book_cover_big.gifIn previous blogs, I explained the principle of the Carnot cycle, focusing mainly on how to obtain work out of heat. The Carnot cycle was designed by Sadi Carnot to understand the workings of steam engines. A basic steam engine contains  a heat source at  high temperature (Th) and a heat sink at a lower temperature (Tl); the engine gives us useful work (W) by extracting heat (Qh) from the heat source and shifting  an amount of heat (Ql) to the heat sink. Carnot showed that the best possible [i] efficiency (defined by W/Qh ) was determined by the temperatures Th  and Tl. We can use Carnot’s cycle in three different applications to calculate  the best possible efficiency of each [ii]. These applications are:

1) Extraction of heat from the heat source for the purpose of converting the heat into work, as in  a steam engine. The efficiency, η = W/Qh , is given by the formula:

η ≤ (T – Tl)/ (Th)

2) Extraction of heat from the low temperature location in order to lower the temperature of that area, or to maintain a low temperature in warmer surroundings – for  example, a refrigerator. The efficiency in this case is defined as Ql /W and shows how much work (W) we must put in to extract an amount of heat (Ql) from the low temperature location. The efficiency (also called Coefficient of Performance, COP) is defined by Ql/W and is given by this equation:

COP ≤ Tl/(ThTl)

3) The third application is where we want to extract heat, Ql, from a low-temperature location  and bring this heat to a high-temperature location;. an example is a heat pump that warms a building with heat extracted from ground water. The efficiency (COPhp) in this case is defined by the ratio Qh/W and is expressed as:

COPhp ≤ Th/(ThTl)

Note: The ≤ symbol in the above equations indicates the best possible efficiencies; in the real world, efficiencies are often considerably lower.

In the table below,  I have given some examples of how these efficiencies  perform  at different temperatures. Thus, for instance, a heat engine running between 293K and 800K cannot achieve a better efficiency than 63%, meaning it can convert into work only 63% of the heat extracted from the heat source.

An important point: a refrigerator and a heat pump can achieve high efficiencies with  small temperature differences between Th and Tl. For instance, a refrigerator with  an efficiency of 10.11, and internal/external  temperatures of  283K and 300K, can expel  an amount of heat more than 10 times greater than the amount of  work put in. However,  this is a bit misleading since pumping heat between two locations of similar temperatures is not all that useful.  In fact, the refrigerator is the top electricity consumer in the average household, despite its impressive COPs calculated below. This is mainly because of  the constant uphill battle of removing the heat that leaks into the refrigerator from the warmer kitchen environment. But other factors play a role as well.

Carnot Efficiencies

    Heat engine Refrigerator Heat pump

Tl

        Th

(Th-Tl)/Th

Tl/(Th-Tl)

Th/(Th-Tl)

293

800

0.63

0.58

1.58

293

700

0.58

0.72

1.72

293

600

0.51

0.95

1.95

293

500

0.41

1.42

2.42

293

400

0.27

2.74

3.74

293

320

0.08

10.85

11.85

293

310

0.05

17.24

18.24

293

300

0.02

41.86

42.86

273

800

0.66

0.52

1.52

273

700

0.61

0.64

1.64

273

600

0.55

0.83

1.83

273

500

0.45

1.20

2.20

273

400

0.32

2.15

3.15

273

320

0.15

5.81

6.81

273

310

0.12

7.38

8.38

273

300

0.09

10.11

11.11

283

293

0.03

28.30

29.30

283

303

0.07

14.15

15.15

283

288

0.02

56.60

57.60

See also: https://secondlawoflife.wordpress.com/2008/10/26/wasting-energy-why/


[i] The best possible efficiency is obtained for a reversible process. Unfortunately, all practical processes are irreversible, meaning that their efficiencies will be less than those calculated from the given formulas. Therefore, in the equations we use the less than or equal to symbol: ≤.[ii] The efficiency is defined by η = (desired output)/(required input). For more details, see J.R. Howell and R.O. Buckius, Fundamentals of Engineering Thermodynamics, 2nd ed., p.337 (1992), McGraw-Hill Inc.

Copyright © 2007 John E.J. Schmitz

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