70
2 Macroscopic Thermodynamics
Table 2.1 Computed and
observed heat engine
efficiencies a
T low T high η max η opt η obs
Power source
/K
/K
/%
/%
/%
Coal-fired steam plant
298 838
64.1 40
36
CANDU nuclear reactor 298 573
48.0 28
30
Geothermal steam plant 353 523
32.3 17.5 16
a Data from Table 2.1 of Ref. [13], with the permission of the
American Association of Physics Teachers and the authors
As pointed out by Callen [7], that η opt depends only upon the temperatures of
the thermal reservoirs and, in particular, is independent of the thermal conductances
σ 1 and σ 2 , is a remarkable result, indicating that optimal efficiency may well be
independent of the mechanism for irreversible heat transfer between an endoreversible engine and its thermal reservoirs. Indeed, Curzon and Ahlborn themselves
recognized this possibility, and carried out a comparison between η max (for a Carnot
engine), η opt for an endoreversible engine, and η obs determined for three realistic
heat engines (power plants). Their comparison is given in Table 2.1.
2.5.4 The Otto and Diesel Engine Cycles
Two thermodynamic cycles that have been proposed as models for first approximations to actual practical engines are the Otto and Diesel cycles. The Otto cycle
provides a relatively crude approximation to the operation of a typical four-stroke
gasoline engine by first compressing the working fluid adiabatically, then heating it
isochorically, followed by an adiabatic expansion (the so-called power stroke), and
finally cooling it isochorically to its initial state. However, the initial compression
step is not quasi-static (hence not isentropic), and the heating step at constant
volume does not account for the internal combustion process. The Diesel cycle first
compresses the working fluid adiabatically (as for the Otto cycle), then heats the
fluid at constant pressure (the combustion step), expands the fluid adiabatically and
finally, cools the fluid at constant volume to its initial state. Generally speaking,
both Otto and Diesel cycles involve four different temperatures, only two of which
are associated with heat reservoirs. All four temperatures would be required to have
associated heat reservoirs were it required to carry out all four steps reversibly.
The Otto Cycle
The Otto cycle is illustrated in Fig. 2.4a by a parallelogram in a ln P vs. ln V plot
[12]. As the initial compression of the working fluid (which is a hydrocarbon-air
mixture in a typical gasoline engine) from (V 1 , P 1 ) to (V 2 , P 2 ) and the expansion
(generated in a typical gasoline engine by internal combustion) from (V 3 , P 3 ) to
(V 4 , P 4 ) are treated as reversible adiabatic processes, we have Q rev
1→2 (fluid) ≡ 0 and
2 Macroscopic Thermodynamics
Table 2.1 Computed and
observed heat engine
efficiencies a
T low T high η max η opt η obs
Power source
/K
/K
/%
/%
/%
Coal-fired steam plant
298 838
64.1 40
36
CANDU nuclear reactor 298 573
48.0 28
30
Geothermal steam plant 353 523
32.3 17.5 16
a Data from Table 2.1 of Ref. [13], with the permission of the
American Association of Physics Teachers and the authors
As pointed out by Callen [7], that η opt depends only upon the temperatures of
the thermal reservoirs and, in particular, is independent of the thermal conductances
σ 1 and σ 2 , is a remarkable result, indicating that optimal efficiency may well be
independent of the mechanism for irreversible heat transfer between an endoreversible engine and its thermal reservoirs. Indeed, Curzon and Ahlborn themselves
recognized this possibility, and carried out a comparison between η max (for a Carnot
engine), η opt for an endoreversible engine, and η obs determined for three realistic
heat engines (power plants). Their comparison is given in Table 2.1.
2.5.4 The Otto and Diesel Engine Cycles
Two thermodynamic cycles that have been proposed as models for first approximations to actual practical engines are the Otto and Diesel cycles. The Otto cycle
provides a relatively crude approximation to the operation of a typical four-stroke
gasoline engine by first compressing the working fluid adiabatically, then heating it
isochorically, followed by an adiabatic expansion (the so-called power stroke), and
finally cooling it isochorically to its initial state. However, the initial compression
step is not quasi-static (hence not isentropic), and the heating step at constant
volume does not account for the internal combustion process. The Diesel cycle first
compresses the working fluid adiabatically (as for the Otto cycle), then heats the
fluid at constant pressure (the combustion step), expands the fluid adiabatically and
finally, cools the fluid at constant volume to its initial state. Generally speaking,
both Otto and Diesel cycles involve four different temperatures, only two of which
are associated with heat reservoirs. All four temperatures would be required to have
associated heat reservoirs were it required to carry out all four steps reversibly.
The Otto Cycle
The Otto cycle is illustrated in Fig. 2.4a by a parallelogram in a ln P vs. ln V plot
[12]. As the initial compression of the working fluid (which is a hydrocarbon-air
mixture in a typical gasoline engine) from (V 1 , P 1 ) to (V 2 , P 2 ) and the expansion
(generated in a typical gasoline engine by internal combustion) from (V 3 , P 3 ) to
(V 4 , P 4 ) are treated as reversible adiabatic processes, we have Q rev
1→2 (fluid) ≡ 0 and