# Origin and verification of e=(Th-Tc)/Th

The point is, the "Law" only applies to cycles, though it is recognized that heat can indeed be converted into work 100% in a "process", such as expanding gas in a cylinder to drive a piston.

The further out the piston is driven the more the pressure drops, as you point out: "until the gas pressure asymptotically approaches zero."

A Stirling engine operates by heating and expanding a gas which raises the internal pressure relative to the external (atmospheric) pressure. The engine ( in the real world) is not operating in a vacuum.

So what happens when the gas does work drives out the piston to let's say double the volume.

Before adding heat to the working fluid the internal and external pressure were in equilibrium. As the gas expands and the heat is converted to work (100%) the internal pressure falls.

According to the ideal gas law, at double the volume, the pressure will be halved (in an isothermal process) so we end up with 0.5 atmosphere internal pressure as the engines piston approaches "bottom dead center" or the full extent of it's expansion stroke.

But at about 3/4 into the expansion, in a Stirling engine the heat input is cut off so the final portion of the stroke is completed without heat input. Momentum carries the piston the final distance. What happens to the pressure and temperature as the gas continues to expand and do additional work adiabatically?

The pressure already dropped down to 0.5 atmospheres.

Additional expansion and adiabatic cooling aside, what volume do we have to return the gas to in order to restore the internal pressure so that it is balanced by the external pressure of 1 atmosphere?

According to the ideal gas law, the volume would need to be reduced to 1/2.

At some point during expansion the external atmospheric pressure is going to overwhelm the falling internal pressure, and with the heat input having been cut off what will happen when the piston reaches bottom dead center?

Atmospheric pressure will drive the piston back, will it not?

In an internal combustion engine we need a flywheel to carry the engine through the compression stroke, but in a Stirling engine this is not necessary. The piston returns by atmospheric pressure.

So, if all the heat has already been utilized during the expansion stroke with 100% efficiency, and the cycle completes by external atmospheric pressure, returning the system to equilibrium or its starting condition, then how is it possible for the "Carnot Limit" to be valid which says the conversion efficiency is limited to say 25 maybe 30% or whatever.

The heat was already fully utilized 100% during the expansion stroke and from there the cycle is competed by atmospheric pressure.

How do we retroactively deduct 75% of the heat input reserved for the "cold reservoir" when it is already gone?
"The cycle completes by atmospheric pressure" means the atmosphere does work on the gas to recompress it. This heats the gas and that heat is wasted. So in the power stroke some of the work done is in effect "lent" to the surroundings and comes back as waste heat in the second part of the cycle. There is no getting away from that: PV work is either done by or done on a gas, depending on whether it is expanding or contracting. See description here: https://chem.libretexts.org/Bookshe...rmodynamics/Thermodynamic_Cycles/Carnot_Cycle

You still owe me a reply on how you are measuring the heat input and the work output, in order to work out the efficiency.

"The cycle completes by atmospheric pressure" means the atmosphere does work on the gas to recompress it. This heats the gas and that heat is wasted. ...
Not wasted at all.

Atmospheric pressure imparts velocity to the piston which converts back to heat at TDC. As with a diesel engine. Basic engine dynamics. The heat generated by compression results in another expansion and contributes power to the next power stroke, where it is converted 100% into work output along with additional added heat. Nothing is "wasted". (Lost or sent or transfered to some mythical "cold reservoir").

Not wasted at all.

Atmospheric pressure imparts velocity to the piston which converts back to heat at TDC. As with a diesel engine. Basic engine dynamics. The heat generated by compression results in another expansion and contributes power to the next power stroke, where it is converted 100% into work output along with additional added heat. Nothing is "wasted". (Lost or sent or transfered to some mythical "cold reservoir").
How are you measuring the heat input and the work output, Tom?

How are you measuring the heat input and the work output, Tom?

So far, I've been mostly using small models with no external load. So the work being done by the "working fluid" is just whatever is necessary to overcome friction at various points: piston/cylinder, bearings, air resistance at the flywheel, energy to generate vibration, noise, work against atmospheric pressure etc. etc.

Impossible, or at least impractical and too expensive and complicated to keep track of and measure all those small outputs.

An alternative is to measure the "waste heat". The unconverted heat left over, as already cited previously.

"This same result can be gained by measuring the waste heat of the engine. For example, if 200 J is put into the engine, and observe 120 J of waste heat, then 80 J of work must have been done, giving 40% efficiency."

My interest is in the feasibility of Tesla's theory or proposal that a heat engine should not pass ANY "waste heat" to a sink if 100% efficient.

So it is simply a matter of measuring the temperature increase at the cold "sink" side of the engine.

If there is no temperature increase above ambient at the "sink", then there can be no heat transfer out of the engine. Not in the form of heat anyway.

In some of my experiments the instrumentation indicated a temperature fall or apparent refrigerating effect. The cold (ambient) side of the engine appeared to get slightly colder (by 1 or 2 degrees) and stay cold (below ambient) through hours and hours of operation with the engine running on hot water or steam.

This does not seem that remarkable. Refrigeration can be heat driven. There is no question of "free energy" as there is energy input a plenty in the form of heat.

In numerous tests, operating the engines on ice, the ice always lasts longer, (takes longer to melt) when used to run a Stirling engine. The running engine is, apparently, in some way reducing the amount of heat reaching the ice.

It would, of course be nice if I could buy or build bigger, power producing engines for experimenting with, but my resources are limited.

One small quality model Stirling engine can easily run $100 or more. All at my own time and expense. I'm continuing to do what I can. I've already covered that. So far, I've been mostly using small models with no external load. So the work being done by the "working fluid" is just whatever is necessary to overcome friction at various points: piston/cylinder, bearings, air resistance at the flywheel, energy to generate vibration, noise, work against atmospheric pressure etc. etc. Impossible, or at least impractical and too expensive and complicated to keep track of and measure all those small outputs. An alternative is to measure the "waste heat". The unconverted heat left over, as already cited previously. "This same result can be gained by measuring the waste heat of the engine. For example, if 200 J is put into the engine, and observe 120 J of waste heat, then 80 J of work must have been done, giving 40% efficiency." My interest is in the feasibility of Tesla's theory or proposal that a heat engine should not pass ANY "waste heat" to a sink if 100% efficient. So it is simply a matter of measuring the temperature increase at the cold "sink" side of the engine. If there is no temperature increase above ambient at the "sink", then there can be no heat transfer out of the engine. Not in the form of heat anyway. In some of my experiments the instrumentation indicated a temperature fall or apparent refrigerating effect. The cold (ambient) side of the engine appeared to get slightly colder (by 1 or 2 degrees) and stay cold (below ambient) through hours and hours of operation with the engine running on hot water or steam. This does not seem that remarkable. Refrigeration can be heat driven. There is no question of "free energy" as there is energy input a plenty in the form of heat. In numerous tests, operating the engines on ice, the ice always lasts longer, (takes longer to melt) when used to run a Stirling engine. The running engine is, apparently, in some way reducing the amount of heat reaching the ice. It would, of course be nice if I could buy or build bigger, power producing engines for experimenting with, but my resources are limited. One small quality model Stirling engine can easily run$100 or more. All at my own time and expense.

I'm continuing to do what I can.
So you are not measuring the work output, nor are you measuring the waste heat output. And since you say nothing about it, my guess is you are not measuring the heat input either. Correct?

You are measuring none of the quantities needed to determine the efficiency of your toy Stirling engine. Instead, you base your challenge to the 2nd Law of Thermodynamics on your failure to detect a temperature increase in the cold sink of your machine. And you really think that is good enough. That is almost funny.

Think about it. If the machine is running on zero load, i.e. just overcoming friction, the amount of work being done is tiny. An ouput of 0.1 Joules/sec, i.e. 0.1Watts, would be able to lift a 10g weight through 1 metre every second. What you have is most likely no more than that. So the heat input and the waste heat output will also be very small. If we assume 10% efficiency, say 1W input and 0.9W waste heat output. Do you really think your measuring system is sensitive enough to pick up a heat flow of <1W?

Get real.

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First of all, this thread is not about my experiments, but what experiments have been conducted previously to verify the accuracy of the "Carnot efficiency limit formula". And in general, the origins and methods for validating the same.

That is, who originated the formula. Who decided on the current interpretation of this temperature ratio. What is the actual basis for such an interpretation. What experiments were performed, by who, when and where?

You can criticize my experiments all you like but that is really a diversion.

If I had answers to the above questions 10 years ago, I never would have felt compelled to do any experiments myself at all.

It is the apparent, complete vacuum of information regarding the origin and history of this "efficiency limit" formula and in particular it's modern interpretation that led me to attempt some amature science experiments of my own.

You don't like my experiments so show me the references to the previous, presumably better, more conclusive experiments by competent scientists.

That would save me a lot of time, trouble and needless expense trying to reinvent the wheel

First of all, this thread is not about my experiments, but what experiments have been conducted previously to verify the accuracy of the "Carnot efficiency limit formula". And in general, the origins and methods for validating the same.

That is, who originated the formula. Who decided on the current interpretation of this temperature ratio. What is the actual basis for such an interpretation. What experiments were performed, by who, when and where?

You can criticize my experiments all you like but that is really a diversion.

If I had answers to the above questions 10 years ago, I never would have felt compelled to do any experiments myself at all.

It is the apparent, complete vacuum of information regarding the origin and history of this "efficiency limit" formula and in particular it's modern interpretation that led me to attempt some amature science experiments of my own.

You don't like my experiments so show me the references to the previous, presumably better, more conclusive experiments by competent scientists.

That would save me a lot of time, trouble and needless expense trying to reinvent the wheel
Ah but this thread is about your experiments. I refer you to your own posts 24, 35, 36 and 40. You even posted a video.

You think the Carnot efficiency limit and the 2nd Law of Thermodynamics must be suspect because you can't detect a heat low of ~1W, from a toy Stirling engine, in your garage.

As for this demand for experiments, I and others pointed out to you many times on the other forum that no one can build a Carnot cycle engine, for obvious reasons. So the only way to show the formula is correct, aside from it being obviously true from the gas laws, is from experience. As I have also pointed out to you many times, power plant designers have spent over a century devising ever more efficient heat engines. None has beaten Carnot cycle efficiency. Ever.

So on the one hand there is 100 years plus of experience, from the best engineering minds on the planet, that agrees with it.

On the other, there's you, in your garage, with a \$100 toy Stirling engine, trying to detect a heat flow of ~1W.

...
As for this demand for experiments, I and others pointed out to you many times on the other forum that no one can build a Carnot cycle engine, for obvious reasons.
First of all, I'm not, and never have "demanded" anything from anyone.

When fist hearing about this Carnot Limit some 15 or so years ago, the first thing I tried to do is read up on its origins and history and expected to find a long chain of research and experimentation leading up to its final acceptance, like any other scientific discovery.

I could find nothing whatsoever of the sort.

Further, it seemed there were various conflicting interpretations. Does the limit refer only to "useful" work output or work done by the working fluid itself.

There seems to be no real consensus.

It could be interpreted consistent with the first law.

Taking the ambient environment as the "cold reservoir" the "hot reservoir" for practical reasons in the real world, more often than not needs to be created or generated, by elevating the temperature from the given, approximately 300°K ambient to whatever, by supplying some quantity of heat in joules.

So, we supply say 2000 joules elevating the temperature of some water to 375°K which is an increase of 20%

Which is also the Carnot Limit 20%

So, we COULD Interpret that as meaning you can only use 20% of all the heat down to absolute zero, or all of the actual 2000 joules used to elevate the temperature 20% from 300° to 375°K

But the modern interpretation is that only 20% of the 2000 joules supplied can be utilized, the other 1600 joules, (at minimum), absolutely MUST by some unidentified universal law, be discarded as "waste heat".

Only 400 of the 2000 joules supplied can be used for work output.

That seems like a pretty extraordinary claim for something based entirely on a mythical, non-existent engine. Having no sound theoretical basis (caloric theory), no empirical evidence, no experiments, nothing really. Just a wild speculation about the meaning of a simple temperature ratio.

What is supposed to be the physical basis for this claim? Why should it be accepted?

The alternative interpretation seems more valid. Take away friction and other loses, a "perfect" heat engine should be able to utilize the 2000 joules supplied and no more.

Why should we accept the hypothesis, based on no evidence whatsoever, that only 20% of the supplied heat is available for work, and not maybe 20.5% or 21% ?

Who made this rather astonishing determination? How was this determination arrived at? What is the actual evidence?

Nobody has built such a machine you say? Not in 200 years?

What proof do you have of that?

Who can say no Stirling engine running on boiling water can't be 21% efficient?

Have we tested every Stirling engine ever built in the past 200 years? Or even one for that matter?

First of all, I'm not, and never have "demanded" anything from anyone.

When fist hearing about this Carnot Limit some 15 or so years ago, the first thing I tried to do is read up on its origins and history and expected to find a long chain of research and experimentation leading up to its final acceptance, like any other scientific discovery.

I could find nothing whatsoever of the sort.

Further, it seemed there were various conflicting interpretations. Does the limit refer only to "useful" work output or work done by the working fluid itself.

There seems to be no real consensus.

It could be interpreted consistent with the first law.

Taking the ambient environment as the "cold reservoir" the "hot reservoir" for practical reasons in the real world, more often than not needs to be created or generated, by elevating the temperature from the given, approximately 300°K ambient to whatever, by supplying some quantity of heat in joules.

So, we supply say 2000 joules elevating the temperature of some water to 375°K which is an increase of 20%

Which is also the Carnot Limit 20%

So, we COULD Interpret that as meaning you can only use 20% of all the heat down to absolute zero, or all of the actual 2000 joules used to elevate the temperature 20% from 300° to 375°K

But the modern interpretation is that only 20% of the 2000 joules supplied can be utilized, the other 1600 joules, (at minimum), absolutely MUST by some unidentified universal law, be discarded as "waste heat".

Only 400 of the 2000 joules supplied can be used for work output.

That seems like a pretty extraordinary claim for something based entirely on a mythical, non-existent engine. Having no sound theoretical basis (caloric theory), no empirical evidence, no experiments, nothing really. Just a wild speculation about the meaning of a simple temperature ratio.

What is supposed to be the physical basis for this claim? Why should it be accepted?

The alternative interpretation seems more valid. Take away friction and other loses, a "perfect" heat engine should be able to utilize the 2000 joules supplied and no more.

Why should we accept the hypothesis, based on no evidence whatsoever, that only 20% of the supplied heat is available for work, and not maybe 20.5% or 21% ?

Who made this rather astonishing determination? How was this determination arrived at? What is the actual evidence?

Nobody has built such a machine you say? Not in 200 years?

What proof do you have of that?

Who can say no Stirling engine running on boiling water can't be 21% efficient?

Have we tested every Stirling engine ever built in the past 200 years? Or even one for that matter?
You can’t build a Carnot engine because, apart from anything else, it would have to run infinitely slowly. You have been told this before.

But I’m getting bored with this shit from you now. You have an idée fixe about it and we are rehearsing old speeches.

I’ll wait for you to come forward with some more interesting history - you are good at that.

You can’t build a Carnot engine because, apart from anything else, it would have to run infinitely slowly. You have been told this before.

But I’m getting bored with this shit from you now. You have an idée fixe about it and we are rehearsing old speeches.

I’ll wait for you to come forward with some more interesting history - you are good at that.
Frankly, your contribution from the start has been "I don't know..."

You have done little more than continually derail the thread and make it about me and my experiments rather than the history and validation of the Carnot Limit equation.

I've reported half a dozen or so of your posts as off-topic in an effort to keep things on track, but no response, and you won't go away

You don't know? That's fine, let someone else answer please and stop littering the topic with attacks on me and my efforts at filling the experimental void.

If a Carnot engine cannot be built and no experiments are possible, then this Carnot engine efficiency limit is really unfalsifiable.

That doesn't make it true.

Tom Booth:

You seem to be using the word "heat" in a non-standard way.

There are two ways to change the internal energy of the working fluid in an engine: (1) bring it into contact with a thermal reservoir of some kind, or (2) use the fluid to do mechanical work, or do some mechanical work on the fluid by applying external forces.

The word 'heat' refers to the first kind of process; the word 'work' refers to the second.
What physical control could that possibly have so as to influencing how the joules of heat supplied to an engine above T(cold) are actually utilized by tbe engine? Why should that in any way limit, say, 10,000 joules of available heat to just 2,000 able to be convertible to work and 8,000 minimum joules that must be tossed out as "waste heat".
An engine utilises the temperature difference between a hot reservoir and the internal temperature of the working fluid to temporarily increase the internal energy of its working fluid, which is the used to produce useful mechanical work. Following that, the working fluid is cooled by placing it in contact with a cold reservoir, which reduces the internal temperature of the fluid back to its initial value. Then the cycle can repeat again.

The efficiency of an engine is basically defined as "what you get out" divided by "what you pay for". What you get out is a certain amount of useful mechanical work. What you pay for is the fuel required to maintain the hot reservoir at a constant temperature, since the engine is constantly extracting heat from it and emitting heat into the cold reservoir.

The Carnot engine is theoretically the most efficient heat engine possible, although it is an 'ideal' engine that cannot be operated in practice. The efficiency of a Carnot engine can be shown to be 1-T_cold/T_hot. Any actual engine that does not emit its 'waste heat' into a cold reservoir at 0 Kelvin (and, by the way, there are no such reservoirs) cannot possibly have an efficiency of 100%. In practice, most actual engines use significantly hotter 'cold' reservoirs than 0 K, so the upper limit on their efficiency is significantly less than 100%.
A more sensible interpretation is that 20% of "all the heat" is what has been supplied, above and beyond the "background energy" or "heat" present all around in the ambient surroundings as well as the given internal energy of the working fluid prior to any additional heat being supplied.
You need to be more careful about your terminology. "Heat" is, by definition, energy transferred from or to a system due to a temperature difference between the system and its environment. "Heat" is not the same as internal energy. There is not "heat present all around in the ambient surroundings". That makes no sense at all as a statement, given the technical definition of "heat".

If you're going to examine the theory of heat engines, it's a good idea to get your definitions straight from the start. Don't you think? You need to be aware of the difference between "heat" and "temperature", between "heat" and "internal energy", and more. Non-experts tend to use those words very loosely, but you're interested in real physics, are you not?
You can only get back what you put in, in perfect accord with conservation of energy.
Again, in any real-world engine, you can't get back everything you put in. What you put in (what you pay for) is fuel to heat the working fluid. For instance, you pay for petrol to run your car engine. But that engine can never be 100% efficient. At best, it could only approach the efficiency of a Carnot engine running between the external air temperature and the internal temperature in the cylinders after ignition of the fuel. You will have noticed that car engines produce 'waste' heat, along with mechanical work. They expel heat to the environment as they operate. So, only part of what you pay for the petrol goes into the mechanical work done in turning the wheels. The rest of your petrol money goes to trying to heat up the air outside the car, which is 'wasted' money as far as you are concerned.
That also harmonizes with my experimental results. Stirling engines are known to be theoretically equivalent in efficiency with the Carnot engine.
Huh? Are you saying the efficiency of a Stirling engine is the same as the efficiency of a Carnot engine?
That then would be complete utilization of the supplied heat. No contradiction there.
The second law of thermodynamics prevents any "complete utilisation of supplied heat". Specifically, it says that no engine can be made whose sole effect is to extract heat from a hot reservoir to convert it all to useful mechanical work.

This is the same law that prevents perpetual motion machines from being real.
A mathematical ratio is just a mathematical ratio. It either represents 20% (all the heat supplied above T(cold) on a scale from T(hot) down to absolute zero or 20% (20% of the heat supplied above T(cold).)
In the formula e=1-T_hot/T_cold, the temperatures must be absolute temperatures (i.e. temperatures in Kelvin units). The formula does not apply when you're using other temperature scales (Celcius, Farenheit etc.)
I see no theoretical basis for or empirical justification for the latter interpretation.
Haven't you read any introductory textbooks on thermodynamics? How long have you been studying this?
Who first decided the 20% should represent only 20% of the heat supplied and not what it actually is; 20% of the thermometer reading between T(hot) and absolute zero.
???
"for 100% efficiency, you would need the sink to be at absolute zero".
Yes. The only value of T_cold in the formula e=1-T_cold/T_hot that makes the efficiency e=1 is T_cold=0 Kelvin.
That only makes sense if the 20% "efficiency" represents 20% of ALL the "heat" or all the internal energy of the working fluid all the way down to absolute zero NOT 20% of the 20% of the heat actually supplied above T(cold).
???

Why do you write "all of the 'heat' or all of the internal energy"? Those two things are very different quantities. Which one are you actually trying to talk about?
IMO someone at sometime just fudged the interpretation of the math because they didn't like the implications if applied to running a heat engine on ice.
It doesn't sound to me like you're in a good position to interpret the maths, given that it seems that you aren't even working with consistent/correct definitions of things like 'heat' and 'internal energy'.
In that case, an engine with an efficiency equivalent to the Carnot engine could, (running on ice), theoretically have available to it all the heat in the ambient surroundings above T(cold).
Again, "all the heat in the ambient surroundings" doesn't make sense as a phrase, given the accepted meaning of the word 'heat'. Heat isn't something that exists in an object. Heat is only ever energy transferred from one thing to another. You seem to be confusing heat and internal energy.

Moreover, the "ambient surroundings" you mention are, in the ideal case, considered to be an unlimited energy reservoir. More specifically, no matter how much energy your engine extracts from the "ambient surroundings" (or emits into them), the assumption is that the temperature of the surroundings does not change. This is what is meant by a "heat reservoir".

When you run your car engine, it is constantly emitting heat into the surrounding air. However, that doesn't tend to raise the temperature of the surrounding air as a whole, since the heat is rapidly spread over a very large volume of air.

Hello James.

You seem rather thoughtful and generally reasonable and considered in your thinking so perhaps we could get somewhere, though I would like to also point out that everything you have asked is nevertheless off topic and does not pertain to the opening question regarding the origin and validation of the formula in question.

I apologize for using the term "heat" in a loose non-technical manner.

"Heat" is a difficult topic. What is it exactly?

The technical scientific definition has, of course, changed from the days of Carnot until now.

In Carnot's time heat was indeed considered some form of actual material substance, specifically a kind of "fluid" that could accumulate like water in "reservoirs".

Why do you, or we, continue to use outdated terminology?

We might ask, what definition of "heat" do we apply when talking about the "Carnot" efficiency?

Carnot believed heat was a fluid; "Caloric". So did Kelvin essentially, in the manner he viewed heat, as a "something" that went THROUGH a heat engine 100% without change or alteration. Without transformation or conversion into another form of energy.

So, not only science itself, but the Carnot Limit as well in its very calculations and mathematical treatment defines "heat" in an ambiguous manner, which is really the crux of the problem.

Let me zero in on one of your assertions above for example:

You wrote above:

"The efficiency of a Carnot engine can be shown to be 1-T_cold/T_hot. Any actual engine that does not emit its 'waste heat' into a cold reservoir at 0 Kelvin (and, by the way, there are no such reservoirs) cannot possibly have an efficiency of 100%."

Can you give any logical reasons for that statement?

Just for example:

I have a Stirling engine that runs on the heat (thermal energy) from a cup of coffee.

I run the engine by placing it on top of the cup of hot coffee.

Heat is transfered from the coffee up through the bottom of the engine, the engine runs.

The engine is converting the heat entering into it that is transfered from the coffee into the engine.

The ambient temperature is approximately 300°Kelvin

The coffee has been heated up in a percolator from ambient at 300° Kelvin to approximately 375° Kelvin (boiling).

How many joules would be required to heat up 1 cup of water from 300°K to 375°K ?

So what is the maximum amount of "heat" that could possibly be transfered from the cup of coffee into the engine?

583,976.61 joules, correct?

Once the temperature of the coffee equalizes with the surroundings transfer will cease, correct?

The joules used to heat the water is equivalent to the joules going into the engine, correct? (Ignoring loses of course)

Now, if the engine were to convert 100% of the "HEAT" (which you clearly defined as the energy TRANSFERED to the engine) into "WORK", all 583,976.61 joules, that would be 100% efficiency. Correct?

The temperature has fallen back down into equilibrium with the surroundings: 300°K

In order to bring the temperature down to absolute zero, however, the engine would have to draw at least 815,352.65 joules out of the cup of coffee. (ignoring the fact that the coffee would absorb heat from the surroundings, we can imagine the coffee cup is perfectly insulated).

If reducing the temperature of the coffee back down to 300°K has already utilized all of the heat supplied (transfered)to the engine, why do we need to proceed all the way down to absolute zero? Another 231,376.04 joules ?

We did define "heat" as referring to only the energy actually transfered correct?

So why does the engine have to convert this additional 231,376.04 joules into work for 100% efficiency? Why is it not enough to convert the 583,976.61 joules actually supplied to raise the temperature of the coffee water from 300°K to 375°K (ambient to boiling).

What's going on here?

What definition of "heat" is this "Carnot efficiency formula" actually using?

Apparently not the modern definition of energy TRANSFERED.

If that were the case, we could stop when the coffee temperature fell back down to where it started before heat was added. 300°k, having utilized the maximum that could actually be transfered 583,976.61 joules.

"The efficiency of an engine is basically defined as "what you get out" divided by "what you pay for". What you get out is a certain amount of useful mechanical work. What you pay for is the fuel required to maintain the hot reservoir at a constant temperature, since the engine is constantly extracting heat from it and emitting heat into the cold reservoir."

This is, or can be, another point of confusion or ambiguity. "the fuel required to maintain the hot reservoir at a constant temperature"

That could mean just about anything. In particular when discussing Stirling engines, because a Stirling engine can run on heat from any source; burning liquid or solid fuel, wood, etc. or solar. Heat from the sun. As already mentioned, a hot cup of coffee or even body heat.

Do we count as "fuel" the electricity used to heat the water to make the coffee, or the coal burned at the power plant to make the electricity, or perhaps the sunshine that went into growing the prehistoric vegetation that eventually became coal?

Maybe the food we ate to generate body heat, or the fuel required to grow the food?

That "fuel" element is a rather wild variable.

Or maybe what you really mean is the actual heat, in joules, or whatever units of measure, that enters the engine, increasing the temperature of the working fluid, regardless of the source?

That is how I've been interpreting it, since anything else, (how much wood is burned, or propane or whatever) is pretty meaningless.

Solar radiation for example. How do we calculate the "cost" of the sunshine warming up our engine so as to accurately determine the efficiency?

Is our "hot reservoir" the surface of the sun? What is. the sun burning for fuel, hydrogen?

You need to be more careful about your terminology. "Heat" is, by definition, energy transferred from or to a system due to a temperature difference between the system and its environment. "Heat" is not the same as internal energy. There is not "heat present all around in the ambient surroundings". That makes no sense at all as a statement, given the technical definition of "heat".

If you're going to examine the theory of heat engines, it's a good idea to get your definitions straight from the start. Don't you think? You need to be aware of the difference between "heat" and "temperature", between "heat" and "internal energy", and more. Non-experts tend to use those words very loosely, but you're interested in real physics, are you not?

???

Yes. The only value of T_cold in the formula e=1-T_cold/T_hot that makes the efficiency e=1 is T_cold=0 Kelvin.

???

Why do you write "all of the 'heat' or all of the internal energy"? Those two things are very different quantities. Which one are you actually trying to talk about?

It doesn't sound to me like you're in a good position to interpret the maths, given that it seems that you aren't even working with consistent/correct definitions of things like 'heat' and 'internal energy'.
...
All those criticism apply to the "Carnot efficiency formula" and the manner in which it is currently interpreted.

Mathematically it is simply the percentage that T-hot is above T-cold.

In my example, which applies to many of my experiments running a Stirling engine on boiling water (or steam from boiling water) 375°K is 20% higher up on the Kelvin temperature scale than 300°K

Obviously, right?

So, logically, to my mind, based on conservation of energy, that 20% is the only energy available to be transfered to the engine.

In other words, I raised the temperature of the water 20% by raising it from 300° to 375° Kelvin

Of course the percentage will be different for different temperatures

Naturally, from 0°K to any temperature is 100% mathematically.

But the standard modern interpretation in all the textbooks, online example problems etc is to first calculate the ∆T as a percentage.

∆T of 75°k = 20% of 375°K right?

The boiling water at 375° (hot "reservoir") is 20% higher in temperature than 300° ambient (cold "reservoir").

Very simple mathematics.

So, as I said, you supply 20% of the heat above the ambient, so you should be able to utilize that 20% but no more

Conservation of energy

But the "Carnot Limit" is currently interpreted as 20% of that 20%

If I raise the temperature 20% to 375°k from 300°k and that required X number of joules then the only heat available is 20% of X not the heat actually transfered, which would be X itself.

Then it is said, that using all of X would still not get us 100% efficiency No we need to use all of X and also all the given "internal energy" all the way down to absolute zero.

Suppose we used 500.000 joules to boil some water. 375°k in a room 300°k ambient

375 is 20% more than 300

But we don't have all of those 500,000 joules available?

Only 20% ?

We supply 500,000 joules of energy to boil water but only 100,000 joules or 1/5 of the energy used to boil the water is convertible to work. ?

20% of the 20% ?

So, the Carnot calculation or whatever you want to call it is saying that "all the heat" includes not only the heat supplied by me to boil the water and transfer to the engine, but also all the "internal energy" already in the water, all the way down to absolute zero!

So I supply 500,000 joules, but have to utilize 900,000 or whatever joules to have 100% efficiency ?

OK, so I'm looking over all this 10 or 15 years ago and saying hold on, who dreamed this up, this is crazy.

So I start researching the history. I come up with a blank

It seems this formula just started appearing in textbooks sometimes after 1900 out of the blue

It did not originate with Carnot.

No Kelvin scale in Carnot's lifetime.

Kelvin?

We've been exploring that here, the Clausius inequality?

The connection seems rather tenuous. Still no Kelvin scale at that time.

Anyway, before I accept this simplistic mathematical ratio as some kind of universal law that is going to limit the efficiency of my engine to 20% for no rational reason, well, I'd like to see some rational reason for this

What physically limits the availability of my 500,000 joules used to boil my water ?

Conservation of energy says I should have all 500,000 joules.

The second law / Carnot limit says "oh no you don't you can only convert, at most, 20% of those 500,000 joules for no other reason than you raised the temperature 20% and for reasons unknown, that 20% is applied to the 500,000 joules used to raise the temperature, so now you are out 400,000 joules deducted and reserved for the "cold reservoir".

Here is your 20% back, of the 20% you supplied, your 100,000 joules out of your 500,000.

So I'm just trying to figure out who ever thought any of this actually made any sense.

Well, Carnot thought heat was a fluid that fell like a waterfall, so raising the temperature 20% means your heat "falls" 20% from 375° down to 300° K which makes some sense at least, but when was it decided that if it takes X joules to go from 300 to 375 K then only 20% of X is convertable to work and the other 80% including all the "internal energy" down to absolute zero needs to go to the "sink". Or "cold reservoir".

I'm saying "internal energy" because how else can you interpret that?

For 100% efficiency, my engine has to utilize MORE heat (energy transfered) than what was supplied?

Where is that EXTRA heat supposed to come from ?

The internal energy that was already there?

So, it is not me confused about the distinction between heat and work and internal energy, it's baked in to the "Carnot efficiency". That is where the confusion originated.

If my engine takes in 10 joules per cycle it must "reject" ALL that actual heat transfered AND all the "internal energy" down to absolute zero to achieve 100% efficiency is what you are actually telling me.

Anyway, I figured an actual experiment should help to sort this out.

What happens to the heat supplied and going into the engine to run it?

Is it #1) going in and "disappearing" as it is converted into another form of energy: mechanical motion, or is it #2) going in, turning the engine like water through a turbine and coming out the other side?

Or, a compromise. Why not a little of both? #3)

Anyway, so far all my experiments appear to indicate that #1 is the actual reality.

That is just an objective observation.

I've run a Stirling engine for hours and hours on near boiling hot water or steam.

Certainly enough heat is going into the engine to keep it running. According to the Carnot efficiency limit, whatever is converted to "work" at that ∆T 5 time more heat should be "rejected" or pass through and out the other side.

For heat to go out of the engine the cold side temperature would need to be above ambient, correct?

You can't really have heat transferring out of the engine without a temperature difference.

But my instrumentation does not show any elevation in the temperature of the cold "sink" side of the engine.

The heat, presumably, (some at least) is going in, because the engine runs.

The cold ambient side however, remains cold.

Actually, it gets just a little bit cooler than ambient.

Maybe 298° or 299°k while all around the ambient temperature is 300°k and the hot side of the engine is on top of a cup of hot water at just under 375°k

Oh, and yes, an "ideal" Stirling engine is theoretically equivalent in efficiency to a Carnot engine.

"Theoretical thermal efficiency equals that of the ideal Carnot cycle, i.e. the highest efficiency attainable by any heat engine"

Anyway, back to the question at hand:

### Origin and verification of e=(Th-Tc)/Th​

Certainly there must have been prior experiments ? Something concrete that established this equation as valid and correct?

Or, maybe some textual historic citation regarding how and why this simple temperature difference is to be interpreted?

How did the modern textbooks arrive at the current interpretation?

Seems odd to me there is not more information regarding the origin other than the assumption that it originated with Carnot, which is obviously false.

Tom Booth:

You seem to be using the word "heat" in a non-standard way.

There are two ways to change the internal energy of the working fluid in an engine: (1) bring it into contact with a thermal reservoir of some kind, or (2) use the fluid to do mechanical work, or do some mechanical work on the fluid by applying external forces.

The word 'heat' refers to the first kind of process; the word 'work' refers to the second.

An engine utilises the temperature difference between a hot reservoir and the internal temperature of the working fluid to temporarily increase the internal energy of its working fluid, which is the used to produce useful mechanical work. Following that, the working fluid is cooled by placing it in contact with a cold reservoir, which reduces the internal temperature of the fluid back to its initial value. Then the cycle can repeat again.

The efficiency of an engine is basically defined as "what you get out" divided by "what you pay for". What you get out is a certain amount of useful mechanical work. What you pay for is the fuel required to maintain the hot reservoir at a constant temperature, since the engine is constantly extracting heat from it and emitting heat into the cold reservoir.

The Carnot engine is theoretically the most efficient heat engine possible, although it is an 'ideal' engine that cannot be operated in practice. The efficiency of a Carnot engine can be shown to be 1-T_cold/T_hot. Any actual engine that does not emit its 'waste heat' into a cold reservoir at 0 Kelvin (and, by the way, there are no such reservoirs) cannot possibly have an efficiency of 100%. In practice, most actual engines use significantly hotter 'cold' reservoirs than 0 K, so the upper limit on their efficiency is significantly less than 100%.

You need to be more careful about your terminology. "Heat" is, by definition, energy transferred from or to a system due to a temperature difference between the system and its environment. "Heat" is not the same as internal energy. There is not "heat present all around in the ambient surroundings". That makes no sense at all as a statement, given the technical definition of "heat".

If you're going to examine the theory of heat engines, it's a good idea to get your definitions straight from the start. Don't you think? You need to be aware of the difference between "heat" and "temperature", between "heat" and "internal energy", and more. Non-experts tend to use those words very loosely, but you're interested in real physics, are you not?

Again, in any real-world engine, you can't get back everything you put in. What you put in (what you pay for) is fuel to heat the working fluid. For instance, you pay for petrol to run your car engine. But that engine can never be 100% efficient. At best, it could only approach the efficiency of a Carnot engine running between the external air temperature and the internal temperature in the cylinders after ignition of the fuel. You will have noticed that car engines produce 'waste' heat, along with mechanical work. They expel heat to the environment as they operate. So, only part of what you pay for the petrol goes into the mechanical work done in turning the wheels. The rest of your petrol money goes to trying to heat up the air outside the car, which is 'wasted' money as far as you are concerned.

Huh? Are you saying the efficiency of a Stirling engine is the same as the efficiency of a Carnot engine?

The second law of thermodynamics prevents any "complete utilisation of supplied heat". Specifically, it says that no engine can be made whose sole effect is to extract heat from a hot reservoir to convert it all to useful mechanical work.

This is the same law that prevents perpetual motion machines from being real.

In the formula e=1-T_hot/T_cold, the temperatures must be absolute temperatures (i.e. temperatures in Kelvin units). The formula does not apply when you're using other temperature scales (Celcius, Farenheit etc.)

Haven't you read any introductory textbooks on thermodynamics? How long have you been studying this?

???

Yes. The only value of T_cold in the formula e=1-T_cold/T_hot that makes the efficiency e=1 is T_cold=0 Kelvin.

???

Why do you write "all of the 'heat' or all of the internal energy"? Those two things are very different quantities. Which one are you actually trying to talk about?

It doesn't sound to me like you're in a good position to interpret the maths, given that it seems that you aren't even working with consistent/correct definitions of things like 'heat' and 'internal energy'.

Again, "all the heat in the ambient surroundings" doesn't make sense as a phrase, given the accepted meaning of the word 'heat'. Heat isn't something that exists in an object. Heat is only ever energy transferred from one thing to another. You seem to be confusing heat and internal energy.

Moreover, the "ambient surroundings" you mention are, in the ideal case, considered to be an unlimited energy reservoir. More specifically, no matter how much energy your engine extracts from the "ambient surroundings" (or emits into them), the assumption is that the temperature of the surroundings does not change. This is what is meant by a "heat reservoir".

When you run your car engine, it is constantly emitting heat into the surrounding air. However, that doesn't tend to raise the temperature of the surrounding air as a whole, since the heat is rapidly spread over a very large volume of air.
Post 45 explains why you can't detect any temperature rise. You are trying to detect a heat flow of the order of 1W.

Good luck with that.

Post 45 explains why you can't detect any temperature rise. You are trying to detect a heat flow of the order of 1W.

Good luck with that.

Merely dismissive opinion / speculation on your part.

I've run experiments of this kind using an 80 watt electrical boiler providing steam to the bottom aluminum heat exchanger plate for hours and hours on end run time with zero thermal transfer through to the cold side heat exchanger.

Regardless of your offhand dismissal of these experimental results, the topic/question of the thread is about PREVIOUS experiments.

What experiments were conducted by Carnot, Kelvin, Joule, Clausius, Faraday, Maxwell, anyone at any time in history by any means, validating that the temperature ratio has the direct effect on heat engine efficiency claimed, in relation to the Carnot efficiency limit formula or temperature ratio?

Your focusing on the inadequacies of my very recent amateur experiments does not address the question regarding the historical picture and apparent lack of any previous experiments of this kind, or something similar, or in the same ballpark validating the current textbook interpretation.

Or, maybe some theoretical physical basis for such an interpretation.

What we have is a completely astonishing claim that contradicts the 1st Law of thermodynamics; conservation of energy, which has been very VERY thoroughly validated, In every other case involving energy conversion, near 100% conversion is possible, less KNOWN, controllable loses, such a as electrical resistance in wires.

Here we have the claim that 80%, 90% or more of the thermal energy supplied to a system is completely barred from any possibility of conversion for absolutely no reason, with no experimental testing or validation, nothing.

To find any kind of theoretical basis for this, we have to go all the way back to Carnot with his assertion about heat being just like a waterfall cascading down from a hot to a cold "reservoir".

That was based on the obsolete Caloric theory.

So what do we have in the intervening decades?

The original theoretical basis has been invalidated.

Why should anyone just accept a dictum from history that has apparently never been tested and validated experimentally?

And why the personal attacks on someone simply trying to conduct some objective experiments in an effort to fill the apparent void in available experimental data

Starting out, in all honesty, my first xperiment was actually intended as a demonstration of the Carnot theory.

I thought blocking the flow of heat to the sink or "cold reservoir" would cause a bottleneck. The engine would quickly overheat and stop.

I was so confident about this result, I video recorded the demonstration to post for my friends on the model engine forum.

But instead of slowing down and stopping, the engine kept going. Actually with the insulation, it started running at a slightly higher RPM.

Puzzled, I posted the video to the forum anyway to see what other model builders thought.

Suggestions were made for improving the demonstration. More insulation. Better insulation, insulation in different locations, more heat, temperature readings.. On and on the experiments continued, year after year, always with the same general outcome.

Failure to demonstrate the validity of the Carnot theory, or the theory in general that a heat engine operates due to the heat FLOWING like water, THROUGH the engine, IN one side and OUT the other. That the heat going into the engine cannot be converted to other forms of energy like any other energy conversion process, but is limited by the ∆T.

So, if my experiments have completely failed to demonstrate the Carnot efficiency limit, then I think it is fair to ask or at least wonder, what was the outcome of similar past experiments?

"Carnot's result was that if the maximum hot temperature reached by the gas is Th, and the coldest temperature during the cycle is Tc, (degrees kelvin, or rather just kelvin, of course) the fraction of heat energy input that comes out as mechanical work , called the efficiency, is

Efficiency = Th-Tc/Th

This was an amazing result, because it was exactly correct, despite being based on a complete misunderstanding of the nature of heat!"

OK, so how do we know NOW that this was "exactly correct"?

Ignoring the fact that the entire above narrative is false. The "Carnot efficiency limit" in its current precise mathematical form, obviously did not originate with Carnot at all, as the Kelvin temperature scale did not exist.

So, where did it originate and how was it finally verified to be "exactly correct" though admittedly: "based on a complete misunderstanding of the nature of heat" ?

Merely dismissive opinion / speculation on your part.

I've run experiments of this kind using an 80 watt electrical boiler providing steam to the bottom aluminum heat exchanger plate for hours and hours on end run time with zero thermal transfer through to the cold side heat exchanger.

Regardless of your offhand dismissal of these experimental results, the topic/question of the thread is about PREVIOUS experiments.

What experiments were conducted by Carnot, Kelvin, Joule, Clausius, Faraday, Maxwell, anyone at any time in history by any means, validating that the temperature ratio has the direct effect on heat engine efficiency claimed, in relation to the Carnot efficiency limit formula or temperature ratio?

Your focusing on the inadequacies of my very recent amateur experiments does not address the question regarding the historical picture and apparent lack of any previous experiments of this kind, or something similar, or in the same ballpark validating the current textbook interpretation.

Or, maybe some theoretical physical basis for such an interpretation.

What we have is a completely astonishing claim that contradicts the 1st Law of thermodynamics; conservation of energy, which has been very VERY thoroughly validated, In every other case involving energy conversion, near 100% conversion is possible, less KNOWN, controllable loses, such a as electrical resistance in wires.

Here we have the claim that 80%, 90% or more of the thermal energy supplied to a system is completely barred from any possibility of conversion for absolutely no reason, with no experimental testing or validation, nothing.

To find any kind of theoretical basis for this, we have to go all the way back to Carnot with his assertion about heat being just like a waterfall cascading down from a hot to a cold "reservoir".

That was based on the obsolete Caloric theory.

So what do we have in the intervening decades?

The original theoretical basis has been invalidated.

Why should anyone just accept a dictum from history that has apparently never been tested and validated experimentally?

And why the personal attacks on someone simply trying to conduct some objective experiments in an effort to fill the apparent void in available experimental data

Starting out, in all honesty, my first xperiment was actually intended as a demonstration of the Carnot theory.

I thought blocking the flow of heat to the sink or "cold reservoir" would cause a bottleneck. The engine would quickly overheat and stop.

I was so confident about this result, I video recorded the demonstration to post for my friends on the model engine forum.

But instead of slowing down and stopping, the engine kept going. Actually with the insulation, it started running at a slightly higher RPM.

Puzzled, I posted the video to the forum anyway to see what other model builders thought.

Suggestions were made for improving the demonstration. More insulation. Better insulation, insulation in different locations, more heat, temperature readings.. On and on the experiments continued, year after year, always with the same general outcome.

Failure to demonstrate the validity of the Carnot theory, or the theory in general that a heat engine operates due to the heat FLOWING like water, THROUGH the engine, IN one side and OUT the other. That the heat going into the engine cannot be converted to other forms of energy like any other energy conversion process, but is limited by the ∆T.

So, if my experiments have completely failed to demonstrate the Carnot efficiency limit, then I think it is fair to ask or at least wonder, what was the outcome of similar past experiments?

"Carnot's result was that if the maximum hot temperature reached by the gas is Th, and the coldest temperature during the cycle is Tc, (degrees kelvin, or rather just kelvin, of course) the fraction of heat energy input that comes out as mechanical work , called the efficiency, is

Efficiency = Th-Tc/Th

This was an amazing result, because it was exactly correct, despite being based on a complete misunderstanding of the nature of heat!"

OK, so how do we know NOW that this was "exactly correct"?

Ignoring the fact that the entire above narrative is false. The "Carnot efficiency limit" in its current precise mathematical form, obviously did not originate with Carnot at all, as the Kelvin temperature scale did not exist.

So, where did it originate and how was it finally verified to be "exactly correct" though admittedly: "based on a complete misunderstanding of the nature of heat" ?
I answered all that many posts ago. There is no value in repeating it. (By the way, this failure to listen and repeating of questions that have been answered is exactly why you were eventually banned from the other forum.)

You are ranting in order to avoid dealing with the reason I have given you for why you can’t detect evidence of heat being rejected from your toy Stirling engine.

But I strongly suggest you deal with that, because you have built your entire challenge to thermodynamics on it.

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I answered all that many posts ago. There is no value in repeating it.
There was little value in it to begin with

Again, nothing more than your biased opinion that has no basis in reality.

And please provide a link to where you provided some prior experiment or theory that forms a basis for establishing the Carnot limit as factual.

Otherwise how about simply leaving the thread and give someone else an opportunity to answer since you obviously don't know and have nothing worthwhile to contribute to the discussion.

I can only assume you must be the owner of this domain. Must be a sad and lonely life sitting around waiting for another victim to wander into your lair.

From the lack of activity, I can only assume you've already driven everyone away, aside from your own sock puppets.

BTW, I never had any desire or intent to "challenge thermodynamics".

The experiments were, as I explained, intended to demonstrate the fact that a Stirling engine would fail to operate if the flow of heat through to the "cold reservoir" were blocked or interrupted.

I've also done similar experiments with high temperature engines and a propane torch as the heat source with the same results.

A propane torch burns at around 2000°F.

Your opinion that there is simply too little heat to possibly be measurable is just plain wrong as well as irrational.

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