Gas Combustion in a Diesel Engine Process:
Modelling and Simulating
OEMO 94/01, August 1994
E.H. Sweerts de Landas
Technische Universiteit Delft Norges Tekniske Hogskole
Faculteit der Werktuigbouwkunde Marinteknisk Avdeling
en Maritieme Techniek
Institutt for Marint Maskineri
Vakgroep 0.E.M.0
Gas Combustion in
a Diesel Engine Process:
Modelling and Simulating
OEMO 94/01, August 1994
E.H. Sweerts de Landas
(
2''
-o'.)
Preface
At the Department of Marine Engineering of the Norwegian Institute of Technology in
Trondheim, Norway, research is being carried out with regard to modelling diesel engines. This research program interested me in view of my intention to specialise in this field. Some
financial help from the Erasmus Student Exchange Program gave me the opportunity of
actually spending the academic year of 93/94 in Trondheim. There I workedon my Fourth
Year Task and Thesis, two obligatoryparts of the study Werktuigbouwkunde (Mechanical
Engineering) at the Delft University of Technology.
The Fourth Year Task was carried out from the beginning of September 1993 until the end of January 1994 and is described in part I of this report. Part II concerns the Thesis. The Thesis was a continuation of the Fourth Year Task and is concluded with a presentation and an exam on August 29th 1994.
In no particular order, I would like to thank the following:
Professor Hans Klein Woud for both his guidance during my studies as well as for coming
to Norway in May to deliver useful criticism.
Professor Hallvard Engja for general guidance during my stay in Norway and for finding the
time to come to the Netherlands formy presentation and exam.
Vilmar iEsray, doctoral student at the Department of Marine Engineering, for pointing me
in the right direction at critical moments. I also thank him for making the journey to the
Netherlands in August.
The Erasmus Student Exchange Program for its financial support.
Hein Sweerts de Landas Rotterdam, August 1994
Preface
Table of Contents
Table of Contents
Preface Table of Contents Abstract ix Nomenclature xi 1 Introduction 1Part I: Fourth Year Task
2 Internal Combustion Engines - Basics 5
2.1 Engine Operating Cycles 5
2.2 Air Standard Cycles 8
2.2.1 Assumptions 9
2.2.2 Cycle Efficiencies 9
2.2.3 Limitations 10
3 Bond Graphs - A General Description 13
3.1 Variables 13 3.2 Bonds 14
3.3 Ports
15 3.3.1 1-Port Elements 15 3.3.2 2-Port Elements 173.3.3 3-Port Junction Elements 19
3.3.4 Fields
203.4 Pseudo Bond Graphs 21
3.5 State Equations 22
3.5.1 Example 1: Mechanical System 22
3.5.2 Example 2: Thermofluid System 25
4 Bond Graph Representation of the Diesel Engine 29
4.1 Clarification of the Bond Graph 29
4.2 Assignment of Causality 29
4.2.1 Turbocharger 29
4.2.2 Air Receiver and Exhaust Receiver 32
. . . . ... . . . . . . . . . .
...
. . ....
. . . . . ....
. . . . ... . . . . . . . . .,4.2.3 Cylinder 32
4.2.4 Crankshaft 33
5 Thermodynamical, Hydraulical and Mechanical Principles 35
5.1 State Variables and Input Variables 35
5.2 General Assumptions and Simplifications 36
5.3 Derivation of the State Equations 37
5.3.1 Angular Momentum of the Turbocharger 37
5.3.2 Mass of the Air in the Air Receiver 39
5.3.3 Internal Energy of the Air in the Air Receiver 41
5.3.4 Mass of the Gas in the Cylinder 41
5.3.5 Internal Energy of the Gas in the Cylinder 43
5.3.6 Fuel/air Equivalence Ratio in the Cylinder 44
5.3.7 Cylinder Volume 45
5.3.8 Crankshaft Angular Momentum 45
5.3.9 Mass of the Exhaust Gas in the Exhaust Receiver 46
5.3.10 Internal Energy of the Exhaust Gas in the Exhaust Receiver 46 5.3.11 Fuel/air Equivalence Ratio in the Exhaust Receiver 47
6 Diesel Engine Simulation Results and Discussion 49
6.1 Steady-State Simulations 51
6.1.1 Cyclical Variations of Engine Variables 51
6.1.2 Correlation of the Variables with Changing Engine Load 54
6.1.3 Rates of Heat Release 58
6.1.4 Heat Flows 63
6.2 Dynamic Response 65
Part
Thesis7 Combustion in a Diesel Engine 71
7.1 Fuel Spray Characteristics . 72
7.1.1 Spray Structure 72
7.1.2 Spray Penetration . 74
7.2 Ignition Delay . 74
7.3 Premixed and Mixing Controlled Combustion 76
7.4 Combustion Modelling 77
8 Adapting the Model 79
8.1 Gas Combustion Model with Glow Plug Ignition 79
8.1.1 Premixed and Mixing Controlled Combustion Rates - a
Two-Zone Model 80
vi Gas Combustion in a Diesel Engine Process: Modelling and Simulating
. . . .
... .
. . . . . . . ....
. . . . . . . . . .Table of Contents vii 8.1.2 Glow Plug Aided Ignition - a Three-Zone Model 87 8.2 Thermodynamic Background of an Open System with Uniform
Pressure 90
8.2.1 Equation Governing the State of the Working Fluid 91
8.2.2 Energy Balance: the First Law of Thermodynamics 92 8.2.3 Matrix Representation of the State Equations 94
8.3 Programming Details 98
9 Simulation Results Obtained with the Combustion Model 99
9.1 Two-Zone Combustion Model Results 101
9.2 Three-Zone Combustion Model Results 102
9.3 Sensitivity of the Model to Changing Combustion Model Parameters 103
9.3.1 Convection Between Glow Plug and Ignition Zone 103
9.3.2 Convection Between Ignition Zone and Combustion Zone
.
1069.3.3 Glow Plug Power 106
9.3.4 Pilot Injection Quantity 106
9.3.5 Rate of Air Entrainment Equation Constant 109 9.3.6 Length of Premixed Combustion Period 110 9.4 Heat Release Correlation with Laboratory Data 110
9.5 Miss-firing 113
10 Conclusions and Recommendations 115
10.1 Conclusions 115
10.2 Recommendations 116
References 119
Appendix A: Derivation of the Basic Nozzle Equation for Transient
One-Dimensional Gas Flow 121
Appendix B: Air Cooler Efficiency 125
Appendix C: Ignition Delay, Combustion Duration and Combustion Rate - Wiebe . 127
Appendix D: Regulator Transfer Function 135
Appendix Er Heat Transfer 139
Appendix F: Zacharias' Equations 143
Appendix G: Two-Zone Diesel Combustion Model - a First Attempt 145
. . . . . .
... ... .
. . . . . . . . . . . .....
. . . . ... ... ...
. . . M . . . . . . . .,,.till
Gas Combustion in ai Diesel Engine Process; Modelling and SimulatingAppendix H: FORTRAN Code L;;., .
,
153Appendix I: Engine Data ao 1.4 9.1 ;pi rsi - sn TA, 173
Abstract
bC
Abstract
The diesel engine has enjoyed and will continue to enjoy much popularity as a result of its wide range of applications - it is a compact power plant with engines ranging from the very small, used in automobiles, to the very large, installed in ships. Not only is the diesel engine the worlds prime mover, stationary operation is another important application for it.
The type of fuel combusted within the diesel engine must fulfil certain requirements. These depend very much on the type (size) of engine, but in general it must be able to auto-ignite
in air at typical end-of-compression
pressures and temperatures of 60 bar and 800K
respectively. Diesel fuel, as its name implies, lends itself well for this purpose. Today however, there is an increased interest in using alternative fuels in diesel engines. A very specific example is the use of crude oil. Using this fuel at the well-head saves a return trip from the refinery. Care for the environment is also becoming increasingly important, hence fuels which produce less pollutants are being investigated. Natural gas is one of them. The combustion of this fuel in a diesel engine forms the core of the research work described in this report.
There are two main aspects to research work. One is the practical aspect, concerningmainly experimental work with the actual engine in the laboratory. The other is more theoretical, utilizing computers to support modelling and simulating efforts. This report is principally related to this second aspect: modelling and simulating of the combustion of gas with glow plug ignition in a diesel engine. It is focused on combustion in a 4-stroke 4 cylinder medium speed turbocharged diesel engine designed for generator operation.
The report has two distinct parts. Part 1 contains a description of an existing model of a diesel engine which is adapted to fit the medium speed diesel engine under consideration. This is done in preparation for PartII, the addition of a model for the combustion of gas to the existing model. The combustion model consists of three zones: the air zone and the combustion zone, characterizing the absence of turbulence within the cylinder, and the ignition zone. The air zone represents the air in the cylinder. Combustion occurs only in the combustion zone. The combustion rate is assumed to be directly related to the rate of
air entrainment by the combustion zone. The rate of air entrainment is in its turn based on
theory concerning free jets in quiescent air. The ignition zone represents the very thin boundary layer surrounding the glow plug and is used to calculate the ignition delay - the glow plug is an ignition aid, gas can not ignite by itself.
x Gas Combustion in a Diesel Engine Process: Modelling and Simulating
Having adjusted a number of heat flow coefficients and combustion parameters, comparisons
are made between simulated results and laboratory data. The rates of combustion for a
number of engine loads all show very good correlation. Furthermore, the resulting model
provided a good basis on which further research - concerning the ignition delay in particular
Nomenclature
Nomenclature
Variables
A
= area
cp. -= specific heat capacity at constant pressure cv = specific heat capacity at constant volume
CD flow coefficient
C
combustion, rate parameter= effort
= internal energy = flow
ifs stoichiometric fuel/air ratio = fuel/air equivalence ratio
Ii, = enthalpy
in= mass
M = torque
= pressure, angular momentum
= power
= work done
= angular displacement = gas constant displacement piston. = temperature V = volume X = pin position = number of cylinder&a
;=- heat transfer coefficient= isentropic efficiency volumetric efficiency
= crank angle
X, = crank mechanism throw ratio, combustion parameter
= pressure ratio = time constant = compressibility factor co = angular velocity
=
= = xiSubscripts
'ac i = after compressor
aac = after air cooler
ace = acceleration
air
= air entering cylinderqr
= air receiver corn = compressor crit = critical cYI = cylinder = differentiator = crankshaft er = exhaust receiverexh = exhaust gas leaving cylinder
f
= fuelflit
= filter
file
= friction' gp = glow plug1
= cylinder number i, integrator = load
last = energy leaving the system
rn = main injection
o = ambient conditions - piston, pilot injection
pre = premixed combustion period
ref = reference'
= surface of cylinder stoich = stoichiometric ratio
= turbocharger tur = turbine
= water
I IF
=
Chapter I: Introduction
1
1 Introduction
The use of alternative fuels in diesel engines is becoming increasingly popular, mainly as a
result of economical and environmental pressures. Natural gas is a fuel which has been
introduced as an alternative to diesel fuel for a number of reasons. It not only reduces
harmful engine emissions, it is also more readily available, and there is more of it, compared to diesel fuel.
The combustion of gas in the cylinder of a diesel engine does however introduce certain
complications.
Gas dynamics play an important role in
the injection system directly influencing the rate of injection. Furthermore, natural gas has poor auto ignition properties, so some ignition aid must be present within the cylinder. Research regarding these mattersis being carried out to obtain a better understanding of them. This research includes both
experimental work and modelling and simulating with the help of computers. In support of
these research efforts, the main task described in this report is to develop
a model and?
computer simulation of the combustion of gas in a diesel engine.The following approach has been used. There are two main parts to the report, a Fourth
Year Task (Part!) and a Thesis (Part II). Pan! encompasses Chapters 2, 3, 4, 5 and 6 and concerns itself mainly with the familiarisation of an existing model of a diesel engine. In
Chapter 2 some background information is given regarding the operation of the reciprocating internal combustion engine. It also gives a simple model of such an engine and thereasons for setting up more accurate engine models. To model any system however, some method
is needed to do this in a structured manner. Such a method, bond graphs, is described in
Chapter 3 followed by
a study of the thermodynamical, hydraulical and mechanicalbackground of the diesel engine. Apresentation of results and a discussion concludes Part I in Chapter 6. Part II contains theThesis work and consists of Chapters
7, 8 and 9. The
implementation of the model of the combustion of gas is discussed in this part. Chapter 7 gives general background information regarding combustion in a diesel engine. Turning more to the actual engine under consideration, a model for the combustion of gas with glow
plug ignition is set
up in Chapter 8.
Results which were obtained with this model are presented in Chapter 9, concluding Part II. Conclusions and recommendations regarding work presented in both parts of the report form the final chapter, Chapter 10. Numbers in brackets Nu, indicate literature references.Part Ii Fourth Year Task
4 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
11
Chapter 2.: Internal Combustion Engines - Basics
2 Internal Combustion Engines
-
Basics
This chapter contains a qualitative description of the operation of the reciprocating internal combustion engine. Internal as opposed to external to indicate that the fuel combusts within
the engine doing work on a piston whose motion is inherently reciprocal Furthermore, a
look is taken at a simple graphical method to represent the complete combustion process both
as a first step to modelling the process as well as an introduction to the problems involved
when doing so.,
2.1 Engine Operating Cycles
Stroke
Figure 2.1: Basic geometry of the reciprocating internal combustion engine
5
6 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
The piston of the reciprocating engine moves back and forth in a cylinder and transmits power through a connecting rod and crank mechanism to the drive shaft, Figure 2.1. The steady rotation of the crank produces a cyclical piston motion. The piston comes to rest at
the top-dead-centre (TDC) crank position and bottom-dead-centre (BDC) crank position when
the cylinder volume is a minimum and maximum, respectively. The minimum cylinder
volume is called the clearance volume, K., and the volume swept out by the piston, being the difference between clearance volume and total volume, V!, is called the displaced or swept
volume, Vd. The ratio of maximum volume to minimum volume is the compression ratio,
et. Most engines operate on what is known as the four-stroke cycle. Each cylinder requires
TC
BC
(a) Intake lb) Compression (c) Expansion
Figure 2.2: Four-stroke operating cycle
four strokes of its piston - two revolutions of the crankshaft - to complete the sequence of
events which produces one power stroke. Both the spark ignited engine (SI engine) and the compression ignition engine (CI engine) use this cycle which is illustrated in Figure 2.2 and
comprises of:
1. An intake stroke, starting with the piston at TDC and ends with the piston at BDC, which draws fresh mixture into the cylinder.
A compression stroke, when both valves are closed and the mixture inside the cylinder is compressed to a small fraction of its initial volume. Toward the end of
the compression stroke, combustion is initiated and the cylinder pressure rises more rapidly.
Chapter 2: Internal Combustion Engines- Basics 7 A power stoke, when the high pressure, high temperature gasses push the piston down. Toward the end of the power stroke the exhaust valve opens to initiate the
exhaust process.
An exhaust stroke, when the remaining burned gasses exit the cylinder. At the end
of this stroke (or more correctly, at the beginning of the next) the exhaust valve
closes and the cycle repeats itself.
Reed spring inlet valve
Exhaust blowdown Scavenging
Transfer
/ ports
Figure 2.3: 711v-stroke operating cycle. A crankcase-scavenged engine is shown.
To obtain a higher power output from a given engine size, and a simpler valve design, the two-stroke engine was developed. Here again the name of the cycle indicates the number of strokes per complete cycle. The two-stroke cycle is also applicable
to both SI and CI engines. Figure 2.3 shows one of the simplest types of two-stroke engine designs. Ports in the cylinder liner, opened and closed by the piston motion, control the exhaust and inlet
flows while the piston is closeto BDC. The two strokes are:
A compression stroke, starting when the inlet and exhaust ports are closed off by the piston, compressing the mixture with toward the end of this stroke the initiation
of combustion. Fresh charge is drawn into the crankcase.
A power stroke when the hot gasses at high pressure do work on the piston until the exhaust ports open (first) followed by the opening of the inlet ports. Most of the burned gasses pass through the exhaust ports during the blowdown process before
8 Gas Combustion in a Diesel Engine Process: Modelling and Simulating compressed in the crankcase enters the cylinder in such a way that it does not flow straight out again via the exhaust ports.
A simple graphical method has been developed which clearly illustrates the processes involved based on air standard cycles. This method will now be discussed.
2.2 Air Standard Cycles
Air standard cycles have been developed to give some insight into the on-going processes during a combustion cycle of an internal combustion engine.
They are useful as an
introduction to the more detailed calculations of engine cylinder pressure diagrams discussedlater (in particular referring to the mathematical representation of the combustion process
within the cylinder).
9 10
11
(c) Duck combustion eye* (d) Modified Atkinson cycle
1-S-9-10-11- I 1 -0-9- ID -12-11
Figure 2.4: Air standard cycles
There are three air standard cycles which are particularly relevant to the reciprocating internal combustion engine and a fourth to a combination of engine and turbine. These are
(Figure 2.4):
(o) Diesel cycle (bl Otto cycle
Chapter 2: Internal Combustion Engines - Basics 9
1 the constant volume combustion cycle (Otto),
2 the constant pressure combustion cycle (Diesel),
3 a combination of these two, the dual combustion cycle (Seiliger) and 4 the modified Atkinson cycle.
2.2.1 Assumptions
Air standard cycles usually assume that the working fluid in the engine is always an ideal gas with constant specific heats and that:
A fixed mass of air is the working fluid throughout the entire cycle, ie. there are no intake and exhaust processes.
The combustion is replaced by a heat transfer process from an external source.
- The cycle is completed by heat transfer to the surroundings until the air temperature and pressure correspond to the initial conditions (this in contrast to the exhaust and intake
processes in an actual engine.)
All compression and expansion processes are internally reversible.
2.2.2 Cycle Efficiencies
All cycles shown in Figure 2.4 include isentropic compression, heat transfer at or near TDC, isentropic expansion and heat transfer at BDC to close the cycle. The work delivered by
such a cycle therefore becomes the difference between transferred heats: efficiency can
subsequently be calculated with ease pi].
From such a calculation, a number of conclusions can be drawn.
The Otto cycle efficiency is independent of the heat transfer to the cycle, whilst the Diesel cycle is dependent on the heat transfer.
Furthermore, it can be said that, for equal compression ratios, the Otto cycle is the most
efficient: it has the highest air standard efficiency. Theoretically true, in practice design limitations will alter this picture. If for example cylinder pressure is a limiting factor, then,
for the same work and thesame maximum pressure:
Thoth.' nSeiliger
no.
Efficiencies increase with increasing maximum cylinder pressure.
Finally by the application of a turbine more work can be extracted from the gas by expansion
from 11 to 12, Figure 2.4. The turbine can be used to drive a compressor, modifying the
cycle further.
2.2.3 Limitations
Although these air standard cycles give useful information of the trends likely to follow from
a change in some engine variable or other and to an extent show why the Diesel cycle is superior to the Otto cycle, they also give extremely optimistic numerical estimates of the output obtainable from an engine. Some obvious reasons for this may be derived from a
study of the assumptions listed earlier, a discussion of which will now follow.
Real gasses have variable specific heats with variations becoming more extreme when the
products of combustion are taken into account.
Furthermore, dissociation at high
temperatures needs to be taken into account (dissociation requires heat). This all results in
lower cylinder temperatures and pressures than those in the standard air cycles.
Heat transfer from the cylinder gasses to the cylinder walls must also be taken into account. Much of it occurs during the combustion process itself and directly reduces the effective heat input and so work output and efficiency of the engine.
In the real engine the simple constant volume heat rejection process is replaced by a gas exchange process which modifies the beginning and end parts of the p-V diagram. A
4-stroke engine cycle also includes the air-pumping loop. This loop can reduce engine output,
especially if the engine is equipped with a carburettor, resulting in throttling losses during
operation below maximum power.
The air standard cycles only say something about the indicated thermal efficiency, that is the
work done by the gas on the piston. The mechanical efficiency of the engine is not taken
into account.
The standard air cycles show quite unrealistic combustion rates and patterns. No real engine
can burn large quantities of fuel instantaneously at TDC, nor can combustion rates be
controlled to give the constant pressure conditions required by the Seiliger and Diesel cycle.It will be clear that the standard air cycles as discussed above the Seiliger cycle being the least unrealistic - are of limited use other than to make the most basic comparisons between
the different cycles. A more fundamental approach is required - achieved by looking more
closely at the underlying cause for the p-V curve, instead of the resulting p-V curve itself.
This approach leads to what may be called a time-marching simulation. By calculating
derivatives of model parameters and subsequently integrating for consecutive time steps, a complete picture is build up of the state of the model as a function of time. A method to set
up such a model is required.
Therefore, before looking more closely at the on-going processes within the diesel engine, a study of a modelling method follows in the next chapter.3 Bond Graphs
-
A General Description
There exist numerous methods by which a number of mathematical equations can be derived of a system which, when solved, give the state of the system unambiguously. One method
which does this in a particularly ordered way is the bond graph method of modelling nap. The computer simulation of the diesel engine to be studied uses this method. It is therefore considered useful to give an introduction of this method here so that a better understanding of the computer simulation can be achieved later.
3.1 Variables
There are four variables which are used by the bond graph method, falling into two
categories, power variables and energy variables.What two of these variables are is best explained using the simple (mechanical) example of the compression of a spring. When the spring is compressed, both a force and a velocity are required. Analogous examples can be given for a hydraulic system (accumulator) and an
electrical circuit (capacitor). In
all these systems the force,
pressure and voltage
respectively, are all called an effort (e(t)), whilst the corresponding velocity, volume flow and current respectively are called a flow (fit)). These variables are called the power variables, the product of which gives the power:
P(t) = e(t)f(t) (3.1)
If these two variables are subsequently integrated over time, the energy variables are obtained. The effort integrated over time gives the generalized momentum p(t):
p(t) =
e(t)dt (3.2)The flow integrated over time gives the generalized displacement q(t):
q(t) = flt)dt (3.3)
3.2 Bonds
A bond is used to connect two components and represents the power flowing intoor out of a component. It gives the following information:
the effort coming from or going into the component it is attached to, the flow coming from or going into the component,
the direction in which the power flow (e x f) is positive, the direction being given by the half arrow as shown below, where the power flows from component A to component B if both the effort and flow are either positive or negative.
A
\ B
the causality of the component, this tells us whether the flow is caused by the effort or whether the effort is caused by the flow. So the causality says something about the input
and output of the component and is represented by a short perpendicular line at one end
of the bond. In the case of the bond represented in the figure below the effort is the input
of component B, because that is where the causal stroke is placed. The power flow
direction has no bearing on the causality.
A
1
BIf one of the two variables can be approximated as being zero however, then the bond
becomes an active bond and is represented by a line with a full arrow as shown below. The output of component A is an effort which is the input of component B. There is no power flow from A to B, the bond has been reduced to a signal.
14 Gas Combustion in a Diesel Engine Process:Modelling and Simulating
3.3 Ports
To be able to model a large number of systems in different energy domains the bond graph method of modelling uses a selection of elements called ports. These shall now be discussed individually.
[Port Elements
A 1-port element has only one power flow going into or coming out from it. There are four basic 1-port elements:
- 1-port resistor, R. The resistor dissipates power. Examples of a resistor are mechanical
dampers, electrical resistors and not-fully-opened valves in hydraulic systems, its bond graph symbol being:
The effort and flow variables are related by a static function, such that:
e =
),(f) (3.4)f = 4),.-1(e) (3.5)
The resistor has no preferred causality. This means that either variable can be used as the input variable.
-
1-port capacitor, C. The capacitor
stores and gives up energy without loss. Examples are mechanical springs, electrical capacitors and accumulators.
Chapter 3: Bond Graphs - a General Description
Here again the energy stored is the time integral of the power which can be rewritten as:
16 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
The energy stored in the capacitor is the time integral of power:
E(t) = 1 e(t)ftt)dt+E0 (3.6)
The constitutive relation of a capacitor however says that q is a function of ie.:
q(e) = 0,(e)
(3.7)e(q) = 0;1(q) (3.8)
so the energy stored can also be written as:
if
E(q) = f e(q)dq+E0
(3.9)q.
As explained earlier, q is the time integral off, so the above equations imply that either e is a function of the time integral off or f is a function of the time derivative of e. This
depends on the chosen causality, resulting in either integral causality or derivative
causality, respectively. The preferred causality for a capacitor is integral causality so theinput of a capacitor should be f resulting in the output e. This preference reflects what happens in 'real life'. For example, a step change in the effort going into a capacitor would result in an infinite flow output which is not desirable when modelling, and is in
any case impossible.
- 1-port inertia, 1. Like the capacitor, the inertia also stores and gives up energy without loss. Examples of an inertia are a mass in a mechanical system, a coil in an electrical system and a fluid flowing through a hydraulic system.
e
\- 1
with:
= At)
iv)) = )
The inertia is the dual of the capacitor, meaning that they have identical constitutive laws
except that the roles of effort and flow are interchanged. Hence, for the inertia to have
the preferred causality of integral causality, the effort must be the input making the flow the output.
- (1-port) effort source and flow source, S. Finally two simple but important 1-ports. Examples for the effort source are batteries and gravity; for the flow source, constant
volume pumps, and a road surface on car suspension.
Se
If the S has the subscript e, then it is an effort source and has as output a constant or time
variable effort which is not dependent on the flow in the ideal case. If it is given the subscript f, then it is a flow source which has as output a flow independent of effort -again in the ideal case. Because theoutput of a source is by definition known there is no
choice as far as the causality is concerned.
3.3.2 2-Port Elements
A 2-port element is an element which simultaneously converts an effort and a flow in such
a way that the power before the element is the same as the power after the element. No energy is lost or stored during the operation. Two 2-port elements need to be defined:
- 2-port transformer, TF.
E(p) = f(p)dp+E0 (3.10)
TF
f2
The function of this 2-port is to carry out the following transformation:
e = me2
1 (3.11)el e,
mfi = (3.12)
The parameter, m, is called the transformer modulus and is in this case constant. If on
the other hand this parameter is not constant, it varies with time for example, then the
modulated transformer is used:
ei
m(t)i
e,
MTF
f:
The causality here is decided upon by elements elsewhere in the system. It should be
noted however that if the effort is the input on one side of the transformer than it must be
the output on the other side. The same will automatically apply for the flow. If this is not immediately possible than it may be necessary to introduce differential causality
elsewhere in the system.
2-port gyrator, GY
fi
el C
\ Y
The function of this 2-port is to carry out the following transformation:
= if2
rJ = e2
The parameter, r, is called the gyrator modulus which for the gyrator is constant.
If
r
varies however the gyrator becomes the modulated gyrator:
el
r(ti
\ MGY
e,
ft
f2Here again, the causality is already decided upon. This time however, the efforts must either both be inputs or both be outputs of the gyrator.
3.3.3 3-Port Junction Elements
Besides 1-port and 2-port elements, bond graphs also make use of 3-port junction elements which are used to represent two types of connections: one where the flow is constant and the effort is split up (like for example a series connection of resistors in an electrical circuit) and one where the effort is constant and the flow is split up (parallel connection in an electrical
circuit). The two types are:
-
0-junction element. Other
names for this element are flow junction and common effort junction. It is represented by:
\ 0 /
(3.13)
(3.14) Chapter 3: Bond Graphs - a General Description
19
e2 f2
Here only three ports are shown, the number of ports can however be increased to create
a 4-, 5- or more-port version of this 3-port. The efforts going into or coming out of this
port are equal:
el =e2=e3 (3.15)
The sum of all the efforts must be zero:
+f2 +f3 = (3.16)
As far as causality is concerned, only one effort can be the input to this element.
- 1-junction element. Also called an effort junction or a common flow junction.
et
e3
N
el
.1; f3
The same rules which were outlined for the 0-junction element apply here except that for this element the flows must all be equal, the sum of all the efforts must be zero and only one flow can be the input.
3.3.4 Fields
A field can be an R-, CL or /-element which has more than one port. For example this R-field:
Its constitutive relation would be:
= r(f)
where Or represents a 4 x 4 matrix in the linear case.
3.4 Pseudo Bond Graphs
Using the components which have been described above already a large number ofsystems in different energy domains can be modelled. There are still a number ofsystems however that can not be modelled using the basic bond graph theory.
An example is the thermodynamic system. It has been suggested to use temperature as the effort variable and heat flow as the flow variable. According to the above theory however, the products of these two variables must be a power which clearly is not the case here since
the flow already is a power. If entropy flow is used as the flow variable this problem is
solved but that does make the mathematics much more complicated. So the heat flow is
indeed used and the resulting bond graph is called a pseudo bond graph.
Thermofluid systems, of which
a large part of the diesel engine is
an example, presentadditional problems in that mass flows, heat flows, work done and momentum must all be
accommodated in the model. Power flow is not a product of just two variables. As a
solution to this problem double bonds are used:
(3.17)
Chapter 3: Bond Graphs - a General Description
3.5 State Equations
The aim of this final part on the bond graph theory is to show how a mathematical model can
be derived using the theory described above. This shall be done using two examples, the
first is a simple mechanical system, the second a thermofluid system.
Once a system has been modelled using bond graphs, the model will have its own specific number (n) of state variables which, when they are all known, will determine the state of the system unambiguously.
The number of state variables will depend on the degree of
complexity of the system in combination with the number of simplifications/assumptions made. This number is generally equal to the number of bonds with integral causality. This
nth-order system can subsequently be represented mathematically by:
1 a single nth-order equation in terms of one unknown variable;
2 n first-order coupled equations in terms of n unknown variables; or
3 various combinations of unknowns and equations of appropriate orders (not necessarily
equal)
Since the second option is particularly suited for use with computer simulations it shall be the one used. The state equations shall be written as follows:
d -
-(x) = fix,u,t)
di
_
x is the vector containing the state variables (unknown variables), u is the vector containing the (known) input variables and t the time.
3.5.1 Example 1: Mechanical System
A mechanical system with two masses. One mass has a force F working on it and is attached to a second mass by a spring arid a damper. This mass is attached to the ground by a spring,
Figure 3.1.
(3.18)
F(t)
Se
Figure 3.1: Mechanical system
5
2
1
Figure 3.2: Augmented bond graph ofmechanical system
The augmented bond graph (the bonds have been numbered, they show the directions in
which the power flows are positive and causality has been assigned) is shown in Figure 3.2.
Note that there is not an 'earth connection', this is not necessary.
The number of state variables are said to be equal to the number of bonds connected to
energy storing elements (I- and C-elements) which have integral causality. So the above
system has four state variables. These are chosen to be the momentums of the
masses (p2 and plo) and generalized displacements of the springs (q5 and q9).
Starting with the constitutive equation for the inertia connected to bond 2, the state equations' are found as follows:
P2 = r-7 Se-es-e7; Se-Y2--RA = Se-q5-R7(f4-4) q5.
= Se'
-R7V21.10) q5 P2 P10 = Se- T-It7 (A similar approach can be used for the other 3 state variables:
(3.19)
mitt
24 Gas Combustion in a Diesel Engine Process. Modelling and Simulating
5
e6+es-e9
= 5+R7(1,f8) q9 C5 9 q5 P2 PIO q9 = C5 12 1W C9
45 = f5
=f3 -f6 = f2 0 = P2 0 /2 /10 (3.20) (3.21) q5 c/9 = C5-9
9=f9
fie
P10 (3.22) Il0When the input variable Se and the state of the system at time t = 0 are known, then the state can be calculated for every t by integration.
3.5.2 Example 2: Thermofluid System
A single element of a thermofluid system, a pressure vessel between for example
a compressor and a turbine, Figure 3.3. Assumption: the mass flows into and out of the vessel are a function of the pressure inside the vessel only.The augmented (pseudo) bond graph is given in Figure 3.4.
p,T
Comp.
Turb.
Figure 3.3: 77zermofluid system
Figure 3.4: Augmented bond graph of thermofluid system
26 Gus Combustion in a Diesel Engine Process:Modelling and Simulating
There are two state variables: the mass of the gas (m or q5) and its internal energy (E or q6) inside the capacitor. When these are known (state at time t = 0) the pressure (p or e5) and temperature (T or e6) can be calculated using the following equations (assuming a perfect gas):
mRT ,r E
P =
=V mc,
See Section 5.2 for some comments regarding these equations. Looking at the bond graph and using the assumption that the mass flows are a function of pressure inside the capacitor, the state equations can be derived as follows:
45 = f5
=f4-f3
= g,(p)-g2(p) (3.23)
The mass flows into and out of the capacitor are subsequently used to determine the energy flows into and out of the capacitor:
=f2-fl
= g1(p)E,--82(p)cpT (3.24)
Where EA is the internal energy of the fluid coming from the compressor.
Chapter 3: Bond Graphs - a General Description 27
4 Bond Graph Representation of
the Diesel Engine
In this section a short description is given of the bond graph of the diesel engine and how it was set up pi. First a model was set up of the diesel engine, Figure 4.1. This model contains all the components of the engine which are to be included in the simulation. The structure of the diagrammatic representation of the diesel engine is also present in the bond
graph, Figure 4.2.
4.1 Clarification of the Bond
Graph
The paired bonds
- one broken, one solid - represent the pseudo bonds.
The bonds
represented by broken lines carry the temperature and internal energy flow of the fluid, the
bonds represented by the solid lines carry the pressure and mass flow of the fluid. The
single bonds which are not paired up with pseudo bonds represent the mechanical parts of the diesel engine carrying the 'normal' bond graph variables, an effort and a displacement
whose product is a power. The bond graph as shown is the correct representation for a one
cylinder engine - for a multi cylinder engine every cylinder will add an extra capacitor, marked cylinder, to the bond graph as well as an extra inlet valve, exhaust valve, piston and crank mechanism.
4.2 Assignment of Causality
Causality has been assigned according to the rules outlined in the previous chapter. Each major component with corresponding causality will now be looked at to get some idea of which relationships need to be established mathematically. The numbers in brackets refer to the numbers of the bonds as shown in Figure 4.2.
4.2.1 Turbocharger
Integral causality has been assigned to the inertia element (32). This immediately implies that, using the rules governing a 1-junction, the angular velocity becomes an input for the
Compressor
Aircooler
01.1
14nwin
Air Receiver
State Variables: ma Ear
Inlet Valve
pa Tar
0><1--21".
Ea
Torques working on Crankshaft:
MI, Mc.u.,M,cc (inbalance)
Turbocharger
State Variable: p,Model of:
Reg. trans. fnc.,
Fuelpump,
Heat release
T,
LaCC(inbalance)
input: X.p q,
.4 QfExhaust Receiver
State Variables: mr, Er,
FExhaust Valve
Figure 4.1: Model of the diesel engine
Pete Ter
Cylinder
State Variables.: rn, EC),,, F qe
Crankshaft
State Variable: pe
30 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
41--m
Turbine
P., P.
Cybanlet 421 TF 131 R
1 1SI
16i-Si Tr
Se Se, Se Se 361; 351I
u30 129 A I, rii.
321-R ----&-il
R
33T T I t t 128 22 I I 1 ii 161 t' t 2,11_1 ,i_ t I 0 ---_,L--Nn0
10--
ri
1-'1-- -',. 0 -21, iR
)- 21- - 0 NI I, T L. ..TN..? T , 43, 42 126 25C
'I.. 1: i 1L411 5.1)A/0' IC
.4C
Figure 4.2- Bond graph representation
of
the diesel engine model'4
t?'
Chapter 4: Bond Graph Representation of Wit Diesel Engine
31 _ 37T 1
R
31 22compressor (34), turbine (31) and mechanical resistance of the turbocharger (33) - all three modelled as a resistor - which will all subsequently then have a torque as output. The effort
sources (29,30,35,36) have by definition an effort as output resulting in the causality as
shown. A resistor has no preferred causality so to determine the causality of the two fields representing the compressor and turbine the causality of the air receiver and exhaust receiver must be determined first.
4.2.2 Air Receiver and Exhaust Receiver
To the four bonds attached to these two capacitors (25,26,42,43) integral causality has been assigned which means that the outputs are efforts. Using the rules governing a 0-junction causality can also be assigned to the bonds connected to the air cooler (40,41), the inlet valve
(1,3), the turbine (27,28) and the exhaust valve (22,24). Causality is also assigned to (39) making the temperature of the cooling water an input of the air cooler. A problem is now encountered since causality can not be assigned to (36,37) by only using the basic rules
governing causality - this is because there is no energy storing element in between the two
resistors representing the compressor and air cooler. A solution round this problem would
be to assume that there is no pressure loss inside the resistor and to estimate the temperature after the compressor, resulting in the causality as shown.
The following can be concluded thus far:
The pressure and temperature 'upstream' and the pressure 'downstream' of the compressor
in conjunction with the speed of the turbocharger determines the mass flow through the
compressor.
- The same can be said for the turbine.
The heat flow to the water in the air cooler and the internal energy of the gas after the air
cooler are determined by the temperatures of the gas before and the gas after the air
cooler, the mass flow of the gas through the air cooler and the temperature of the water.
The temperature and pressure of the gas in the air receiver (and exhaust receiver) are
determined by the energy and mass flow into the air receiver (exhaust receiver).
4.2.3 Cylinder
The cylinder volume has been modelled as a capacitor which implies that, generally
speaking, a flow is an input and an effort is an output when integral causality is assigned toit (7,8,12). Causality can now also be assigned to the bonds going to the inlet valve (2,4), the exhaust valve (21,23), the model of the heat release (5,6) and the bonds going to the piston (12,13). The effort source connected to bond (10) represents the temperature of the cylinder wall, the flow source connected to (11) is the work done by the gas. The energy
32 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
-flow E to this -flow source is equal to the energy -flow represented by (12), pV. These -flows can however not be coupled by for example a transformer due to the fundamental differences
between a 'normal' bond and a pseudo bond.
More conclusions:
The mass flow and energy flow through the inlet valve and exhaust valve are determined by the pressures and temperatures upstream and downstream of the valves.
The output of the model of the rate of heat release is an energy flow and a mass flow.
The energy flow from the gas in the cylinder to the cylinder wall is a function of the
temperatures of the gas and the cylinder wall.
The input into the crank mechanism from the cylinder is an effort.
The temperature and pressure of the gas inside the cylinder is determined by the mass flow and energy flow into the cylinder, the cylinder volume and (not shown in the bond graph) the fuel/air equivalence ratio which determines to an important extent the properties of the gas (this also applies for the exhaust receiver).
4.2.4 Crankshaft
Since the mass of the piston - an inertia - is coupled to the crankshaft - also an inertia - via a rigid mechanism a choice has to be made as to which inertia is assigned integral causality. Here the crankshaft has integral causality (20) and the piston has differential causality (15). Final conclusions regarding causality:
- The engine has as output a torque (19).
The mechanical resistance (18) as well as the torque needed to drive secondarycomponents (44) is a function of engine speed.
The resistance between piston and cylinder lining is a function of the speed of the piston (14).
The input to the model of heat release (46) is a fuel mass flow.
Chapter 4: Bond Graph Representation of the Diesel Engine 33
Chapter 5: Thermodynamical, Hydraulical and Mechanical Principles 35
5 Thermodynamical, Hydraulical and Mechanical
Principles
5.1 State Variables and Input Variables
In this section, equations which determine the state of the diesel engine are set up. As was discussed in Chapter 3, these state equations shall be writtenas:
d
(x) = f(x,u,t)
dt
Where I is the vector of state variables, TA is the vector of input variables and t the time.
The vector of state variables will include the following variables:
= fprnaEae,Tricyl.i...z,y1,2,3 >E.cy1,1 .z,F .z,(1 e,p e,M ,E eF eri
p, = turbocharger angular momentum mar = mass of the gas inside air receiver
Ear = internal energy of the gas inside air receiver
m11 = mass of the gas inside cylinder i
= variables governing regulator
Eod = internal energy of the gas inside cylinder i = fuel/air equivalence ratio in cylinder i
ge = crankshaft angular displacement (to calculate cylinder volumes, determine
opening of valves and injection timing)
Pe = crankshaft angular momentum
mer = mass of the gas inside exhaust receiver
= internal energy of the gas inside exhaust receiver
= fuel/air equivalence ratio in the exhaust receiver
The following input variables are required:
U = [po,ToTs,T.,,M,,cor,Xee]
Po -= ambient pressure Y
-To = ambient temperature
7; = cylinder wall temperature
= cooling water temperature in air cooler = engine load
reference engine speed
Xrcf = reference pin position
The order in which the state variables appear in the vector containing them corresponds with the order in which the state equations are derived.
5.2 General Assumptions and Simplifications
The following general assumptions and simplifications which are inherent to the bond graph
used, Figure 4.2, should be kept in mind when looking at the derivation of the state
equations and subsequent results in the following section:.Assuming a perfect gas, pV = mRT and E = mc,,T, then:
T=
mcv(F,7) and: mR(F,7)TP =
VWith the temperature, T, expressed as the absolute temperature the medium has zero enthalpy and internal energy at absolute zero. This is in line with the choice made by
Zacharias, Appendix F and oh Gas kinetic energy has been assumed negligible. T and cv
are calculated using an iteration procedure. An estimate for T will result in a value for cv using Zacharias' formulas. T is then calculated using the formula above and cv is
calculated again, just so often until T is sufficiently accurate.
Dynamic effects in canals through which gas flows are not taken into consideration. 36 Gas Combustion in a Diesel Engine Pmcess: Modelling and Simulating
Mixing of gasses with different temperatures and compositions in the cylinders and the receivers is assumed to be so fast that the composition of a gas is homogenous inside these volumes.
There is no heat loss due to radiation to the
environment from the engineor its
components.
Assumptions and simplifications which
are relevant to a particular state equation
arediscussed when the time arises.
5.3 Derivation of the State Equations
With the help of the aforementioned
-choice of state variables and the causality
considerations - and a number of thermodynamical, hydraulical and mechanical principles the state equations shall now be derived.5.3.1 Angular Momentum of the
Turbocharger, p,The state of the turbocharger is fully determined by state variable, its angular momentum,
p given by:
= 1,0.71
(5.1)
Using the bond graph, Figure 4.2,
as an aid it is possible to set up
an equation whichbalances all the torques working on the turbocharger shaft:
Pi = MturMtfric Mc State Equation I (5.2)
This is the state equation of the turbocharger, in which:',
Atm, is the turbine driving torque which can be determined by using:
111134r = r tuAs.turn 1 (5.3)
t
Chapter 5: Thermodynamical, Hydraulical and Mechanical Principles
37
-In this equation, li1wt. is the mass flow through the turbine which can quite accurately be
approximated by modelling the turbine as an orifice and subsequently using the basic nozzle equation for transient one-dimensional gas (Appendix A):
rnwr = CDA Per
gr.)
VRTer
Per
where r
air =*saris the isentropic enthalpy drop given by:
Per
Ahisaur = c Ter[1-() "
]P pc)
In these equations the isentropic efficiency, and the flow coefficient CD can be found from the graph shown in Figure 5.1. In this graph the turbine blade-speed ratio We° is given by:
rww,
O
Where rb, is the mean turbine blade radius.
Mt jne is the frictional torque given by the manufacturer which can be modelled as being a
percentage of Mr.
Mc is the compressor torque, given by:
Alcom = COM.Ahis.cofcnvcam 1
nixorn (4) t
(5.6)
which is equivalent to that for the turbine. Here the variable titcon, can be found by first finding
from the compressor performance map shown in Figure 5.2 (momentary
pressure ratio and turbocharger speed are known), correcting for the ambient temperature To, and subsequently using the gas equation pV = mRT. The compressor isentropic efficiency,can also be found using the compressor performance map, the compressor volumetric 38 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
(5.4)
(5.5)
CDT nIT
as 3.9
Ms.. has already been determined using the method described above.
1 I i i iI I 1 [ _ ., , f s
- -
... ---1-r-.--rr.,-..-_-r-r.z..-.-
:
_----
.
i ---ni 11 .rsb"T..- th.... +-..
T T_
% :::1 X.--...
1.N.Icet , , Ii
: 1 /at
/
_\1`\
. .-.,\ , I \ Y;..\ IJ
.7
_ ... ,\A ii_ J II 4,[
,,v.,,
°tamer 5: Thermodynamical, Hydrauhcal and Mechanical Principles
_39
0.0 0 4
4,2.C
----
3Y
ca
a.. I I .;21
tilc
Figure 5.1: Turbine isentropic efficiencyand flow coefficient against turbine blade speed
efficiency, is assumed to be constant and given by the manufacturer'.
0014,
mpuv-60c
Surging (which occurs when the pressure ratio across the compressor is larger then the
maximum pressure ratio possible for the momentary turbocharger speed) must also be accounted for.. This is done by making the volume flow zero once surging occurs.
5.3.2 Mass id the Air in the Air
Receiver,The state equation governing the mass of 'the aft inside the air receiver is:
Filar
rhro,E mar)
State Equation. ,2 (5.7]3
\
45
1
3
1 . 5
2 .
Gas Combustion in a Diesel Engine Process: Modelling and Simulating
' Vi,, r=1 rine -12..71MWM raw ilt 3.0 l.a 0.0 3.0 1.4ei /r-3s-1
Figure 5.2: Compressor performance map
Matra represent the mass flows to the cylinders (of which there are z). For each cylinder this flow must be calculated. This can again be done quite satisfactorily making use of the 'basic nozzle equation':
= CDAL Par
(7r)(5.8)
VRTar
in which .21-. = Par.
Reverse flow can be modelled by exchanging the pressures upstream and downstream and making the expression for the mass flow negative. The flow coefficient, CD j, and the
geometrical valve area, Ai, vary with crank angle. It is the crank angle which of course
determines the valve lift. 40
-5.3.3 Internal Energy of the Air in the Air Receiver,
EarEar = ECM C Eair,t State Equation 3 (5.9)
1
Eaar is the energy flow coming from the air cooler. The energy flow from thecompressor is known so, from this, the energy loss due to the air being cooled in the air cooler must be subtracted. This 'loss' can be modelled in several ways. In this model this is done by defining an air cooler efficiency, based on the airflow through the air cooler, see Appendix B.
r = a-brizrcyn
which subsequently gives the temperature after the air cooler as:
T = Tar-(Tor-T,,,) airc
and:
= nicovn.cp.T.,
1
E
car,i =2. if
air.i'(T +T
ar cyl-i)+1-f I(5.11)
(5.12)
taw., represent the energy flows to the cylinders. For each cylinder the following equation is used:
(5.13)
In which Incur., has been determined above. Should reverse mass flow take place than the energy flow adjusts accordingly.
5.3.4 Mass of the Gas in the Cylinder,
Pic),The state equation governing the mass of gas inside each cylinder will be the following
(leaving out the subscript i):
Chapter Thermodynamical, Hydrauhcal and Mechanical Principles 41
r
Assumptions
There is no mass flow out of the cylinder via the cylinder rings.
- When fuel combusts it is added on to the total mass present in the cylinder.
224,-01,4o
11+ er
'filt
Figure 5.3: Filter and Regulator
1+Tds
The values for the time constants are derived in Appendix D. The equations representing the
transfer function need to be written as a series of first order equations. These are:
= mair+mfmeth State Equation 4 (5.14)
?hair has already been determined, rich, can be determined using a method which is analogous to the one used for Mar.,
MI represents the rate at which fuel is converted to combustion products and is dependent on
the model for the rate of combustion of the fuel. The model which shall (at first) be used is a single Wiebe model. How this model works is explained in depth in Appendix C.
The Wiebe model uses as input the total mass of fuel injected during one combustion cycle. This is the output of the fuel pump which is regulated by the regulator. The regulator's task is to compare the actual engine speed with a reference engine speed and subsequently alter
the pin position of the fuel rack as determined by the regulator's transfer function. The modelling of this transfer function is a rather ad-hoc task since seldom is this function known. In this model a regulator which is made up of a filter and a PID transfer function is used. The block diagram of this regulator is given in Figure 5.3.
Xie
42 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
115 1 110 + T wfilt K
Assumption
There is no heat transfer from the engine to the air on
entrance into the cylinder. this
will result in 'trapped' conditiontemperatures which are slightly too low.2 30 2
YEA 7 rid:((WRWieD-27fili YITTfillY2.
,921i = Y2 93 = 74 State Equation 5 (5.15) State Equation 6 (5.16) State Equation 7 (5.17)
Eai, has been determined during the discussion of the air receiver
(bearing in .mind the assumption above), Ego, can be found in a similar Way.
With these state variables the value ofAX can be found as follows: AX K(I+r4s)(Y2+9
= K(y3+1dy2-124-rdY;11
0.181
7,
To obtain the actual pin. position,, AX needs to be added to a reference pin position ice, see Figure 5.3.,
St = Xref+AX (5.19)
5.3.5 Internal' Energy of the Gas in
the CYlinder,Energy balance:
try/ = 10,
P
State Equation 8(.20)
Chapter 5: Thermodynamical, Hydraulical and Mechanical Principles 43
-Y1
Ecy, =
with
is the heat energy released by the fuel and is determined by the model for the rate of heat release.
Ow: is
the rate of heat transfer through the cylinder walls and piston and
can be
approximated by:Oiasi = aTlic (ReAs(T-Ts)+cA(T4-Ts4) (5.21)
A derivation of this equation as well as values for the constants can be found in Appendix E.
Pp, is the work done by the piston. In that the cylinder volume is directly related to the
crank angle, the work done by the piston shall be expressed in terms of the cylinder pressure and the crank angle (because the crank angle is used later as well), as follows:
=P'
= pApS= pAin(q)4,
(5.22) singecosq, m(q) = Rsinge+RX (5.23) \11-Vsin2q,5.3.6 Fuel/air Equivalence Ratio in the Cylinder, Fe.
The fuel/air equivalence ratio is an important state variable as it will greatly effect the
properties of the gas inside the cylinder and in the exhaust system. It is used in a number of equations derived by Zacharias which are elaborated upon in Appendix F. The fuel/air
equivalence ratio is given by:
----(
m,JmF=
fs
where ingas represe
inside the cylinder.
(5.24)
sr the mass of the gas inside the cylinder minus the mass of the fuel
This expression, when differentiated once becomes (Appendix J):
1+Ff,(1 m
f f
cyt s5.3.7 Cylinder Volume, V
State Equation 9 (5.25)ee./7
ctfi'
The volumes of the gas in the cylinders are of course important variables when calculating the cylinder states. Because however a rigid mechanism connects all the pistons, only one volume needs to be a state variable. If one volume is known then all the others are known as well. To make things slightly simpler though the crank angle is used as the state variable.
Lie = State Equation 10 (5.26)
Here qe is the displacement of the crankshaft (crank angle) and We is the angular velocity of
the crankshaft.
5.3.8 Crankshaft Angular Momentum,
peAn approach similar to the one used for the turbocharger can be used for the crankshaft, that is, finding the state equation by balancing the torques working on the crankshaft.
Pe few,
pe is the angular momentum of the crankshaft.
PC=E(
, e State Equation 11 (5.27)Mot, is the torque working 9n the 4rankshaft for one cylinder which can be broken down into several components:
Chapter 5: Thermodynamical, Hydraulical and Mechanical Principles 45
M1 = m(q
cY1. e,t.)(A_Dcyl. -Ffru- arc ,cylin which Ffrk. is the force due to friction between cylinder and piston which is assumed to be proportional to the piston speed, S , and is the force needed to accelerate or decelerate the combined mass of the piston and part of the connecting rod, equal to IpS. The relation
between s and qe is known and described above so S and S can be found in terms of qe by differentiation.
A 1e fric is the frictional torque which is assumed proportional to the engine speed and includes
the torque needed to drive the fuel pump, camshaft, etc.
MI is the load torque, an input variable.
5.3.9 Mass of the Exhaust Gas in the Exhaust Receiver, me,
Identical approach to the one used for the air receiver. The mass flows into and out of the
exhaust receiver have already been determined. Addition will result in the state equation
governing mass of the exhaust gas in the exhaust receiver.
fil
Eril
State Equation 12 (5.28)1
5.3.10 Internal Energy of the Exhaust Gas in the Exhaust Receiver, En
Assumption
- There is no heat transfer from the exhaust gasses to the cooling water as the gas passes
through the exhaust passage. This should result in exhaust receiver temperatures which
are too high.
The energy flow from the cylinders to the exhaust receivers can be derived in a similar way
to the energy flow from the air receiver to the cylinders, the energy flow to the turbine is known. Here again the state equation can be derived by summing the two energy flows.
46 Gas Combustion in a Diesel Engine Process: Modelling and Simulating
l+f F E(Fcy1,1-Fer)rhexh,i
1
1 +f F5 ,y1.i
State Equation 14 (5.30)
7.
ter = E te_th
tzar
State Equation 13 (5.29)1
5.3.11 Fuel/air Equivalence Ratio in the Exhaust Receiver, Fe,
A change of the fuel/air equivalence ratio in the exhaust receiver will occur if the exhaust gasses coming from the cylinders have a fuel/air equivalence ratio which is different from
the one in the exhaust receiver. This change is calculated as follows (AppendixJ):
