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3 Second Law and Thermodynamic Cycles: the Carnot Efficiency

3 Second Law and Thermodynamic Cycles: the Carnot Efficiency

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Chemistry of Sustainable Energy

the concept of theoretical efficiency and waste energy. A mechanical heat engine is a

device that converts heat into work as in, for example, an automobile engine. There

are four stages in any mechanical heat engine:

1.Compression

2.Heat intake

3.Expansion

4.Heat rejection

There are a multitude of engine designs that vary the ways in which these steps

are carried out, typically by describing the four steps in the form of a system consisting of a frictionless piston and an ideal gas, as described later for the Carnot engine.

Because the system involves an ideal gas, the ideal gas law is our equation of state.

The surroundings are a heat reservoir that provides the heat to be converted into

work by a cyclic, reversible (vida infra) process.

The Carnot engine is based on the Carnot cycle, devised by Sadi Carnot (1796–

1832). In order to understand the thermodynamics of the Carnot cycle, we have to

understand the concept of reversibility. A reversible process simply means that whatever is “done” can be “undone” with no loss along the way. Thus a reversible process

is made up of infinitesimal steps, each one of which can be reversed to its previous

state without any deviation in any way. In a reversible cycle, a system and surroundings begin at an initial state, proceed through the steps of the cycle, and end at a final

state which is identical to the initial state.

A representation of the reversible Carnot cycle relating the thermodynamic quantities of heat, work, energy, and entropy is shown in Figure 3.1a and b. Note that

representations a and b focus on the exact same cycle but simply present different

relationships (pressure–volume vs. temperature–entropy).

(b)

Pressure (P)

2

P2

qin

P3

P4

wnet

V2

4

1

V1

qout

V3

Volume (V )

qin

TH

3

P1

Absolute temperature (T )

(a)

V4

2

3

wnet

TL

1

4

qout

Entropy (S)

FIGURE 3.1  The Carnot cycle (a) as a function of pressure and volume and (b) as a function

of absolute temperature and entropy.

Thermodynamics

59

• Step 1 → 2 represents adiabatic compression: the system is perfectly insulated so that there can be no exchange of heat with the surroundings (dq = 0).

Keep in mind that this does not imply that there is no change in temperature: because the heat cannot escape into the surroundings, the temperature

increases. Because the change in entropy for a reversible process is related

to heat and temperature by

dS = dqrev/T(3.3)

and dqrev = 0, there is no change in the quantity of entropy (Figure 3.1b). The

volume decreases as the piston is compressed and as a result, the pressure

increases (Figure 3.1a).

Step 2 → 3 represents heat addition. The insulation is removed and the piston

expands while heat is drawn from the surroundings into the system (qin). The

temperature is constant (see Figure 3.1b), making this an isothermal step. (The

pressure decreases, the volume increases and as a result, the entropy is increased.)

Step 3 → 4 is expansion yet again (volume increase, pressure decrease)

yet again, but with insulation reapplied so that heat exchange is prevented,

making this an adiabatic expansion. As a result, the work that is done to

expand the piston is done at the expense of the internal energy from the

gas, with the result that the temperature drops. Entropy again stays constant

because no heat is drawn from the surroundings.

Finally, step 4 → 1 is isothermal heat rejection. The insulation is removed

and heat exchange between the system and surroundings is once again

allowed. The piston is compressed as the temperature is held constant, so

the excess heat flows into the surroundings. This is wasted heat (as indicated by qout) and the amount is given by the shaded area.

The net amount of work done by the Carnot cycle is indicated by the area

bounded by the four steps of the cycle (wnet).

It can be shown that the net total work for the reversible cycle is given by

wnet = (TL − TH)R ln (V4/V1)(3.4)

where TH and TL represent the maximum and minimum absolute temperatures (K)

and V4 and V1 are the initial and final volumes of the step 4 → 1 in the cycle. Since

TL –TH is negative and V4 > V1, the net work (wnet) is invariably negative.

What is the significance of the Carnot cycle? Since the net work is negative in

this ideal process, the Carnot engine represents the maximum possible work for any

given expenditure of energy for a mechanical heat engine. Even in this perfect system some heat (qout) must be dissipated to the surroundings with the result that the

­corresponding amount of work produced is less than the amount of energy expended

to produce it. This consequence of the second law cannot be overstated:

If you are a newcomer to thermodynamics, note carefully the implications of this

­concept. There are limits on the ability of even a flawless heat engine to convert heat

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Chemistry of Sustainable Energy

Extraction

Transport

Energy losses

Combustion

Electricity generation

Electricity distribution

Available

work

Electricity use

Processing

FIGURE 3.2  Degradation of energy resulting in losses in the availability of work. (From

Winterton, N. 2011. Chemistry for Sustainable Technologies. A Foundation. Cambridge,

UK. Reproduced by permission of The Royal Society of Chemistry.)

into useful work. No amount of technical innovation or political intervention will further improve the machine’s performance beyond this ideal. (Tester et al. 2005, p. 89)

How, then, does this fit in with the first law? The quantity of energy is constant

(first law)—but the quality is not. Energy is degraded in every energy conversion

process and entropy is generated (second law). Figure 3.2 presents this concept in a

very practical sense: the energy value associated with energy derived from the sun

and ultimately transformed into fossil fuels, then electricity, degrades significantly

at each step of energy conversion.

The maximum efficiency (η) of a heat engine (the Carnot efficiency) is represented mathematically by

η = wnet/qin = TH − TL/TH or η = 1 − TL/TH(3.5)

where the temperature is expressed in degrees Kelvin. Note the significance of the

temperature limits: a heat engine produces work because of the difference in the

temperature limits; the larger the difference in temperature, the higher the efficiency.

An apt analogy is the work provided by a water wheel. Just as water flows from a

higher level to a lower level to produce work in a water wheel, so heat flows to produce work in a heat engine. In each case, the amount of work is related to the difference in energy (heat energy or potential energy). In both examples, the efficiency

approaches 100% only at the extreme, that is, as the low temperature approaches

absolute zero for the heat engine, and in an infinite separation of water levels for

the water wheel. If there is no differential, the efficiency is zero and the system is at

thermodynamic equilibrium.

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Thermodynamics

(b)

3

2

qin

wnet

4

1

qout

Volume (V )

Absolute temperature (T )

Pressure (P)

(a)

qin

2

1

wnet

3

4

qout

Entropy (S)

FIGURE 3.3  The Otto cycle (a) as a function of pressure and volume and (b) as a function

of absolute temperature and entropy

Mechanical heat engines are ubiquitous in energy conversions. Numerous

other thermodynamic cycles exist on which several engine designs are based (e.g.,

Brayton, Otto, Diesel, Stirling, Ericsson, Rankine). The Otto cycle (Figure 3.3a

and b) is the model for many automobile engines and consists of a four-stroke

cycle: adiabatic compression (1 → 2), constant-volume heating (2 → 3), adiabatic expansion (3 → 4), and constant volume cooling (4 → 1) (Fenn 1982). The

efficiency of a typical gasoline-fueled internal combustion engine is about 30%;

a diesel engine is more efficient at ≈45% (U.S. Department of Energy Energy

Efficiency and Renewable Energy 2003).

Improving engine efficiency is an active area of research. A more recent incarnation is the “X2 rotary engine” whose designers claim can deliver an efficiency of 75%.

This engine, based on the “high efficiency hybrid cycle” design, combines features

of the Otto, Diesel, Rankine, and Atkinson cycles and is closely related to the rotary

engine once featured in Mazda automobiles. The key to the new design is constant

volume combustion plus overexpansion of the burning fuel/air mixture. This allows

all of the fuel to combust, capturing almost all of the energy as work (Szondy 2012).

The Carnot efficiency for a natural gas-fired power plant is 63% based on the

lower heating value (LHV) for natural gas (Higman and van der Burgt 2003) and

49% for a nuclear reactor operating at 300°C (and cooling to 20°C). Given the theoretical limits to energy conversion based on the physical laws of thermodynamics,

what can we do? One approach is to operate energy generation devices at even higher

temperatures, as in next-generation nuclear power plants (Chapter 9). Another tactic

is to either minimize the generation of waste heat or make use of it to improve the

efficiency. There are ways we can eliminate the mechanical middlemen and convert

fuel directly into electricity, as in voltaic cells: without the intermediacy of thermal

waste, these devices are not limited by the Carnot efficiency … although they must

still obey the second law!

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Chemistry of Sustainable Energy

3.4  EXERGY AND LIFE-CYCLE ASSESSMENT

We will look at specific applications of thermodynamics in regard to the efficiency of

energy conversion processes throughout the text. Although the first law of thermodynamics requires that energy is conserved, the second law shows that there is a limit

to the amount of useful work that can be produced, thanks to entropy. The limited

energy corresponding to this maximum is known as exergy which, like energy, has

the units of Joules. This concept is more familiar in the realm of chemistry as Gibbs

free energy. Thus, like free energy, the amount of exergy depends entirely upon the

reference environment of which it is a part. For example, a beaker of water at 100°C

on a 0°C day has a greater exergy content than it does on a 35°C day—as we would

expect, given the discussion of heat engine efficiencies earlier (Dincer and Rosen

2007). Exergy is the degraded energy whose amount is proportional to the amount

of entropy produced: it is the amount of energy that is actually available to do work.

Thus, in Equations 3.4 and 3.5, “wnet” really represents the exergy of the process.

Exergy as a broadly applied (and somewhat fuzzily defined) concept plays an

important role in sustainability analyses, having been applied to ecosystems, industrial systems, and even in economic studies in order to assess their sustainability

(Dewulf et al. 2008). It is a critical facet of life cycle assessment (LCA), a “cradle-tograve” approach to assessing the impact of any process on the environment (where

environment is environment writ large: the system plus surroundings). Exergy is used

because it is the practical, accurate reflection of real, irreversible technology. Thus

a prototypical LCA would take into account the inputs and outputs into and from a

system in terms of exergy and materials. The extraction, processing, manufacture,

distribution, use, and disposal or recycle of raw material inputs are all accounted for.

All outputs are similarly considered, including waste emissions and any byproducts

that are formed. While an in-depth discussion of LCA is not germane to this chapter,

they are important and monumental undertakings that attempt to capture the big

picture of sustainable processes and sustainable development, and are crucial to the

development of any sustainable system.

OTHER RESOURCES

Books

Kirwan, A. D. 2000. Mother Nature’s Two Laws: Ringmasters for Circus Earth. Lessons on

Entropy, Energy, Critical Thinking, and the Practice of Science. River Edge, NJ: World

Scientific.

Fenn, J. B. 1982. Engines, Energy, and Entropy. New York: W.H. Freeman and Company.

REFERENCES

Dewulf, J., H. Van Langenhove, and B. Muys et al. 2008. Exergy: Its potential and limitations

in environmental science & technology. Environ. Sci. Technol. 42:2221–2232.

Dincer, I. and M.A. Rosen. 2007. Chapter 1—Thermodynamic fundamentals. In Exergy, 1–22.

Amsterdam: Elsevier.

Thermodynamics

63

Energy Efficiency and Renewable Energy. Just the Basics—Diesel Engine. 2003. Washington,

DC, U.S. Department of Energy, http://www1.eere.energy.gov/vehiclesandfuels/­

technologies/fcvt_basics.html.

Higman, C. and M. van der Burgt. 2003. Gasification. Burlington, MA: Elsevier (Gulf

Professional Publishing).

Szondy, D. 2012. Liquid piston unveils 40-bhp X2 rotary engine with 75 percent thermal

­efficiency. Gizmag, http://www.gizmag.com/liquidpistol-rotary/24623/

Tester, J.W., E.M. Drake, and M.J. Driscoll et al. 2005. Sustainable Energy. Choosing Among

Options. Cambridge, MA: MIT Press.

Winterton, N. 2011. Chemistry for Sustainable Technologies. A Foundation. Cambridge, UK:

RSC Publishing.

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