to outperform.

An imperfect engine should eject MORE waste heat than the theoretically perfect Carnot engine. Never less

We are not talking about no heat at all, or some quantity of heat too small to measure. The Carnot formula calculations demand that there be an enormous amount of waste heat from even the "perfect" carnot engine.

Any of our imperfect engines should eject more waste heat than what is calculated for a Carnot engine.

At the typical ∆T we run our little LTD Stirling engines (on a cup of hot water), a "perfect" Carnot engine would have an efficiency of about 20% meaning 80% of the heat from our cup of hot water would arrive at the cold side of a "perfect" Carnot engine.

If the Carnot formula is any kind of true and accurate representation of a real limit on heat conversion to work, our little LTD (or any other heat engine) should have more, not less "waste heat" arriving at the sink.

[Tom Booth]

I can see the logic of the efficiency limit based on the ∆T on the absolute scale, I think.

Considering that cold, generally speaking is relative. 300°k is not the absence of all heat relative to 400°k

What we have really is two heat sources one hotter than the other.

So if we think about this as a kind of tug of war, instead of degrees in temperature we have men pulling on a rope over a center line.

Polish_20230715_002404814.jpg (407.23 KiB) Viewed 20128 times

If the force on either side is the same, the 600 men pulling on the rope just cancel each other out.

So if we heat out working fluid in the engine to 400°k we do not have 400 units of work output because that is working against the existing 300°k atmospheric heat on the other side of the piston, so our "efficiency" cannot be greater than 25%

But is this actually valid reasoning?

The balance of 300°k ambient on both sides already exists. That balance of heat is a given.

If we add 300 men to one side of the rope we have the pulling force of all three hundred men, not just 25 men.

In the same way if we tip the balance of heat in our heat engine by adding 500,000 joules, we have the work potential of all 500,000 joules that we used to change the balance. Whatever exists below that, if it is in balance and cancels, it's irrelevant.

So I think it is perfectly valid to consider ambient as zero degrees in our calculations because in reality it has already canceled itself out.

And for a normal efficiency calculation it works out perfectly. It is only when using the Carnot formula that there is a discrepancy.

[MikeB]

Tom,
The problem with treating ambient as being equivalent to zero, is 'pressure' doesn't agree with that at all - gas pressure at 300K is 75% of that at 400K (for the same volume) ...

[Tom Booth]

Tom,
The problem with treating ambient as being equivalent to zero, is 'pressure' doesn't agree with that at all - gas pressure at 300K is 75% of that at 400K (for the same volume) ...

What matters is pressure differential.

300 equally strong men on each side of a tug of war cancel each other out.

If I balance a scale with a ten pound weight on each side. Adding one additional pound to one side will read one pound on the scale the same as if there were zero pounds or 20 pounds on each side of the balance at the start.

Ambient (atmospheric temperature and pressure) is the starting point on either side of the piston.

[Tom Booth]

The principle of ballance of pressure is what allows a Stirling engine to operate in the first place.

Although the pressure in the engine rises and falls continuously this has little to no effect on the displacer because the pressure above and below the displacer is always in balance and so the pressure on the displacer is effectively zero so that it can be moved up and down or side to side, with negligible work input.

[Tom Booth]

At any rate, the actual known experimental measurements taken from my LTD Stirling engine indicate an efficiency of:

99.62%

Granted, the experimental setup was less than perfect but:

The heat input was certainly not well balanced in terms of the potential work output of the little model engine. That is, heat input of 85 watts from continuously boiling water was likely an excessive amount of heat.

We don't know with any high degree of certainty the actual heat intake to the engine, heat and work output, but, from the available information at our disposal so far.

99.62%

We have to in some way account for a great deal more "waste heat" escaping to or leaving the sink, and/or determine why, or in what way the supplied heat is failing to pass through and into the engine.

Certainly my experiments are flawed in a number of ways and we should endeavor to make corrections, but to just assume that a discrepancy of such magnitude is irrelevant or can be flippantly dismissed or explained away without some better experiment having ever been conducted would be rather presumptuous IMO.

Until some further better, testing and/or experiments can be conducted, I take this as the best experimental evidence we have to go by.

The actual measured efficiency of a more or less typical LTD Stirling engine looks like around

99.62%

Perhaps the efficiency could be improved with better load balancing.

[VincentG]

Tom, I wonder if you did a control test with a stationary engine, and perhaps one with just a block of Styrofoam.

Can you share how you are measuring power output to calculate efficiency?

That efficiency implies that the engine is generating 80 watts of power. Hard to believe.

[Tom Booth]

Tom, I wonder if you did a control test with a stationary engine, and perhaps one with just a block of Styrofoam.

Can you share how you are measuring power output to calculate efficiency?

That efficiency implies that the engine is generating 80 watts of power. Hard to believe.

For the purposes of "Carnot efficiency" any heat not "rejected" to the sink is generally considered as going towards "work".

In that sense it is not really a matter of power output but the general work involved on the part of the working fluid to expand and drive the engine to keep it running.

At least in terms of any possibility of running an engine with a "cold hole" on ambient heat, the critical consideration is after heat is converted to work or any other form of energy including sound vibration, friction and so forth how much heat is actually left to be dumped into the sink.

With these small Stirling engines running with no load, power output in terms of "useful work" is, well, if there is any potential at all for power output of that sort, to light lights or generate electricity or something, in most cases, that's not being recorded or even much considered and is really irrelevant.

If the engine is using up however many joules of heat to keep itself turning, the heat is still being used in a way that does not result in its deposit at the sink.

In other words, if I'm running a Stirling engine on ambient heat by using ice, or possibly cold water, how much heat is actually passing through the engine to the water or ice to warm it up and is that really absolutely necessary?

I did do several experiments with "controls" just running one engine on ice and leaving another (actually the same engine at a different time) engine idle on the ice, to see if the ice under the running engine melted at the same rate, faster or more slowly than ice under an engine just sitting still.

The outcome was that the ice used to run the engine took significantly longer to melt by about 15% as I recall.

I'm thinking now that more meaningful results could be obtained more quickly using cold water.

[Tom Booth]

Anyway that 99.62% number was based on stephenz's calculations:

viewtopic.php?f=1&t=5547&start=135#p19721

Why might this be significant?

Well, I've run engines under styrofoam, silica aerogel blankets etc. And I've been told repeatedly that these insulations are no better insulating material than air and so it makes no difference.

In this case the aluminium plate is not insulated by any potential conductive material that might carry heat away better than air.

Probably a water bath sink would generate more meaningful numbers as it has more heat capacity than aluminium and there is an accepted standard method for measuring joules of heat used to raise the temperature of water.