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3 Second Law and Thermodynamic Cycles: the Carnot Efficiency
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:
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).
Volume (V )
Absolute temperature (T )
FIGURE 3.1 The Carnot cycle (a) as a function of pressure and volume and (b) as a function
of absolute temperature and entropy.
• 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
Chemistry of Sustainable Energy
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
Volume (V )
Absolute temperature (T )
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!
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.
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
Fenn, J. B. 1982. Engines, Energy, and Entropy. New York: W.H. Freeman and Company.
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.
Energy Efficiency and Renewable Energy. Just the Basics—Diesel Engine. 2003. Washington,
DC, U.S. Department of Energy, http://www1.eere.energy.gov/vehiclesandfuels/
Higman, C. and M. van der Burgt. 2003. Gasification. Burlington, MA: Elsevier (Gulf
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: