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Final Technical Report
Development of Low-Irreversibility Engines
Work Sponsored by the
Global Climate and Energy Project
at Stanford University
Submitted by: C. F. Edwards, Assoc. Prof.,
P. A. Caton, S. L. Miller,
M. N., Svrcek, and K.-Y. Teh,
Graduate Researchers
September 5, 2006
- i -
Table of Contents
Page
Introduction 1
Background 1
Organization of the Report 2
Section 1 Late-Phase HCCI Combustion Experiments 4
Section 2 Theoretical Optimization of Adiabatic Reactive Engines 9
Section 3 Numerical Optimization of Non-adiabatic Reactive Engines 16
Section 4 Reactive Engine Architecture 20
Section 5 Limitations imposed by conventional work extraction devices 23
Section 6 Architectural Requirements for Unrestrained-Reaction Engines 29
Section 7 Architectural Requirements for Restrained-Reaction Engines 33
Summary and Future Plans 37
Publications 38
Contact 38
- 1 -
Introduction
An engine is a device that converts some fraction of the energy in a resource into
work. The most work that can be developed by a particular engine design is its reversible
work. The irreversibility of an engine is the difference between the reversible work that
it could develop and the actual work that it performs; it is the lost work. In this project,
we investigate the potential to design and implement engines with significantly reduced
irreversibility, and thereby, improved efficiency.
The relevance of this work to the objective of GCEP is that significant improvements
in efficiency are one of the most effective approaches to reducing greenhouse-gas
emissions. Since the approach we take is fundamental and comprehensive, it enables
improvements with fuels (such as existing hydrocarbons and possible future hydrogen
fuel), and with engines (such as those for transportation, propulsion, and electrical power
generation).
Background
Improvement of engine efficiency requires efforts in two areas: engine architecture
and implementation. By architecture we mean the basic operating cycle or strategy of the
engine (e.g., Brayton versus Stirling). By implementation we mean the degree of
perfection with which the architecture is realized in practice (e.g., the degree to which gas
expansion is isentropic). Our focus is on the former aspect. It is based on the realization
that current internal combustion engines—gasoline, Diesel, and gas turbine—have been
designed based upon an incorrect premise that they are subject to limitations based on the
Carnot efficiency. This misconception persists because these engines are often modeled
as heat engines subject to Carnot limitations, when in fact, they are chemically reactive
engines which are not subject to Carnot proscriptions.
The Carnot misconception has led to erroneous conclusions about the architecture of
efficient combustion engines. The most serious of these is that it is necessary to make the
peak temperature in the cycle as high as possible (so as to improve the Carnot limit).
That this is not correct is confirmed by experiences with homogeneous-charge
compression-ignition (HCCI) engines—engines that achieve higher efficiency than their
SI counterparts while reducing the peak temperature. The ultimate example of both the
inapplicability of the Carnot criterion and the benefits of low-temperature reaction is the
fuel-cell—a reactive engine with first-law efficiency potential in excess of 90% when
operated at low temperatures.
But even before the details of an engine architecture are considered, it is worth
investigating the work potential of the resource that is to be utilized by the engine. This
is the available energy or exergy of the resource. The exergy of a chemical resource is
found by permitting heat and work to be exchanged between the resource and its
environment, and by permitting diffusive and reactive interactions between the two. The
reversible work that can be achieved under these most-general conditions is the exergy X,
given by the expression:
- 2 -
0
0 0 0( )rxn
X U PV T S G G= + − − − ∆ (1)
where G0 = Σ Ni µi0 and ∆G0
rxn = Σ µi0(ν''i − ν'i). The expression indicates that the
absolute upper bound on the work that can be achieved from a resource is dependent
upon both the thermal and the chemical state of the resource with respect to the
environment. For the case of a chemical resource in thermal and mechanical equilibrium
with the environment, the chemical exergy reduces to the difference between the
chemical potential (Gibbs function) of the resource before and after it has reacted and
diffused to become part of the environment, all at T0 and P0.
Unfortunately, it has proven difficult to construct engines which come close to
extracting this limiting amount of work. This is due to a number of complications
including the inability to produce perfect devices (turbines, membranes, catalysts) and the
inability to produce devices without ancillary complications (e.g., water management in
PEM membranes, ionic conductivity in SOFC membranes) that force design choices that
are non-optimal (e.g., operation of a fuel cell at elevated temperatures). These latter
choices lead to engine architectures with what are referred to as option losses—
reductions in the reversible work potential from the exergy limit by choice. In this case,
even if perfect devices are realized to implement the engine architecture, the work
developed will be less than the exergy limit since the architecture does not take advantage
of all possible interactions with the environment. Analysis of detailed engine designs to
determine their work-producing capabilities is the subject of Second-Law Analysis.
It is in the domain between the bounds of what is possible from an exergy standpoint
and those of conventional engine cycles that we seek to improve engine performance.
Stated another way, we propose to improve reactive engine performance by seeking
engine architectures that reduce option losses. This final report documents our findings
over the course of the research project.
Organization of the report
Part I
This report is organized in two parts. The first part, consisting of Sections 1 through
3, examines the following hypothesis:
By restraining the chemical reactions via simultaneous energy extraction in the form
of work from a combustion engine, the exergy destruction can be reduced.
Premised on the fact that the higher theoretical efficiency achievable in a fuel cell
system as compared to a combustion engine is due to higher exergy destruction during
unrestrained combustion reactions, the hypothesis was proposed for the latter
(combustion) system based on its apparent analogy to fuel cell operation, in that when
energy is extracted in the process of reaction, peak temperatures can be held at much
lower values than combustion devices, and entropy generation is correspondingly
reduced.
- 3 -
Three approaches were used to investigate the hypothesis. Firstly, Section 1 presents
the results from late-phase HCCI combustion experiments. The results show that
delaying combustion into the expansion process enables an increase in engine efficiency
not because of reduced combustion irreversibility, but due to a substantial decrease in
heat transfer losses. In other words, by use of simultaneous energy extraction to reduce
peak temperatures, late-phase HCCI strategy allowed us to reduce exergy loss via heat
transfer, not exergy destruction due to combustion.
Secondly, Section 2 provides an analytical proof which shows that exergy destruction
due to combustion is minimized in an adiabatic, homogeneous piston engine when
combustion takes place at the minimum allowable volume. This optimal result is
general—independent of the underlying reaction kinetics—and reduces, in
thermodynamic terms, to the principle of equilibrium entropy minimization. It proves
that the hypothesis put forth above is incorrect for adiabatic systems.
Thirdly, Section 3 discusses the numerical results for the problem of maximizing
work output from a non-adiabatic combustion engine—an extension to the adiabatic
system considered in the previous section. The optimized combustion reaction always
occurs close to the clearance volume, not unlike the optimal solution for an adiabatic
system. The results point to heat transfer at elevated temperatures as the dominant loss
term being minimized, reminiscent of the conclusion drawn from the HCCI experiments.
Furthermore, the results suggest that the potential for piston engine efficiency
improvement via piston motion optimization is marginal.
The results from these three sections reflect a fundamental difference between
engines driven by restrained reactions (e.g., electrochemical reactions in a fuel cell)
versus unrestrained reactions (e.g., combustion), which invalidates the hypothesis
constructed based on the simple analogy drawn between the two.
At the same time, the results also indicate that the general topic of efficient engine
architecture—applicable to both restrained- and unrestrained-reaction engines—may be
approached from the perspective of exergy management. The second part of this report,
consisting of Sections 4 through 7, develops in some detail a systematic approach to
understanding the design and architecture of simple-cycle engines via exergy
management.
Part II
In Section 4, we present a system which identifies three essential processes occurring
within a (chemically) reactive engine: resource preparation, reaction, and work
extraction. For illustration, we apply this framework to decompose the operations of the
Diesel and gas turbine engines, the PEM fuel cell, as well as the SOFC-gas turbine
combined cycle.
Section 5 identifies the limitations imposed by conventional work extraction devices
on engine architecture, namely: (a) only three types of engine work extractors are in wide
use; and (b) the optimal extraction strategy is dictated by the environmental state. The
consequent mismatch with the combustion process is also discussed.
- 4 -
Keeping the focus on combustion systems, Section 6 examines the available options
in preparation (or “positioning”) and reaction of the chemical exergy resource (fuel) with
these limitations in mind. The optimal strategy of equilibrium entropy minimization
(from Section 2) applies readily in such systems.
Lastly, Section 7 elaborates on the architectural requirements for efficient engines
driven by restrained chemical reactions: catalysts and conductors to enable selective
chemical, matter and heat interactions within the reversible limits, as defined—based on
the kinetic-theory viewpoint—by a pair of oppositely-directed processes at the molecular
level. This framework is used to describe the (electrochemical) fuel cell system and its
non-electrochemical counterpart—a reversible, reactive, expansion engine concept.
Section 1: Late-Phase HCCI Combustion Experiments
(The experimental platform used in this research and the results are described in detail in
Pubs. 1 and 2.)
Late-phase homogeneous charge compression ignition (HCCI) shows promise as one
means to implement a low-irreversibility engine. The basic premise is that by combining
the combustion and expansion processes, energy may be extracted during the combustion
process and overall losses may be reduced. This is in some ways analogous to a fuel cell
in that since energy is extracted in the process of reaction, peak temperatures are held at
much lower values than traditional devices, and entropy generation may be
correspondingly reduced.
The studies undertaken utilize a single cylinder, variable compression ratio engine
equipped with variable valve actuation (VVA) as depicted in Figure 1. The use of VVA
permits us to tailor the composition and thermal state of the gases in the cylinder by
mixing controlled amounts of hot, burned gases (from the previous combustion cycle)
with the fresh air-fuel charge. In doing so, the engine can be made to operate as either a
spark-ignition (SI) or homogeneous-charge, compression-ignition (HCCI) engine.
Propane was chosen as the fuel so as to provide typical hydrocarbon kinetics but
without the possibility of inhomogeneity inherent in liquid fueled engines. It was
premixed with the air well ahead (~1 m) of the engine to insure a homogeneous charge.
The equivalence ratio was held at 0.95. Compression ratio was set to 15:1—high enough
to provide a broad range of HCCI operation, low enough to avoid excessive heat losses.
Two sets of intake valve profiles were used to enact late phase combustion. In the
first set, the intake valve was opened at its nominal (SI) opening time, held at full lift, but
then closed prematurely (as required to obtain late phasing). In the second set, the intake
valve was held open throughout its nominal (SI) period, but the lift was reduced to adjust
phasing. In both cases, the exhaust valve was held open throughout the intake stroke so
that by adjusting the intake valve alone the residual fraction (mass fraction of burned gas)
could be varied.
- 5 -
Figure 1: Engine used to study late-phase combustion. Closed-loop, fully flexible electrohydraulic valve
actuation is used to enable exhaust reinduction for initiating late-phase HCCI.
In-cylinder pressure and exhaust emissions are used to determine indicated engine performance metrics.
As illustrated in Figure 2, both sets of valve profiles were capable of initiating
combustion with phasing after top dead center (ATC). The full-duration, partial-lift
strategy was capable of achieving a wider range of residual mass fractions, but the
partial-duration, full-lift strategy was able to achieve overall later combustion timings.
Note that in either case, increasing the residual fraction causes an increase in the extent to
which combustion can be delayed.
Figure 3 shows the effect of combustion phasing on the indicated thermal efficiency
of the cycle. Regardless of the strategy employed, delaying combustion into the
expansion process enables a significant increase in efficiency. As phasing is delayed
from top center to ~15°ATC efficiency rises from ~30% to ~43%. At a given
combustion phasing, some differences due to the choice of valve profiles are evident, but
these are generally smaller effects than the overall change with phasing.
Figure 4 shows the effect of combustion phasing on exhaust temperature as measured
immediately downstream of the exhaust port. For clarity, only data from the full-lift,
partial duration tests are shown. The indicated thermal efficiency is also plotted for
comparison. These data indicate that the rise in efficiency does not come at the expense
of the exhaust enthalpy. This is consistent with a hypothesis that since the combustion
has been delayed, some of the ability to extract energy via expansion work has been lost.
The results are inconsistent with the hypothesis that delaying combustion can reduce
irreversibility without the occurrence of other effects.
- 6 -
Figure 2: Effect of residual fraction on phasing of HCCI combustion as indicated by the crank angle at
which peak pressure occurs. The black line corresponds to full-lift, partial-duration intake valve profiles.
The red line is from full-duration, partial-lift profiles.
Figure 3: Effect of combustion phasing on indicated thermal efficiency.
The black line corresponds to full-lift, partial-duration intake valve profiles.
The red line is from full-duration, partial-lift profiles.
- 7 -
Figure 4: Comparison of the effect of combustion phasing on efficiency and exhaust temperature.
Only results for full-lift, partial-duration intake valve profiles are shown.
Figure 5 resolves the inconsistencies by showing that the increase in efficiency is due
to a drastic reduction in heat losses. Although our objective in studying late-phase
combustion has been to reduce combustion irreversibility, these results show that the
single biggest effect is to reduce overall heat losses (and their concomitant irreversibility)
from the system. This reduction stems from having lowered the peak temperature in the
cylinder (by late phasing) such that the driving potential for losses is significantly
reduced. An additional component of this reduction may be a reduced convective heat
transfer rate due to reduced turbulence (laminarization) during the expansion stroke.
Taken in combination, Figure 4 and Figure 5 suggest that additional improvement in
efficiency via late-phase HCCI combustion is possible, since the gases have still not
undergone optimal expansion. This possibility was explored by using elevated geometric
compression ratios combined with dynamic control of valve timing to alter the effective
extent of compression and expansion. Tests were conducted to determine if an efficiency
benefit could be realized from increasing the compression ratio and delaying the intake-
valve closing so as to increase the extent of expansion relative to compression. However,
no significant change in efficiency was observed using the existing experimental
platform—the additional expansion was offset by a slight increase in thermal losses with
an increased compression ratio—although an engine optimized for HCCI operation with
reduced thermal losses might realize an efficiency benefit with this approach; see Pub. 2
for details.
Figure 6 confirms that the reduction of heat loss using late-phase combustion is not
peculiar to the full-lift, partial-duration valving strategy. Essentially similar results are
obtained with either strategy at a fixed combustion phasing.
- 8 -
Figure 5: Comparison of the effect of combustion phasing on efficiency and heat transfer losses.
Only results for full-lift, partial-duration intake valve profiles are shown.
Figure 6: Effect of combustion phasing on heat loss for both sets of valve profiles.
The black line corresponds to full-lift, partial-duration intake valve profiles.
The red line is from full-duration, partial-lift profiles.
- 9 -
Section 2: Theoretical Optimization of Adiabatic Reactive Engines
(The adiabatic reactive engine model, analysis and results are described in detail in Pubs.
3 and 4.)
In this study, we investigate the possibility of reducing the irreversibility of
combustion engines by extracting work during an adiabatic combustion process. The
question posed is whether, by suitable choice of the volume-time profile, the combustion
and work-extraction (expansion) processes can be combined in an optimal fashion to
improve efficiency. In keeping with our ideal design investigations, the analyses are
performed assuming an adiabatic enclosure. While the experimental results show that
this is not what happens in experiments, this analysis provides a key stepping stone in
understanding what is possible in an optimal-design sense. It is also important for
understanding what is happening in the engine tests, since in the experiments the effects
of heat transfer and reduction of combustion irreversibility cannot be deconvolved.
Optimal Control Problem Description and Solution
Recognizing that entropy-free boundary work is the only mode of energy transfer
allowed for this model system (where heat and mass transfers are absent, as are the
associated entropy flows), the change in system entropy must equal the entropy
generation due to chemical reactions. In other words, the adiabatic ideal engine model
isolates chemical reactions in the system as the sole source of entropy generation. We
seek to minimize the total entropy generation—which equals the overall change in system
entropy over some time interval [0, tf]—through optimal control of the piston motion.
To avoid quenching and incomplete reaction, we also require the gas mixture to react
to near completion as defined by the entropy difference between its instantaneous
thermodynamic state and the corresponding constant-UV equilibrium state, labeled (·)eq.
This is the thermodynamic state which the system would have reached had it been
isolated with internal energy U and volume V fixed at the instantaneous values, and with
its atomic composition unchanged. By the second law, the total system entropy is
maximized at this equilibrium condition, i.e.,
( , , ) ( , , )eq eq
S U V S U V ′≥N N (2)
for all possible mixture compositions N′ with the same atomic composition, including the
actual composition N(t) of the system at the instant. We also note that Gibbs’ equation
reduces to
d d deq eq eq
T S U P V= + (3)
by the definition of equilibrium state.
The revised optimal control problem, written in the standard form, is
- 10 -
0 0 0
0
,0
0
min ( , , ) ( , , )
subject to ( , , ) , (0)
( , , ), (0)
, (0)
( , , ) ( , , )
f
ff
gen t t
i i i i
max
eq eq t tt t
min
S S U V S U V
U P U V q U U
N f U V N N
V q V V
q q
S U V S U V
V V
ε
=
==
= −
= − ⋅ =
= =
= =
≤
≤ +
≥
N N
N
N
N N
ɺ
ɺ
ɺ
(4)
In the absence of the minimum volume constraint V ≥ Vmin, it can be shown that the
Hamiltonian of Problem (4) reduces to
d
( ) (1 ) ,d
eq
eq
P P SH q
T tα α
−= − ⋅ − + (5)
where the multiplier α ≤ 0. Therefore, the switching function is −α(P − Peq)/Teq, and by
Pontryagin’s maximum principle, the optimal bang-bang control that maximizes the
Hamiltonian is
*if ,
if .
max eq
max eq
q P Pq
q P P
− <=
+ > (6)
Since the constant-UV reaction of common fuel/air mixtures is highly exothermic, the
initial mixture pressure P0 is lower than its corresponding equilibrium value Peq,0.
Therefore, the first stage of the optimal control strategy would be to compress the
mixture at the maximum rate, until P = Peq. The mixture is then expanded at the
maximum rate as the system approaches complete reaction, i.e., Seq − S → 0.
The analysis can be extended to solve Problem (4) with active minimum volume
constraint V ≥ Vmin, yielding the optimal bang-bang control
*
if and ,
0 if ,
if .
max eq min
min
max eq
q P P V V
q V V
q P P
− < >
= =+ >
(7)
The first three stages of the optimal control that minimizes the entropy generated in an
adiabatic reactive piston engine subject to the minimum volume constraint are therefore:
1. rapid compression of the reactant mixture to the minimum allowable volume;
2. maintaining the mixture at the minimum volume until the system pressure equals
its constant-UV equilibrium value; followed by
- 11 -
3. rapid expansion as the system approaches complete reaction.
Equilibrium Entropy Minimization as the Optimal Strategy
Absent the minimum volume constraint, it can be shown that the entropy difference L
= Seq − S is a Lyapunov function. This means that the constant-UV equilibrium
thermodynamic states are stable (in the Lyapunov sense) and that the equilibrium states
serve as attractors for the system at any point on the state space (U, V, N). In this case,
the equilibrium Gibbs’ relation, Eq. (3), combined with the first law, dU = −PdV, reduces
the Hamiltonian to
d d d d
(1 ) .d d d d
eqS S L S
Ht t t t
α α α= − + = − (8)
At any time t, one has no control over the instantaneous rate of entropy change dS/dt for
the adiabatic system, which is set by the chemical kinetics and the state at that time.
Instead, the maximum principle calls for choosing the control input that minimizes dL/dt,
which is a function of the rate of volume change q (the control input) as well as the state
(U, V, N). This, in turn, means that the optimal control minimizes the equilibrium
entropy Seq, to which the system is attracted.
In retrospect, the reason that equilibrium entropy minimization is optimal is
straightforward. The objective in Problem (4) to be minimized is the final entropy S(tf),
required to be some distance ε away from its constant-UV equilibrium. Since ε is defined
a priori, it follows that the optimal control strategy calls for minimizing the
corresponding equilibrium entropy Seq.
For an adiabatic system, the relationship dUeq = dU = −PdV = −PdVeq holds when the
subscript (·)eq refers to the constant-UV equilibrium. This means that Gibbs’ equation at
equilibrium, Eq. (3), can be written as
d (1 )d .eq
eq eq
PT S U
P= − (9)
The optimal strategy of equilibrium entropy minimization therefore translates to
either compression (and dU > 0) of the reacting gases if the pressure ratio Peq/P is greater
than unity, or expansion (dU < 0) if Peq/P < 1.
Numerical Examples
A series of three numerical examples are used to illustrate the results discussed above.
Some of the model parameters used in the non-adiabatic numerical optimization study are
retained, e.g., the engine geometry, compression ratio CR = 13, engine speed ω0 = 1800
rpm, and thus the maximum allowable rate of volume change qmax. The initial
temperature T0 was chosen such that ignition occurs very close to the clearance volume
Vmin = V0/CR (top-center) of the slider-crank engine. The reaction completion parameter
ε was fixed at 10−4
L0, where L0 = Seq(U0, V0, Neq,0) − S(U0, V0, N0) = 1.3 kJ/kgmix-K is the
difference in entropy between the initial state and its corresponding equilibrium. The
GRI-Mech 3.0 natural gas combustion mechanism is used in the simulations.
- 12 -
10.50 0.30 0.34
+1
−1
0
+1
−1
0
+1
0
1/CR0
1/CR
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
t/tc
U* v(
t)−
U0 (
MJ/
kg
mix
)
q*v(t)/qmax
1−ξ*v[t]
V*v(t)/V0
1
10−4
10−2
1
10−50
10−25N*
fuel,v(t)/Nfuel,0
S
10.50 0.30 0.36
+1
−1
0
+1
−1
0
+1
0
1/CR0
1/CR
1
0
0.5
1
0
0.5
1
0
0.5
U*(t
)−U
0 (
MJ/
kg
mix
)N*
fuel(t)/Nfuel,0
q*(t)/qmax
1−ξ*[t]
V*(t)/V0
1
0
0.5
1
10−4
10−2
1
10−50
10−25
t/tc
NX
10.50 0.50 0.55
q(t)/qmax
+1
−1
0
1
0
1/CR
1
0
1
0
0.5
1
10−4
10−2
1
0
0.51−ξ[t]
−1
V(t)/V0
Nfuel(t)/Nfuel,0
U(t
)−U
0 (
MJ/
kg
mix
)
1
10−50
10−25
t/tc
Figure 7: Comparison of control input q, states V, U, Nfuel, and the entropy-based extent of reaction “deficit” 1−ξ = L/L0 as functions of time t (normalized by the
period tc = 1/30 sec).
• Left (Columns 1 and 2 / black / labeled (·)v*): the optimal solution for Problem (4) without the minimum volume constraint;
• Center (Columns 3, 4 / blue / labeled (·)*): the optimal solution for Problem (4) with the minimum volume constraint; and
• Right (Columns 5, 6 / red / - ): the slider-crank solution, which is feasible but non-optimal.
- 13 -
Figure 7 compares the feasible—but non-optimal—slider-crank solution (in red) to
the optimal solutions for Problem (4) with the minimum volume constraint (in blue,
labeled (·)v*) and without (in black, labeled (·)
*). The plots on the second and the fourth
columns highlight the optimal control inputs and the state responses near the bang-bang
control switches, marked S (for Switching), or in the volume-constrained case, N and X
(for eNtering and eXiting the constraint) on the volume profiles (row 2 on Figure 7).
The optimal solutions shown are computed based on the analytical results, Eqs. (6)
and (7), which are not explicitly dependent on the choice of the reaction mechanism.
However, the detailed kinetics affects the instantaneous mixture composition,
temperature, and thus pressure, which in turn determines the optimal switching time. For
instance, in the case without the volume constraint, the reactant mixture is compressed
beyond the volume ratio of CR = 13 before ignition (indicated by the precipitous decrease
in fuel (propane) mole number Nfuel), leading to pressure P —as determined by the
reaction mechanism—equaling the corresponding equilibrium pressure Peq, at which
point the product gases are expanded.
The function L = Seq − S being Lyapunov also means that the optimal extent “deficit”
plots (row 5, columns 1 through 4 on Figure 7) can be interpreted as showing the optimal
final times tf,v*(ξf) and tf
*(ξf) (on the horizontal axes) as monotonically increasing
functions of the parameter ξf (on the vertical axes) for the terminal inequality constraint
L(tf) = Seq(tf) − S(tf) ≤ (1 − ξf) L0.
Figure 8 compares the entropy generation from the optimal systems to that from the
non-optimal, slider-crank system. Given an extent parameter ξ (or correspondingly, a
reaction completion parameter ε = (1 − ξf) L0), the figure shows that Sgen,sc ≥ Sgen* ≥ Sgen,v
*
when all three systems react to that same extent.
Figure 8: Comparison of the entropy generated by the optimal systems versus that by the non-optimal
slider-crank system, plotted as functions of the extent “deficit” 1−ξ = L/L0. The Sgen,sc line is not shown on
the left plot because it is not distinguishable from the Sgen* line at the given vertical scale.
For the volume-constrained system, the constraint becomes active at the point labeled N.
The first inequality Sgen,sc ≥ Sgen* is simply the definition of an optimal solution. The
slider-crank line Sgen,sc (red, dashed) begins to deviate noticeably from the optimal Sgen*
line (blue, dash-dotted) at ξ > 0.3, as highlighted on the right. This is because it takes the
- 14 -
slider-crank system longer to complete the compression stroke, allowing relatively more
time for partial oxidation (and entropy generation) of the fuel/air mixture prior to
ignition. However, subsequent difference between the two quantities is miniscule, since
ignition—which accounts for most of the entropy generated—occurs at essentially the
same minimum (clearance) volume for both systems. At extent of reaction ξ = 0.9999 (ε
= 10−4
L0), the “excess” entropy generated from the slider-crank system as compared to
the optimal system is less than 0.1 J/kgmix-K and is not distinguishable on the vertical
scale used on the left.
The second inequality Sgen* ≥ Sgen,v
* arises because the set of admissible control
inputs—i.e., the set of acceptable piston motion profiles {q(t)} —for Problem (4)
including the minimum volume constraint is a subset of control inputs {qv(t)} for the
variant problem excluding the constraint. In other words, there is a “richer” set of control
inputs which solve the variant unconstrained problem. Therefore, the corresponding
(minimum) objective functions are related by
* *
,{ ( )} { ( )}min min
v
gen gen gen gen vq t q t
S S S S= ≥ = (10)
The right plot of Figure 8 shows that the two optimal entropy generation lines overlap
until the minimum volume constraint becomes active (labeled N), at which point they
deviate. Most of the “excess” entropy generated from the volume-constrained optimal
system compared to the unconstrained one, Sgen*− Sgen,v
* is due to the ignition occurring at
a larger volume for the former (=Vmin) as compared to the latter (<Vmin). At extent of
reaction ξ = 0.9999, the difference is about 50 J/kgmix-K.
The general conclusion of this analysis—that equilibrium entropy minimization is the
optimal strategy—is best depicted on internal energy/entropy (U, S) diagrams as shown
on Figure 9. The figure shows the optimal U, S trajectories as well as the corresponding
U, Seq equilibrium trajectories for both the volume-constrained and unconstrained
systems. The latter trajectories, shown as dash-dotted lines, indicate that the reacting gas
mixture is indeed being manipulated such that the equilibrium entropy is constantly
decreasing, both at the initial reactant stage when the system pressure is far below its
corresponding equilibrium pressure (upper plot of Figure 9) as well as after ignition when
the system pressure overshoots its equilibrium value and the reaction approaches
completion (highlighted on the lower plots of Figure 9).
A Suboptimal Solution for Expansion Work Maximization
We note that a terminal constraint, either in the volume form
( ) ,f f
V t V≥ (11)
or in terms of pressure,
( ) ,f f
P t P≤ (12)
- 15 -
needs to be appended to Problem (4) to ensure full expansion of the high pressure
combustion product gases to either some final volume Vf (e.g., symmetric expansion to
the initial volume, Vf = V0) or final pressure Pf (e.g., optimal expansion to the
atmospheric pressure, Pf = 1 atm). However, we are thus far unable to solve the extended
optimal control problem because adding either Constraint (11) or (12) would invalidate
the control synthesis reported here. In particular, we are unable to argue with
mathematical rigor that the function L = Seq − S is Lyapunov when either constraint is
introduced.
0.8130.8120.811
X
0.6
0.4
0.8
0.2
U−
U0 (
MJ/
kg
mix
)
S−S0 (kJ/kgmix-K)
With Vminconstraint
Q
to Q0
0.7600.759
0.6
0.4
0.8
0.2
U−
U0 (
MJ/
kg
mix
)
0.758
S
S−S0 (kJ/kgmix-K)
No Vminconstraint
Q
to Q0
0
Optimal
U, S trajectories
1.00.50
0.5
−0.5
1.0
−1.0
U−
U0 (
MJ/
kg
mix
)
S−S0 (kJ/kgmix-K)−0.5 1.5
U, Seq traj.
N X
Q
Q0
S
Figure 9: The internal energy/entropy (U, S) diagrams for adiabatic systems given optimal piston motion
profiles. The actual trajectories (solid) and the corresponding constant-UV equilibrium trajectories (dash-
dotted) are shown for systems with active minimum volume constraint (blue) and without (black).
Suboptimal trajectories when the symmetric expansion constraint is added are also shown (dotted lines).
Nevertheless, an adequate suboptimal solution for the optimal control problem with
full expansion can be devised based on the results obtained thus far. The control would
follow the optimal strategy per Equation (6) or (7), and then react at constant volume to
reach the minimum equilibrium entropy state (labeled Q—for eQuilibrium—on Figure
9). The equilibrated mixture can then be expanded quasi-statically to some final state Vf
or Pf.
- 16 -
Figure 9 illustrates the state trajectories using this suboptimal control for a symmetric
expansion constraint Vf = V0, shown as dotted lines horizontally across and vertically
down to the final states labeled Q0. More important than the detailed trajectories is the
recognition that the final entropy of the optimal and fully-expanded system is bounded
above by the value Seq at equilibrium state Q. In other words, the entropy generation for
the full-expansion optimal system will not exceed that for the original optimal system
(considered in Problem (4)) by more than the quantity ε, which is very small for common
fuel/air systems, as the numerical examples above exemplify.
Section 3: Numerical Optimization of Non-adiabatic Reactive Engines
(Details of the model, the numerical algorithm, and the preliminary optimization results
are described in detail in Pub. 5.)
In this section, the analysis of optimal adiabatic reactive systems reported above is
extended to non-adiabatic systems. The purpose of the analysis is to determine the extent
of engine efficiency improvement when the work extraction process is optimized.
The model developed to explore this question includes a single-step combustion
mechanism for propane combustion, and an empirical heat transfer term to account for
the main source of energy dissipation from the system. The model also adopts
specifications from the single-cylinder engine used in the HCCI research (where the heat
transfer observations were made; see Section 1). The problem of maximizing expansion-
work output is then cast as an optimal control problem for a constrained dynamical
system with volume V, temperature T, and composition (mole numbers) Ni of the gas
mixture in the engine as state variables. The rate of volume change dV/dt , which is
proportional to the piston velocity, is the control variable. The conventional slider-crank
piston velocity profile is used to initialize the gradient-based local optimization
calculations.
Figure 10 shows a set of control and state points which constitute a solution set for
the optimal control problem. Compared to the slider-crank solution, the optimized piston
motion yields increase in expansion work output of 2.6%. A feature unchanged from the
slider-crank solution is that ignition still occurs close to the clearance volume Vinit/CR at
optimality (even though flexibility is built into the optimization algorithm to allow
ignition away from the clearance volume). This maximizes system pressure, which
maintains high expansion work output. On the other hand, the attendant high system
temperature would also keep the heat loss rate high.
Despite that, the engine efficiency improves as a consequence of lower total heat
transfer loss, achieved by shortening the time spent at elevated temperatures. This is in
turn accomplished by delaying the combustion phasing well beyond the midpoint (TC of
the slider-crank) and then expanding the product gases at the maximum allowable rate.
Figure 10c also indicates that at optimality, the final temperature of the gas mixture is
slightly higher as compared to the conventional case. Therefore, the efficiency increase
cannot be attributed to reduction in exhaust (waste) enthalpy, but is due to lower heat
transfer losses. This is reminiscent of results obtained in past studies of thermally
- 17 -
insulated (low heat rejection) diesel engines. The explanation is straightforward: even
though heat loss from the system is reduced, not all the available energy retained is
extracted as work. Consequently, some amount is wasted in the form of exhaust gas at a
higher temperature. Full utilization of the exergy that is retained would require the use of
secondary expansion and/or heat recovery devices that are not considered in the present
model.
10−12
0 0.2 0.4 0.6 0.8 1
Ex
ten
t o
f re
acti
on
, ξ
Time, t (normalized by tf = 2πω−1)
d.
10−8
10−4
100
tign
5
1
Tem
per
atu
re, T
(n
orm
aliz
ed b
y T
init)
c.
4
2
3
0
6
0 0.2 0.4 0.6 0.8 1
Time, t (normalized by tf = 2πω−1)
Tw = Tinit
1
1/CR
0 Vo
lum
e, V
(n
orm
aliz
ed b
y V
init)b.
∆texp
−1Rat
e o
f v
olu
me
chan
ge,
q
(no
rmal
ized
by
qm
ax)
a.
0
+1
Figure 10: The optimal solution set (including control q, states V, T and ξ) as functions of time.
The discrete control and state points are plotted at the grid points (shown as crosses) and at mid-intervals
(shown as dots). For comparison, the slider-crank solution used to initialize the calculation is also shown
as solid lines.
The sensitivity of the optimal solution to three model parameters was also considered.
Figure 11 shows similar optimal piston motion and extent of reaction profiles for the
same engine operating at speeds ω = 1800 rpm and 2000 rpm: slow compression to the
clearance volume at which point ignition occurs, followed by fast expansion to the initial
volume. Figure 12 indicates that the optimal expansion work output increases with
engine speed (solid line labeled “vary ω”). However, the slider-crank work output
increases more over the same range of speeds. As a result, the efficiency gain via piston
motion optimization diminishes from 2.6% to 1.6%.
- 18 -
1800 rpm
2000 rpm
1/CR
0
Volu
me,
V
(norm
aliz
ed b
y V
init) 1
a.
10−12
0 0.2 0.4 0.6 0.8 1
Exte
nt
of
reac
tion, ξ
Time, t (normalized by tf = 2πω−1)
b.
10−8
10−4
100
2000 rpm
1800 rpm
Figure 11: Comparison of the optimal solution sets for maximum expansion work output from a piston
engine operating at two engine speeds, showing similar piston motion profiles of slow compression to the
clearance volume—at which point the ignition occurs—followed by fast expansion to the initial volume.
Ex
pan
sio
n w
ork
ou
tpu
t (d
imen
sio
nle
ss)
4.35
Engine speed (rpm)
Optimized (vary ω)
Initial (slider-crank)
200019001800
4.40
4.30
4.45
4.25
(vary tf )
Figure 12: Variation of optimal work output with engine rotational speed (solid line labeled “vary ω”),
showing diminishing gain from slider-crank solution as speed increases. Optimal work output does not
depend on the cycle time tf if the maximum piston linear speed is fixed (dashed line labeled “vary tf”). The
average values and 95% confidence intervals shown are calculated based on 20 optimal solution sets.
- 19 -
For a given engine geometry, the maximum allowable rate of volume change (or,
equivalently, the piston linear speed) varies directly with engine speed, qmax ∝ω, whereas
the cycle time varies inversely with speed, tf ∝ω−1
. Therefore, increasing the engine
speed shortens the (total) cycle time tf over which heat transfer occurs, as well as
reducing the minimum time ∆V/qmax needed for the piston to displace some volume ∆V
during expansion, when heat loss rate is high due to high post-combustion gas
temperature.
A second set of calculations was used to de-convolve these two effects by studying
the sensitivity of the optimal solution to cycle time while the maximum piston speed
remains fixed. (As rotational speed ω increases and cycle time decreases, the maximum
piston speed may be fixed, to some extent, by changing the slider-crank geometry, i.e.
increasing the ratio of connecting rod to throw lengths.) Figure 12 shows that the optimal
work output is approximately constant under these conditions (line labeled “vary tf”).
Figure 13 shows how expansion work output varies with the mean wall temperature
Tw used in the heat transfer model. Higher wall temperature reduces the temperature
difference and therefore the rate of heat transfer loss from the hot post-combustion gas
mixture to the cylinder wall. The optimized piston motion takes advantage of this effect,
such that the overall expansion work output increases with the wall temperature.
In contrast, the initial (slider-crank) solutions show lower sensitivity to wall
temperature changes. In fact, the expansion work output decreases slightly as wall
temperature increases within the range considered in this set of calculations. The reason
is that, within this range of wall temperatures, the slider-crank solutions yield over-
advanced combustion phasing. Therefore, the reduction in heat transfer losses due to
higher wall temperature further advances the phasing to before TC, which lowers the
expansion work output. Within the range of wall temperatures considered, the efficiency
gain via piston motion optimization varies from 2.3% to 2.9%.
Expan
sion w
ork
outp
ut
(dim
ensi
onle
ss)
4.35
Wall temperature, Tw (K)
Optimized
Initial (slider-crank)
410400390
4.40
4.30
4.45
4.25
Figure 13: Variation of optimal expansion work output with mean wall temperature Tw.
The average and 95% confidence intervals shown are calculated based on 20 optimal solution sets.
- 20 -
The results from these sensitivity studies, taken together, support the assertion that
heat transfer at elevated temperatures is the dominant loss term being minimized in these
optimization calculations. However, when compared to the conventional slider-crank
piston profile, the increase in expansion work output via piston motion optimization
never exceeds 3%. Considering the marginal potential for efficiency improvement—in
particular when weighed against the engineering challenge of designing such an optimal
engine—we decided not to continue the numerical piston motion optimization effort.
For the range of model parameters considered in the sensitivity studies, we also
observed that the optimized combustion reaction always occurs close to the clearance
volume. This suggests that unrestrained combustion reaction at the minimum allowable
volume, while lossy in the exergetic sense, is likely to be as good as it gets. It is also
reminiscent of the adiabatic system optimization results obtained previously: regardless
of the dynamics involved, the optimal piston motion for an adiabatic reactive system is
bang-bang (moving at either maximum compression or expansion speed) or, when
constrained by the geometry, remains stationary (no work extraction, i.e., reaction at
constant volume).
Section 4: Reactive Engine Architecture
An engine is a device that converts the exergy of a resource into work. We confine
our attention to the space of reactive (chemical) exergy resources, and aim to develop a
way to bridge between this abstract notion of resource exergy and the concrete limitations
set forth by second-law analysis of specific engine designs. It is, in effect, an attempt to
systematically enumerate the possible choices that can be made so as to make the engine
architecture design problem less one of inspiration and invention, but more one of
systematic analysis of possibilities and capabilities.
In this section, we present a classification system that categorizes the processes
occurring within a reactive engine. We assert that all engines employ three essential
processes that occur at least once, but sometimes more than once, in their architecture:
1. Preparation/positioning of the resource: This involves creation of the correct
composition and phase as required by the chemical reaction process as well as
positioning of the state of that resource so as to optimize chemical reaction and
work extraction.
2. Reaction to an extractable form: This requires that the chemical resource be
activated by some means (e.g. thermally or electrically) so as to open pathways
for reaction, followed by either conversion to products (chemical species with
minimum achievable bond energy but high sensible energy) or to another
chemical form that is more easily coupled to extraction than the original resource.
3. Extraction of work from the resource: This is the essential conversion step
wherein the exergy of the resource is reduced and work is done by the engine.
For example, consider the Diesel engine. Two resources are selected for use: a
hydrocarbon fuel and atmospheric air. The air resource is prepared by compressing it to
- 21 -
high pressure and temperature. The high temperature ensures that thermal activation of
chemical reaction will occur at the time of fuel injection. The choice of a high
compression ratio ensures that the post-reaction resource (high-pressure, high-sensible-
energy gases) are well positioned for extraction by the expansion process. The fuel
resource is also prepared by pressurization to the required injection pressure. Reaction to
an extractable form occurs following fuel injection. Because of the positioning of the
thermal state of the air and the reactivity of the fuel, autoignition occurs (a thermal form
of activation) and some part of the total exergy of the resources is transferred into a more
extractable, thermo-mechanical form of exergy in the products. Extraction of work from
the resource occurs on the expansion stroke of the engine in the form of boundary work.
Since this takes place in a batch process, this work is computed in terms of the integral of
PdV for the expansion process.
The gas turbine engine exhibits essentially the same architecture as the Diesel but
does so using a steady flow process. Air and a hydrocarbon fuel remain the resources of
choice, and both are prepared by compression to high pressure. Unlike the Diesel,
however, flame holding can be used to provide chemical activation in the gas turbine
engine. This relaxes the requirements on fuel properties and post-compression air
temperature relative to the Diesel engine, that can be used to advantage in enhanced
designs using intercooling and reheating. The essential processes, however, remain the
same: reaction takes place such that the exergy is transferred from primarily chemical to
primarily sensible, and extraction occurs by doing boundary work on the rotors of the
expansion turbine. Since this takes place in a flowing process, the work output is
computed in terms of the integral of VdP for the expansion process.
The PEM fuel cell provides an example that accentuates both the similarities and
differences between engines. Here the resources are hydrogen and air. These are
positioned by compression (pumping) and humidification. A key difference between the
fuel cell and the combustion engines cited above is that the former does not use a
reaction stage that transforms the resources into products, but instead it accomplishes a
transformation into a more extractable species. In the case of a PEM fuel cell, hydrogen
is transformed into protons and electrons. The key here is the formation of free (in the
conduction band of platinum) electrons that provide the ability for efficient work
extraction when transferred to a motor. Another important difference is that unlike
combustion engines that discharge their depleted resource (products) to the environment,
this is not possible in the case of electrons. Thus, a second reaction stage is required—
one that can provide a sink for electrons from the motor and convert the electrons into a
stable form that can be discharged in a charge-neutral fashion to the environment. This
electron sink and conversion takes place on the cathode where oxygen, protons, and
electrons react to form water. At both electrodes the chemical reactions are activated by
a combination of thermal and electrical processes, with the relative contributions
depending on the current density demanded by the load and the temperature of the cell.
A final observation is that the actual conversion to work does not occur in the fuel
cell, but in a motor that is distinct from the cell. As such, a fuel cell is not an engine but a
chemical transformation device that is capable of sourcing and sinking a very special
- 22 -
chemical species, the electron. The combination of a fuel cell and an electric motor,
however, does constitute an engine.1
One of the reasons that fuel cells are considered to be very different from combustion
devices is this requirement to have two half-cell reactions in order to be able to create a
potential difference. We note here that this is not unique to fuel cells but is inherent in
any energy system that uses a closed cycle for its working fluid. For example,
considering the case of thermal energy conversion in a Rankine cycle, the steam (or other
vapor) is supplied to the turbine at high pressure and enthalpy by the boiler system, but is
removed by a low pressure (via low temperature) condenser system before being
recycled. While it is true that the steam could be discharged to the environment (whereas
electrons cannot) it is more advantageous to use a closed cycle where the resource is
managed at both the input and output of the extraction device (in this case the steam
turbine).2 A similar situation exists in chemical energy systems that use carrier species
for intermediate reactant transport or conversion. (Black liquor in the pulp/paper cycle
and chemical looping for oxygen separation before combustion are two examples that
may also be considered.)
Even more complicated systems such as the SOFC/gas turbine combined cycle using
natural gas and air as resources can be broken down into a sequence of these three
architectural components: preparation, reaction, and extraction; see Table I.
1 Another reason for emphasizing that it is the fuel-cell-plus-motor combination that constitutes an engine
is to maintain a fair comparison of efficiencies with other devices. While electric motors can achieve
efficiencies on the order of 90%, they do incur additional energy loss that is often disregarded in the
discussion of fuel cell efficiency. The analogous situation for a piston engine would be to quote indicated
efficiencies in place of brake values. In fact the ratio of the two—the mechanical efficiency of the piston
engine—is usually of order 90%, essentially the same as that of the electric motor.
2 Another example of a two-sided thermal energy system, which has been proposed for use with advanced
nuclear energy systems (HTGR), is the closed Brayton cycle, with helium as the working fluid.
- 23 -
Table I: Decomposition of the SOFC/gas turbine combined cycle.
Reformer → → Fuel cell + motor → → Gas turbine
Preparation/positioning of the resource
• Mix fuel (natural gas)
with steam, preheat
• Compress fuel gas
• Preheat and compress
air
• Vitiated anode gas
mixed with air
Reaction to an extractable form
• Thermal activation
(with Ni catalyst)
• Natural gas → syngas
• CO + H2O → H2 + CO2
(shift reaction)
• Electrical activation
(with Pt catalyst)
• H2 + O2−
→ H2O + 2e−
(anode)
• 0.5O2 + 2e− → O
2−
(cathode)
• Thermal activation (by
flame holding)
• Combustion (conversion
from chemical to
sensible energy)
Extraction of work from the resource
• Electrical work:
electrons moving
through motor windings
• Boundary work: V dP
flowing, expansion
work
Section 5: Limitations imposed by conventional work extraction devices
The framework presented in the previous section decomposes the essential elements
of a generic reactive energy conversion system into three types: preparation, reaction, and
extraction. Following this framework, it is apparent that the third component, i.e., the
extraction device, is the critical one in any energy system. The preparation and reaction
stages are absolutely necessary and present significant challenges, but their role is simply
that of providing impedance matching between the exergy resource and the extraction
stage. We assert that it is our ability to extract work that sets the key limitations on
engine design.
In this section, we identify two specific aspects of engine work extractors that place
limits on the optimal engine architecture, namely: (a) only three types of work extractors
are in wide use; and (b) the optimal extraction strategy is dictated by the environmental
state. The consequent mismatch with the combustion process is also discussed.
Short List of Work Extractors in Wide Use
A survey of work-producing devices over the course of history shows a rather
surprising result: That we have only three types of extraction devices in engines in wide
commercial use.
- 24 -
1. Impulse machines: Devices that convert kinetic energy to work, such as the water
wheel and wind turbine.
2. Expansion machines: Devices that convert thermal energy to work using either
batch or flowing expander, such as the piston/cylinder and gas turbine.
3. Electrical machines: Devices that use paired sources and sinks of electrically
charged species to do work by virtue of magnetic-field interactions. While this is
most commonly accomplished using electrons (due to the ease of conduction in
metals), any charged species may be used in this manner. (Recall, e.g., the MHD
efforts of the 1970s.)3
The short list brings three ideas to mind: Firstly, while we know of no osmotic or
surface-tension devices which convert, respectively, chemical or interfacial potential
energy to work, we have yet to systematically explore all possibilities to develop an
alternative type of extraction device upon which improved engines might be built. We
defer discussion of this issue for the time being, and focus on requirements for optimal
design using existing extraction approaches—in particular, extractors based on fluid
expansion.
Secondly, we note that kinetic energy also provides an entropy-free mechanism for
energy transfer not in the form of work. While generation of kinetic energy is often used
visibly in certain types of engines (e.g., turbojets and rocket motors), it is also used
internally as an intermediate carrier within other energy-transformation devices (e.g., the
flywheel of a piston engine and the nozzles of a gas turbine).
Thirdly, since the list is so short, and the requirements for the preparation and
reaction stages are entirely dictated by the extraction device, it may be productive to
organize the discussion of efficient engine design by working backward from these few
options to the appropriate architecture for the overall machine. That is the approach we
have adopted in this section and the next.
Before moving to a discussion of what is possible using these devices, it is important
to make one additional observation about the methods we use to develop work: None of
the machines that actually produce the work do so with simultaneous chemical reaction.
In the case of the gas turbine and piston engines, combustion is completed (or at least it is
so intended) before the expansion process begins. Even the fuel cell—often spoken of as
having work extraction during reaction—actually accomplishes the reactions separately
from the work extraction. And rocket motors, perhaps the best known case of reaction
3 The question might be asked is why we do not include potential energy machines in this listing. For
example, a pulley with a weight might be considered a machine that uses high-altitude mass as a resource
to do work. While we admit that this could be considered as another type of extractor, we would argue that
it is of no practical importance to our analysis. A more relevant argument could be made about
hydroelectric power systems where potential energy is converted to work. However, since our discussion is
confined to the extraction device, this reduces to the problem of an impulse machine since the potential
energy of the resource is first converted to kinetic energy during its fall before being extracted by the
turbine.
- 25 -
during expansion (shifting in equilibrium composition with thermal state, if not active
burning) are designed with the intent that reaction should not be coupled with extraction.
The point of interest here is that all of these systems, except the fuel cell, are cases
where the extraction strategy is intended to be adiabatic, and as such the proof of
optimality without simultaneous reaction holds for these cases (see Section 2). However,
in counterpoint, this raises the question of whether any serious exploration has been
devoted to systems in which heat transfer is not only permitted (e.g., via passive losses)
but in fact used to our advantage in the reaction process. The former situation is already
discussed in detail in Sections 1 and 3 in connection with optimization of HCCI
combustion phasing and piston velocity profile, respectively, in non-adiabatic piston
engine systems. The latter is better viewed as a method to achieving reversible chemical
transformations than as a way to building better engines—although the two objectives
may merge in the development of reversible chemical engines.
Work Extraction and the Environmental State
Limiting our attention to work extraction devices based on fluid expansion, the
challenge of optimizing the relevant device efficiencies may be divided into three parts:
1. Theoretical device performance: What are the theoretical capabilities of an
expansion-based extraction device, assuming ideal implementation?
2. Actual device losses: How well does the device perform relative to its theoretical
limit? In other words, is it lossy?
3. Positioning the device with respect to environmental interactions: What is the
best way to operate the extractor with respect to the environment?
The first two issues are considered in conventional energy system design process:
How should, or could, a system work? How well does it actually work? The third is less
obvious and ties back to the discussion in Section 4 regarding the requirement of two
half-cell reactions in the fuel cell: For a system with one input and one output, there
exists a choice of how those connections are made with respect to the environment. In
traditional combustion engines, the working fluid is discharged to the environment as
“exhaust.” In a fuel cell, this is not possible due to the need to maintain overall charge
neutrality while generating an electrical potential. In heat engines such as the Rankine
and closed Brayton cycles, neither connection of the expander is made directly to the
environment; the states at both connections are positioned so as to provide the optimum
differential conditions for the expander. The subsystems that set those states instead
make the connections to the environment.
This question of whether to discharge a working fluid directly to the environment is
critical both for optimization of system performance and for environmental reasons. The
environmental aspects are well known, e.g., direct health effects, photochemical smog
and climate change. The system optimization impacts, on the other hand, are more
subtle. We note that exergy is defined as the reversible work that can be extracted from a
resource when it is allowed to interact with an environment; see Eq. (1). The way that
- 26 -
the optimal performance is realized is for the resource to be transformed reversibly to the
exact conditions of the environment itself; it must be cooled, expanded, and reacted to the
concentrations of species present in the environment in order to achieve this maximum,
reversible work. This implies that any matter that is released in a state that is not in
complete equilibrium with the environment still has the potential to do additional work
and thereby to improve efficiency. Stated in an inverse manner, any matter that is
discharged with non-zero exergy has the ability to drive environmental change. For these
reasons, the optimal situation, both environmentally and for efficiency, would be to
discharge matter only at the conditions of the environment.
If the choice is made not to discharge matter (e.g., in the case of closed cycles)
choices still need to be made with respect to environmental interactions. The reason is
that regardless of the overall system architecture, entropy that enters the system with the
energy resource (be it heat or matter) or that is generated within the system (due to device
imperfection) must be discharged. This can take place only via heat or matter.
Therefore, precluding matter transfer would necessitate heat interactions with the
environment, which must be managed in order to achieve high system efficiency. Such is
the case, for example, with conventional steam cycles where the pressure is lowered
below atmospheric to improve extraction in the turbine. This design choice is limited,
however, by the ability to condense steam and thereby maintain the condition of low exit
pressure. This pressure is ultimately set by the temperature of the entropy sink (via heat
transfer) available in the environment.
A final possibility to consider is whether it might be desirable, even if not required, to
manage both the inlet and outlet states of the extraction device at some state that is away
from that of the environment. Reasons to consider this possibility include material
temperature and stress limitations as well as the ability to realize near-adiabatic
conditions. Consider, for example, the case of the gas turbine engine. It is well known
that the efficiency of this engine improves as the pressure ratio is increased (improving
the ability to expand the gas and extract work). Unfortunately the ability to use higher
pressure ratios is limited not only by turbomachinery performance but by the turbine inlet
temperature. However, instead of starting the cycle with an environmental inlet
condition—air at 1 bar, 295 K, consider the alternative of pre-cooling the air to 100 K,
perhaps as a consequence of some other process that needed to take place at cryogenic
conditions. Now it would be possible to improve the performance of the gas turbine
since the complete cycle could be shifted to lower temperatures and therefore use a
higher temperature swing while still maintaining the same peak temperature limitation.
Whether this benefit outweighs the cost of the work required to obtain the low
temperature remains to be seen. In some cases, for example the steam turbine, the benefit
clearly outweighs the disadvantages. The point here is that such cases must be
considered in a general evaluation of how best to position the extraction.
Mismatch of Expansion Machines with the Combustion Process
The discussion above relates the first and third issues listed at the start of the previous
subsection, i.e., the theoretical device performance when operating optimally with respect
to the environment, and points to the mismatch between expansion machines and
conventional combustion process.
- 27 -
Consider, for instance, the combustion of a stoichiometric mixture of propane and air,
initially at room temperature T0 = 300 K. Figure 14 shows the volume and pressure ratios
of the combustion products plotted against the temperature ratio (referenced to product
gas mixture at T0 = 300 K, P0 = 1 atm). The figure shows that the adiabatic combustion
product temperature is so high (~2200 K, or temperature ratio ~7.5) that it would require
a volume expansion ratio in excess of 500:1 to be able to expand the gas back to the
reactant temperature. Due to the even weaker dependence of gas temperature on
pressure, a pressure ratio in excess of 5000 would be required to successfully expand this
mixture to the environmental temperature T0. These values should be contrasted with
typical values for piston engines (10 to 20 in volume ratio) and gas turbine engines (up to
50 in pressure ratio).
Adiabatic
Stoich.
Combustion
Today's
Materials
Today's Gas Turbines
Today's IC Engines
Propane
Helium
Pre
ssure
and V
olu
me
Rat
ios
103
102
101
100
1 2 3 4 5 6 7 8 9 10
Temperature Ratio
Pressure Ratio
Volume Ratio
Figure 14: Volume and pressure ratios as functions of temperature ratio (reference T0 = 300 K, P0 = 1 atm)
for the isentropic expansion of stoichiometric propane/air mixture combustion products (solid lines) and of
helium gas with fixed specific heat ratio k = 5/3 (dash-dotted lines).
One approach to providing a better match might be to use a working fluid with a
better specific heat ratio. The dash-dotted lines on Figure 14 show the expansion
performance when a monatomic gas such as helium (specific heat ratio k = 5/3) is used as
the working fluid. While the consequent expansion ratios can be significantly reduced—
20:1 in volume for a temperature ratio of 7.5—one is now committed to operating the
system as a heat engine, not an internal combustion engine, and ductile materials to
withstand the 2200 K hot-side temperature at pressure do not exist.
In fact it is ductile material working temperatures that impose many limits on engine
performance. In addition to the obvious examples like the gas turbine temperature limit
cited above, material temperature limitations are also felt in other ways. One of these is
the need for piston engine cooling. Given that the intermittent combustion process
reaches temperatures well above material melting points, it is no surprise that active
cooling is used to provide thermal management for piston engines. The issue that arises,
however, is that under these conditions, the heat loss from the working fluid through the
- 28 -
combustion process and expansion stroke is significant and we do a poor job of realizing
the ideal process of an adiabatic (and reversible) extraction process. (About 1/3 of the
heating value of the fuel is lost to engine coolant.) We note that if peak flame
temperatures in the piston engine were reduced significantly, say to 1500 K, then it might
be possible to achieve a batch expansion process that was more nearly adiabatic than we
have at present.
It is the same issue of excessive heat loss that sets the upper limit on the compression
(and hence expansion) ratio in some Diesel engines. Large heat transfer losses are
usually associated with poor surface-to-volume ratio, but the root cause remains the
requirement to cool the load-bearing walls of the cylinder. In this connection it is worth
considering whether it may be possible to reduce the peak combustion temperature such
that heat loss is less significant. One way to do this is to operate away from
stoichiometric equivalence ratios. We note that a peak temperature of ~1500 K chosen to
reduce heat loss would correspond to equivalence ratios in the vicinity of either 0.5 on the
lean side, or just over 2.0 on the rich side. The corresponding temperature ratio ~5 would
require an isentropic expansion ratio in the vicinity of 100:1—still well above typical
values. In fact, in order to reduce the volume ratio to ~20, the temperature ratio would
need to be reduced to about 3:1 (peak temperature of 900 K) with corresponding
equivalence ratios of less than 0.25 or greater than 3.5.
From a traditional engine point of view, these extreme equivalence ratios present a
number of problems. One obvious problem is getting reactions to occur at such low
temperatures, although catalysis may be used in some situations. Another is low power
density. An engine that processes four times the stoichiometric amount of air will
produce relatively little output per unit of energy that is cycled internally to support the
compression process. (In gas turbine engines this is referred to as the back-work ratio.)
If an attempt is made to reduce the amount of work committed to air compression—for
example by operating far to the rich side—then it will be necessary to provide some sort
of bottoming cycle to the engine in order to consume the fuel.
The point of this discussion should be clear by now: conventional combustion
temperatures are poorly suited to gas expansion given the range of volume and pressure
ratios over which we can achieve adiabatic, reversible expansion. If power density is not
to be sacrificed (by running at very lean or dilute conditions, for example) then some
form of either bottoming or topping cycle is needed in order to obtain efficient energy
conversion. This is done routinely in electric power generation systems where a steam
cycle is used to bottom a gas turbine engine. In this application the steam cycle can be
executed in nearly ideal fashion due to the excellent match between the turbine exhaust
temperature and the isentropic expansion pressure ratios available using heated steam.
For example, efficiencies in excess of 60% have been reported for the GE “H” system
combined cycle power plant. Another example is the combination of the solid oxide fuel
cell used as a topping cycle for a gas turbine engine. This provides a particularly
interesting example in that not only does the SOFC require afterburning due to fuel
utilization limitations—hence the combustor of the GT cycle is a necessity—but the
vitiation of the fuel gas mixture by the fuel cell before being combusted for the gas
- 29 -
turbine provides an almost ideal match to its expansion requirements. Combined cycle
efficiencies in excess of 70% have been reported for this configuration.
Three approaches may be taken to address this mismatch between our ability to
execute adiabatic, reversible expansions and the process of combustion:
1. Compound expansion processes: We can continue to optimize systems such as the
combined cycle power plant (NGCC or IGCC) so as to maximize extraction
despite the mismatch;
2. Improved expansion devices: We can investigate ways to extend the range of
expansion processes to significantly higher volume or pressure ratios while
maintaining (near) adiabatic condition. This might require materials development
or consideration of using sub-ambient temperature conditions (e.g., combined
with cryogenic air separation) to access wider temperature ranges within existing
material limitations;
3. Modification of the reaction process: As illustrated by the SOFC/gas turbine
example, it is possible to extract work from the fuel gas stream in a fashion that
has low overall exergy loss. The possibility that we can interact with the fuel
either during preprocessing (a partial transformation with work production) or
during the final conversion to products in such a way as to improve extractability
in the gas expander is worth considering.
Section 6: Architectural Requirements for Unrestrained-Reaction Engines
The previous section focused on the limited options for work extraction which are
currently in wide use, and the resulting mismatch with the combustion process. In this
section, we retain the focus on the combustion engine and examine the two remaining
architectural components, i.e., the preparation and reaction stages of the engine system
since, upon making choices with respect to the product expansion and its connection to
the environment, what remains is to define the best way to position the resource for
reaction such that the effectiveness of the work extraction process is maximized.
For instance, we would like to operate at states that minimize inadvertent exergy loss
(e.g., heat loss) but allow for extraction efficiencies near the limits of the device. One
additional concern is that in many cases the architecture for the reaction process is itself
lossy. This is exemplified by the use of unrestrained combustion reactions, and the
associated exergy destruction, that is common in all existing piston and gas turbine
engines. We would like to consider whether these architectural irreversibilities can either
be reduced (low-irreversibility designs) or eliminated (reversible reaction designs) and
whether we can accomplish this while matching results of the reaction process to the
expansion process.
Figure 15 illustrates the situation, showing a property surface for water in the space of
internal energy U, entropy S, and volume V. (This is the surface described by the
fundamental relation of thermodynamics.) The reason for choosing water as the base
species is that it is useful for illustrating both the properties of a typical working fluid (in
- 30 -
steam cycles) as well as for illustrating a typical product of a chemical reaction (between
hydrogen and oxygen). Note that the surface depicted is one which shows the properties
of water in chemical equilibrium, that is, the effects of dissociation have been included in
the construction of the surface.
Figure 15: The U-S-V surface of water in chemical equilibrium with minor species (e.g., OH).
The importance of this surface is that it shows the range of possible paths by which
the exergy of this resource can be coupled in a reversible manner to the outside world.
For example, the red line indicates an adiabatic, reversible path that extends from the
state of the environment (298 K, 1 atm) up to the maximum pressure considered for the
surface (about 104 atm). Note that the red curve corresponds to the intersection of the
property surface with a plane at constant entropy. If our objective is to use isentropic
expansion as a means to extract exergy from this resource, then the goal of the
preparation/positioning and reaction stages is to transform our resource in such a way as
to make the working fluid land on this isentrope. In the case of a heat-recovery steam
generator this is done by providing a well matched heat exchange process and choice of
feed pressure to the boiler. In the case of an internal combustion engine, this matching
must be provided by the combustion process following reactant positioning.
Figure 16 and Figure 17 show the relationship of reactant and product surfaces for the
case of stoichiometric hydrogen and air. The product surface, therefore, contains
nitrogen as well as water, but dissociation of the water is still permitted in calculation of
the surface. The reactants, however, are depicted with a frozen composition, consistent
with a traditional combustion process. Although Figure 16 shows a view that is more
nearly normal to the two surfaces, it is Figure 17A that shows the essential feature of
these surfaces—that they do not intersect. Recall that for a process to be reversible, it
must take place in such a way that it corresponds to a continuous path along the property
surface. No jumps or discontinuities are permitted. It is not surprising, then, to see that
- 31 -
the reactant surface does not connect to the product surface under any conditions—some
form of jump (irreversibility) is always required.
Figure 16: Thermodynamic property surfaces for stoichiometric mixture of hydrogen and air, in blue; and
the equilibrium product mixture, in red.
Figure 17: The thermodynamic property surfaces for stoichiometric H2/air and equilibrium product
mixtures (same conditions as Figure 16), in three-dimensional view (subplot A), as well as on the three
bounding planes (subplots B, C, and D).
- 32 -
Also shown in the figures is the process path for the ideal, reactive Otto cycle. Best
seen in Figure 17A, this begins with an isentropic compression along the frozen reactant
surface to a high internal energy condition, followed by a jump to the product surface at
the same volume and energy (but with significantly increased entropy). From this
landing point on the product surface, expansion then occurs until the volume returns to its
original value. Two observations can immediately be made: The first is that the cycle
terminates with a significant amount of internal energy still retained in the product gases.
This corresponds to the high exhaust temperatures commonly experienced with SI
engines. The second is that the expansion process takes place in a plane that is parallel
to, but significantly displaced from, the process connecting to the environmental state.
This underscores our previous point that there exists a poor match between stoichiometric
combustion processes and an optimal expansion (at least one that is open to the
environment).
Figure 17B through Figure 17D show projections of these processes onto the three
bounding planes. Particularly apparent in Figure 17B is that the cold reactant mixture
exists at a higher entropy than the products (at the conditions for an environmental
expansion) even before combustion. Equally important, Figure 17C shows that there are
no conditions within the current bounds of temperature and pressure where a constant-UV
combustion process can provide a link to the environmental expansion isentrope. In
order to make the reactant surface overlap the isentrope, a sub-ambient-temperature
starting condition is needed. (Recall the previous discussion about positioning the
expansion process.)
Alternatively, one may choose to accept the positioning of the expansion process (as
determined by the reaction), and instead minimize the entropy generation due to
combustion. The desirability of this approach stems from the Gouy-Stodola theorem
which states that the exergy destroyed in a process is equal to the product of the
environment temperature and the entropy generation. As might be surmised from Figure
17A, the entropy generation is minimum at high internal energies where the two surfaces
approach each other. The precise conditions can be established by evaluating the
distance between the surface in the entropy direction over a range of U and V conditions.
Figure 18 gives this result over the range of conditions in the vicinity of the depicted Otto
cycle. (This range of conditions is the dashed rectangle on Figure 17C.) The figure
shows that entropy generation is minimized as you move to the upper right on the U-V
plane—to higher reactant internal energies and volumes.
A final note is that the black line on Figure 18 shows the path in U-V space along
which the compression process occurs. As the reactants are compressed into a small
volume (moving to the left) the entropy generation due to combustion is diminished.
This observation is a generalization of the optimization results obtained in Section 2, and
is consistent with the improvement in Otto cycle efficiency as the compression ratio
increases.
- 33 -
7580
100
100
120120
140
log(V/Vd)
U −
Ud (
MJ/k
mo
l fuel)
Contour plot of Sgen
(kJ/kmolfuel
−K)
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1240
260
280
300
Figure 18: Contour plot of entropy generation during constant-UV combustion.
Section 7: Architectural Requirements for Restrained-Reaction Engines
In the previous two sections, the thermodynamic conditions associated with
unrestrained chemical reactions are translated into architectural requirements for efficient
combustion engines. In this section, we turn our attention to engines driven by restrained
chemical reactions, typified by those occurring in a PEM fuel cell. The goal of this study
has been to explore the design of a more efficient, reversible expansion engine
architecture modeled after the fuel cell/motor electrochemical engine architecture, and to
better understand the similarities and differences between these engines and conventional
combustion engines.
The PEM Fuel Cell/Motor
Section 4 provides a description of the basic electrochemical engine architecture: the
fuel cell uses two chemical reactors, each with a restrained, reversible reaction (at least at
very low loads) to set up the differential electron source/sink pair necessary to operate an
electric motor. In the case of a PEM (proton exchange membrane) fuel cell, the reactions
are, respectively,
+
2
+ 12 22
H 2H 2 at the anode; and
2H 2 O H O at the cathode
e
e
−
−
+
+ +
⇌
⇌
An abstract representation of the system is shown in Figure 19 (left).
At the anode, an equilibrium is set up between gaseous (molecular) hydrogen, protons
in the electrolyte, and electrons in the conduction band of platinum. The mechanism in
the forward direction consists of: (a) hydrogen undergoing dissociative chemisorption
onto the platinum catalyst, (b) the electrons being pulled into the conduction band by the
ionic cores of the platinum, thus allowing (c) the proton to enter the electrolyte.
A similar equilibrium is set up between the protons, electrons, molecular oxygen and
water at the cathode. From a mechanistic perspective: (a) protons from the anode diffuse
across the electrolytic semipermeable membrane (e.g., Nafion) to the cathode, where (b)
they combine with oxygen adsorbed onto the platinum catalyst. The cathode is now at a
higher voltage than the anode, thus creating a positive electrical potential for electrons at
the anode. (c) The electrons moves to the cathode through its own semipermeable
- 34 -
membrane—the conduction band of the metal catalyst and external wiring—so as to (d)
combine with the proton and oxygen to form water.
Figure 19: Abstract representation of an electrochemical engine (PEM fuel cell/motor; left) and
its non-electrochemical (expansion) counterpart (right).
The conditions for equilibrium are, respectively,
+2
+2 2
gas, a Nafion, a a
H H
Nafion, c c gas, c gas, c1O H O2H
2 2 at the anode; and
2 2 at the cathode
e
e
µ µ µ
µ µ µ µ
−
−
= +
+ + =
ɶ ɶ
ɶ ɶ
where the electrochemical potential i
µɶ = µi(T, Pi) + ziFφ is a function of the chemical
potential µi of species i (at temperature T, partial pressure Pi), its charge zi, and the
electrical potential φ of the electrode (denoted by superscript “a” or “c”). Combining the
two equations and recognizing that, at equilibrium, the electrochemical potential of the
proton is the same across the its membrane ( Nafion, a Nafion, c
H Hµ µ+ +=ɶ ɶ ), we obtain
2 2 2
gas, a gas, c gas, c a c c a1H O H O2
2( ) 2 ( )rxn e eG Fµ µ µ µ µ φ φ− −−∆ = + − = − = −ɶ ɶ (13)
which is the maximum (reversible) work output for the engine, coinciding with the Gibbs
free energy −∆Grxn of the overall reaction, H2 + ½O2 → H2O, and with the Nernst
potential φ c −φ a
of the fuel cell.
An Isothermal Reversible Expansion Engine Concept
Based on the thermodynamic features of the fuel cell described above, an analogous
engine driven by restrained reactions, but which eschews electrical work extraction—
relying on expansion machines instead—is conceptualized as shown on Figure 19 (right).
Mechanistically, reactant AB enters the engine at the “half cell” labeled 1 on the
diagram (HC-1) and reacts, perhaps in the presence of a catalyst, to form species A and
B. Species A diffuses to HC-2 through a semipermeable membrane, inducing an increase
- 35 -
in the partial pressure of species B at HC-1. Species B then flows to HC-2 where it
combines with species A and reactant C to form the product ABC. As species B expands
from high partial pressure PB1 to low partial pressure PB
2, work is extracted through an
expansion machine.4
To obtain the maximum work output for this engine, consider again the conditions for
equilibrium at the two half cells:
1 1 1
AB A B
2 2 2 2
A B C ABC
-1: AB A B, ;
-2: A B C ABC,
HC
HC
µ µ µ
µ µ µ µ
+ = +
+ + + + =
⇌
⇌
where the chemical potential of species i is given by µi = µi0(T, P
0 ) + RT ln(Pi/P
0 ) and
depends on its partial pressure Pi.
Again, combining the equilibrium equations and recognizing that the partial pressure
of species A is constant everywhere (so µA1 = µA
2), we find that the maximum work
output for this isothermal engine is
1 2 2 1 2 1 2
AB C ABC B B B Bln( / )rxn
G RT P Pµ µ µ µ µ−∆ = + − = − = (14)
The logarithmic term ln P in the expansion engine—replacing the electrical potential
term φ in the fuel cell—has consequences for the physical implementation of this
expansion engine. The pressures required to balance a typical combustion reaction are
much larger than we currently use in expansion engines and would be very difficult to
realize in practice. Figure 20 shows the magnitude of −∆Grxn which can be balanced by
pressure ratios up to 1000. These magnitudes are much lower than standard combustion
reactions indicating that it is unlikely that such energetic mixtures could be used in this
engine design.
Essential Elements of a Reversible Engine
While we could potentially start to look for reactants, products and semipermeable
membranes that would allow us to build this type of isothermal and reversible expansion
engine, the real benefit of this research has been in understanding how fuel cells and
other restrained reaction systems fit into the broader context of reactive engines.
In particular, our discussion of both the fuel cell/motor system and its non-
electrochemical analog above is prefaced with the notion that the reactors (half cells)
within each system are at equilibrium, so the restriction to low load is crucial. More
precisely, it is a restriction that the net current requirement must be very close to (or
below) the exchange current of the reaction taking place at each half cell. Upon
reflection it can be seen that this reversible limit of operation corresponds to other
reversible limits in thermodynamics: reversible heat transfer and reversible matter
4 Work extraction from the flow of species A from HC-1 to HC-2 may also be possible. This would be
comparable to extracting work from a proton current in the fuel cell (imagine a motor with Nafion
windings). Note that the resulting reversible engine work output still equals −∆Grxn; it is merely
apportioned to two separate extractors (see Eqs. (13) and (14)).
- 36 -
transfer. In all cases, the requirement for approaching the reversible limit is that the flux
required approach that due to inherent thermal processes (i.e., diffusion). The exchange
current density, thermal diffusion of energy (heat), or matter (species diffusion) are all
manifestations of the fact that a non-zero temperature state carries with it the potential to
support small net exchanges with extremely low entropy generation relative to the
transfer.
0 200 400 600 800 10000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18r
Pressure Ratio
- ∆r G
(e
V/m
ole
cu
le fu
el)
Figure 20: −∆Grxn versus pressure ratio.
Considered in this way, we arrive at the requirements for a general reversible, non-
electrochemical reaction system (not necessarily isothermal, as described above), which
look much like those for the fuel cell:
1. Catalysts must be used to ensure that desired reaction pathways are open over the
time scale of interest, and that undesirable pathways are closed;
2. Heat conductors (fins if you like) must be available to interact thermally with the
material throughout its bulk;
3. The ability must exist to do boundary work as the resource flows through the
system. This might be visualized as though the reactor was the inner fluid volume
of a gas turbine expander; and
4. Selective matter conductors which provide the ability to reversibly add or remove
species from the system. Within the PEM fuel cell, electrons are moved through
conducting fibers (wires) that are a direct analog of the heat-conducting fins
proposed above, whereas in the case of protons, it is the nanoscale water channels
formed by interaction of sulphonated side-chains in Nafion that act as the
conductor.
For all four interactions—chemical reaction as well as heat, work, and matter transfers—
the requirement of reversibility is that the agent enabling the interaction be so finely
dispersed as to be within the diffusion length scale of the gas resource. (This is
analogous to requiring sufficiently fine dispersion and contact proximity in the vicinity of
- 37 -
the triple junction of a fuel cell.) In effect, the outer walls are permitted to move in order
to allow energy extraction by work, while the inner surfaces are webbed with conductors
and catalyst in order to permit a homogenous thermal and chemical state within the
system.
For the engine architecture typified by the fuel cell and its expansion counterpart
discussed above, the half-cells do not actually do work but instead provide for reversible
species transfers of reaction intermediates (electrons and ions in a fuel cell, or uncharged
species in the expansion analog). There is no need in these systems to allow moving side
walls (to do work). There is also no need to actively manage the thermal state (the heat-
conducting elements can be passive).
Lastly, we reiterate that reversibility can be approached so long as transfer rates or
reaction velocities are not far in excess of the thermal exchange limits. Clearly, if higher
loads are imposed upon such systems, irreversibilities associated with the gradients
necessary to raise the rates of transport will be incurred—just as in the case of the fuel
cell.
Summary and Future Plans
The efficiency potential of engines remains far from theoretical limits. Due to the
pervasiveness of chemical-energy-to-work transformations, recovery of a significant
fraction of destroyed or unused work potential could make one of the largest
contributions to the reduction of greenhouse gas emissions.
A key result of this project, as discussed in Sections 1 through 3, is that we now
understand what is required to develop a simple-cycle combustion engine with 75%
efficiency: a significant reduction in irreversibility during reaction, achievable only by
going to extreme states of internal energy. Since internal energy is intimately linked with
temperature, this means that exergy loss due to heat transfer must be significantly
reduced to enable the low-irreversibility option. Although a number of materials-
associated approaches have been pursued in the past—and should continue to be pursued
and deployed—an additional option, use of ultra-fast compression and expansion, holds
promise for providing additional benefits beyond those based solely on materials. We
have undertaken a study to establish whether such a strategy is possible using a
laboratory compression-expansion machine capable of producing the required states for
such an engine. If successful, this work will pave the way for the design of ultra-efficient
engines.
In Sections 4 through 6 of this report, we began to develop a fundamental
understanding of the architectural requirements for a simple-cycle engine driven by
unrestrained reactions and their implications on the engine efficiency, bridging the
abstract notion of resource exergy and the second-law limit for specific engine design.
We intend to continue a systematic exploration of the architectural design space for
restrained-reaction engines (discussed in brief in Section 7) and compound-cycle engines.
- 38 -
Publications 1. Caton, P.A., Song, H.H., Kaahaaina, N.B. and Edwards, C.F., 2005. “Strategies for Achieving
Residual-Effected Homogeneous Charge Compression Ignition Using Variable Valve Actuation,”
SAE Paper 2005-01-0165.
2. Caton, P.A., Song, H.H., Kaahaaina, N.B. and Edwards, C.F., 2005. “Residual-Effected
Homogeneous Charge Compression Ignition with Delayed Intake-Valve Closing at Elevated
Compression Ratio,” International Journal of Engine Research, 6(4), pp. 399–419.
3. Teh, K.-Y. and Edwards, C.F., 2006. “An Optimal Control Approach to Minimizing Entropy
Generation in an Adiabatic Internal Combustion Engine,” in Proceedings of the 45th IEEE
Conference on Decision and Control (CDC).
4. Teh, K.-Y. and Edwards, C.F., 2006. “An Optimal Control Approach to Minimizing Entropy
Generation in an Adiabatic IC Engine with Fixed Compression Ratio,” IMECE2006-13581, in the
Proceedings of the 2006 ASME International Mechanical Engineering Congress and Exposition
(IMECE2006).
5. Teh, K.-Y., and Edwards, C.F., 2006. “Optimizing Piston Velocity Profile for Maximum Work
Output from an IC Engine,” IMECE2006-13622, in the Proceedings of the 2006 ASME
International Mechanical Engineering Congress and Exposition (IMECE2006).
Contact
C.F. Edwards: [email protected]