thermodynamics - What's the efficiency of real heat engines?

Real heat engines always have lower efficiency than the Carnot efficiency.

I wonder how efficient real engines can be? Can their efficiency get anywhere near the Carnot-limit?

Answer

The most efficient heat engines are invariably the biggest and slowest working. For a steam turbine, "slowest working" means having many turbine stages, so that work is extracted from the steam as it "slowly" expands in many stages, doing a small amount of work against many turbine stages. The high thermal stability of a very big system means that a wide difference between the upper and lower reservoir temperatures can be upheld, and thus a high potential Carnot efficiency. A reciprocating engine's efficiency is generally improved by making it run very slowly: one or two hertz maximum.

The Steam-Electric Power Station Wiki page states actual efficiencies of big steam electricity plants of between 33% and 48%. Assuming the steam turbine can exhaust at, say 100C (373K), if the 48% were near to the Carnot efficiency, this would mean an upper reservoir temperature $T_{max}$ given by:

$$1-\frac{T_{min}}{T_{max}} = \eta \Leftrightarrow T_{max} = \frac{T_{min}}{1-\eta} = \frac{373}{1-0.48} = 720K$$

This is somewhat below what contemporary technology can superheat steam to; from the article:

V Ganapathy, "Superheaters: design and performance", Hydrocarbon Processing, July 2001

I glean rough temperatures of 1300K (2000F) as being within the reach of a radiant superheater. This would imply a Carnot efficiency of

$$\eta = 1-\frac{T_{min}}{T_{max}} = 1-\frac{373}{1300} = 71\%$$

So it would seem that even at these high efficiencies, we're working rather below the Carnot efficiency. It would be good to get the input from an energy technologist at this point to firm some of these figures up.

The Earth's biggest internal combustion engine is the Wärtsilä-Sulzer RTA96-C, a two-stroke, fourteen cylinder 750MW monster used to power the ship Emma Maersk. The manufacturer's specifications state a thermal efficiency of greater than 50%, which they define explicitly as the work output divided by the free energy of the fuel burning reaction. As we have seen above, this would imply an upper reservoir temperature of the order of $700K$ if it were reaching Carnot efficiency, which is still rather cooler than the likely initial temperature of the combustion products.

A closer reading of the Ganapathy reference cited above and some more thought on this interesting question leads to the following comments.

1. 
   

   The steam output temperature of a modern powerstation superheater is likely to be of the order of $850{\rm K}$ to $900{\rm K}$ ($1100^o{\rm F}$) and this should be the figure we take as our upper reservoir temperature. I was taking the upper reservoir temperature to be that of the radiation in the superheater ($1300{\rm K}$), thinking that the difference between the gas and radiation temperature is an inefficiency that needs to be included. However, presumably we can think of the furnace as a system that is closed aside from the input of heat and the output of steam and that no other, or little other, energy is lost from the furnace system. Coincidentally, $850{\rm K}$ is also the temperature that the modern stainless steels used in turbine blades can withstand long term without creep (see the "Efficiency" section of the Steam Engine Wiki Page). Also, one could legitimately argue that the question could be taken to be asking for the efficiency of the turbine alone, and not of the furnace - turbine system. This being so, $T_{max}=850{\rm K}$ would be a reasonable assumption.

   
   
   


2. 
   

   Modern steam turbines actually have lower reservoir temperatures below $100^o{\rm C}$: they are sealed and their later stages work below atmospheric pressure. So the lower reservoir temperature $T_{\min}$ is more like $30^o{\rm C}$ : let's say $300K$.

   
   

With these figures, our turbine's potential Carnot efficiency would be:

$$\eta = 1-\frac{T_{min}}{T_{max}} = 1-\frac{300}{850} = 65\%$$

which, for a system on the upper end of the scale of the [Wikipedia's $33\%$ to $48\%$ estimates]((http://en.wikipedia.org/wiki/Steam-electric_power_station), implies a working efficiency (work output compared to the Carnot efficiency) of

$$0.5/0.65 = 77\%$$

So I would suggest this is a fairly good answer and as near to an answer as you're going to get on this forum, unless we hear from an energy technologist. So steam turbines do rather well. Interestingly, if we use the "experimental" Novikov formula you cited, we foretell an efficiency under these conditions of

$$1-\sqrt{\frac{300}{850}} = 41\%$$

so this is a little pessimistic for the modern steam turbine, which is the paragon of modern efficiency in heat engines, with a great deal of research being input to the sophisticated computer control of supercritical superheaters and furnaces.