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Clearing the Carnot Confusion



by Tom Napier,   May 27, 1998



Something I often hear from maverick engine designers is that 

the thermodynamic efficiency equation applies only to Carnot 

Cycle engines.  Since their engine doesn't use a Carnot Cycle the 

law doesn't apply to it.  Over-unity performance, here we come!



So what does the thermodynamic efficiency really mean?



Simply put, it is the ratio of the available heat energy to 

the input heat energy.  It is Qa/Qi where Qa is the available 

energy and Qi is the input energy.  If all the input heat energy 

could be converted to mechanical energy, Qa would equal Qi and 

the thermodynamic efficiency would be 100%.  Since you can't 

convert more than "all" of the input energy, the thermodynamic 

efficiency can never exceed 100%.  Note that this statement is a 

simple matter of arithmetic, it makes no reference to how the 

conversion is done; in other words, no conceivable engine can 

have a thermodynamic efficiency of greater than 100%.



How high a thermodynamic efficiency can I get?



Thermodynamic efficiencies are generally much less than 100%.  

Qa/Qi can be rewritten as (Qi-Qo)/Qi where Qo is the heat energy 

leaving the system.  It represents the energy which cannot be 

converted into useful work.  Since we are dealing with ratios we 

don't need to consider conversion factors or how much energy we 

are talking about, the ratio is the same regardless of how much 

energy is passing through the device or how we measure it.  Thus 

we can replace "quantity of heat energy" with "temperature" 

provided heat energy is proportional to temperature.  The 

temperature scale to which this applies is the "absolute" scale 

in which zero degrees represents zero heat energy.  You can 

convert Fahrenheit temperatures to absolute temperatures by 

adding 459.69 degrees.



Thus the thermodynamic efficiency of a heat engine is given by 

(Ti-To)/Ti where Ti and To are the input and output temperatures, 

measured on the absolute scale.  If you have a furnace at 1200 F 

and a river at 60 F, the best thermal efficiency you can ever 

achieve by running a turbine between them is 68.7%.



So thermodynamic efficiencies can be quite low?



Yes, an important fact which can be derived from the 

thermodynamic efficiency equation is that when the input and 

output temperatures are the same, the available heat energy is 

zero.  (This restates the Second Law of Thermodynamics.)  No 

matter how much heat energy something contains, you can't use any 

of it unless there is something colder to which you can transfer 

heat energy.  For example, there are gasses which boil at room 

temperature but you can't run an engine from them without having 

something much colder around to condense the gas back to a 

liquid.



Why are engines less efficient than thermodynamics says?



The equation which calculates thermodynamic efficiency makes 

no reference to the type or the construction of the conversion 

device.  It is a statement of how much input energy is available 

to it.  The engine itself may be quite efficient or it could be 

quite inefficient, but now we are talking about the mechanical 

efficiency of the engine.  This is a function of how much heat 

energy it wastes through bad insulation or how much friction 

there is in its moving parts.



There is scope to improve the mechanical efficiency of heat 

engines, and much effort has been directed to that aim in the 

last two centuries, but the best we can do is to approach 100%.  

Mechanical efficiency can never quite reach 100%.



The total efficiency of a heat engine is thus its mechanical 

efficiency multiplied by the thermodynamic efficiency calculated 

from its input and output temperatures.  Both are less than 100% 

so their product is always less than 100%.  Efficiencies greater 

than 100% are a chimera.



So where does Carnot come in?



Poor Sadi Carnot (1796-1832) would be be turning in his grave, 

if he could hear all the weird things people say about Carnot 

Engines and Carnot Cycles. Sadi Carnot wrote only one scientific 

paper during his short life but it virtually founded the science 

of thermodynamics.



He was the first person to suggest that the efficiency of a 

real heat engine should be referenced to a hypothetical engine 

which wasted no energy.  He said, we can't build a 100% 

(mechanically) efficient engine but if we could, how much heat 

energy could it use?



This "Carnot Engine," being perfectly efficient, would work 

equally well in both directions.  It would convert the energy 

difference between a hot source and a cold sink entirely into 

mechanical energy with no losses of any kind.  Equally, if you 

put mechanical energy into it, it would convert every bit of that 

energy into a heat energy difference.



Thus the expression "Carnot Engine" describes any heat engine 

which is perfectly efficient.  It is impossible for an engine to 

be more efficient than a Carnot Engine simply because "Carnot 

Engine" is the name given to the most efficient possible engine.  

A Carnot Engine is 100% efficient.  To look for a more efficient 

engine is like looking for an integer between 1 and 2, by 

definition it does not exist.







What is the Carnot Cycle and why won't it really work?



Carnot then went on to show, at least in principle, how to 

build this perfectly efficient engine.  It would operate by a 

series of compressions and expansions of gas in a cylinder which 

is known as the "Carnot Cycle."  This is a really remarkable 

achievement.  It's as if Albert Einstein had not only discovered 

the equation for the conversion of matter into energy but had 

also published the plans for a fusion reactor!



In the first part of the Carnot Cycle, heat flows into the gas 

in a cylinder.  The gas expands and pushes a piston, doing work 

on it.  The gas remains at the same temperature, the temperature 

of the source, during this expansion.  Now right away you can see 

that it is not very practical to build this 100% efficient engine 

since, if the gas is at the same temperature as the source, heat 

won't flow into it.  The gas has to be slightly cooler than the 

thing heating it otherwise no heat will flow.  This means that 

either the engine has to be a bit less than 100% efficient or it 

has to move infinitely slowly.



Another problem is that somehow the motion of the piston has 

to be synchronized so that the gas expands at exactly the right 

rate for the temperature to remain constant.  None of this crank 

and connecting rod stuff here, you would need computer control.



So the gas has expanded at a constant temperature.  Now we 

remove the hot source from the cylinder and let the gas expand 

some more with no heat input.  This still does some work on the 

piston so the gas cools down as it expands.  Eventually it will 

reach the temperature of the cold sink.  When it does we move the 

sink into contact with the cylinder and start pushing the piston 

in.  This means doing a little work on the piston but not very 

much since, as the gas starts to heat up, it loses heat to the 

sink and so remains at the same temperature.  Again the rate of 

heat transfer is infinitely slow in a 100% efficient engine.



In the last part of the cycle the cold sink is removed and the 

gas in the cylinder is allowed to heat up as the piston 

compresses it further.  No heat energy is gained or lost but the 

mechanical energy put into the piston shows up as the higher gas 

temperature.  When the gas is as hot as the source, the source is 

reapplied to the cylinder and the cycle repeats.



When the gas is expanding it is absorbing heat energy from the 

source, when it is contracting it is giving up heat energy to the 

sink.  This means that the gas does more work on the piston 

during its outward stroke than is needed to compress it during 

the inward stroke.  The net energy output is why we built the 

machine.







Are real engines as efficient as a Carnot Engine?



Carnot postulated a perfectly efficient engine and then showed 

that one existed, at least in theory.  This does not imply that a 

fairly efficient engine cannot be built using other cycles.



However, real engines are built with materials which don't 

insulate perfectly, don't transfer heat without a temperature 

drop and which have friction.  That is, all real engines are less 

efficient than the hypothetical Carnot Cycle engine, no matter 

how they are made.



 Some, like the Stirling engine work by heating and cooling 

the same mass of gas over and over again, just as the Carnot 

engine does.  Engines of this type tend to be rather slow and 

inefficient since it is difficult to heat and cool a mass of gas 

quickly without having a large temperature difference between the 

source and the gas.  Other engines, such as the steam engine, use 

the fact that gases condense to liquids under some conditions of 

temperature and pressure.  The hot gas drives the piston outwards 

and when the gas cools to a liquid the piston can be driven back 

in with less energy.  This cycle is called the Rankine cycle and 

although it is much more practical than a Carnot engine, its 

efficiency is considerably lower.



Turbine engines, which use rotating rather than reciprocating 

parts, tend to have a higher mechanical efficiency.  They are 

also capable of running with a higher inlet temperature and so 

may have a higher thermodynamic efficiency.  They are still bound 

by the same thermodynamic limits as any other engine.



Thus when someone tells you that his engine doesn't run on a 

Carnot cycle, you can be sure he is telling the truth.  If he 

tells you this makes it more efficient than a Carnot Cycle engine 

he is exposing his ignorance of what a Carnot Cycle engine is.