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

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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:

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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.

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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.

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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.

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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.

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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.

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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: cfe@stanford.edu