AN INVESTIGATION INTO THE USE OF A COMPRESSED
AIR LOADING MECHANISM AS A MEANS OF IMPROVING
THE EFFICIENCY OF A FOUR STROKE SPARK IGNITION
INTERNAL COMBUSTION ENGINE
John Bruce Mackay
A research report submitted to the Faculty of Engineering and the Built Environment, University of the
Witwatersrand, Johannesburg, in partial fulfillment of the requirements for the degree of Master of Science
in Engineering.
Johannesburg, 2011
DECLARATION
I declare that this dissertation is my own unaided work. It is being submitted to the degree of Master of
Science in Engineering to the University of the Witwatersrand, Johannesburg. It has not been submitted
before for any degree or examination to any other University.
............................... (Signature of Candidate)
.......... day of ...................... year ..............
ii
Abstract
Internal combustion (I.C.) engines typically exhibit a characteristic efficiency profile which varies with
operating load and engine speed, and it is widely known that the operating efficiency is poor under low
loading conditions. The objective of this project is to investigate whether an energy storing and recover-
ing process, involving compressing air and subsequently using it for propulsion, could be used to achieve
better overall efficiency. An engine so modified would operate in two alternate modes. When using fuel,
the engine operates as close to maximum efficiency as practicable, with the excess of engine output
over driving requirements being absorbed by air-compression loading - driving an external compressor,
charging air into a receiver. Later, under low driving requirements, this air is expanded - using the engine
cylinders - as a source of propulsion. Heat transfer from the exhaust gases to the stored compressed air
is used to improve engine efficiency. Through modelling and simulation, an overall efficiency improve-
ment of 10% over standard engine operation is predicted to be realisable by applying this modification,
and scope exists to further improve this figure through improved heat recovery from exhaust gases and
improved loading capability.
iii
ACKNOWLEDGEMENTS
I would like to thank Dr Bailey-McEwan for his guidance during the course of the research
project.
iv
Contents
1 Introduction 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Required features of a compressed air augmentation system . . . . . . . . . . . 2
1.3 Existing applications of the compressed air-augmented I.C. engine . . . . . . . 3
1.4 Project objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Outline of research report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Background 8
2.1 How an internal combustion engine works . . . . . . . . . . . . . . . . . . . . 8
2.2 Feasibility of using compressed air to store and release energy . . . . . . . . . 9
2.2.1 Efficiency definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Standard air analysis as an analytical tool for this project . . . . . . . . . . . . 11
2.4 Compressed air system model . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 First law-based differential equation model . . . . . . . . . . . . . . . 12
2.4.2 Second law analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Sources of irreversibility in the compression-expansion processes . . . 18
2.4.4 Differential equation model applied to a compression & expansion stroke 19
2.5 Differential equation model of the Otto cycle . . . . . . . . . . . . . . . . . . 21
2.5.1 First law model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.2 Application of the second law to the dynamic Otto cycle model . . . . 23
2.5.3 Sources of irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.5 Efficiency mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.6 Moving toward a more realistic engine model . . . . . . . . . . . . . . 28
2.5.7 Transient effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.8 Summary: dynamic Otto cycle model . . . . . . . . . . . . . . . . . . 29
2.6 More accurate modelling of feasibility of C.A. augmentation of an engine . . . 30
3 Application of the compressed air system to engines in existing vehicles 32
3.1 Overview: retrofitting to existing vehicles . . . . . . . . . . . . . . . . . . . . 32
v
3.2 Using compressed air to propel the engine without fuel . . . . . . . . . . . . . 32
3.2.1 Using an external compressor to expand the air . . . . . . . . . . . . . 32
3.2.2 Using the engine to expand the compressed air . . . . . . . . . . . . . 33
3.3 Performance requirements if compressed air propels the engine . . . . . . . . . 35
3.4 Varying torque generation when compressed air propels the engine . . . . . . . 42
3.4.1 Irreversibility due to inlet of compressed air into the cylinder . . . . . . 43
3.5 Using compressed air in conjunction with fuel . . . . . . . . . . . . . . . . . . 46
4 Enhanced compressed air system with heat recovery from exhaust gases 51
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Required performance improvement . . . . . . . . . . . . . . . . . . . 52
4.2 Heat exchange from the exhaust gases to the compressed air . . . . . . . . . . 53
4.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Heat exchanger design . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.3 Heat transfer coefficient for turbulent flow through the exhaust pipe (R1) 58
4.2.4 Heat transfer coefficient for forced convection over heat exchanger area (R2) 59
4.2.5 Heat transfer coefficient for natural convection (R3) . . . . . . . . . . . 61
4.2.6 Heat transfer solution algorithm . . . . . . . . . . . . . . . . . . . . . 61
4.2.7 Performance improvement due to heat transfer . . . . . . . . . . . . . 62
4.3 Energy storage system description . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 External compressor specification . . . . . . . . . . . . . . . . . . . . 64
4.3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3 System parameters for the compressed air storage model . . . . . . . . 69
4.3.4 Results obtained from the compressed air storage model . . . . . . . . 69
4.3.5 Overall efficiency: combining performance factor and loading capacity 74
4.3.6 Receiver sizing and time taken to reach final receiver pressure . . . . . 77
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Conclusions and recommendation 80
5.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vi
A Finite-time thermodynamics A.0
A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.0
A.2 Internal Combustion Engine Model . . . . . . . . . . . . . . . . . . . . . . . . A.1
B Sample calculation for the differential equation model B.0
C Heat transfer sample calculations C.0
C.1 Heat transfer coefficient from exhaust gas to heat exchanger . . . . . . . . . . C.0
C.2 Heat transfer coefficient from heat exchanger surface to receiver gases . . . . . C.1
C.3 Heat transfer via natural convection . . . . . . . . . . . . . . . . . . . . . . . C.2
D Sample calculation for the compressed air storage model D.0
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List of Figures
1.1 Typical engine efficiency mapping . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Four stroke combustion cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 A graphical representation of the relationship between speed, load and efficiency 10
2.3 Illustration of a crank shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Otto cycle simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Dynamic Otto cycle model: Engine efficiency mapping . . . . . . . . . . . . . 28
2.6 Viability of loading an internal combustion engine - improved model . . . . . . 30
3.1 Schematic diagram of the compressed air system . . . . . . . . . . . . . . . . 36
3.2 Illustration of possible torque demands made by the driver . . . . . . . . . . . 37
3.3 Typical power-torque-speed curve of a Toyota 4ZZ-FE engine . . . . . . . . . 38
3.4 Propulsion by compressed air: cylinder pressure & P-V diagram . . . . . . . . 39
3.5 Propulsion by compressed air: work done vs. receiver pressure . . . . . . . . . 42
3.6 Propulsion by compressed air: receiver temperature vs. receiver pressure . . . . 43
3.7 Propulsion by compressed air: useful work lost due to irreversibility . . . . . . 45
3.8 The effect on cylinder gas temperature with compressed air injection . . . . . . 47
3.9 P-V & T-V diagrams for a standard & modified Otto cycle . . . . . . . . . . . 48
4.1 Energy storage schematic overview . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Annual saving due to efficiency improvement . . . . . . . . . . . . . . . . . . 54
4.3 Heat exchanger - conceptual drawing . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Heat transfer analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Exhaust gas temperature and cylinder air mass during an exhaust stroke . . . . 58
4.6 External air compressor P-V diagram . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 External compressor size specification . . . . . . . . . . . . . . . . . . . . . . 65
4.8 Heat transfer as a function of heat exchanger size . . . . . . . . . . . . . . . . 70
4.9 Performance factor vs compr. vol. and max receiver pressure, GR 1:1 . . . . . 72
4.10 Performance factor as a function of external compression volume and gear ratio 73
4.11 Overall system efficiency for a range of driver load requirements . . . . . . . . 78
A.1 Finite-time thermodynamic model - Power and efficiency curves . . . . . . . . A.4
A.2 TS diagram representation of an Otto cycle . . . . . . . . . . . . . . . . . . . A.5
viii
List of Tables
2.1 Engine model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Engine configuration parameters . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Dynamic Otto cycle model: Magnitude of sources of irreversibility . . . . . . . 27
2.4 Feasible conditions where loading will improve operating efficiency (revised) . 31
3.1 Using an external compressor to load an IC engine . . . . . . . . . . . . . . . 34
3.2 Simulation summary, compressed air injected into cylinder during combustion. 50
4.1 Feasible conditions under which partial loading will improve efficiency . . . . 75
4.2 Extracts from the efficiency-load mapping . . . . . . . . . . . . . . . . . . . . 77
A.1 Sample data for the FTT model . . . . . . . . . . . . . . . . . . . . . . . . . . A.4
ix
Nomenclature
β = Volumetric coefficient of thermal expansion
γ = Specific heat ratio
η = Engine efficiency
θ = Crank angle
λ = Thermal conductivity
µ = Viscosity
Ξ = Exergy
ρ = Density
τ = Torque
ω = Angular velocity
a = Crank radius
A = Area
Ath = Orifice area
B = Cylinder bore
b = Gap between heat exchanger fins
c¯ = Mean piston speed
Ch = Higher heat capacity for a stream in a heat exchanger
Cl = Lower heat capacity for a stream in a heat exchanger
cd = Discharge coefficient, drag coefficient
cv = Specific heat constant volume
d = Diameter
E = Total energy
F = Force
Gr = Gear ratio
g = Gravitational acceleration constant
H p = Heat input
HV = Heating value
x
h = Enthalpy
¯h = Coefficient of heat transfer
Ir = Work lost due to irreversibility
L = Length of fin in vertical flow direction
l = Con rod length
m = Mass
N = Rotational speed (rpm), number of teeth on a gear wheel
NPr = Prandtl number
NRe = Reynolds number
m f = Mass of fuel
rp = Pressure ratio
p = Pressure
pr = Relative pressure
Q = Heat
R = Ideal gas constant
Ra = Rayleigh number
Rht = Resistance to heat transfer
s = Entropy
T = Temperature
t = Time
U = Internal energy, overall heat conductance
um = Mean free stream velocity
V = Volume (m3)
v = Specific volume (m3/kg)
W = Work
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Subscripts
c = Cold, Combined
cpr = Compressor
cyl = Cylinder
d = Downstream
drv = Drive
e = Exhaust
f = Final, fuel
f mech = Mechanical friction
f ric = Friction
f pipe = Friction in pipe
h = Hot
ht = Heat transfer
hr = Heat released
h = Higher, hot
i = inlet, initial
l = Lower
mx = Mixing
o = outlet
p = Products, Process
pl = Process loaded (effective)
r = Reactants
rec = Receiver
req = Required
supl = Supplied
th = Throttling
u = Upstream
v = Volumetric
xii
w = Wall
0 = Dead state
x = Compression control variable, used to denote cylinder volume when the inlet
valve closes or opens
z = Expansion control variable, used to denote cylinder volume when the exhaust
valve closes or opens
1,2 = State 1, 2 etc
Superscripts
k,n = number of species of products and reactants respectively
xiii
Glossary of Terms
ABDC = After bottom dead center
A/F = Air fuel ratio
BDC = Bottom dead center
BT DC = Before top dead center
BSFC = Brake specific fuel consumption
C.A. = Compressed air
f mep = Friction mean effective pressure (kPa)
I.C. = Internal combustion
idle = Term used to describe the state of the engine when fully throttled and disen-
gaged from the gear box
LMTD = Log mean temperature difference
RPM = Engine speed (revolutions per min)
T DC = Top dead center
VV T = Variable valve timing
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1 Introduction
Increasing oil prices and a growing concern about global warming places pressure on the auto-
motive industry to improve internal combustion engine technology such that fuel consumption
and green-house gas emissions are minimised. To this end, players in the automotive market
have responded with numerous innovations including alternative fuels, dynamic valve train sys-
tems, exhaust gas recirculation, use of light-weight engine components, hybrid technology, and
advanced transmission systems [1, 2, 3]. Hybrid engines are typically designed as follows: an
internal combustion engine is mechanically coupled via a planetary gearset to an electric motor,
which allows for regenerative braking and for the engine to be operated closest to the point of
optimum efficiency [4]. Continuously varying transmissions (CVT) allow the engine to be op-
erated along an optimum torque-speed trajectory thereby decreasing overall fuel consumption,
despite a CVT being typically less efficient than a standard manual gearbox [5]. With this in
mind, it must be stated that every engine has a specific fuel consumption characteristic which is
a function of engine torque and speed. In spark ignition engines, throttling is used as a means
of controlling the amount of work done per cycle [6], and hence the output of the engine. En-
gine efficiency decreases with throttling [7] and by default, engine efficiency is lowest when
idling. This is attributed to pumping losses1 increasing with throttling [8], as the inlet manifold
pressure of a throttled engine is lower than atmospheric pressure. Exhaust gas recirculation
(EGR) has achieved limited success in reducing throttling losses [2], although the amount of
exhaust gas that can be recirculated is restricted2 [9], and therefore the amount by which throt-
tling losses can be reduced is also limited. Dynamic valve train systems allow for early intake
valve closing (EIVC) to greatly reduce throttling losses [1].
The aim of this project is not to re-invent the hybrid engine, as the electrical motor is more
efficient acting as the energy transformation and reuse mechanism than any thermodynamic
power cycle; nor is it to supersede the use of CVTs. The concept is rather to be developed as a
retro-fit to an existing engine for the primary reason that scope exists to improve the operating
efficiency of spark ignition vehicles already on the road. Electrical hybridisation of an exist-
ing vehicle or the replacement of a standard transmission system with a CVT has installation
1Pumping losses are defined as the work done to move air or exhaust gases into or out of the cylinder.
2The amount of exhaust gas that can be recirculated is limited because the A/F ratio is limited, to ensure normal combustion.
1
prohibitive practicality and cost implications, hence the investigation into the more practicable
alternative of compressed air.
1.1 Problem Statement
The modern internal combustion engine is designed for acceleration - it is designed to produce a
certain maximum torque, and is most efficient when operating at a value near to this maximum.
However motor vehicles are required to operate over a wide range of conditions, and therefore
the engine may spend a significant proportion of time operating at a point with an efficiency
lower than the optimum efficiency of the engine. An efficiency mapping of a typical spark
ignition internal combustion engine is shown in Figure 1(a), demonstrating that the region of
optimum efficiency lies at an operating point of approximately 85 % of optimum torque at an
engine speed which is half of the maximum engine speed. Engine efficiency is particularly low
when torque requirements are low. The proposed mandate for this project is to improve the
efficiency of the spark ignition engine when torque requirements are low. Figure 1(b) shows
an efficiency mapping obtained using a differential equation model (see Section 2.5, page 21),
and shows the same basic efficiency characteristics as the actual efficiency mapping - which
is that the engine has a low operating efficiency when highly throttled. It is proposed that the
engine be loaded with a compressed air process when torque requirements are low, and that
the compressed air be used independently of or in conjunction with a combustion cycle. The
loading mechanism is analogous to an electrical generator-battery-motor energy storage and
re-use system in this regard.
1.2 Required features of a compressed air augmentation system
Due to the space restrictions of the engine bay, the size (and therefore the energy storage capac-
ity) of the device will be limited. For this reason, the potential efficiency gain obtained using
a variety of both external and internal loading mechanisms should be quantified. A belt driven
compressor might be used to load the engine externally, or cylinders within the engine might be
used to do this air compression, thus loading the engine internally. The loading process must
not adversely affect the operation of the engine - the operational characteristics of the loaded
engine, such as torque output for a particular throttle setting, must not differ from conventional
2
(a) Actual engine efficiency mapping obtained from [5]
 0.05
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 0.4
 0.45
 0.2  0.4  0.6  0.8  1
En
gi
ne
 e
ffi
cie
nc
y
Inlet manifold pressure (bar)
Engine efficiency mapping
999 rpm
1954 rpm
2909 rpm
3864 rpm
4819 rpm
5774 rpm
(b) Efficiency mapping obtained from a differential equation
model
Figure 1.1: Typical engine efficiency mapping
engine characteristics. The efficiency of the compressed air system must be determined for all
candidate loading mechanisms, and this is done using both the first and second laws of thermo-
dynamics. Naturally no process is free of irreversibility, and a method involving regeneration,
particularly the transfer of heat energy from the exhaust gases into the compressed air system,
is presented to further increase the overall engine efficiency as much as is possible. (Scope also
exists for the loading mechanism to be used as a regenerative braking device). The presentation
of the proposed solution begins with an investigation into compressed air vehicle applications.
1.3 Existing applications of the compressed air-augmented I.C. engine
Before any analysis can be undertaken, the question of whether an internal combustion engine
can be powered by compressed air must be addressed. To this end, a brief literature survey
was conducted to find relevant examples. The literature survey shows that success has been
achieved in this field, with compressed air first used as a motive source at the beginning of the
last century. Advances in this field have been made since then.
The concept of using compressed air to drive an engine is not novel. One of the earliest
applications of a compressed air engine is the Beaumont engine, which was developed as an
alternative power source for a locomotive steam engine in the 1880’s. The principle behind this
engine was to introduce compressed air into a sequence of cylinders, starting with the cylinder
3
of the smallest area, until all useful work was extracted from the air. One problem noted with
this application was the extremely low temperatures that cylinders and pistons would reach.
This was rectified by applying heat to the air as it was expanded into the cylinder [10]. This
example of compressed air engine avoids uncontrolled expansion. However, a separate specially
designed compressed air expansion device would have to be connected to the engine in order to
expand the air in this fashion.
The earliest application of compressed air to an automotive reciprocating four stroke engine
was achieved by John Broderick in 1912 [11]. His idea was to enable the driver to use either
compressed air or gasoline to propel the vehicle. He also made provision for the engine to
be used as an air compressor, as air would be drawn into the cylinder and then forced into a
compressed air storage tank through a specially modified exhaust valve. The inlet and exhaust
valve timing was altered such that actuation would be achieved twice as often as in the case of
a four-stroke engine (i.e. if the inlet valve were to open at 0◦ and close at 180◦, the exhaust
valve would open part way through the compression stroke and close at 360◦, allowing air to
be compressed into a receiver on the compression stroke). In essence this engine modification
not only allowed the car to be powered by compressed air, but also converted it into a hybrid
vehicle as otherwise wasted energy could be captured and stored. Cylinders within the engine
are used to compress air (and load the engine). This approach was discarded as a possible
retrofit option because internal combustion engines typically have large clearance volumes (the
engine under investigation has a compression ratio of 10 and clearance volume of 44 cm3, see
Table 2.2). This would mean that the engine (operating as a compressor) would have a poor
volumetric efficiency, and perform very poorly as a compressor. A four stroke engine would
also not be properly balanced if one or more of the cylinders were taken out of conventional
four stroke operation, without a significantly complicated control algorithm to ensure that every
half revolution delivered the same net amount of torque.
Simington invented a modification for the internal combustion engine which allowed it to
run on compressed air. This was achieved through a device that allowed compressed air to be
injected into the correct cylinder at the correct time through the spark plug holes [12]. The mod-
ified engine would still have an inlet and compression stage, both of which being unnecessary in
a compressed air engine as the charge of air/fuel mixture need not be inducted or compressed.
4
This was later achieved by Stricklin through another modification, by replacing the original cam
shaft with a new cam shaft: an additional cam is added at an angular displacement of 180◦ to
original cam, such that the valve operates once per revolution of the crank shaft [13]. This idea
is more fully explored in Section 3.5. The concept of a hybrid internal combustion/compressed
air engine has been taken further in recent years through the use of electronic valves. In May
2006, Miller and Froloff were issued a patent for an engine which could be configured to be
powered by gasoline or compressed air, could use the compressed air to improve the volumetric
efficiency of the combustion process, and could capture braking energy by running the engine
as a compressor. All of the modes of operation are obtained by dynamically reconfiguring the
engine valve timing [14].
MDI, a French automotive company, have developed an air powered vehicle, and are about
to begin producing this vehicle on a commercial scale [15]. This engine offers two modes of
operation: the Mono-Energy operating system where the compressed air drives a piston engine
equipped with an active chamber and an expansion cylinder; and the Dual-Energy operating sys-
tem where a burner is used between the compressed-air tank and the engine, in order to achieve
a continuous combustion below 900 K, which heats the air to increase its pressure before being
injected into the engine. A recent comparative study was conducted where this compressed air
engine performs similarly in terms of energy efficiency with an electric vehicle, and exceeds
the performance of hybrid vehicles, however the range and maximum speed of this engine are
limited. The charging and usage efficiency of the MDI engine running in Mono-Energy mode
is stated to be between 62 - 70%, and 43 - 60% respectively, while the charging and usage
efficiency of an electric vehicle is stated to be between 64 - 82 %, and 65 - 87 % respectively
(depending on battery type) [16]. This does show that the practical efficiency of a compressed
air powered vehicle is still poorer than an electically powered vehicle, but compressed air does
allow opportunities for energy regeneration from exhaust gases (if applicable) - see Section 4.2.
The Cargine-Engineering company have developed a pneumatic engine hybrid which facili-
tates waste energy capture from braking and exhaust heat. Steam is generated by allowing water
to come into contact with exhaust gases via a heat exchanger. This steam is used to drive the
engine during every alternative power stroke, as the engine is based on the Hedman combus-
tion cycle, which allows the engine to be powered by compressor strokes alternated with power
5
strokes [17].
1.4 Project objectives
The proposed mandate of the project is summarised by the milestones listed below. These mile-
stones represent the core objectives of the project.
Core objectives:
1. To propose a number of loading mechanisms, such that the energy differential between
the amount of work per cycle required for driving and the work per cycle delivered at the
point of optimum efficiency can be stored. These loading mechanisms can be internal or
external to the engine.
2. To develop a macroscopic control routine of the compressed air system such that engine
is always correctly loaded despite the states of the air receivers.
3. To derive models of the compressed air system so that irreversibilities like heat transfer,
the mixing of gases at different temperatures, throttling and friction can be taken into
account. This also allows the states of the air in the cylinder and receivers to be estimated.
4. To develop a model of a spark ignition combustion cycle so that the efficiency of such
an engine can be quantified based on the operating conditions of the engine. Various
operation related data is also drawn from this model, for instance the amount of energy
consumed during a typical exhaust stroke.
5. To use the above mentioned models to analyse process efficiency as affected by engine
speed and work done on or by the compressed air system, as well as the effect of injecting
compressed air into an engine cylinder part way through a combustion process.
6. To investigate the potential for using heat transfer from the exhaust gases to improve
process efficiency through regeneration.
1.5 Outline of research report
The background section (Chapter 2) begins with a description of how an internal combustion
engine works, drawing attention to the characteristically poor part load efficiency (where the
6
engine is throttled). A solution is then proposed to the problem of poor part load efficiency,
which is to load the engine by an air compressor storing compressed air into a receiver to
be used later. A differential equation model of the storage and usage of this compressed air
is presented. The aim of this model is to illustrate the effect of irreversibility on the overall
feasibility of a compressed air augmentation system. Effects such as friction, heat transfer and
non-steady flow are then taken into account using a more detailed differential equation based
compressed air model.
A differential equation model of an Otto cycle model is then presented and used to generate
an efficiency table for a typical engine, and this table is used to show where augmenting an
engine with a compressed air system would be most viable.
A detailed description of how the compressed air will be used is then given (Chapter 3).
Compressed air usage constraints and parameters are specified to ensure that the compressed
air is used as efficiently as possible. These constraints are then related to the performance
requirements of a typical internal combustion engine, and a balance between these two design
objectives (efficient usage of compressed air and meeting internal combustion engine output
torque levels) is reached. The efficiency of compressed air usage is presented and discussed.
The way in which compressed air is stored is then discussed (Chapter 4), together with
how heat transfer from the exhaust gases will be used to improve overall process efficiency. A
design for a heat exchanger is proposed, and the analysis of how heat transfer will occur into
the compressed air is presented. The design for the air compressor is also proposed, taking into
account engine loading requirements (such that the engine operates at a better efficiency) as
well as the physical space limitations of the engine bay. The overall performance improvement
of the modification is presented.
Chapter 5 reviews the research report, critically reviews the report and its limitations, and
suggests where further work is needed.
7
2 Background
In this chapter an overview of how an internal combustion (IC) engine works is given, together
with reasons for poor part load efficiency. The feasibility of using compressed air as a source
of motive power in an engine is presented, with emphasis on quantifying process efficiency
by including an isentropic efficiency factor. A differential equation based model of such a
compressed air process is then presented with the intent of quantifying the effect on process
efficiency of various sources of irreversibility present within the system. An air standard Otto
cycle is then presented, and used to determine the operating region for which loading an engine
with a compressed air process would be feasible. This model is then improved on using a
differential equation based model, and the same feasibility analysis is repeated.
2.1 How an internal combustion engine works
A typical four stroke spark ignition internal combustion engine works as follows:
1. Intake stroke: A fresh mixture of fuel and air (charge) is drawn through the open intake
valve while the piston moves from top dead center position to bottom dead center position.
2. Compression stroke: The intake valve is closed and the charge is compressed by the piston
doing work on the charge. When the piston is close to top dead center, ignition is initiated
causing the charge to burn resulting in an increase in the temperature and pressure of the
air within the cylinder.
3. Expansion stroke: The air within the cylinder expands and does work on the piston, until
the piston reaches bottom dead center.
4. Exhaust stroke : The burnt gases are then pushed out through the open exhaust valve while
the piston moves from bottom dead center to top dead center.
The thermal efficiency of a spark ignition engine with a compression ratio of 10 modelled as a
standard ideal Otto air cycle is approximately 60% and is a function of the compression ratio
and the specific heat of the air alone [19][pg 378]. This value is calculated for when the engine
is operating with the inlet throttle wide open and is greater than the efficiency when the throttle
is partially closed.
8
Figure 2.1: Four stroke combustion cycle [18]
In order for the engine to be useful, the amount of output power of the engine must be controlled,
and this is done by limiting the supply of fuel to the engine. Spark ignition engines require the
stoichiometric ratio between fuel and air to be maintained within flammable limits [20, 21].
This is done by regulating the amount of air supplied to the engine with a throttle valve. The
engine must do work against this valve, increasing the pumping loss area on the PV diagram
relative to the amount of work produced by the cycle. The residual exhaust gas fraction in
the cylinder also increases as the load decreases [21], and these two factors contribute toward
inefficient engine operation during low load conditions.
2.2 Feasibility of using compressed air to store and release energy
The proposed solution to this problem is to unthrottle and load the engine such that it operates
at a point nearer to the point of optimum efficiency. Figure 2.2 shows that engine efficiency
increases as BMEP increases.
It is proposed that the engine be loaded either externally, by an external compressor attached
to the drive output of the engine via a belt and pulley arrangement; or that the engine be loaded
9
Figure 2.2: A graphical representation of the relationship between speed, load and efficiency for a standard
engine, and an engine loaded such that it operates at the point of greatest efficiency for a particular speed
[21]
internally, using cylinders within the engine to load the engine sufficiently. Compressed air will
be the mechanism by which energy will be stored and released as needed.
2.2.1 Efficiency definitions
The efficiency of using (or storing) compressed air by expanding it from (or compressing it into)
a storage tank with a piston-cylinder arrangement is now discussed. Consider the injection of a
quantity of high pressure air into a cylinder from a high pressure receiver. During the expansion
an amount of work will be done on this cylinder’s piston, thereby decreasing the internal energy
and mass of the air left in the receiver.
Energy out = −he∆me (2.1)
The expansion process is not isentropic, and in practice an isentropic efficiency factor is used
to account for irreversibility present in the system. The work done during an expansion process
of the air in a cylinder is given by h1 − h2, where h1 and h2 correspond to the initial and final
enthalpy respectively. The maximum amount of work obtainable without external heat input
achievable by the system is given by h1 − hs2, and this is for a case where the air expands
isentropically to state 2s. Using the above, the isentropic efficiency of an expansion process is
10
given by:
ηe =
h1 −h2
h1 −h2s
(2.2)
If air was to be compressed between two states, the isentropic efficiency of the compression
process would be given by:
ηc =
h2s −h1
h2 −h1
(2.3)
2.3 Standard air analysis as an analytical tool for this project
Examples of work done to analyse Otto [22] and refrigeration cycles [23] based on standard air
analysis techniques have been used successfully, where polytropic models are used to describe
compressed air processes and isentropic efficiency factors are used to account for irreversibility.
Such models of compression and expansion processes are useful, as good approximations of the
amount of work done or absorbed during a process, as well as of the final state of the system,
can be obtained. These models predict the behaviour of a compressor which is operating at
steady state, with the initial and final temperatures of the process given by the average inlet
and exhaust temperatures of the compressor [23]; but this has limited benefit when dealing
with coupled processes. For instance: when modelling the discharge of air from a cylinder at
one state into a receiver at another state, the receiver and cylinder cannot be considered as one
system with the total volume of the system being equal to the sum of the cylinder and receiver
volumes, because the states for the cylinder or receiver may not necessarily be equal, and the
effects of the flow restriction between the receiver and cylinder, heat transfer and friction cannot
be included.
The air standard analysis approach relies on estimates of the isentropic efficiency of each
process to have been determined beforehand by experiment. The approach taken for this project
was to estimate the isentropic efficiency numerically using a differential equation based method
developed by Bejan [24].
2.4 Compressed air system model
In this section a compressed air process is described by a differential equation which is then
used in conjunction with the second law of thermodynamics to characterise the amount of
11
useful work destroyed due to irreversibility and hence determine overall process efficiency.
This approach does not characterise the entropy generated during the compression or expansion
strokes, but rather the effect on entropy due to heat transfer, mixing of gases at different states,
and friction. The differential equation modelling approach has been applied to combustion
engines [25, 26, 27, 28] and to refrigeration compressors [29].
2.4.1 First law-based differential equation model
Extensive use of the differential equation modelling approach has been used to model internal
combustion engines. Most of these models are derived as a first step to proposing a form of
control algorithm. An assortment of papers from which insight into cylinder state dynamics
was drawn are introduced briefly below. Differential equation models of the cylinder state were
(possibly still are) too complex for microprocessor technology used for control to compute, but
serve to describe processes for further analysis. Works have thus attempted to abstract these
models such that they are compatible with existing automotive engine controller computational
capabilities. Research avenues include: a control oriented perturbation model of a differential
equation model describing cylinder pressure [30], a non-linear sliding observer [31], and an
approximation of cylinder pressure using the maximum-likelihood method [32]. The differen-
tial equation modelling techniques presented in the above mentioned papers are based on the
first law of thermodynamics and form the basis for the differential equation model used for this
project. The temperature and pressure in the cylinder are governed by Equations 2.5 and 2.9,
which are then integrated numerically. The rate of change of pressure in the cylinder (Equa-
tion 2.5) was arrived at by the differentiation of pV = mRT with respect to the angular position
of the crank shaft. The differential equation model is integrated with an integration step size of
pi/50 rad of a crankshaft revolution.
Cylinder pressure:
pV = mRT
(2.4)
12
Differentiating with respect to time:
p˙V + ˙V p = R(m˙T + ˙T m)
p˙ = R(m˙T +
˙T m)− ˙V p
V
(2.5)
Modifying the partial differential equation such that it can be integrated with respect to the
angular position of the crank shaft (this applies to all further differentials with respect to time):
d p
dθ =
d p
dt ×
dt
dθ (2.6)
Charge temperature: The rate of change of the cylinder temperature was obtained from the first
law of thermodynamics, shown below:
˙U = ˙Q− ˙W +him˙−hem˙
(2.7)
using : U = mu,
m˙u+mu˙ = ˙Q− p ˙V +him˙−hem˙
(2.8)
using : ˙T = 1
cv
u˙,
˙T = 1
mcv
[ ˙Q− p ˙V +him˙−hem˙−um˙] (2.9)
Mass flowrate [33, pg 907]: . Where m˙ is positive if the inlet pressure p1 is larger than the
cylinder pressure p2.
m˙ =









CDAthP1√
RT1
γ 12
(
2γ
γ+1
)
γ+1
2(γ−1) if P2P1 >
2
γ+1
γ+1
2(γ−1)
CDAthP1√
RT1
(
P2
P1
)
1
γ
[
2γ
γ−1
(
1− P2P1
γ−1
γ
)
]
1
2
if P2P1 ≤
2
γ+1
γ+1
2(γ−1)
(2.10)
Receiver pressure and temperature are obtained from Equations 2.5 and 2.9 respectively with ˙V
as zero because the receiver volume is assumed constant.
Receiver pressure:
p˙ = R[m˙T +
˙Tm]
V
(2.11)
13
Figure 2.3: Illustration of a crank shaft for the derivation of Equation 2.16
Receiver temperature:
˙T = 1
mcv
[ ˙Q+him˙−hem˙−um˙] (2.12)
Heat transfer (cylinder): Heat transfer between the cylinder and the engine walls is given by
the Annand expression [25, 34, 28] :
˙Qht = Acyl
[
0.3λN
0.7
Re
D
(Tcyl −Tw)+1×10−8(Tcyl 4 −T 4w)
]
(2.13)
Where:
λ = Thermal conductivity (W/K.m)
D = Cylinder bore (m)
NRe = Reynolds Number
Tcyl = Temperature of air in the cylinder (K)
Tw = Temperature of cylinder wall (K)
The surface area of the combustion chamber is derived below:
Area = Ach +Ap +piB(l+a− s) (2.14)
Where (a, l and s are depicted in Figure 2.3):
14
Ach = Cylinder head surface area (m)
Ap = Piston crown area(m)
a = Connecting rod length (m)
l = Crank radius (m)
s = Position of piston pin to crank centre (m)
s = acos(θ)+
√
l2 − (asin(θ))2 (2.15)
Resulting in [33, pg 44]:
Area = pi(B/2)2 +pi(B/2)2 +piB(l+a−acos(θ)+
√
l2 − (asin(θ))2) (2.16)
The Reynolds number (Re) is given by ρDc¯/µ, where c is the mean piston speed and ρ, µ are
the density and viscosity of the gas respectively.
Mechanical friction: Mechanical friction is incorporated into the model according to an em-
pirical approach developed by Rakopoulos et al [26], which describes the relationship between
engine speed (N), peak cylinder pressure (pmax) and friction. A similar model was proposed by
Heywood, but the effect of peak cylinder pressure on friction is not included [33]. Friction is
given in bar and must be multiplied by the rate of change of cylinder volume to determine the
amount of power required to overcome mechanical friction.
f mep = 0.137+0.005pmax +0.0108N (2.17)
Fluid friction: Pipe friction is included in the model through the Darcy-Weisbach equation
[35, pg 271] (Equation 2.18 below) and has to be calculated iteratively, as pipe friction and
mass flow rate are dependent on one another. Pipe friction and mass flow rate are calculated
according to the following algorithm:
• Step 1. Assuming pipe friction is zero, calculate mass flow rate
• Step 2. Calculate pipe friction using this flow rate
• Step 3. Recalculate flow rate incorporating the effects of pipe friction
• Step 4. Repeat the above until mass flow rate converges
15
p f ric =
64
Re
1
D
ρV 2
2
(2.18)
Thermal conductivity (λ) [36] and fluid viscosity (µ) [37] are given by the following equations:
λ = 1.0184×10−4T −3.9333×10−4 (2.19)
µ = (18.27×10−6)
( 411.15
T +120
)( T
291.15
)1.5
(2.20)
The following engine parameters are assumed [38]:
Table 2.1: Engine model parameters
Cylinders 4
Bore (m) 0.08
Compression ratio 10
Clearance volume (m3) 4.407×10−5
Swept Volume (m3) 3.967×10−4
Comment: This first-law model has merit in that time-varying system states can be predicted
based on the parameters specified. However, the system should be modelled as closely to a
real system as possible, in that aspects such as heat transfer and the mixing of gases at two
different states are taken into account; as well as the effects of pipe friction. A means to measure
the degradation of useful energy (available for work) due to such real world irreversibilities is
obtained from the second law of thermodynamics.
2.4.2 Second law analysis
Second law analysis will be used to evaluate how much useful energy (available for work) might
be degraded by various sources of irreversibility present in the system during a particular com-
pression or expansion stroke. Process efficiency (affected by the degradation of useful energy)
may be affected by various operational requirements as well as changing system states. Exam-
ples were found in literature which take this approach in quantifying the second law efficiency
16
[25, 29] of a thermodynamic process. First law analysis is used to describe the variation in
the state of the system according to the differential equation model presented in Section 2.4.1,
and the amount of useful energy which has been degraded is quantified using exergy analysis
techniques. The exergy of a system in a given state is defined as the maximum amount of useful
work which could be obtained from the combination of the system and the surrounding atmo-
sphere, as the system goes through reversible processes to equilibrate with the atmosphere [25].
Irreversible processes give rise to a ”lost” work term which is quantifiable in terms of the en-
tropy produced, through a relationship known as the Gouy-Stodola theorem; where the amount
of energy made unavailable to do work due to irreversibility is proportional to the amount of
entropy produced by a process. The proportionality constant is dependent on the system of
interest [24, pg 24].
˙Wlost = T0 ˙Sgen (2.21)
Where:
T0 = constant of proportionality, taken to be the temperature of the surroundings (300 K)
˙Sgen = rate at which entropy is produced during a process.
The compressed air loading process is dynamic in the sense that the conditions surrounding the
system change, and the model provides a means for the effect of the changing surroundings on
process efficiency to be quantified. Using Equation 2.21, one can quantify the amount of useful
work lost due to heat transfer, mixing of air streams at different states, and friction; all of which
are present in the loading process under investigation. The amount of ”lost” work can be used
to quantify a process efficiency, which is defined as the amount of work absorbed or released
during a complete a process (compression, expansion etc) taking into account the amount of
useful energy (available for work) degraded during this process, divided by the work required
for that a process if it were to occur without irreversibility. This efficiency is known as the
second law efficiency (Equation 2.22), as given by Bejan [24].
ηII = 1−
˙Wlost
˙Wmax
(2.22)
17
Where:
˙Wmax = ∑
in
m˙(h−T0s)−∑
out
m˙(h−T0s)−
d
dt (E −T0S) (2.23)
2.4.3 Sources of irreversibility in the compression-expansion processes
The following relationships are based on work done by Bejan [24]. Sources of irreversibility
present in the envisaged compression or expansion processes are the mixing of air streams at
different states, pressure drops due to friction between air and piping, mechanical friction and
heat transfer. Loss of useful energy due to friction between the cylinder walls and piston rings
is given by Equation 2.24.
Note:
Tw = Cylinder wall temperature
T0 = Temperature at the reference state
R = Ideal gas constant
m˙ = Mass flow rate
Pu, Tu = Up stream pressure and temperature
Pd, Td = Down stream pressure and temperature
˙Ir f mech = Irreversibility due to mechanical friction
˙Ir f pipe = Irreversibility due to pipe friction
˙Irmx = Irreversibility due to mixing
˙Irth = Irreversibility due to throttling
s = Entropy at the inlet
s′ = Entropy at the outlet
h = Enthalpy at the inlet
h′ = Enthalpy at the outlet
˙Ir f mech = ˙Wf mech
(
1− Tw −T0
Tw
)
(2.24)
Friction between the gas and pipe tubing results in a pressure drop and an associated loss of
useful energy given by Equation 2.25 [24]. In addition to pipe friction, a pressure drop occurs
18
across the engine valves, which varies with engine speed [39]. Steady flow has been assumed.
˙Ir f pipe = T0Sgen = T0mRln
( pin
pout
)
(2.25)
A key design aim of this project is to avoid uncontrolled expansions (mixing of air at different
pressures) as much as possible, but the mixing of air streams at different temperatures seems
unavoidable (air will be discharged into a receiver and mix with its contents not necessarily at
the same state as the air exhausted from the compressor), with the loss in useful work arising
from such an event given by Equation 2.26.
˙Irmx = m˙T0Cp
(
ln
(T1
T2
)
− T1 −T2
T1
)
(2.26)
A certain amount of uncontrolled expansion is expected as valve actuation will not be ideal,
and the corresponding loss in useful work is given by Equation 2.27. The expansion (when
mixing gases at different pressures) is assumed to be isenthalpic. The reference state is taken
to be 300 K, 1× 105 pa. The irreversibility directly associated with the act of compressing
or expanding the air within the cylinder is not included in this model. This could have been
modelled using finite element thermodynamics, but this was beyond this project’s scope. These
processes have been modelled as isentropic.
˙Irth = T0m˙(s′− s) (2.27)
Where: h′ = h
Alternatively one could evaluate the loss of available work due to mixing by using the following
expression [28]:
˙Irth = T0m˙
[
cpln
(Tu
Td
)
−Rln
( pu
pd
)]
(2.28)
2.4.4 Differential equation model applied to a compression followed by an expansion
stroke
An idea of compressed air process efficiency is obtained from the dynamic model presented
above. Air is compressed from a volume of 4.407× 10−4 m3 and a state of 1 bar, 300 K to a
volume of 4.407×10−5 m3 and a state of 23 bar, 700 K, and then expanded to the initial volume
and a state of 0.92 bar, 275 K. Heat transfer and friction are both included in the simulation,
19
hence the difference between initial and final states. The process was modelled in Octave - a
mathematical simulation package similar to Matlab.
The process efficiency is 67 %. The compression mechanism modelled here is isentropic,
so the actual efficiency of a compression stroke followed by an expansion stroke is expected to
be lower than 67%.
20
2.5 Differential equation model of the Otto cycle
The air standard approach to Otto cycle analysis has been developed and expressions for max-
imum power and efficiency as functions of various parameters have been obtained. The results
obtained using this approach are not realistic and so have not been used, but the methodology
is presented in Section A of the Appendix.
The differential equation based model of a conventional spark ignition internal combustion
engine allowed investigation into the use of high pressure air injection to improve the efficiency
of the existing combustion cycle directly. Such a model would also allow a more accurate
efficiency mapping of an engine to be obtained (more accurate than an air standard analysis
approach). However this mapping would only provide a qualitative understanding of engine
efficiency; the efficiency of a combustion engine would be determined through experiment for
most accurate results.
Other operational related data required for the analysis of various aspects of this project
are also drawn from the differential equation model. This includes work done during a typical
expansion, compression, intake, or exhaust stroke; as well as the exhaust gas state and outlet
mass flow rate used for heat transfer analysis.
2.5.1 First law model
When simulating a combustion process, the rate of heat release is dependent on the rate at which
fuel is burned, which is approximated by a Weibe function [33, pg 390]:
mb = m f
(
1− exp
[
−a
(θ−θ0
∆θ
)m+1])
(2.29)
Where a and m are constants used to match the Weibe function to experimentally obtained data
on to burn profiles. Real burn fraction curves have been fitted to a Weibe function with a = 5
and m = 2 [28, 40]. The amount of the heat released is given by:
˙Qhr = QLHV m˙bηb (2.30)
where:
21
QLHV = Lower heating value
m˙b = Mass burn rate
ηb = Burn efficiency
Combustion efficiency can be included into the model using an empirical model [28]:
ηb = 0.9(4.6509λ−2.0764λ2) (2.31)
where λ represents the excess air in the cylinder, but it has been assumed that the air fuel mixture
is stoichiometric so λ = 0, and ηb = 0.9 accordingly.
˙T = 1
mCv [
˙Qhr − ˙Qht − p ˙V +him˙−hem˙−um˙] (2.32)
Valve and ignition timing settings are as shown in Table 2.2. The differential equation model
provides a means to observe the effect of valve and ignition timing on engine operational effi-
ciency. For instance: altering the ignition timing has an effect on the amount of work done,
because the point at which cylinder pressures peak changes accordingly. Closing the inlet
valve after BDC results in the induction of more air into the cylinder, and experimentation
with this model indicates that the inlet valve should close at 35◦ after BDC at an engine speed
of 6000 RPM.
The breathing characteristics are dependent on the structure of the engine and the inertia
of the air, and have been included into the model through an empirical model of volumetric
efficiency (Equation 2.33), based on work done by Williams [41]. Other empirical methods of
characterising intake flow rate were found in [42, 43, 44].
ηv(n, p) = 0.597071+n5.15605×10−5 + p9.30417×10−7 (2.33)
Where:
n = engine speed
p = peak cylinder pressure
The efficiency model is expanded using exergy analysis, so that the effect of certain irreversibil-
ities on process efficiency can be quantified. This approach is based on work done by Rakopou-
los [25] and Ribeiro et al [28]. P-V and T-V diagrams of a typical simulation are presented in
Figure 2.4.
22
Table 2.2: Engine configuration parameters [38]
Bore 0.08 m
Stroke 0.077 m
Burn duration 0.628 rad
Ignition 10◦ BTDC
Inlet valve open TDC
Inlet valve close 15◦ ABDC
Exhaust valve open BDC
Compression ratio 10
Air fuel ratio (A/F) 14.7
 0
 5
 10
 15
 20
 25
 30
 5e-5  1e-4  1.5e-4  2e-4  2.5e-4  3e-4  3.5e-4  4e-4  4.5e-4
Pr
es
su
re
 (b
ar)
Volume (m3)
Otto cycle - PV diagram
(a) Pressure-volume diagram
 500
 1000
 1500
 2000
 2500
 3000
 5e-5  1e-4  1.5e-4  2e-4  2.5e-4  3e-4  3.5e-4  4e-4  4.5e-4
Te
m
pe
ra
tu
re
 (K
)
Volume (m3)
Otto cycle - TV diagram
(b) Temperature-volume diagram
Figure 2.4: Otto cycle simulation results
2.5.2 Application of the second law to the dynamic Otto cycle model
The second law of thermodynamics is used to obtain an efficiency model of the engine when it
is operating at a constant load and speed. The analysis presented in Section 2.4.3 is extended to
encompass irreversibility and associated loss of useful work due to combustion and the exhaust
of hot air into the surroundings (known as gas transfer). This is based predominantly on work
done by Rakopoulos and Heywood [33, 45]. The approach taken by Rakopoulos is to model
the combustion processes according to first law principles, going as far as to include detail
about the nature of the chemical reactions taking place within the cylinder. This aspect of
combustion modelling has not been included in the model presented here. A Weibe function
23
is used to describe the combustion process as indicated above, without taking into account the
irreversibility due to the change in chemical species of the reactants and products. Modelling of
the burn profile as an empirical function (i.e. Weibe function) is the generally accepted practice,
as the combustion profile can be fitted to experimental data [26].
2.5.3 Sources of irreversibility
Irreversibility due to the combustion process was evaluated by Ribeiro et al [28] according to
Equation 2.34 below. During experimentation they noted that the irreversibility due to combus-
tion evaluated according to this method was much less than irreversibility due to valve throttling
for example, which is not consistent with other results obtained by Heywood [33, pg 795]. Us-
ing Equation 2.34, the amount of ”lost” work due to combustion is approximately 14 J per
cycle, which is not correct. Irreversibility due to combustion accounts for about a third of the
total irreversibility of the engine [33].
˙Ircomb =
k
∑
i=1
m˙pispi −
n
∑
i=1
m˙risri (2.34)
where:
m = mass
s = entropy
p = products
r = reactants
k,n = number of species of products and reactants respectively
Instead of using the method described above, the beginning and end states of the combustion
process are used to determine the destruction of exergy due to the irreversibility associated with
combustion, in accordance with Rakopoulos and Heywood [25, 33]. The change in exergy due
to combustion is given by Equation 2.35 below. Using Equation 2.35, the amount of ”lost” work
due to combustion is approximately 213 J per cycle (with engine unthrottled, and configured
according to Table 2.2). This is more in line with the results presented in the above mentioned
literature.
Ξ2 −Ξ1 =−T0m(s2 − s1) (2.35)
24
Irreversibility due to gas transfer is taken into account with Equation 2.36 [25], and represents
the loss of useful energy during the exhaust process.
dΞ = dm[h−h0 −T0(s− s0)] (2.36)
Aspects which have been included in, and omitted from the differential equation model are
listed below.
Aspects included in the dynamic Otto cycle model
• Uncontrolled expansion.
• Irreversibility due to combustion
• Heat transfer between cylinder walls and cylinder contents.
• Mechanical friction, through an empirical model.
• Flow through valves.
• Engine breathing characteristics (through a simple empirical model).
• The effect of valve and ignition timing.
Aspects not included in the Otto cycle model
• Nature of the combustion products during exhaust. Chemical potential energy will be
present in combustion products which are subsequently exhausted to the atmosphere.
• Exergy generated during the reaction process.
• Residual exhaust gases in the cylinder.
• Heat transfer between the reactants and products.
2.5.4 Discussion
A combustion cycle was configured according to the parameters in Table 2.2, and the simulation
results are presented in Table 2.3. The simulation was performed for two throttled conditions
(inlet manifold pressure of 0.3 bar and 0.5 bar respectively) for a speed of 1000 RPM, and
for two unthrottled conditions (the inlet manifold pressure is 1 bar), for an engine speed of
1000 RPM and 3000 RPM. The sum of the amounts of energy made unavailable to do useful
25
work due to irreversibilities and the net work output of the engine should equal the amount of
energy released during the combustion, which although very close (1245 as opposed to 1324),
is not quite the case as shown in Table 2.3. This is attributed to some expressions for work lost
due to irreversibility including assumptions about the ambient environment.
For instance when evaluating ”lost” work due to mechanical friction and heat transfer, it is
assumed that cylinder wall temperature is 400 K and that ambient air temperature is 300 K - in
other words, the irreversibility is scaled by constant factors which may have been incorrectly
selected.
In Table 2.3, the net irreversibility due to uncontrolled expansion (as evaluated by Equa-
tion 2.27) decreases as the amount by which the engine is throttled increases. For an increase
in throttling, further investigation reveals that the irreversibility due to uncontrolled expansion
increases during the intake stroke, but then decreases during the exhaust stroke as low peak
cylinder pressures result in lower cylinder pressures at the end of the expansion stroke - and
therefore results in less irreversibility due to uncontrolled expansion during the exhaust stroke.
Peak pressures are less because less fuel-air mixture has been combusted. Pumping work under
throttled conditions is higher than under unthrottled conditions as expected, and this is observ-
able by plotting work done against angular displacement for a particular throttling condition.
Comparing the cases where the unthrottled engine operates at 1000 RPM and 3000 RPM:
heat transfer decreases as engine speed increases as there is less time for heat transfer to oc-
cur. Irreversibility due to uncontrolled expansion increases with engine speed as the pressure
difference across the inlet valve increases. There is less time available for charge mixture to
be drawn into the cylinder, therefore irreversibility due to combustion decreases because there
is less charge to be combusted. Friction irreversibility increases with piston speed, and irre-
versibility due to gas transfer is higher because the energy content of air in the cylinder after
expansion is larger, in turn because less heat transfer out of the cylinder has occurred.
2.5.5 Efficiency mapping
The engine efficiency mapping was generated through simulation (see Figure 2.5), but does not
display the behavioral characteristics common to real engines, particularly a decrease in engine
efficiency at high load, and limited increases in efficiency with engine speed due to improved
26
Table 2.3: Dynamic Otto cycle model: Magnitude of various sources of irreversibility.
Two throttling conditions are presented for partially throttled conditions (throttled such that the inlet man-
ifold pressure is 0.3 bar and 0.5 bar) for an engine speed of 1000 RPM. Two unthrottled conditions are
presented (inlet manifold pressure is 1 bar), for the engine speeds of 1000 RPM and 3000 RPM respec-
tively. The engine was set up as in Table 2.2. It is assumed that the ambient pressure is 1 bar.
Throttling Work Mechanical Gas Heat Uncontrolled Comb- ∑ Net heat
condition (J) friction(J) transfer(J) transfer(J) exp (J) ustion(J) (J) input(J)
unthrottled (1 bar) 434 108 274 280 29.5 211 1245 1324
low load (0.5 bar) 153 87 122 175 6.2 107 650 674
idle (0.3 bar) 40 68 69 126 2.3 66 371 411
unthrottled 3000 rpm 207 276 307 218 37 202 1247 1273
breathing characteristics. High load at high speed results in poor combustion. An example of
an actual fuel consumption model is shown in Figure 2.2, and one can see that in this instance
maximum efficiency is at about 80% of maximum torque, with a global optimum efficiency
at an engine speed which is 50% of maximum. Considering the optimum engine operating
point as a starting point: at a constant load, increasing speed from this point results in increased
friction losses, increasing load at a constant speed increases air flow limitations, and the mixture
is subsequently enriched [33].
Another factor not included in the model is cylinder turbulence, particularly the effect that
this has on heat transfer and fuel mixing, resulting in better mixing and burning of fuel [46],
despite a decrease in the volumetric efficiency. The model does provide an understanding of
engine efficiency magnitudes at a range of operating conditions, but where these above ignored
phenomena begin to take effect, it is inadequate.
The engine is most efficient when completely unthrottled according to the model derived
for this project. Each efficiency value is calculated under the assumption that the engine is
operating at steady state. Engine efficiency is defined below:
η = work done per cycle
work done per cycle+work lost to irreversibility per cycle
27
 0.1
 0.15
 0.2
 0.25
 0.3
 0.35
 50  100  150  200  250  300  350  400  450
Ef
fic
ie
nc
y 
(%
)
Work output (J)
Engine efficiency Vs Work at an engine speed of 1000 rpm
Figure 2.5: Dynamic Otto cycle model: Engine efficiency mapping
2.5.6 Moving toward a more realistic engine model
Typically BSFC mappings are determined through experiment [33, 47]: engine torque, speed and
fuel consumption will be measured over a wide range of operating conditions, and the mapping
will be generated accordingly. A method (developed by Goering and Cho) has been developed
to generate efficiency mapping models without conducting many tests. In this method empirical
models of torque, friction, and thermal efficiency are used to generate the mapping [47]. This
mapping approach lends itself well to the intended future focus of the project, as a wide variety
of engines may be fitted with the loading mechanism, and an efficiency mapping test would
probably need to be done on each of the candidate engines. The efficiency mapping method
developed by Goering and Cho is outlined below:
ηengine =
piτ rpm
30HVlm˙ f
= ηtherm
1+ f mepbmep
(2.37)
After equating the above:
m˙ f =
piτ rpm(bmep+ f mep)
30bmepHVl
BSFC = m˙ f
Out put power
(2.38)
where:
28
bmep = Brake mean effective pressure
f mep = Friction mean effective pressure
rpm = Engine speed in rev/min
m f = Mass of fuel
BSFC = Brake specific fuel consumption
HVl = Lower heating value of the fuel
2.5.7 Transient effects
No attempt has been made at modelling the efficiency of the engine at an accelerating transient
state. A recent study has shown that during fast accelerating transients, measured fuel con-
sumption is up to twice as much as during the corresponding steady-state load conditions [48].
The transient fuel consumption model presented in the above mentioned reference is based on
the sum of the static fuel consumption properties of an engine and a transient fuel consumption
correction factor, which is based on measured engine performance data. Given that fuel effi-
ciency is low under accelerating transient conditions, it may prove beneficial to use compressed
air to accelerate the engine, which is proposed as a topic for future work. One would begin
by assuming that fuel consumption under accelerating transient conditions is 50% higher than
normal (i.e. efficiency is 50 % poorer), and then determine engine efficiency if compressed air
is used as means of accelerating the engine.
2.5.8 Summary: dynamic Otto cycle model
This section develops the models used to simulate two modes of operation of an I.C. engine:
conventional operation with fuel, and propulsion by compressed air. The dynamic Otto cycle
model was used to predict the amount of work associated with each of the four strokes of a
combustion cycle, and allowed the amount and state of exhaust gases exiting an engine dur-
ing an exhaust stroke to be evaluated. Expressions relating maximum power and efficiency to
operational parameters such as engine speed and throttle setting (amongst others) were also
developed from this model.
29
2.6 More accurate modelling of feasibility of compressed air augmentation of an
engine
This model is used to demonstrate where loading the engine with the compressed air system
could be applied. Table 2.4 indicates that the effectiveness of the modification decreases as
the combined compressed air storage and usage efficiency decrease, and serves to show that
the efficiency gain decreases as the amount of required loading decreases. Consider the engine
operating with an inlet pressure of 0.42 bar, producing 100 J of work per cycle at an efficiency
of 17%. If the engine were fully loaded, 360 J of energy would be available to be used by
the compressor. Of course, as noted in Section 2.4, not all this energy will be available for
reuse. For the purposes of the explanation, assume that the combined efficiency of the air
compression-expansion process is 70%; 252 J of energy is available for reuse, and the effective
efficiency (ηpl) of the engine after taking this energy into account is 27% (which is greater than
if the engine were operated partially loaded). The region of feasibility has been summarised in
Figure 2.6.
Figure 2.6: Viability of loading an internal combustion engine for various compression-expansion process
efficiencies and different required loading conditions
30
Table 2.4: Feasible conditions where loading will improve operating efficiency (revised)
ηc 1 0.9 0.8 0.7 0.6 0.5
Load IP ηp W Hp ER ηpl ER ηpl ER ηpl ER ηpl ER ηpl ER ηpl
(bar) (J) (J) (J) (J) (J) (J)
2 0.05 0.03 10 384.62 450 0.35 405 0.31 360 0.28 315 0.25 270 0.21 225 0.18
4 0.28 0.05 20 400.00 440 0.35 396 0.31 352 0.28 308 0.25 264 0.21 220 0.18
9 0.32 0.09 40 444.44 420 0.35 378 0.32 336 0.28 294 0.25 252 0.22 210 0.19
13 0.35 0.12 60 491.80 400 0.35 360 0.32 320 0.29 280 0.26 240 0.23 200 0.20
17 0.38 0.15 80 533.33 380 0.35 342 0.32 304 0.29 266 0.26 228 0.23 190 0.20
22 0.42 0.17 100 574.71 360 0.35 324 0.32 288 0.29 252 0.27 216 0.24 180 0.21
28 0.46 0.20 130 637.25 330 0.35 297 0.32 264 0.30 231 0.27 198 0.25 165 0.22
35 0.51 0.23 160 698.69 300 0.35 270 0.32 240 0.30 210 0.28 180 0.26 150 0.23
41 0.56 0.25 190 763.05 270 0.35 243 0.33 216 0.31 189 0.29 162 0.27 135 0.25
48 0.61 0.27 220 823.97 240 0.35 216 0.33 192 0.31 168 0.29 144 0.27 120 0.26
54 0.66 0.28 250 886.52 210 0.35 189 0.33 168 0.32 147 0.30 126 0.28 105 0.27
61 0.70 0.30 280 949.15 180 0.35 162 0.33 144 0.32 126 0.31 108 0.29 90 0.28
67 0.75 0.31 310 1013.07 150 0.35 135 0.34 120 0.32 105 0.31 90 0.30 75 0.29
74 0.79 0.32 340 1075.95 120 0.35 108 0.34 96 0.33 84 0.32 72 0.31 60 0.30
80 0.85 0.33 370 1138.46 90 0.35 81 0.34 72 0.33 63 0.33 54 0.32 45 0.31
87 0.90 0.33 400 1201.20 60 0.35 54 0.34 48 0.34 42 0.33 36 0.33 30 0.32
93 0.94 0.34 430 1264.71 30 0.35 27 0.34 24 0.34 21 0.34 18 0.34 15 0.34
100 0.99 0.35 460 1325.65 0 0.35 0 0.35 0 0.35 0 0.35 0 0.35 0 0.35
31
3 Application of the compressed air system to engines in existing
vehicles
3.1 Overview: retrofitting to existing vehicles
Efficiency mappings released by manufacturers[21] as well as the simulation results of the
previous chapter show that a typical spark-ignition internal combustion engine is least efficient
when the amount of torque required by the engine is low. For existing vehicles, this is the
reason for proposing that a compressor be used to load the engine when it is producing torque
below a certain threshold, by compressing air into a receiver, allowing it to operate in a more
efficient zone (as throttling could be decreased). The compressed air will then be used to propel
the engine in some way at a later stage: either in place of conventional fuel, or in conjunction
with it.
This chapter contains a description of how the compressed air will be used, the performance
requirements of the compressed air augmentation system, how the torque developed by this
system will be controlled, and expected efficiency of usage of the compressed air. The idea of
using compressed air in conjunction with conventional fuel is presented and reasons are given
why this avenue of investigation is not pursued.
3.2 Using compressed air to propel the engine without fuel
3.2.1 Using an external compressor to expand the air
An external compressor could be used to expand compressed air in order to supplement driving
requirements. Bearing in mind that this external compressor will be small in order to fit into
the engine bay, the amount of torque that this external compressor can produce will be limited.
This means that the external compressor will have to work in conjunction with the IC engine to
produce the amount of torque required by the driver. The overall efficiency of such a process
would have to be quantified, and this is done by means of a numerical example, assuming that
the external compressor is sized such that it can load the engine by 30 N.m (94 J stored per
stroke), that the performance factor of the energy storage process is 2.4 through the effects of
regenerative heat transfer (see Section 4.2.7 for more details on performance factor and regen-
erative heat transfer). The specifications of the external compressor have arisen from analysis
32
presented in Chapter 4. The isentropic efficiency of compressed air usage will be assumed to
be 0.8 [49].
The reader is referred to Table 3.1. Driver requirements at the time of loading are assumed
to be 51 N.m (160 J per stroke), and the external compressor will provide 94 J of additional
loading per stroke. Heat transfer into the system will mean that 225 J is effectively stored.
When it comes to using the stored energy: 94 J can be supplied by the external compressor,
but only 75 J is useful (due to an isentropic efficiency of 0.8), and the engine must supply the
remainder of the requirement. The efficiency of this process is evaluated by calculating the net
amount of energy delivered by the IC engine (160 J + 85 J) plus the surplus energy stored as
compressed air which was not available to be used for propulsion due to expansion capacity
limitations 3 of the external compressor, but could be used in future expansion strokes. This
figure is then divided by the energy input as fuel.
This modification results in a performance improvement over standard operation, and it is
practical to implement as no modification to the valve system of the engine is required. The
weakness of this modification is that the engine is still required to operate under compressed air
expansion mode, at a reduced torque output and subsequently a poor operating efficiency. For
this reason, using the engine to expand the compressed air is also considered.
3.2.2 Using the engine to expand the compressed air
Compressed air will be used in place of conventional fuel under certain operational conditions.
When these conditions are met, normal four stroke combustion processes will stop (no more
fuel will be injected, and conventional valve operation will stop), and the engine will be used
to expand compressed air directly. This will be done through a separate valve and pipe system
connected to a receiver containing the stored compressed air, as shown in Figure 3.1. The
conventional inlet valves would no longer operate as normal - because air will no longer be
drawn into the cylinder from the atmosphere, but rather injected directly into the cylinder from
a receiver as compressed air. The inlet valves would have to be closed and then disengaged
during compressed air operation by modifying the way in which the camshaft actuates these
3The amount of energy available for later use is: (225−94)×0.8, or the energy not used after storage, multiplied by the usage
efficiency.
33
Table 3.1: Using an external compressor to load an IC engine, and expand the compressed air to reduce the
work requirement on the engine at a later stage.
Modified Unmodified
Driver requirement 160 J 160 J
Energy storage
Amount of loading 94 J 0 J
Amount of energy stored 225 J 0 J
Engine work requirement 160 + 94 J 160
IC engine efficiency 0.28 0.23
Energy consumed as fuel 775 J 695 J
Energy usage
Energy supplement from ext. compr. 75 J 0 J
Engine work requirement 160 - 75 J 160 J
IC engine efficiency 0.15 0.23
Energy consumed as fuel 584 J 695
Overall efficiency 160+85+0.8×(225−94)775+584
160+160
695+695
= 0.26 = 0.23
valves - possibly by shifting the camshaft forward so that it no longer pushed on the valve stems.
Compressed air would have to be admitted through its own valve system. The conventional
exhaust valves would also not be able to be used because the exhaust valve will have to operate
twice as often, Section 3.3 contains further details.
A four cylinder four stroke engine [38] is the candidate engine type for retrofitting with the
compressed air modification. This means that at any point in time, two pistons will be moving
towards top dead center and two pistons will be moving towards bottom dead center, as each
cylinder must perform one of the four strokes (intake - downward stroke, compression - upward
stroke, expansion - downward stroke, exhaust - upward stroke). This allows two cylinders to be
used to expand compressed air concurrently. Air can only be expanded on a downward stroke,
and four downward strokes will occur per revolution.
The essential structure of this modification is to connect a small auxiliary compressor to the
34
drive shaft on the engine, which will be responsible for compressing air into the receiver. When
receiver pressure is sufficient, the engine will be able to enter compressed air mode (indicated
by Figure 3.14), where all four conventional strokes cease, and compressed air is allowed to
enter cylinders currently at the beginning of an expansion stroke. The charging of compressed
air into the receiver will also cease, as there is no longer a need to alter the torque requirement
of the engine through loading.
Driver requirements (inferred from the accelerator pedal position) will not be constant, and
may at times be larger than what the compressed air system can produce. For these reasons,
an overall control philosophy must be defined in order to avoid some undesirable operating
conditions. For instance, if the driver requirements are quite close to what the compressed air
system can supply (the dotted line labelled B in Figure 3.2), the system will only supply the
maximum torque available. If driver requirements increase even more and exceed this level
(the line labeled A), then compressed air usage will stop and the engine will again run on
conventional fuel. A time dead band is imposed so that if torque requirements drop suddenly, the
engine will not revert back to compressed air operation unless the time dead band has expired
(i.e. the engine has been operating on conventional fuel for some time). This is to prevent
excessive switching of operation between the two systems.
3.3 Performance requirements if compressed air propels the engine
This augmentation system is intended for a typical motor vehicle as a retrofit to improve per-
formance, and it should therefore be able to do (in part) what the engine could do under normal
operation. Considering an engine that produces 70 kW at 6000 rpm, one can determine that the
torque at this speed is:
Torque = 70×10
3
2pi×100
= 111.4 N.m (3.1)
The maximum torque produced by the engine varies with engine speed (see Figure 3.3 for an
example of a 4ZZ-FE Toyota engine), with peak torque produced typically just over halfway
4The external compressor is omitted from this drawing because it will not run while the engine is propelled by compressed air
35
Figure 3.1: Schematic diagram of the envisaged compressed air augmentation system.
between the maximum and minimum speed of the engine. The variation with speed is not
excessive; in the diagram used as an example, engine torque varies from a minimum of 100 N.m
to a maximum of 120 N.m, and for the remainder of this section it is assumed that the datum
engine torque of our sample engine is 110 N.m. It must be noted that this value of torque is
measured when the engine is completely unthrottled. The total amount of work done by the
engine to produce 110 N.m of torque over one revolution is 110×2pi = 691 J.
torque = work per revolution/2pi (3.2)
Bearing in mind that this is a four stroke, four cylinder engine; two expansion strokes occur
for every revolution (when operating on conventional fuel), which means that 345 J of work is
done per expansion stroke (this is the net amount of work done). The pumping losses incurred
by the other cylinders have already been included within Figure 3.3, where the torque is the
net torque output of the engine. Under the normal four stroke internal combustion mode of
operation, each cylinder performs one of the four stroke processes (either intake, compression,
36
Figure 3.2: Illustration of possible torque demands made by the driver. Line A represents the torque require-
ment at which the engine will revert to conventional operation. Line B represents the maximum torque that
the augmented system can provide. Line C represents the minimum amount of torque that the augmented
system can provide.
expansion, exhaust), at any instant in time. The piston moves downward on the intake and
expansion strokes, and it moves upwards on the compression and exhaust strokes. This can be
used to advantage when expanding the compressed air, as two cylinders could work in parallel
to produce the required amount of work. For example, if the driving requirements at the time
are such that 110 N.m of torque is required, each piston would have to do 345 J of work per
expansion stroke (if using conventional fuel). However, if using compressed air two cylinders
can be combined to produce this amount of work, so each cylinder would produce 172.5 J of
work. The benefit of this is that lower operational pressures are required for the compressed air
retrofit.
This value of work done per expansion stroke gives an indication of the kind of pressures
that would be required for this engine retrofit to operate. Uncontrolled expansions must be
avoided as much as possible (to ensure that the combined compressed air storage and usage
efficiency be as high as possible), which means that the pressure of the air within the cylinder at
the end of an expansion stroke must be sufficiently high enough to overcome the back pressure
due to the exhaust pipe (say 1.5 bar gauge). The requirement that the air pressure in the cylinder
be approximately 1.5 bar at the end of the expansion sets the maximum cylinder pressure at the
beginning of the expansion. The estimation of what this pressure could be is done assuming
37
Figure 3.3: Typical power-torque-speed curve of a Toyota 4ZZ-FE engine [50].
that the compressed air is modelled as an ideal gas, and the expansion process is modelled as
an isentropic expansion process:
p2
p1
=
(V1
V2
)γ
p2 = p1
(V1
V2
)γ
= p1
(4.407×10−4
4.407×10−5
)1.4
= 1.5×105 ×25.11
= 37.66 bar (3.3)
This figure is the start of the top curve in Figure 3.4a. The amount of work done if air is allowed
to expand from this pressure to 1.5 bar can also be estimated (assuming an isentropic expansion
38
(a) Implication of cylinder geometry on initial cylin-
der pressure
(b) Proposed P-V diagram for the expansion of compressed air
Figure 3.4: Propulsion by compressed air: maximum cylinder pressure & proposed P-V diagram for the
cylinders expanding compressed air.
process):
work = p1V1 − p2V2
1− γ
= (1.5×10
5)(4.407×10−4)− (37.66×105)(4.407×10−5)
1−1.4
= 249.65 J (3.4a)
work = p(V1 −V2)
= 1.01×105(4.407×10−5 −4.407×10−4)
=−40.05 J (3.4b)
One must also take into account that the other side of the piston will be exposed to roughly at-
mospheric pressure5 [51], and this will decrease the net amount of work done on the crank shaft
by 40 J, as shown by Equation 3.4b. The result of the sum of Equation 3.4a and Equation 3.4b
indicates the operational range of the compressed air system.
The fact that the engine (with its fixed geometry) is used to expand the compressed air
poses a challenge because if air at an initial pressure less than 37 bar is expanded, the air
5A PCV valve system is used to regulate crank case pressure as a high crank case pressure leads to oil leakage from seals and
gaskets. The pressure conscious valve (PCV) regulates the crank pressure by allowing air to flow into the inlet manifold.
39
pressure will always drop to some value below atmospheric pressure, which will result in an
energy wastage when the exhaust valve opens as air at atmospheric pressure would rush into
the cylinder. Figure 3.4a illustrates this point - only the curve with an initial pressure of 37 bar
expands to approximately 1.5 bar. All other initial pressures will expand to some other value and
cause irreversibilities. The exhaust valve could be opened when cylinder air pressure equaled
atmospheric pressure in order to circumvent this wastage of energy. Another option is to lower
the operational pressures of the system, but allow compressed air to flow from the receiver into
the cylinder for a longer duration (see Figure 3.4b). The cylinder pressure and compressed air
inlet duration would be carefully selected to ensure that cylinder pressures were still close to
atmospheric at the end of the expansion stroke. The disadvantage of lowering receiver pressure
is that the engine torque range that can be reached by the compressed air system is reduced.
The advantages of lowering the receiver pressure are:
a. The pressure difference between the air in the cylinder at the beginning of the compressed
air injection and the pressure in the receiver would be lower, resulting in lower energy
wastage due to throttling.
b. A cheaper external compressor would be required. The external compressor would need
to supply compressed air at a lower pressure, so its internal components would not need
to be quite as robust.
c. Lower receiver pressure results in a lower receiver temperature, and this improves the
capacity for heat transfer from the exhaust gases to occur.
d. Lower receiver pressure means that the receiver does not have to be built as strongly.
Looking at Figure 3.4b, the expansion processes have a period where cylinder pressure is
constant with volume, and another where the cylinder pressure changes with volume. The
points at which the cylinder pressure begins to decrease are the points where the valve between
the receiver and the cylinder is closed (labelled x,y & z). The work done during this stroke is
given by Equation 3.5 below. The maximum and minimum work done per stroke was plotted
against receiver pressure (see Figure 3.5), and it will be seen that the work done per stroke
increases rapidly up to a point (approximately 10 bar), after which the curve begins to flatten
off. Maximum work done per stroke corresponds to 1-x-2 on Figure 3.4b, and minimum work
40
done per stroke corresponds to 1-5-2. This minimum amount of work does not create a problem
for the driver who wants a very low torque output, because the losses incurred with expand-
ing compressed air are actually higher than this minimum amount of work. See Section 3.4.1
following.
work = p1(Vx −V1)+
p2V2 − p1Vx
1− γ −1.01×10
5(V2 −V1) (3.5)
Where:
p1 = Receiver pressure
Vx = Cylinder volume at which the inlet of compressed air stops
V1 = Cylinder volume at top dead center
p2 = Cylinder pressure at bottom dead center
V2 = Cylinder volume at bottom dead center
γ = Specific heat ratio
The receiver temperature is also a function of receiver pressure, and a graph of receiver temper-
ature vs. receiver pressure has been generated (Figure 3.6) based on the assumption that the air
can be modelled as an ideal gas, and that the compression process is isentropic (Equation 3.6):
Tf
Ti
= p f
pi
(γ−1)/γ (3.6)
Where:
Tf = Final cylinder air temperature
Ti = Initial cylinder air temperature
p f = Final cylinder air pressure
pi = Initial cylinder air pressure
γ = Specific heat ratio
Figure 3.6 shows that the relationship between receiver temperature and pressure is roughly lin-
ear, but a lower receiver pressure results in lower receiver temperature and allows for improved
heat transfer between the exhaust gases and the air within the receiver. Heat transfer will take
place between a heat exchanger (through which exhaust gases will flow), and the air in the re-
ceiver because the heat exchanger is located in the receiver. This heat exchanger is considered
in Section 4.2.
41
Figure 3.5: Propulsion by compressed air: variation of work done per stroke vs. receiver pressure.
3.4 Varying torque generation when compressed air propels the engine
A mechanism must be implemented to control how much torque is produced. Referring back
to Figure 3.4b, there are three dotted expansion lines beginning at points x, y and z. Each of
these points corresponds to a cylinder volume at which the valve between the compressed air
receiver and the cylinder is closed, and the expansion process then begins. Cylinder air pressure
will reach atmospheric pressure before the end of the expansion stroke, so the exhaust valve of
the cylinder must be opened in order to prevent the cylinder drawing a vacuum and absorbing
energy from the crank shaft unnecessarily. The point at which the inlet valve closes can be
calculated from Equation 3.7:
Vx =
required work per stroke× (1− k)+ p1V1(1− k)− p2V2
−kp1
(3.7)
Mass will flow out of the receiver during compressed air usage, decreasing both the pressure
and temperature of the air in the receiver. The decrease in pressure will have an effect on
the maximum torque that the engine can provide when being propelled by compressed air, as
indicated in Figure 3.5.
42
Figure 3.6: Propulsion by compressed air: variation of receiver temperature vs. receiver pressure.
3.4.1 Irreversibility due to inlet of compressed air into the cylinder
Energy wastage due to uncontrolled expansion will be large when air at receiver pressure is
allowed to flow into the cylinder at atmospheric pressure. This loss will be minimised if the
inlet pressure is reduced, thereby reducing the pressure difference across the inlet valve. The
amount of irreversibility can be determined considering that the throttling across this valve is an
isenthalpic process, and work lost due to throttling can be calculated from Equation 2.28. The
compressed air has been modelled as an ideal gas, so the temperature does not change across
the expansion valve [52, pg 145] i.e. the Joule-Thomson co-efficient has been assumed to be
zero.
The air pressure within the cylinder will rise as the air enters the cylinder, resulting in a time
varying pressure across the inlet valve. This can be calculated using the model described in
Section 2.4 (the equations of which are summarised in the equation block below), with cylinder
volume fixed and heat transfer into or out of the cylinder ignored. Mass flow rate through the
valve is given by Equation 2.10.
43
p˙ = R(m˙T +
˙Tm)
V
(2.5)
˙T = 1
mcv
[him˙−um˙] (2.9)
˙Irmx = m˙T0Cp
(
ln
(Tu
Td
)
− Tu −Td
Tu
)
(2.26)
˙Irth = T0m˙
[
−Rln
( pu
pd
)]
(2.28)
Once through the valve, the air will mix with air already in the cylinder. The temperature of
this air is determined by assuming that the air is modelled as an ideal gas initially at 1000 K
and was expanded from a volume of 44.07 cm3 to 440.7 cm3, resulting in a final temperature6
of 400 K. In reality this final temperature may be considerably lower (see Section 4.2.7 - heat
transfer from the exhaust gases does not occur fast enough to raise the receiver temperature to
1000 K, even though exhaust gas temperature is near this value), and therefore the irreversibility
due to mixing of gases at different temperatures will be less than the amount calculated here.
This calculation gives the worst case scenario.
The irreversibility due to mixing gas at different temperatures is given by Equation 2.26. A
plot has been generated of work lost due to irreversibility for various values of receiver pressure,
and is shown in Figure 3.7. If a cylinder pressure of 12 bar is selected (see Section 4.3.4 for
details), 202 J of work would be done per expansion stroke with no irreversibilities present,
9.3 J of energy will be wasted due to the irreversibilities of throttling and mixing gases at
different temperatures; 25 J of energy will be lost due to friction (calculated according to the
model described in Section 2.4.1, where maximum cylinder pressure is specified at 12 bar, and
engine speed is 1000 RPM), 19.1 J will be lost because air is expanded to 1.5 bar and not to
atmospheric pressure, and 108.6 J of work is done during this stroke - resulting in a stroke
6 T2
T1 =
(
V1
V2
)1−1.4
= 1000×10−0.4 = 398 K
44
efficiency of 53.7%.
work = work done at particular receiver pressure
−work done by piston against the atmosphere
−work lost due to f riction
−work lost due to throttling
−work lost due to uncontrolled expansion
= 202−40−25−9.3−19.1
= 108.6
(3.8)
η = 108.6
202
= 53.7% (3.9)
Figure 3.7: Propulsion by compressed air: variation of useful work lost due to irreversibility (specifically
irreversibility due to throttling and the mixing of air at different temperatures) vs. receiver pressure.
45
3.5 Using compressed air in conjunction with fuel
This section gives an overview of the investigation into using compressed air in conjunction
with conventional fuel. Compressed air could be injected directly into the cylinder at some
point during a combustion cycle, in much the same way as Simington has done [12]. The
question of when the compressed air should be injected now arises; it can be done prior to,
during, or after the combustion process. If the compressed air injection is done prior to ignition,
the ensuing combustion will be very lean (i.e. the air-fuel ratio will be greater than 1). This has
both positive and negative ramifications: combustion efficiency is improved and heat transfer
loss is decreased, but flame speed is reduced [53] and nitrogen oxide (NO) emissions increase
[54]. Fuel-air mixture of a sufficient richness must surround the spark plug to ensure ignition,
which may place limitations on injecting the compressed air prior to combustion, as such an
injection may make the fuel air mixture too lean for ignition. This approach is possible, but the
added control complexity eliminates this option as a potentially feasible retrofit candidate. It
may be possible to inject compressed air into the cylinder during the combustion, although the
complex relationship sure to exist between the compressed air wave and the flame front would
best be investigated experimentally as opposed to mathematically, as the differential equation
model developed for the analysis of cylinder states may not provide an accurate description of
the process7. Additionally, combustion must not be arrested by the injection of compressed
air, although this is unlikely as the auto ignition temperature of petrol is 246◦C [55], and the
temperature of the air within the cylinder (after compressed air injection) is predicted to be
still greater than the auto ignition temperature of the mixture, as shown by Figure 3.8, which
was generated using the Otto cycle model from Section 2.5 but with compressed air allowed to
enter the cylinder 0.23 rad after combustion had been initiated. The remaining alternative is to
inject the compressed air into the cylinder after or during the final stages of combustion so as to
supplement the amount of fuel injected into the cylinder per cycle.
Compressed air injection will also reduce the exhaust gas temperatures which may decrease
the potential for efficiency improvements through regeneration. Exhaust gas pressure will be
7A finite element or multidimensional burn model approach is probably required for meaningful analysis here. Combustion is
modelled with a Weibe function in this work, which is empirical in nature - based on measurements of an actual combustion process
[33, pg 145].
46
higher than in standard operation, causing more irreversibility due to uncontrolled expansion
during the exhaust stroke. This can be seen in Figure 3.9 where final cylinder gas temperatures
of the system with compressed air injection are lower than those of the standard cycle, and
cylinder gas pressures are higher than those of the standard cycle. Irreversibility due to mixing
of gases at different temperatures would increase.
500
1000
1500
2000
2500
0 2 4 6 8 10 12
Te
m
pe
ra
tu
re
 (K
)
Angular position (rad)
Standard cycleModified cycle
Figure 3.8: The effect on cylinder gas temperature with compressed air injection. Cylinder gas temperature
drops when compressed air is injected, but not to a level lower than the auto ignition temperature of the fuel
air mixture.
The mixture of oxygen rich air with hot exhaust gases may also have an impact on the
amount of emissions produced by the engine, as unreacted CO may then react with the O2 to
form CO2. This would need to be verified with an emissions analyser. Additionally, inject-
ing compressed air into the cylinder post-combustion will increase the amount of free oxygen
present in the exhaust gases, resulting in less NO gases being reduced into N2 and O2 by a typ-
ical gasoline type catalytic converter [56]. A NO absorber (commonly used for diesel engine
emission control due to the characteristically lean operation of such engines) may be required
in order to meet emission regulations.
Using the compressed air in conjunction with conventional fuel was investigated by per-
forming simulations in which the differential equation models of the Otto cycle derived in the
previous chapter were modified, so that compressed air was injected into the cylinder towards
47
05
10
15
20
25
30
35
40
50 100 150 200 250 300 350 400 450
Pr
es
su
re
 (b
ar)
Volume (cm3)
Compressed air injectionStandard Otto cycle
(a) P-V diagram
 500
 1000
 1500
 2000
 2500
 5e-5  1e-4  1.5e-4  2e-4  2.5e-4  3e-4  3.5e-4  4e-4  4.5e-4
Te
m
pe
ra
tu
re
 (K
)
Volume (m3)
Compressed air injectionStandard Otto cycle
(b) T-V diagram
Figure 3.9: P-V and T-V diagrams for a standard Otto cycle and an Otto cycle with compressed air injection.
The injection pressure and temperature are 40 bar and 600 K respectively, and the injection duration is
pi/15 rad. The model used to analyse the compressed air system is presented in Section 2.5.
the end of the combustion process and used to generate Figure 3.8 and Figure 3.9. The re-
sults of these simulations show an improvement in operational efficiency over a standard cycle,
but the receiver pressures required were very large (simulations show a range from 40 bar -
75 bar, depending on the current load requirement of the engine - for the required amount of
compressed air to be forced into the cylinder in the time available), and so a very powerful ex-
ternal compressor would be required to generate the required pressures. An additional negative
factor was that the differential equation models were not comprehensive enough to simulate
the interaction between combustion flame fronts and the compressed air - thereby affecting the
combustion process without fully simulating the effects thereof. So this is definitely not recom-
mended as a retrofit option, but simulations of compressed air injection into the cylinder were
done, and the results are presented in Table 3.2 - where the amount of compressed air injected
during the combustion process is varied for a particular throttle setting (in this case yielding
an inlet manifold pressure of 0.55 bar). The engine is still loaded with an external compressor
which supplies air to the receiver. Air is compressed adiabatically assuming an isentropic effi-
ciency of 0.80, resulting in a work input per kg of compressed air of 690 kJ/kg.
Note - air temperature if compressed isentropically from atmospheric conditions to 40 bar is
859 K.
48
compression work = h(859)−h(300)
= (887.23−300.19)/0.80
= 690.6 kJ/kg (3.10)
The overall engine efficiency must be found iteratively as the energy source of the external air
compressor is the engine itself; the overall efficiency is used to determine the equivalent fuel
cost (input heat cost) of compressing a certain quantity of air. A sample efficiency calculation is
shown in Equation 3.11 below, on the bases of 80% of the work input to the compressor being
used to create the pressure rise. The amount of energy added to the system as compressed air
is determined by multiplying the mass of air injected by the work input of the compressed air
(Equation 3.10 above).
ηengine =
work out put
energy input
= work out put to drive sha f t +ηc(work extracted by compressor)
heat input +air energy input/ηengine
(3.11)
For the calculation below, the value are the final ones obtained by iteration, causing both sides
of the equation to balance.
air energy input = 1.5×10−4 ×690×103
= 103J
η = (269−183)+0.8×183
746+103/0.1734
= 17.34% (3.12)
As seen from Table 3.2, injecting compressed air into the cylinder directly results in a very
limited efficiency advantage for the torque output range which will be useful to the driver. The
results are good for a very low output range (less than 50 J per stroke) but this is not very useful
to the driver. This is a complex modification with high compressed air pressures, and will not
be considered as a candidate retrofit option for this project.
49
Table 3.2: Simulation summary - compressed air injected into the cylinder during combustion. The work
input of compressing air from atmospheric conditions to receiver pressure is 690 kJ/kg. The engine is
loaded by the amount of 183 J per stroke to produce compressed air at 40 bar. Net heat input is 746.4 J.
This table was generated using the modified dynamic Otto cycle model.
Air injection duration (rad) pi/12.5 pi/15 pi/17.5 pi/20 pi/25 pi/30
Irreversibility due to :
Gas transfer (exhaust)(J) 153.6 147.56 144.24 142.18 139.86 131.5
Heat transfer (J) 43.8 43.89 44.32 44.69 45.25 45.0
Throttling (J) 21.27 16.7 14.14 12.58 10.86 9.3
Mixing (J) 34.48 25.8 19.76 15.41 9.94 7.6
Combustion (J) 101.4 101.4 101.4 101.4 101.4 101.4
Gross work done (J) 304.0 269.8 247.6 232.4 214 203.0
Net work done (J) 121 86.8 64.6 49.4 31 20
Injected air mass (kg ×104) 2.1 1.5 1.1 0.82 0.51 0.4
Overall efficiency (%) 16.72 17.34 18.69 19.0 19.32 18.7
Unmodified engine efficiency (%) 19.6 15.8 12.92 10.6 7.3 5
50
4 Enhanced compressed air system with heat recovery from ex-
haust gases
4.1 Overview
An external reciprocating compressor will be used to store compressed air into a receiver, be-
cause the pressure ratio of the compressor will vary with the state of the receiver, and therefore
the point at which the outlet valve must open will change. This can be overcome by using the
type of reed valve commonly found in small refrigeration compressors. The reed valve opens
automatically once the pressure within the cylinder is greater than the pressure in the receiver
[57, pg 141]. The engine drive shaft and compressor must also be able to be disconnected
(through an electrical clutch mechanism similar to those used for car air conditioners) when
driver torque requirements are high and loading is not needed. The throttling of the engine
would also have to be automatically controlled such that the engine is automatically unthrottled
when it is loaded with the external compressor. Figure 4.1 shows a schematic diagram of the
system.
The efficiency of compressed air usage is low (53.7% as shown in the previous chapter), and
the use of heat from the exhaust gas is investigated as a way of improving the overall process
efficiency. The exhaust gases present a source of ”free” energy in the sense that under normal
conditions exhaust gases are not used by the engine (if considering a normally aspirated engine,
as in this research report). It is envisaged that heat from the exhaust gases would be transferred
into the receiver, thereby increasing the temperature of the air therein, and ultimately the useful
work output of the system. As an illustration, consider heat transfer to an ideal gas stored in a
receiver (of constant volume) initially at 9 bar and 562 K, such that the final temperature of the
receiver is 800 K:
p2 = p1 ×
T2
T1
= 9× 800562
= 12.8 bar (4.1)
51
Assuming a constant pressure expansion from the receiver into the cylinder:
work = p(V1 −V2)
work1 = 9×105(4.407×10−5 −4.407×10−4) (without heat recovery)
= −356.3 J (4.2)
work2 = 12.8×105(4.407×10−5 −4.407×10−4) (with heat recovery)
= −507.6 J (4.3)
The above illustration shows that the potential for work from the receiver is far larger because
the receiver pressure is higher, so overall process efficiency can be improved using heat from
the exhaust that would otherwise be wasted. The heat exchanger would be placed in the receiver
to maximise the time available for heat transfer to take place.
A model was created to simulate how the system will be affected by certain system param-
eters. This model is created assuming that air is an ideal gas, and all compression processes are
isentropic, and therefore serves to demonstrate the performance of the system. The model does
allow for the effects on process efficiency of a number of system parameters to be observed,
including:
• Exhaust gas temperature and mass flow rate
• Heat exchanger area
• External compressor size
• Receiver size
• Gear ratio between the external compressor and the engine
This chapter describes how the model was created, how heat transfer from the exhaust gas is
expected to occur, and how effectively the energy storage system mitigates the losses incurred
when using compressed air to propel the engine.
4.1.1 Required performance improvement
An indication of the performance required by the improvement must be explored in order to
specify a design of the heat exchanger and external compressor. The modification will have a
certain outlay cost but would yield an efficiency improvement. It is assumed that the vehicle will
52
Figure 4.1: Energy storage schematic overview.
be driven between 1500 - 2500 km per month, the fuel price is taken to be R 10/l and the fuel
consumption of the unmodified engine is taken to be 10l/100km. This data is used to calculate
how much money would be saved over a year for a range of monthly driving distances, and
this has been summarised in Figure 4.2. The feasibility of this modification ultimately depends
on the cost of the retrofit and the envisaged life span of the vehicle, but for the purposes of
this investigation it is assumed that an efficiency improvement of 10% is required for economic
feasibility.
4.2 Heat exchange from the exhaust gases to the compressed air
4.2.1 Overview
Heat from the exhaust gases will be used to make up for the irreversibilities incurred when
expanding the compressed air, however this heat transfer must take place between the exhaust
gases and the compressed air, and therefore requires the use of a heat exchanger. Heat must
be transferred into the compressed air of the receiver to gain an increase in process efficiency
(through the increase in receiver pressure). The engine has been designed with a specific back
pressure from the exhaust system in mind, and the heat exchanger must not affect this ex-
cessively. Fouling will also build up along the exhaust pipe over time, which will affect the
53
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
1600 1800 2000 2200 2400 2600
An
nu
al
 s
av
in
g 
(R
)
Monthly distance (km)
Eff = 10%
Eff = 15%
Eff = 20%
Figure 4.2: Annual saving due to efficiency improvement.
54
effectiveness of the heat exchanger as well.
4.2.2 Heat exchanger design
In deciding on which type of heat exchanger would best suit this application, a number of
possible exchanger options were investigated. These include[58, pg 13]:
• Double-pipe heat exchanger - which is made up of two concentric pipes, where one fluid
flows through a center pipe and the other flows through the annular space between the
pipes.
• Shell and tube heat exchanger - which is made up of a bundle of tubes mounted in a
cylindrical shell. One fluid flows in the tubes and the other fluid flows over the tubes,
within the shell.
• Plate type heat exchanger - which consists of a number of stacked plates between which
the hot fluid and cold fluid flow.
It was decided that the heat exchanger should be placed in series with the exhaust pipe and have
minimum frictional resistance so as to affect the system as little as possible. For this reason it
was decided that the heat exchanger be a pipe with an extended surface area (achieved by adding
fins to the pipe). As stated earlier, the heat exchanger will be placed in the receiver, and the air
inlet from the external compressor will be directed such that forced convection takes place over
a portion of the surface of the heat exchanger. Heat transfer via natural convection will occur
from the remainder of the heat exchanger surface, and this will allow heat exchange to take place
even when no compressed air is flowing into the receiver. Figure 4.3 is a conceptualisation of
the heat exchanger.
The rate at which heat transfer can occur into the receiver air must also be considered. There
are four components which act to inhibit heat transfer, a representation of which is shown in
Figure 4.4. R1 is the resistance to heat transfer between the exhaust gas and the heat exchanger,
R2 is the resistance between the heat exchanger surface and the air within the receiver with
forced convection present, R3 is the resistance between the heat exchanger surface and the air
within the receiver with only natural convection present, and R4 is the resistance due to the
metal of the pipe - and is omitted from this analysis since its resistance to heat transfer is much
55
Figure 4.3: Heat exchanger - conceptual drawing
lower than the other resistances. The derivation of each of these resistances to heat transfer is
presented in Sections 4.2.3 to 4.2.5 below.
Each source of resistance to heat transfer is a complex function of variables, all of which
change with the operating state of the system. For example, the heat transfer coefficient be-
tween the exhaust gases and the heat exchanger increases as the engine speed increases due to
increased mass flow and turbulence in the exhaust pipe; heat transfer from the surface of the
heat exchanger due to forced convection decreases as the flow rate of air over the surface de-
creases, yet heat transfer due to both forced and natural convection increases as the pressure in
the receiver increases.
Exhaust gases will have to be diverted around the heat exchanger when compressed air is
used to power the engine, otherwise heat transfer from the receiver into the exhaust gases will
occur, and process efficiency will decrease.
Heat transfer into the air within the receiver is dependent on the overall heat transfer coeffi-
cient (U ), the surface area of the heat exchanger which is in contact with the air in the receiver,
and the log mean temperature difference (LMTD - Equation 4.5) between the temperature of
the exhaust gases and the temperature of the air in the receiver. The heat exchanger is mounted
in the receiver, so there is no cold receiver inlet (t1) and hot receiver outlet (t2), rather just a re-
ceiver air temperature8 (trec) when evaluating the LMTD between the exhaust gas and receiver
8Receiver air temperature will not be uniform, but has been assumed so to simplify modelling complexity.
56
Figure 4.4: Heat transfer analogy - each resistor represents a source of resistance to heat transfer.
air.
˙Q =U ×A×LMTD (4.4)
Where:
LMT D = (T1 − t2)− (T2 − t1)
ln T1−t2T2−t1
= (T1 − trec)− (T2 − trec)
lnT1−trecT2−trec
(4.5)
And:
A = Area of the heat exchanger in contact with the air in the receiver (m2)
T1 = Exhaust stream inlet temperature (K)
T2 = Exhaust stream outlet temperature (K)
t1 = Cold stream inlet temperature (K)
t2 = Cold stream outlet temperature (K)
trec = Temperature of the air in the receiver (K)
LMTD = Log mean temperature difference (K)
U = Overall heat transfer coefficient (W/m2K)
57
1200
1300
1400
1500
1600
1700
9.5
BDC
10 10.5 11 11.5 12
TDC
12.5
Te
m
pe
ra
tu
re
 (K
)
Angular Position (rad)
(a) Exhaust gas temperature
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
9.5
BDC
10 10.5 11 11.5 12
TDC
12.5
M
as
s 
(kg
)
Angular Position (rad)
(b) Mass of the exhaust gas
Figure 4.5: Exhaust gas temperature and cylinder air mass during an exhaust stroke. The sharp fall in
exhaust gas temperatures is due to the rapid decompression of the cylinder during the blow-down phase of
the exhaust stroke, also seen by the rapid decrease in cylinder mass in 5(b). The simulation was done for
the engine described in Section 2.5, but with an inlet pressure of 0.6 bar (typical driving requirement) -
producing approximately 70 N.m of torque (gross).
4.2.3 Heat transfer coefficient for turbulent flow through the exhaust pipe (R1)
The descriptions of turbulence from Foust et al[59, pg 152] are based on turbulence that is fully
developed - occurring where there is no further change in the velocity pattern of the fluid with
the progression of the fluid. Exhaust gas flow will be pulsed (due to four cylinders exhausting
gas sequentially), and simulation results from Section 2.5 (reproduced as Figure 4.5) show that
mass flow during each stroke will also not be constant (due to the blow down and cylinder
pumping phases of the exhaust stroke) but an averaged value for exhaust gas mass flow is
assumed. The mass flow rate is estimated based on the current air density in the cylinder, the
displacement of the cylinder, and the time it will take for the cylinder to displace all the gas
inside of it (this is related to the engine speed - 1/RPM gives the period in minutes for two
strokes, one ”up” stroke and one ”down” stroke).
58
¯m˙ = ∆m∆t
where :
∆m = ρ(Vi −Vf ) (mass o f air charged per stroke)
∆t = 30
N
(period in seconds f or one stroke) (4.6)
The model used to describe how much heat is transferred from the exhaust gas is described
below. It is based on theory presented in Foust et al [59, pg 168] and Incropera et al [60, pg
354]. The convective heat transfer coefficient for turbulent flow in circular tubes is dependent
on the Reynolds number and the Prandtl number (characterising the dependence of heat transfer
on the fluid properties). This value of the heat transfer coefficient is taken for the case where
cooling of the fluid is occurring (heating of the fluid is modelled using a different exponent for
the Prandtl number).
¯h = 0.023λd (NRe)
0.8(NPr)0.3
= 0.023λd
(umdρ
µ
)0.8(cpµ
λ
)0.3
(4.7)
Where:
d = Diameter of the exhaust pipe (m)
um = Mean free stream velocity (m/s)
¯h = Heat transfer coefficient from exhaust gases to the heat exchanger surface (W/m2.K)
λ = Thermal conductivity of the exhaust gas (W/K.m)
µ = Viscosity of the exhaust gas (N.s/m2)
cp = Specific heat of the exhaust gas (kJ/kg.K)
4.2.4 Heat transfer coefficient for forced convection over heat exchanger area (R2)
Heat transfer due to forced convection will occur where the air is directed over the heat ex-
changer surface. This is not expected to occur over the entire surface - rather just a portion
59
of it. The size of this portion is dependent on how the compressed air inlet into the receiver
is designed - one could flare this inlet more to allow the air to pass over more segments, but
this in turn will decrease the velocity (because there will now be more space through which the
same amount of air will flow) and lower the heat transfer coefficient due to forced convection,
ultimately yielding a heat transfer coefficient comparable to that of natural convection.
Equation 4.8 below is valid for a Reynolds number less than 5×105 and a Prandtl number
greater than 0.6 [61]. Air is directed along the length of the heat exchanger surface, and no
attempt has been made to analyse how the air flow will rise as it progresses along the length of
the heat exchanger surface, as it has been assumed that the air will move much faster along the
surface of the heat exchanger than it will rise from the surface of the heat exchanger.
NRe =
Lρum
µ
NPr =
cpµ
λ
¯h = λ(NRe
1
2 )(NPr
1
3 )
L
(4.8)
60
4.2.5 Heat transfer coefficient for natural convection (R3)
By placing the heat exchanger inside the receiver, heat transfer will also take place when no
forced convection occurs. In this case, there are many adjacent fins (much like a heat sink for a
microchip), and so the standard relationships for heat transfer due to natural convection cannot
be used. Heat transfer relationships due to natural convection specific to heat sinks have been
used [62, 63, 64, 65].
h = λ Ra b
24 L
(
1 − e− 35Ra b
)0.75
Ra =
ρ g β cp b3 ∆ T
µ λ (4.9)
Where:
b = Gap between fins (m)
Ra = Rayleigh number
β = Volumetric coefficient of thermal expansion (K/m3)
∆T = Temperature difference from surface to air (K)
L = Length of fin in vertical flow direction (m)
4.2.6 Heat transfer solution algorithm
In the elementary design of the heat exchanger, the outlet temperature has to be solved for
iteratively, which is done according to the following algorithm:
1. Guess a value for the outlet temperature of the exhaust gas from the heat exchanger. In
this case the initial guess was made according to T2 = trec + T1−trec2 , because it is known
that the outlet temperature must lie somewhere between the exhaust gas inlet temperature
and the receiver temperature.
2. Calculate the UA and LMT D and hence the amount of heat transferred.
3. A new value for the outlet enthalpy (he) and hence temperature is then calculated, based
on the known mass flow rate, inlet enthalpy, and previously calculated heat transfer value
61
using Equation 4.10.
he =
− ˙Q+him˙i
m˙i
(4.10)
4. Steps 1. to 3. are repeated until T2 converges, which is reached when the value of T2 from
the previous iteration differs from the value of T2 by a small margin, i.e. T2old −T2 < 0.1
At times it was found that the solution became unstable (in which case the outlet temperatures
from the heat exchanger never converge). This was found to happen when the algorithm guessed
a heat exchanger outlet temperature which was less than the temperature of the air in the receiver
- which is impossible. In order to prevent such instability, the process would be repeated (from
step 1 above), but now step 3 was changed as follows: T2 = T2 − T2−T2old2 . This has the effect of
slowing down the rate of the convergence of the algorithm, and eliminated solution instability.
Sample calculations for each of the three types of heat transfer have been included in Appendix
C.
4.2.7 Performance improvement due to heat transfer
Heat transfer into the receiver has the effect of raising the temperature and pressure of this air,
thereby reducing the amount of work that the external compressor is required to do in order to
raise the pressure of the receiver to the maximum level. Performance of this process can be
quantified by comparing the amount of work required to raise receiver pressure by a certain
amount when no heat is transferred into the system (i.e. with no heat exchanger), to the case
where heat is transferred into the system. This comparison results in a factor which gives an
indication of how effective the system will be. For example, if no heat is transferred into the
receiver 69.4 kJ of work would have to be done to raise the pressure of a receiver from 6 bar
to 12 bar. If heat is transferred into the receiver, 58.8 kJ of work is required to increase the
receiver pressure from 6 bar to 12 bar, and the performance of the system can be quantified as:
per f ormance f actor = Work done without heat trans f er
Work done with heat trans f er (4.11)
= 693.8588.9
= 1.18
62
To put this figure into perspective, if 100 J of energy were to be released from the receiver,
84 J would have actually have been expended by the external compressor, and 26 J would
be ”recovered energy” which would mitigate some of the other irreversibilities incurred while
using the compressed air to propel the engine. For example if 100 J were to be released using
a compressor with an isentropic efficiency of 50 %, then the overall efficiency of the process
(from energy storage to energy usage) would be:
ηc = per f ormance f actor×ηusage (4.12)
= 1.18×0.5
= 59%
63
4.3 Energy storage system description
4.3.1 External compressor specification
As mentioned above, loading is achieved using an external compressor. Ideally the engine
would be loaded by the difference between what is required by the driver and the output torque
at which the engine operates most efficiently. This external compressor has to fit inside the
engine bay so that it can be driven off the engine drive shaft, and so space limitations exist. The
aim was to specify a compressor roughly similar in size to a car air conditioning system com-
pressor, or alternator - so an estimate for a practical dimension was taken to be approximately
10-15 cm long, 10-15 cm high, and 5-10 cm wide.
The external compressor must be designed to compress air at atmospheric pressure up to
receiver pressure, at which point the valve between the receiver and external compressor will
open, and compressed air will flow into the receiver. The cylinder volume at the end of the
compression stroke will be approximately zero, and at this point an inlet valve will open and air
at atmospheric pressure will be drawn in. An example of such a process is shown in Figure 4.6,
and the work required by this process is given by Equation 4.13.
Vb = Vi/
( prec
pin
)1/γ
work = prec(Vf −Vb)+
precVb − pinVi
1− γ (4.13)
The initial compressor volume can be calculated from Equation 4.14.
Vi = (τ×pi)/(Prec(−1/X)+ (Prec/X −Pin)/(1− γ)) (4.14)
Where:
X =
(Prec
Pin
)
1
γ
Using:
γ = Specific heat ratio
Prec = Receiver pressure
Pin = Inlet pressure
64
Figure 4.6: External air compressor P-V diagram for charging receiver to various pressures.
Figure 4.7: External compressor size specification.
It is also desired that the loading of the engine be as constant as possible. If using a re-
ciprocating compressor, work input is only required for half of a revolution of the external
compressor; this being on the upward stroke - the downward stroke draws fresh air into the
cylinder chamber for compression. For this reason, the external compressor should comprise
of two cylinders, with the pistons out of phase by 180◦ on the shaft driving the compressor, so
that one cylinder will require some work input from the engine at all times - thereby providing
a more constant source of load for the engine.
4.3.2 Model description
The performance of the energy storage system is a function of the amount of work done on
the air in the receiver, and the amount of heat transferred into it, which are functions of many
different system parameters. The objective of developing the model was to allow the effect of
certain parameters on the energy storage system to be investigated. Such parameters include:
65
• Exhaust gas temperature and mass flow rate
• Heat exchanger area
• External compressor size
• Receiver size
• Gear ratio between the external compressor and the engine
• Engine speed
This allows estimation of the key measures of performance of the system including:
• Heat transfer into the receiver
• Exhaust gas temperatures after passing through the heat exchanger
• Total work done on the receiver
• Mass added to the receiver
• Number of compression strokes required to fill the receiver
• Final receiver temperature
• Maximum receiver pressure
• Heat exchanger area
Some of these factors are predetermined and are out of the bounds of control by the intended
modification, yet their impact on system performance is still of interest. Examples of such
factors are:
• Exhaust gas temperature and pressure
• Compressed air inlet temperature and pressure
Each stroke of the compressor must raise the pressure of air at atmospheric pressure to receiver
pressure (denoted a-b in Figure 4.6), and then push it into the receiver (denoted b-c in Fig-
ure 4.6). The work required to perform this action is dependent on the receiver pressure, and
is given by Equation 4.15 below. The equation is idealised for the purpose of simplification, as
receiver volume is not taken into account when calculating the work required per compression
stroke. In reality, receiver pressure will not be constant while air is forced into it, but if the re-
66
ceiver volume is assumed to be much larger than the cylinder volume9, then this rise in pressure
is small, in which case Equation 4.15 provides a good approximation for the work required.
The temperature of the air in the cylinder at point b is estimated by assuming that the gas
is ideal, compression is adiabatic, and is determined by Equation 4.16. The volume at which
the pressure of the air in the cylinder reaches the pressure of the air in the receiver is deter-
mined using Equation 4.17. The density of the air in the cylinder is given according to Equa-
tion 4.18. The amount of mass added to the receiver per compression stroke is calculated using
Equation 4.19. The final receiver temperature after a compression stroke is calculated using
Equation 4.20. The equations listed in the above paragraph are used by a program (Octave) to
simulate compression into the receiver.
9If the receiver is at a pressure of 6 bar, then the amount of mass added to the receiver per stroke is: ρcpr(V cpr f −V cpri) =
6×105
8314
28.97 500
(2.2×10−4−2.2×10−5)= 8.27×10−4 kg. If the receiver has a volume of 0.5 m3 (and an initial mass of 2.09 kg), resulting
in a new pressure of 6.016 bar.
67
work = pb(Vb −Vc)+
paVa − pbVb
1− γ (4.15)
Tb = Ta
( pb
pa
)
γ−1
γ (4.16)
Vb =
Va
(
pb
pa
)
1
γ
(4.17)
ρb =
pb
RTb
(4.18)
∆m = ρb(Vb −Vc) (4.19)
u f =
hin ×∆m+∆Q+uim× reci
∆m+mreci
(4.20)
Where:
Va = Cylinder volume at bottom dead center (m3)
Vb = Cylinder volume at which cylinder pressure equals receiver pressure (m3)
Vc = Cylinder volume at top dead center (m3)
pa = Atmospheric pressure (bar)
pb = Pressure of the air within the receiver (bar)
Ta = Atmospheric temperature (K)
Tb = Temperature after compression (to receiver pressure) (K)
ρb = Density of the air in the cylinder once at receiver pressure (kg/m3)
∆m = Mass of air added to the receiver per compression stroke (kg)
mreci = Initial mass of the receiver (kg)
ui = Initial specific internal energy of the receiver (kJ/kg)
u f = Final specific internal energy of the receiver (kJ/kg)
hin = Enthalpy at the inlet to the receiver (kJ/kg)
∆Q = Heat transfer into the receiver (kJ)
γ = Specific heat ratio
68
4.3.3 System parameters for the compressed air storage model
A number of the parameters which affect the performance of the modification will vary with
the requirements of the driver at the time, and therefore average values over the expected range
have been used for the simulation. Exhaust gas temperature is a function of a number of factors,
namely throttle position and engine speed, and simulations from the engine model presented in
Section 2.5 show exhaust gas temperature to vary between 900 K −1300 K, so this is assumed
to be 1100 K. The mass flow rate of the exhaust gas depends on the engine speed and the
pressure in the cylinder when the exhaust valve opens. The model shows that this pressure
ranges from between 3− 5 bar for a range of operating condition, and is assumed to be 4 bar
for this simulation. The engine speed is selected to be 2000 RPM, because it is envisaged that
the modification will be most feasible under low torque and engine speed requirements. Air will
enter the external compressor at a pressure and temperature of 1.01 bar and 300 K respectively.
A sample calculation for this model has been included in Appendix D.
4.3.4 Results obtained from the compressed air storage model
4.3.4.1 Heat exchanger area parameters Two components of the size of the heat ex-
changer play a dominant role: referring back to Figure 4.4, R1 and R2//R3 are the resistance to
heat transfer between exhaust gas and the heat exchanger body, and the heat exchanger surface
and the air within the receiver respectively. R1 is dependent on the length of the exhaust gas
pipe through the heat exchanger, and can be increased by increasing the number of 180 degree
bends through it. R2//R3 is dependent on the surface area of the heat exchanger in contact with
the air in the receiver, namely the number of fins of the heat exchanger. A simulation for a range
of these parameters was completed - from 1-9 passes of the exhaust pipe, and for each of these
permutations, 1-50 fins. The simulation shows that the amount of heat transferred increases
with increasing surface area, but does so in a diminishing manner (Figure 4.8). The number of
passes is selected to be five, and the number of fins is selected to be forty for this reason.
The limiting factors for this design decision are the back pressure exerted by the pipe bends
onto the engine, and whether the heat exchanger will fit into the receiver. The effects of the
back pressure could be mitigated by removing the silencer. Typically, the back pressure in the
exhaust pipe is about 1.5 bar - and so the back pressure due to the heat exchanger should not
69
02000
4000
6000
8000
10000
5 10 15 20 25 30 35 40 45 50
H
ea
t t
ra
ns
fe
r (
J)
Number of fins
Heat transfer vs heat exchanger area
1 x pass
3 x pass
5 x pass
7 x pass
9 x pass
Figure 4.8: Heat transfer into the receiver while compressing from an initial pressure of 6 bar to a final
pressure of 12 bar as a function of heat exchanger size.
exceed this value. Assuming that the pressure drop through the heat exchanger will be mainly
dependent on the number of 180 degree bends of the exhaust pipe, the pressure drop can be
evaluated as follows [66]:
∆pb = 0.5KρV 2 (4.21)
Where:
pb = Pressure drop due to the bend (pa)
K = Bend constant associated with the type of bend (1.6)
ρ = Gas density (kg/m3)
V = Gas velocity (m/s)
Calculations show that each bend and each length of pipe will contribute about 448 pa
and 168 pa towards the total pressure drop respectively. A configuration of five passes will
70
contribute 4×448+5×168 = 2632 pa of pressure loss, which should not negatively affect the
performance of the engine, and a silencer could still be used.
4.3.4.2 Effects of definable parameters on system performance:
1. Increasing the gear ratio between the engine drive shaft and external compressor will al-
low the external compressor to load the engine more, but it will also cause the compressor
to rotate faster. This results in a larger mass flow rate through the compressor and im-
proves the heat transfer coefficient between the incoming air and heat exchanger surface.
However, it also means that the air is moving much faster over the heat exchanger, and
there is less time available for heat transfer to take place.
2. A larger compressor swept volume will allow the external compressor to load the engine
more effectively, but it also means that the mass flow rate across the heat exchanger will
be greater, resulting in less time for heat transfer to take place, as in point 1 above.
3. Increasing maximum receiver pressure will allow the engine to be loaded more effectively,
but the temperature of the compressed air flowing across the heat exchanger will also be
greater, and allow less opportunity for heat transfer.
A range of maximum receiver pressures were simulated for a range of external compressor
swept volumes, and these results are shown in Figure 4.9, at a gear ratio of 1:1 between the
engine and external compressor. The trend observed is that the performance factor will decrease
as the swept volume and maximum receiver pressure increase, and for this reason a maximum
receiver pressure of 12 bar will be selected. Note: the minimum receiver pressure is limited to
6 bar, as by Equation 4.13, the loading capacity at a lower pressure and for the swept volume
selected in Section 4.3.5 below would not load the engine sufficiently.
The effect of the gear ratio between the external compressor and engine was then investi-
gated, the results of which are shown in Figure 4.10, with increases in the gear ratio resulting in
a degradation in performance factor. Both the gear ratio between the external compressor and
the engine, and the swept volume of the external compressor increase the amount of loading
that the compressor can exert on the engine.
71
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
150 200 250 300 350
Pe
rfo
rm
an
ce
 F
ac
to
r
Compressor Volume,Vi (cm3)
Performance Factor vs Compressor size, Gear Ratio = 1
Pmax = 10 bar
Pmax = 12 bar
Pmax = 14 bar
Pmax = 16 bar
Pmax = 18 bar
Figure 4.9: Performance factor as a function of external compression volume and maximum receiver pres-
sure, for a gear ratio of 1:1.
4.3.4.3 General Observations Simulations show that the final temperature of the receiver
(once it has reached maximum pressure) is less than the exhaust gas temperature, which shows
that there is still potential for heat transfer to take place. Once the receiver reaches 12 bar,
the external compressor would be disengaged, and forced convection across the heat exchanger
surface would cease. The coefficient of heat transfer due to natural convection would then
become the controlling resistance to heat transfer. Heat transfer due to natural convection will
still occur from the heat exchanger into the air in the receiver once the external compressor
has been disengaged from the engine, but this is not ideal because once the system has reached
maximum pressure, the air within the receiver must be used to propel the engine so that receiver
pressure will be at a minimum and the loading process can begin again. The rate of heat transfer
due to natural convection will also be very low, and will contribute minimally to raising receiver
pressure. Two other effects on the performance factor were noted, but are not controllable:
• The performance factor improves as the exhaust gas temperature increases, as heat transfer
72
1.5
2
2.5
3
3.5
150 200 250 300 350
Pe
rfo
rm
an
ce
 F
ac
to
r
Compressor Volume, Vi (cm3)
Performance Factor vs Gear ratio
GR = 1.0
GR = 1.5
GR = 2.0
GR = 2.5
GR = 3.0
GR = 3.5
Figure 4.10: Performance factor as a function of external compression volume and gear ratio, for a reciever
pressure range of between 6 and 12 bar.
increases.
• The performance factor improves as the engine speed decreases, as more time is available
for heat transfer to occur (as the external compressor will be rotating slower and take
longer to compress the receiver to maximum pressure).
73
4.3.5 Overall efficiency: combining performance factor and loading capacity
It has been shown in Table 2.4 that loading the engine with an external compressor where the
overall efficiency of compressed air usage and storage is 60%, is viable up to a load of 48% of
full torque. This table was generated under the assumption that the engine could be loaded by
the difference between the current torque requirement of the driver, and the maximum torque of
the engine. This is not possible with the external compressor being considered due to loading
capacity limitations, and simulations show that the performance factor is significantly degraded
when high loading capacity is required. A new table was generated accordingly, where the
engine is loaded by a fixed amount - chosen10 for the purposes of this explanatory example in
this case to be 60 N.m. This equates to a 188 J of work per stroke:
workper rev = torque×2pi (4.22)
= 376 J
workper stroke = workper rev/2 (4.23)
= 188 J
The first five columns represent the engine only - and are the same as the first five columns
of Table 2.4. The sixth and seventh columns (at a combined compression-expansion [usage]
efficiency of 70%) represent the net work requirement (per stroke) of an engine loaded by the
external air compressor, and the efficiency of the engine when operating at this load point. The
eighth and ninth, and tenth and eleventh columns are at a combined usage efficiencies of 60%
and 50% respectively. Consider the sixth row of the table where the work requirement of the
driver is 100 J. The engine is loaded by 188 J, so the net load requirement of the engine (column
labelled W+L) is 288 J, and the engine has an efficiency of 30% when operating at this point.
If the combined efficiency of storing and using the 188 J of energy stored by the external
compressor is 70%, then 0.7× 188 = 131.6 J of energy is available to be reused to propel the
10Fixed in order to generate the table. In reality the amount of work absorbed by the compressor will depend on receiver pressure
(will vary up to a maximum of 73 N.m in this case)
74
Table 4.1: Feasible conditions under which partial loading by an external compressor will improve effi-
ciency. This table is similar to Table 2.4 except that the engine is loaded by a fixed amount (188 J in this
case), and not the difference between driver requirements and the maximum torque.
ηc Loaded engine 0.7 0.6 0.5
W ηW HP W+L ηW+L ER η ER η ER η
(J) (J) (J) (J) (J) (J)
10 0.03 792.00 198 0.25 131.6 0.18 112.8 0.16 94 0.13
20 0.05 832.00 208 0.25 131.6 0.18 112.8 0.16 94 0.14
40 0.09 844.44 228 0.27 131.6 0.20 112.8 0.18 94 0.16
60 0.12 918.52 248 0.27 131.6 0.21 112.8 0.19 94 0.17
80 0.15 957.14 268 0.28 131.6 0.22 112.8 0.20 94 0.18
100 0.17 960.00 288 0.3 131.6 0.24 112.8 0.22 94 0.20
130 0.20 1025.81 318 0.31 131.6 0.26 112.8 0.24 94 0.22
160 0.23 1087.50 348 0.32 131.6 0.27 112.8 0.25 94 0.23
190 0.25 1145.45 378 0.33 131.6 0.28 112.8 0.26 94 0.25
220 0.27 1236.36 408 0.33 131.6 0.28 112.8 0.27 94 0.25
250 0.28 1288.24 438 0.34 131.6 0.30 112.8 0.28 94 0.27
75
engine. The overall efficiency of the engine is:
η = energy out
energy in
= driver requirements + energy available f rom compressed air(W +L)/ηW+L
= 100+131.6
288/0.3
= 0.24 (4.24)
Comparing Table 2.4 with Table 4.1, it will be seen that the efficiency improvement of the
engine has been degraded, but is still more efficient than the unmodified engine. The decrease
in efficiency can be explained by the fact that the engine is no longer always operating at its
point of maximum efficiency, but is still operating at a better efficiency than it would have been
if it were to operate unloaded. The model is now at the point where the overall efficiency of the
engine is a function of the amount by which the engine is loaded, and the performance factor of
the system. This is presented in Figure 4.11. Table 4.1 is used as a basis of this figure, where
the efficiency is plotted against the amount of loading, for a particular compressed air usage and
storage efficiency (this is the performance factor × efficiency of compressed air usage). Three
distinct driver load requirements are selected corresponding to 60 J, 100 J, 160 J per stroke,
and each driver load requirement results in a family of curves. The curves marked with a ”+”,
”-”, and ”*” correspond to driver requirements of 60 J, 100 J, and 160 J respectively. The
unmodified engine efficiencies are also indicated for each family of curves; the horizontal lines
marked with a ”+”, ”o”, and ”*” are the unmodified engine efficiency for driver requirements
of 60 J, 100 J, and 160 J respectively. This figure is used to show the efficiency range for
a range of possible loading options and driver requirements, and allows the final specification
of the system to be defined for a desired performance outcome. It was stated above that a
performance requirement of 10% over an unmodified engine is required.
Referring to Figure 4.11, 150 J of loading is required to ensure a 10% efficiency improve-
ment when driver requirements are 160 J (family of curves marked with ”*”). This must be with
a compressed air usage and storage efficiency of 1.2, which corresponds to a performance fac-
tor of 2.4 assuming a compressed air usage efficiency of 0.5. A lower performance factor will
theoretically require more loading to achieve the same efficiency improvement. The required
amount of loading is obtained by looking at the horizontal line marked with a ”*”, adding 0.1,
76
and reading off the required amount of loading from the curve corresponding to a compressed
air efficiency of 1.2.
The swept volume and gear ratio can be specified using Figure 4.10. A performance factor
of 2.4 will intersect the curve corresponding to a gear ratio of 1:1 and 1.5:1 at a swept volume
of 250 cm3 and 157 cm3 respectively. Selecting the greatest amount of loading for the lowest
of the two gear ratios: i.e. a swept volume of 250 cm3 and a gear ratio of 1:1 which results
in a loading capacity of between 83−116 J as the receiver pressure increases from 6−12 bar
(from Equation 4.13). From Figure 4.11, the modified engine efficiency will be between 0.29
and 0.31, depending on the current amount of loading.
This does not reach the performance improvement target of 10%. However, these parameters
do compare better at the other target regions. For instance, a driver requirement of 100 J per
stroke will result in a modified engine efficiency of between 0.26 and 0.29, and at a driver
requirement of 60 J per stroke will result in a modified engine efficiency of between 0.21 and
0.25. Table 4.2 was generated by taking extracts from Figure 4.11.
Table 4.2: Extracts from the efficiency-load mapping. Note: compressed air efficiency represents the per-
formance factor multiplied by the compressed air usage efficiency.
Driver requirements 60 J 100 J 160 J
Base efficiency 0.12 0.17 0.22
Required efficiency (10% improvement) 0.22 0.27 0.32
Compressed air efficiency
1.2 90 J 140 J 150 J
1.1 98 J 160 J 170 J
1.0 110 J na na
0.9 127 J na na
0.8 144 J na na
4.3.6 Receiver sizing and time taken to reach final receiver pressure
The receiver will be sized as 0.06 m3 (equal to the average petrol tank), and this has been done
because of the space limitations of a typical motor vehicle. The amount of time required to
77
0.1
0.15
0.2
0.25
0.3
0.35
0.4
60 80 100 120 140 160 180
Ef
fic
ie
nc
y
Loading from external compressor (J)
Efficiency vs external loading, at three driver load requirements
CA Eff = 1.20
CA Eff = 1.00
CA Eff = 0.80
Figure 4.11: Overall system efficiency for a range of driver load requirements, for various amounts of
external loading and a range of performance factors.
increase the pressure of this receiver from 6 bar to 12 bar can be calculated assuming that the
engine is operating at 2000 RPM and that the gear ratio between the engine and the external
compressor is 1. If 288 strokes are required to increase the pressure of the receiver from 6 bar
to 12 bar, and two compression strokes occur per revolution, then 2282×2000 × 60 = 3.42 s will
pass before the receiver reaches 12 bar and compressed air can be used to propel the engine.
This is clearly not a desirable time span as the system will have to rapidly move between com-
pressed air storage and usage modes of operation. The effect of switching so rapidly between
compressed air storage and compressed air usage has not been analysed, and is left for future
work. Increasing the receiver volume to 0.5 m3 will result in a 28.5 s compressed air storage
time period, but this size of the receiver is not practical to be installed as a retrofit. Decreasing
the swept volume of the compressor to 150 cm3 does not significantly lengthen the charging
time of the receiver, and reduces the loading capacity. Increasing the maximum receiver pres-
sure also does not significantly increase the charging time, and reduces the performance factor
78
of the system. Receiver sizing has proved a major weakness of this retrofit.
4.4 Conclusion
In the above sections it has been shown that the concept of energy storage and reuse through
compressed air can work, and has significant advantages when combined with heat transfer
from some source of ”waste” heat energy. It is also subject to significant constraints, in that
the way in which the compressed air is used can seriously degrade the feasibility of the energy
storage and reuse system - as seen when using the internal combustion engine to expand the
compressed air. The internal combustion engine is not designed to expand compressed air at
12 bar (for example) to atmospheric pressure - it has been designed to expand gases at much
higher temperatures and pressures, and has been built accordingly.
Target efficiencies for two of the three investigated regions of operation have been reached
with the proposed design. There is still room for improvement of with regard to heat transfer,
as final receiver temperature is less than exhaust temperature when the receiver has reached
maximum pressure.
79
5 Conclusions and recommendation
The intended mandate for this project is to improve the overall operating efficiency of a spark
ignition internal combustion engine, by loading the engine with an energy storage device when
it is operating at an efficiency lower than optimum. The engine tends to operate more efficiently
when unthrottled, so such loading is used to allow the engine to operate with less throttling.
The stored energy is later reused to propel the engine. The loading process could be external or
internal to the engine, where cylinders within the engine are used to compress air into a receiver.
This loading mechanism was discarded as a potential application option due to the associated
impracticalities. An external loading mechanism (external compressor) was used accordingly,
but was subject to the space limitations of the engine bay, and the engine could not be loaded by
the full difference between driver requirements and the torque produced when operating at the
point of optimum efficiency, so the full efficiency benefit of this modification was not realisable.
The engine was dynamically modelled (on a differential basis), and this engine model was
used to create an efficiency mapping (presented as a table) which would indicate whether load-
ing an engine was feasible based on the combined efficiency of the energy storage and reuse
processes. This dynamic model also formed the basis for the analysis of how efficiently com-
pressed air could be used to propel the engine, which was fairly low (approximately 53%). The
operational parameters of the system were also examined and specified to achieve feasible op-
erating pressures for the system, these being between 6 and 12 bar. An excessively low receiver
pressure is undesirable because little useful torque can be produced by the engine, and an exces-
sively high receiver pressure is undesirable because a multistage external compressor would be
required. The idea of using compressed air in conjunction with the combustion process is also
presented, but was not pursued further because an external compressor capable of producing
very high pressures was required.
The use of heat transfer from the exhaust gases to improve process efficiency was also
investigated, and is actually required to make this compressed air augmentation system feasible.
The heat transfer process can still be improved, because the final temperature of the air within
the receiver once the final pressure (pressure at which the compressed air storage process will
cease, and the compressed air usage process will begin) is reached, is still less than the exhaust
gas temperatures. The receiver size has been limited to 0.06m3 due to space limitations within
80
the vehicle, which makes for impracticably short compressed air loading and usage cycle. The
compressed air augmentation system has been shown to be feasible, but the constraints imposed
by applying this technology to an existing motor vehicle are numerous.
5.1 Future work
The idea of using compressed air to propel the engine under transient conditions could be ex-
plored, to determine whether compressed air could be used more effectively for acceleration
than conventional fuel. The effect of switching frequently between loading the engine with
an external compressor and using compressed air to propel the engine has not been addressed
and is left for future work. The idea of disengaging the compressor from the engine at some
intermediate pressure, so that heat transfer to the compressed air is allowed to continue before
compressed air is used to propel the engine, could also be explored.
Yet another topic for future work includes applying the compressed air energy storage and
reuse mechanism elsewhere, such as a peak load generator system at a power station or indus-
trial plant with waste heat generation. Base load would be used to store compressed air into
a receiver, and this compressed air would be later expanded when peak demand was required.
Heat which would normally be expelled to the atmosphere via the cooling towers etc. could be
used to improve the system performance - in the same way as the exhaust gases are used when
applying the system to an internal combustion engine. The sizes of both the heat exchanger and
the receiver would no longer be design constraints, and the way in which the compressed air is
expanded could also be optimised for the application so that the usage efficiency is higher, be-
cause the constraint of having to use an existing internal combustion engine would be removed
(one would design a compressed air expander specifically).
81
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A Finite-time thermodynamics
A.1 Overview
The differential equation model based on the Otto cycle now gives an analyst an idea of the
effect of irreversibility on process efficiency, but is still not very useful as an analytical tool,
as a time consuming numerical simulation must be performed before any results are obtained.
One of the required outputs of this project is to evaluate process efficiency for a particular
set of system conditions; and a possible avenue of investigation may be found in the field
of finite-time thermodynamics. This tool allows a process to be abstracted into two distinct
parts: an internally reversible part, and an irreversible part that includes interactions with the
surroundings (i.e. heat transfer between the surroundings and the working fluid) [67]. This
methodology has been successfully applied to the Otto cycle [68], reciprocating engines [69],
the CurzonAhlborn heat engine, which is based on the Carnot cycle [70], the Diesel engine [71],
and numerous other thermodynamic power cycles including the Stirling cycle, Joule-Brayton,
Atkinson, Ericsson and Lorentz cycles [72]. From the previously mentioned applications of
finite-time thermodynamics, it is apparent that this tool has only been applied to cycles, while
the internal loading process may not necessarily operate in a cycle; but no evidence was found
in the literature indicating why this thermodynamic tool can not be applied to a process not
operating in a cycle, and an example of finite-time thermodynamics applied to such a process is
known to Dr Paul Roberts [73] - a visiting lecturer at the University of the Witwatersrand. Finite
time thermodynamics is particularly useful in characterising the upper operating limit of real
cycles where the rate at which work is done becomes an important factor, [69]. Carnot analysis
is based on a reversible process, and so all states visited during a process are equilibrium states,
and so does not provide an accurate description of real power cycles. A quasistatic process is
carried out at an infinitesimally slow rate and as stated by Qin et al: ”a finite amount of work
produced by the engine over an infinite amount of time delivers no power” [74]. The concept
of endoreversibility has been used to derive abstracted models of heat engines for the purpose
of optimisation.
A.0
A.2 Internal Combustion Engine Model
The combustion process can be modelled according to differential equation models which prove
particularly useful to the analysis of engine control projects. One of the objectives of modelling
the internal combustion process is to obtain an efficiency mapping which would vary with en-
gine speed and load, and finite-time thermodynamics is used to obtain such a mapping as signif-
icantly less computational resources are required to solve a finite-time model than a differential
model.
As mentioned above, finite time thermodynamics is used in the optimisation of power cy-
cles; and in a sense, the compressed air system is similar to the Joule cycle - as air will be
compressed, and heated (to improve overall processes efficiency) and finally expanded. It may
prove beneficial to analyse this cycle using finite time thermodynamics, and as a precursor to
this analysis, finite time thermodynamics is used to characterise the efficiency of a spark ignition
internal combustion engine (of which numerous examples exist). Finite time thermodynamics
has been successfully applied to Otto and Diesel cycle models [68, 71] and will be used to
obtain a qualitative description of the efficiency mapping of an internal combustion engine.
Much work has been done by Qin et al on the modelling of reciprocating heat engines as
endoreversible cycles [75]; and by Blank et al [69] and Angulo-Brown et al [67, 76] in which
engines are modelled as non-endoreversible cycles. The concept of endoreversibility is used
extensively in finite time thermodynamics; an endoreversible process is internally reversible but
externally irreversible - i.e. irreversibility is due to heat transfer to the working fluid, no entropy
is generated by the working fluid when producing work [77], which has applicability limits as all
realistic power cycles produce entropy. This leads to the concept of non-endoreversible cycles,
where entropy is produced when work is done by the working fluid. For an endoreversible
cycle:
Q1
T1w
= Q2
T2w
(A.1)
For a non-endoreversible cycle [76]:
σi =
Q1
T1w
− Q2
T2w
= (1−R) Q2
Tw2
A.1
Combining the above:
Q1
T1w
= R Q2
T2w
(A.2)
Where R is a measure of the departure from an endoreversible process. Expressions for power
and efficiency were obtained from [67, 75] :
P(R,K1,K2) =
CV 2(R− r1−γv
K1 +K2r1−γv
−µv2 (A.3)
η(R,K1,K2) = 1−
1
R
(
r1−γ + µv
2
Cv2
(K1 +K2r1−γ)
)
(A.4)
Where:
K1 = Inverse of the rate of heat transfer (k1) where k1 = dTdt for a constant volume heating
process
K2 = Inverse of the rate of heat transfer (k2) where k2 = dTdt for a constant volume cooling
process
rv = Maximum volume over the minimum volume
Cv1 = Heat capacity at constant volume, during the combustion process
Cv2 = Heat capacity at constant volume, during the exhaust process
µv = Power lost due to friction, where µ is the coefficient of friction and v is the average
piston speed.
K1 and K2 are related to the time taken for heat transfer to occur to and from the working fluid
respectively, which implies that these variables are a function of the engine speed. Power lost
due to friction is also a function of the engine speed, so equations A.3 and A.4 provide a power
and efficiency relationship as a function of engine speed and the irreversibility factor R. The
effect of loading on the Otto cycle is taken into account by including the term: (Patm −Pin)∆V ,
where ∆V is the displaced volume and Pin is the inlet pressure which is less than atmospheric
pressure due to throttling; as the degree to which an engine is loaded is determined by the extent
to which it is throttled. Equations A.3 and A.4 were modified to include the effects of throttling
in the model. The work done per cycle is given by the net heat transfer to the system, as per the
first law of thermodynamics.
A.2
Wc =Cv(T3 −T2)−Cv(T4 −T1)− (Patm −Pin)∆V
The power per cycle is given as follows, where µv2 is the power lost to friction.
P = Wc
τc
−µv2 (A.5)
Where τc is the cycle time of the process. The compression and expansion processes are mod-
elled as instantaneous adiabats [67] and have no contribution to the cycle time of the process,
which is determined by the engine speed. For instance, an engine speed of 3600RPM will have
a cycle time of 33.33ms, and it is assumed that the time spent on each of the heat transfer
processes is half of the cycle time [68]. The cycle time is then represented as follows:
τc = K1(T3 −T2)+K2(T4 −T1)
Where K1 = t12\(T3 − T2) and K2 = t41\(T4 − T1) obtained assuming that heat transfer to and
from the working fluid occurs at the rate given by dTdt = 1K1 and
dT
dt = 1K2 respectively, and
assuming that t12 = t41 = τc\2. The efficiency of the cycle is given by:
η = PQin\τc (A.6)
Where Qin =Cv(T3 −T2).
Efficiency and power curves plotted for a range of engine speeds and throttle settings are shown
in Figure A.1. The results are qualitatively correct in that efficiency decreases as power in-
creases. Although similar, this behavior is not strictly the same as the behavior typically demon-
satrated by finite time thermodynamic models of power cycles; which is calculated for a fixed
engine speed to yield an expression for power per cycle against efficiency. It is well known
that cycle efficiency is maximum when the power produced per cycle is zero, and that cycle
efficiency is less than maximum efficiency at maximum power [72]. Although qualitatively
correct, this finite-time model of the Otto cycle can be greatly improved; for instance some
researchers have taken into account the variation of specific heat with temperature [68], and
Burzler incorporated the irreversibility due to combustion in a diesel cycle [71]. Table A.1
shows sample data for a throttle setting corresponding to an inlet pressure of 0.8 bar, and an
engine speed of 3600 RPM.
A.3
 0.2
 0.25
 0.3
 0.35
 0.4
 0.45
 0.5
 0.55
 1000  2000  3000  4000  5000  6000
Ef
fic
ie
nc
y
Engine speed (rpm)
Efficiency vs Engine speed
0.4 bar
0.45 bar
0.5 bar
0.55 bar
0.6 bar
0.65 bar
0.7 bar
0.75 bar
0.8 bar
0.85 bar
0.9 bar
0.95 bar
1 bar
(a) Otto cycle efficiency for a number of different throttling
conditions
 0
 2000
 4000
 6000
 8000
 10000
 12000
 14000
 16000
 1000  2000  3000  4000  5000  6000
Po
w
er
 (W
)
Engine speed (rpm)
Power vs Engine speed
0.4 bar
0.45 bar
0.5 bar
0.55 bar
0.6 bar
0.65 bar
0.7 			bar
0.75 bar
0.8 bar
0.85 bar
0.9 bar
0.95 bar
1 bar
(b) Power generated for an Otto cycle for a number of throttling
conditions
Figure A.1: Finite-time thermodynamic model - Power and efficiency curves
Table A.1: Sample data for the FTT model, for a throttle setting corresponding to an inlet pressure of 0.8 bar,
and an engine speed of 3600 RPM
T1 = 300 K K1 = 8.5×10−6
T2 = 756 K K2 = 21.6×10−6
T3 = 2699 K µ = 12 kg.s−1
T4 = 1071 K rv = 10
P = 7713 W η = 0.49
This method shows promise but is not used to characterise process efficiency due to the large
discrepancy between actual and predicted engine efficiency.
A.4
Figure A.2: TS diagram representation of an Otto cycle
A.5
B Sample calculation for the differential equation model
The following sample calculation is used to demonstrate how the differential equation models
have been handled. The equations are solved with the LSODE solver, which is a function
native to the OCTAVE mathematical simulation package. The sample calculation is for the
cylinder drawing air in from a manifold at a pressure of 0.5 bar. The temperature of the air
(in the manifold and the cylinder) is 300 K. As the piston moves downwards, the pressure and
temperature of the air within the cylinder decrease (as seen by the negative partial differential
for each).
m˙ =









ηvAthPtank√
RTtank
γ 12
(
2γ
γ+1
)
γ+1
2(γ−1) if PcylPtank >
2
γ+1
γ+1
2(γ−1)
ηvAthPtank√
RTtank
(
Pcyl
Ptank
)
1
γ
[
2γ
γ−1
(
1− PcylPtank
γ−1
γ
)
]
1
2
if PcylPtank ≤
2
γ+1
γ+1
2(γ−1)
(B.1)
2
γ+1
γ+1
2(γ−1)
= 2
2.4
2.4
2(0.4)
= 0.578
pcyl
ptank
= 5.499×10
4
5.5×104
= 0.99
∴ m˙ =
ηv ×pi
(
0.02
2
)2
×5.5×104
√
28.97×300
1.4 12
(2×1.4
1.4+1
)
1.4+1
2(1.4−1)
= 2.497×10−4 kg/s
ηv = 0.597071+N ×5.51605×10−5 + pcyl ×9.30417×10−7
= 0.597071+ 104.7×60
2pi
×5.51605×10−5 +(5.5×104)×9.30417×10−7
= 0.7
B.0
Vcyl = Vc ×
[
1+0.5
(V d +Vc
V c
−1
)( l
a
+1− cosθ−
( l
a
2
− (sinθ)2
)0.5
)
]
= 4.407×10−5
[
1+0.5
(3.967×10−4 +4.407×10−5
4.407×10−5 −1
)
...
( 0.1
0.0385 +1− cos(0.0146)−
( 0.1
0.0385
2
− (sin(0.0146))2
)0.5)]
= 4.4099×10−5 m3
˙Vcyl =
Vc
2
×
(Vd +Vc
Vc
−1
)
×
[
ωsin(pi−θ)+
(
0.5× ...
( l
a
2
− (sin(pi−θ))2
)−0.5)
× (−2sin(pi−θ)cos(pi−θ))×ω
]
= 3.967×10
−4
2
×
(3.967×10−4 +4.407×10−5
4.407×10−5 −1
)
× ...
[
104.7sin(pi−0.0146)+
(
0.5×
( 0.1
0.0385
2
− (sin(pi−0.0146))2
)−0.5)
× ...
(−2sin(pi−0.0146)cos(pi−0.0146))×104.7
]
= 4.199×10−4 m3/s
Acyl = Ach +Ap+piB(l+a−acos(θ)+
√
l2 − (asin(θ))2)
Ap = pi(B/2)2
= pi(0.08/2)2
= 5.02×10−3 m2
Ach = Ap
Acyl = 2×5.02×10−3 +pi×0.08× (0.1+0.0385−0.0385× cos(0.006)+
√
0.12 − (0.0385sin(0.006))2)
= 0.001
B.1
R = 8314
28.97
= 286.99
m = p×Vcyl
R×T
= 5.499×10
4 ×4.4075×10−5
286.99×300
= 2.815×10−5 kg
ρ = 5.499×10
4
278.9×300
= 0.638 kg/m3
NRe =
ρ× ω2×pi ×D
µ
= 0.638×
104.7
2pi ×0.08
1.84×10−5
= 3544.4
˙Qht = Acyl
[
0.3λNRe
0.7
D
(Tcyl −Tw)+1×10−8(Tcyl 4 −T 4w )
]
= 0.001
[
0.3×0.026
(3544
0.08
)0.7
(300−300)+1×10−8(3004 −3004)
]
= 0kW
B.2
˙T = 1
mCv
[
˙Q− p ˙V +him˙−hem˙−um˙
]
= 1(2.815×10−5)(718)
[
0− (5.499×104)(4.199×10−4)+ ...
(300.19×103)(2.497×10−4)−0− (214.07×103)(2.497×10−4)
]
= −75.99 K/rad
p˙ = R(m˙T +
˙Tm)− ˙V p
V
d p
dθ =
d p
dt ×
dt
dθ
= R(m˙T +
˙Tm)− ˙V p
V
/ω
= 286.98(2.497×10
−4 ×300−75.99×2.815×10−5 −4.199×10−4 ×0.5×105)
4.4099×10−5 ×104.7
= −4531.4 pa/rad
B.3
C Heat transfer sample calculations
C.1 Heat transfer coefficient from exhaust gas to heat exchanger
The following calculation serves to demonstrate how the heat transfer coefficient from the ex-
haust gases would be calculated, assuming the following parameters:
Texh = 900 K Trec = 300 K pexh = 1.5 bar cp = 1120.8 J/kg.K d = 0.04 m
λ = 0.06 W/K.m µ = 3.89×10−5 kg/m.s L = 1 m m˙ = 0.02
Note: p is the pressure in the exhaust pipe, d and L are the diameter and length of the segment
of the heat exchanger through which the exhaust gas passes, m˙ is the mass flow rate through the
exhaust pipe, and the remainder of the properties are considered at a temperature of T = 900 K
ρ = P
RT
= 1.5×10
5
8314
28.97 900
= 0.58 kg/m3
Apipe = pi×0.022
= 1.25×10−3 m2
¯m˙ = ρ(Vi −Vf )30/N
= 0.58(4.407×10
−4 −4.407×10−5)
30/2600 Assuming an engine speed o f 2600 RPM
= 0.02 kg/s
Note: the exhaust gas air will actually be pulsed rather than steady - hence the averaged value
above is used. The outside diameter of the exhaust pipe is 0.04 m.
C.0
um =
m˙
ρApipe
= 0.02(0.58)(1.25×10−3)
= 27.4 m/s
NRe =
dρum
µ
= (0.04)(0.58)(27.4)(3.89×10−5)
= 16345
¯h = 0.023λd
(umdρ
µ
)0.8(cpµ
λ
)0.3
= 0.0230.0810.04
(27.4×0.04×0.58
3.89×10−5
)0.8(1120×3.89×10−5
0.06
)0.3
= 75.9 W/K.m2
¯h×A = (75.9)(pi×0.04×1)
= 9.53 W/K
C.2 Heat transfer coefficient from heat exchanger surface to receiver gases
Tw = 900 K Trec = 300 K p = 1.5 bar cp = 1005 J/kg.K
λ = 0.026 W/K.m µ = 1.84×10−5 kg/m.s L = 1 m Ncpr = 1000 rpm
ρ = 1.5×10
5
8314
28.97 300
= 1.74 kg/m3
m˙ = ∆m
1/N
= 2×ρ(Vi −Vf )
1/Ncpr
(two cylinders discharge air per revolution)
= 2×1.74(2.2×10
−4 −2.2×10−5)
30/1000
= 22.96×10−3 kg/s
C.1
um =
m˙
A
= 22.96×10
−3
8×0.01×0.08
= 3.58 m/s
NRe =
Lρum
µ
= (1)(1.74)(3.58)
1.84×10−5
= 97170
NPr =
cpµ
λ
= (1005)(1.84×10
−5)
0.026
= 0.707
¯h = λ(NRe
1
2 )(NPr
1
3 )
L
= 0.026(97170
1
2 )(0.707 13 )
1
= 6.81 W/K.m2
¯h×A = (5.15)(8)(2× (1×0.08)+ (0.01×1)
= (6.81)(8)(0.17)
= 9.26 W/K
C.3 Heat transfer via natural convection
Tw = 900 K Trec = 300 K p = 1.5 bar cp = 1005 J/kg.K b = 0.01 m
λ = 0.026 W/K.m µ = 1.846×10−5 kg/m.s L = 0.08 m
C.2
ρ = 1.5×10
5
8314
28.97 300
= 1.74kg/m3
Ra = ρgβcpb
3∆T
µλ
= (1.74)(9.8)(1/300)(1005)(0.01
3)(900−300)
(1.846×10−5)(0.026)
= 7.21×104
h = λRa
24b
(
1− e− 35bRa
)0.75
= (0.026)(0.01)(7.21×10
4)
(24)(0.01)
(
1− e−
35
(0.01)(7.21×104)
)0.75
= 1.6 W/m2.K
¯h×A = (1.6)(16×0.03×1)
= 0.512 W/K
C.3
D Sample calculation for the compressed air storage model
Vb = Va/
( prec
pin
)1/γ
= 2.2×10−4/
( 6×105
1.01×105
)1/1.4
= 6.16×10−5 m3
work = prec(Vf −Vx)+
precVx − pinVi
1− γ
= 6×105(2.2×10−5 −6.16×10−5)+ (6×10
5)(6.16×10−5)− (1.01×105)(1.1×10−4)
1−1.4
= −60.64 J (D.1)
torque = 2×60.64
2×pi
= 19.3 N.m (D.2)
The workings of the model can be demonstrated by considering a single compression stroke
of the external compressor. Consider the receiver initially at a pressure of 6 bar, 300 K. One
D.0
compression stroke of the external compressor will add 1.65×10−4 kg of air into the receiver:
Tb = Ta
( pb
pa
)
γ−1
γ
= 300
( 6×105
1.01×105
)
1.4−1
1.4
= 499 K
Vb =
Va
(
pb
pa
)
1
γ
= 2.2×10
−4
(
6×105
1.01×105
)
1
1.4
= 6.15×10−5 m3
ρb =
pb
RTb
= 6×10
5
8314
28.97 499
= 4.19 kg/m3
∆m = ρb(6.15×10−5 −2.2×10−5)
= 4.19(Vb −Vc)
= 1.65×10−4 kg
The state of the receiver will change as this air is added, and heat will be transferred into the
incoming air as it blows across the heat exchanger:
ui = 214.07×103 J/kg
mi =
(6×105)(0.5)
8314
28.97 300
= 3.4 kg
∆Q = ˙Q×∆t
∆t = 30GR
N
= 30
1
2.5
2000
= 6×10−3 s
D.1
The rate of heat transfer to the air is determined according to the method described in Sec-
tion 4.2.6, and for this case was determined to be 2622 W . The heat transferred into the air is
determined by multiplying the rate of heat transfer by the time over which the stroke occurs:
∆Q = ˙Q×∆t
= 2622×6×10−3
= 15.7 J
u f =
hin ×∆m+∆Q+ui×mreci
∆m+mreci
= (501.99×10
3)(1.65×10−4)+15.7+(214.07×103)(3.4)
1.65×10−4 +3.4
= 214.25 kJ/kg
Trec = 300.25 K
prec = 6.01 bar
work = pb(Vb −Vc)+
paVa − pbVb
1− γ
= 6×105(6.15×10−5 −2.2×10−5)+ (1.01×10
5)(2.2×10−4 − (6×105)(6.15×10−5)
1−1.4
= 60.6 J
D.2