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Department of
Mechanical Engineering
Report EM/99/10
Complete Analytical Model of a
Single-Cylinder Diesel Engine for Non-linear
Control and Estimation
Yahya H. Zweiri, James F. Whidborne and Lakmal D. Seneviratne
Department of Mechanical Engineering
King's College London
Strand, London WC2R 2LS, UK
email: [email protected]
23 November 1999
Abstract
A non-linear model for a single-cylinder diesel engine is developed. The model can be used as an engine
simulator to aid diesel engines control system design and fault diagnostics. The model does not use
any empirical inputs, and accurately predicts in-cycle variations of engine parameters during transient
response. The model treats the cylinder strokes and the manifolds as thermodynamic control volumes by
using the �lling and emptying method, solving energy and mass conservation equations with submodels
for combustion, heat transfer, instantaneous friction and dynamic analysis. Inertia variations and pis-
ton pin o�set are taken into consideration. In-cycle calculations are performed at each crank-angle, so
the correct crank-angle of ignition, speed variation, amount of fuel and air as well as the fuel burning
rate are predicted. The model is validated using experimentally measured cylinder pressure and engine
instantaneous speeds, under transient operating conditions, giving good agreement.
Keywords: Diesel engine modelling, simulation, dynamic modelling, thermodynamic modelling,
transient response, fueling control, engine diagnostics.
Notation
c: compression ratio
d: cylinder bore diameter [m]
h
f
: forced heat transfer convective coe�cient
h
for
: enthalpy of formation of the hydrocarbon
h
n
: natural heat transfer convective coe�cient
h
o
: stagnation enthalpy
}: ratio length
i: subscript to identify di�erent entries to the control volume
j: subscript to identify surfaces with di�erent rates of heat transfer
m
a
: air mass
m
c
: crankshaft mass
m
f
: mass of fuel
m
p
: piston mass
m
r
: connecting rod mass
m
0
: mass ow rate
m
0
f
: fuel rate
m
0
fb
: burned fuel mass rate
m
0
fburn
:fuel mass burning rate
p
atm
: atmospheric pressure(101 KPa)
p
i
: indicated pressure [Pa]
r: crank radius [m]
u: speci�c internal energy
y: piston displacement [m]
A: heat transfer area [m
2
]
A
m
: exhaust manifold area [m
2
]
A
p
: piston area [m
2
]
A
v
: valve area [m
2
]
C
d
: discharge coe�cient
C
p
: speci�c heat
D: damping coe�cient [Nm/(rad/s)]
D
i
: inner exhaust manifold diameter
D
o
: outer exhaust manifold diameter
F : equivalence ratio
F
f
: friction force [N]
G(�): geometric function
ID: ignition delay
J : moment of inertia of crankshaft, ywheel, main gear and rotating part of connecting rod [Kgm
2
]
J
c
: moment of inertia of crankshaft [Kgm
2
]
J
R
: moment of inertia of connecting rod [Kgm
2
]
J
1
: moment of inertia of dynamometer rotating parts [Kgm
2
]
K: thermal conductivity
L: connecting rod length [m]
M : piston, rings, pin and small end of connecting rod mass [Kg]
P : pressure
P
d
: down stream pressure
P
u
: up stream pressure
Q
ht
: heat transfer
R: gas constant
S: coupling sti�ness
1
T : temperature [K]
T
D
: damping torque [Nm]
T
f1
: piston-ring assembly friction torque [Nm]
T
f2
: crankshaft bearing friction torque [Nm]
T
f3
: valve train friction torque [Nm]
T
f4
: pumping losses torque [Nm]
T
f5
: pumps friction torque [Nm]
T
i
: indicated torque [Nm]
T
Lj
: engine load torque [Nm]
T
m
: exhaust manifold temperature [K]
T
r
: reciprocating inertia torque [Nm]
T
rad
: apparent radiant temperature [K]
T
S
: torsional sti�ness torque [Nm]
T
w
: cylinder wall temperature [K]
V
d
: Displacement volume [m
3
]
�: connecting rod angle [rad]
�: piston pin o�set [m]
�: crankshaft angular position [rad]
�
1
: dynamometer angular position [rad]
�: the dynamic viscosity of the oil [Ns/m
2
]
�: density [Kg/m
3
]
�: connecting rod angle when piston is at TDC [rad]
": apparent grey-body emissivity
�: kinematic viscosity
& : volumetric coe��cient
$: average exhaust ow rate
1 Introduction
The modelling of diesel engines has received much research attention. The models have been used for
engine simulation and design. In addition, with the recent rapid advance of microprocessor technology,
it has become feasible to use these models for real time applications, such as state estimation, control
and fault diagnostics. However, there needs to be further research in modelling techniques, to improve
predictive accuracy, before models can be used for estimation and control tasks.
Several investigators have studied diesel engine thermodynamic modelling. The investigations can be
divided for two categories:
i) Linear engine models relying on empirical data [1, 2, 3]: These models link steady-state experi-
mental data representing engine thermodynamics and gas ow, with simple dynamic models of the
mechanical components. The major disadvantage of such quasi-linear models is their heavy reliance
on experimental data. Furthermore, they poorly represent of the transient response of the engine.
ii) Non-linear engine simulation models using the Filling and Emptying Method, and the Method
of Characteristics [4, 5, 6, 7, 8]: These thermodynamic engine models have been developed for
engine design and performance predictions, and, few have been developed for diesel engine control
[9, 10, 11, 12]. In [9], a mean value model was used. The model consists of several empirical
algebraic and �rst order di�erential equations in order to achieve real time implementation. The
main drawback of this type of model is the requirement of empirical test data to �t model equations.
Diesel engine models for electronic control development were developed in [10, 11]. The in-cylinder
combustion process was included by using a single zone model, and a �lling and emptying modelling
technique. In [10] several modi�cations to the model are proposed to cut down the computational
time and commercial bureau rate cost. These models provide a tool for engine simulation but is not
suitable for diagnostic formulations and in-cycle crankshaft angular speed variations predictions.
2
Hybrids of the two engine models, a mean torque production model and a simpli�ed cylinder-by-
cylinder model, are investigated, in order to provide control engineers tools for developing control
and diagnostic algorithms [12]. The implementation of the mean torque model required empirical
test data to �t the model equations, and the simpli�ed cylinder-by-cylinder model is not able to
accurately predict in-cycle variations of engine parameters.
Some investigators have studied engine dynamic modelling, describing the relationship between the
net engine torque and the angular speed of the crankshaft, [13, 14, 15, 16]. None of these studies include
the inertia variations of the crankshaft assembly, and the piston pin o�set.
In this paper a thermodynamic and dynamic based diesel engine model is developed and used to predict
the in-cycle variations of the engine states. The model treats the cylinder strokes and the manifolds
as thermodynamic control volumes by using the �lling and emptying method, solving energy and mass
conservation equations with submodels for combustion, heat transfer, instantaneous friction and dynamic
analysis. The proposed model does not require any empirical inputs, and has taken into consideration
piston pin o�set and inertia variation of the crankshaft assembly.
The in-cycle calculations are performed at each crank-angle. The model has been implemented in
MATLAB[17]/SIMULINK[18].
The paper is arranged as follows. First, the engine and dynamometer model, which is composed of the
engine thermodynamic model, the engine dynamic model, the friction torque model and dynamometer
dynamic model, are formulated on a crank-angle basis. Next the implementation is described, followed
by some simulation results to show the model behaviour and validation. Finally, there is a discussion and
some conclusions are drawn.
2 Model Analysis
A general engine model will consist of both thermodynamic and dynamic equation. The model inputs will
be the air and fuel ow rates, and the load on the engine. Then, given the de�ned engine parameters, the
model should predict the crankshaft speed variations. The thermodynamic model presented in this paper
is based on the �lling and emptying method. Three thermodynamic control volumes are considered; the
inlet manifold, the cylinder and the exhaust manifold. A schematic is shown in Figure 1. Equations for the
conservation of mass and energy, gas property relations and perfect gas law are solved in the crank-angle
domain. A single zone combustion model is used for simulating the theoretical fuel burning rate. The
cylinder is treated as one control volume with homogeneous temperature and pressure. The non-linear
dynamic model in [19], including instantaneous friction components model [20] is improved to include
the computation of the inertia variation of crankshaft assembly, and the time domain is transformed to
the crank angle domain.
2.1 Equations for Conservation of Mass and Energy
The �lling and emptying method treats the cylinder and each manifold as a thermodynamic control
volume. The �rst law of thermodynamic for an open system can be written as:
U
0
= Q
0
�W
0
+
X
i
H
0
0i
; (1)
where subscript i denotes di�erent entries to the control volume and (
0
) represents derivatives with respect
to crank angle (�). Equation 1 can be written as:
(mu)
0
=
X
j
Q
0
j
� PV
0
+
X
i
h
0i
m
0
i
; (2)
where j denotes surfaces with di�erent rates of heat transfer, Q includes heat released by combustion, h
0
is the stagnation enthalpy, hence kinetic energy implicitly accounting for, u is the speci�c internal energy,
P and T are control volume pressure and temperature respectively and m is the mass.
3
?
- -
�^
Inlet manifold
Exhaust manifold
.
.
.
.
.
.
.
.
.
.
.
.
.
.
U
�
r
-
�
Q
0
ht
Q
0
ht
�
-
�
�
m
0
f
-�
d
B
E
L
C
A
D
Figure 1: Single-cylinder diesel engine
Di�erentiating the left-hand side of Equation 2 gives
mu
0
+ um
0
=
X
j
Q
0
j
� PV
0
+
X
i
h
0i
m
0
i
; (3)
The speci�c internal energy u is assumed to be a function of temperature T and the equivalence ratio
F only. Since dissociation e�ects are small [10], the in uence of pressure is neglected. Thus
u = u(T; F ) (4)
A curve �t of combustion product data, as a continuous function is used [4], based on absolute zero
temperature of the elements:
u =
K
1
(T )�K
2
(T )F
1 + f
s
F
(5)
where
K
1
(T ) = :692T + 39:17� 10
�6
T
2
+ 52:9� 10
�9
T
3
� 228:62� 10
�13
T
4
+ 277:58� 10
�17
T
5
and
K
2
(T ) = 3049:39� 5:7� 10
�2
T � 9:5� 10
�5
T
2
+ 21:53� 10
�9
T
3
� 200:26� 10
�14
T
4
The gas constant R is given by
R =
:278 + :02F
1 + f
s
F
(6)
where f
s
is the stoichiometric fuel air ratio.
This technique has the advantage of automatically predicts a reduction of speci�c heat release by
combustion at richer than stoichiometric fuel air ratio, which occur during transient operation. Thus
Q
0
= Q
0
ht
+m
0
f
h
for
(7)
4
where Q
0
ht
is the heat transfer rate, m
f
is the mass of fuel burnt, and h
for
is the enthalpy of formation
of the hydrocarbons (fuel), given by [21]:
h
for
= 2:326[ � 19183+ :5(T
fuel
� 537)] (8)
is the low heating value of the fuel.
Assuming that the gases involved behave as perfect gases, PV = mRT , where R is the gas constant.
Using Equations 4 and 7, Equation 2 becomes
T
0
=
2
4
0
@
X
j
Q
0
htj
+
X
in
h
oin
m
0
in
�
X
out
h
0out
m
0
out
+m
0
f
h
for
� um
0
1
A
1
m
�
RT
V
V
0
�
@u
@F
F
0
3
5
�
@u
@T
: (9)
Mass conservation for both air and fuel is required. Considering the total mass (fuel and air) gives
m
0
=
X
in
m
0
�
X
out
m
0
+
X
m
0
f
(10)
The total mass is given by
m = m
a
+m
bf
(11)
where m
a
is the mass of the air and m
bf
is the mass of the burnt fuel. The fuel-air equivalence ratio is
de�ned as
F =
f
f
s
(12)
where f is the fuel air ratio. Thus, the mass of the burnt fuel m
bf
in total mass m of air and burnt fuel
is given by
m
bf
=
mf
s
F
1 + f
s
F
(13)
From Equations 11, 12 and 13, the term F
0
in Equation 9 can be found;
F
0
=
�
1 + f
s
F
m
� �
1 + f
s
F
f
s
m
0
bf
� Fm
0
�
(14)
2.2 Equations for Ports and Valves Mass Flow Rates
A one-dimensional model for ow through a valve (or port) using the analogy of an ori�ce having an
equivalent area, is used. Applying the energy equation from upstream to downstream for isentropic steady
ow, and assuming that the inlet velocity is negligible, for subsonic ow
P
d
P
u
>
�
2
+1
�
(
�1
)
, the mass ow
rate has the form,
m
0
= C
d
A
v
P
u
v
u
u
t
�
2
� 1
�
1
RT
u
"
�
P
d
P
u
�
2
�
�
P
d
P
u
�
+1
#
(15)
where C
d
is the discharge coe�cient, is the speci�c heat ratio and A
v
is the valve or port area.
For sonic ow
P
d
P
u
�
�
2
+1
�
(
�1
)
, the mass ow rate has the form
m
0
= C
d
A
v
P
u
v
u
u
t
RT
�
2
+ 1
�
(
+1
�1
)
(16)
5
2.3 Equations for Combustion
Many combustion models have been discussed in [5, 4, 21]. The single zone model proposed in [10] is used
here, because it is widely accepted and can be used to predict both premixed and di�usion burning. The
ignition delay (ID) equations as function of cylinder pressure and temperature can be used to calculate
the start of the combustion [5].
ID = 3:45
�
P
101:3
�
�1:02
e
2100
T
(17)
Z
�
ign
�
inj
d�
ID
= 1: (18)
where �
inj
is the angle at injection and �
ign
is the angle at ignition The normalised premixed burning
rate is given by:
m
0
fpre
= k
p1
k
p2
�
k
p1
�1
norm
�
1� �
k
p1
norm
�
k
p2
�1
(19)
while the normalised di�usion burning rate is given by:
m
0
fdiff
= k
d1
k
d2
�
k
d1
�1
norm
e
�k
d2
�
k
d2
norm
(20)
where �
norm
is non-dimensional angle, increasing from 0 at start of combustion to 1 at the end of
combustion. The parameters k
p1
, k
p2
, k
d1
and k
d2
have been correlated with fundamental factors having
the strongest in uence on the combustion rate. The best �t values of these parameters for a range of
direct injection diesel engines was presented in
k
p1
= 2 + 1:25� 10
�8
(ID �N)
2:4
(21)
k
p2
= 5000 (22)
k
d1
=
14:2
F
:644
(23)
k
d2
= :79� k
:25
d1
(24)
The fraction of the total fuel that goes to the premixed burning, f, is given by [10]:
f = 1� :926F
:37
ID
�:26
(25)
Assuming complete combustion in 125 crank angles, the normalised angle is de�ned as:
�
norm
=
� � �
ign
125
: (26)
Finally, the fuel burning rate is given by [10]:
m
0
fnorm
= fm
0
fpre
+ (1�f)m
0
fdiff
(27)
m
0
bf
=
m
f
m
0
fnorm
�
com
(28)
where �
com
is the combustion period in degree.
2.4 Equations for Cylinder Heat Transfer
An instantaneous convective heat transfer is de�ned by,
Q
0
ht1
= h
con
A
v
(T � T
wall
) (29)
where h
con
is the heat transfer coe�cient based on forced convection and is given by [22].
h
con
= 130V
�:06
P
:8
T
�:4
(N
p
+ 1:4)
:8
(30)
6
where N
p
is the mean piston speed and T
wall
is the cylinder wall temperature. The primary sources
of radiative heat transfer in a diesel engine are the high temperature burned gases and soot particles.
Following work in [23], the instantaneous radiant heat transfer can be expressed as:
Q
0
ht2
= "�A
v
(T
4
rad
� T
4
wall
) (31)
where " is the apparent grey-body emissivity, � is Stephan-Boltzman constant and T
rad
is the apparent
radiant temperature. The adiabatic ame temperature of the slightly greater than stoichiometric (F =
1:1) zones of hydrocarbon-air combustion products. The apparent radiant temperature is calculated as
the mean of the adiabatic ame temperature and the average bulk gas temperature, i.e.
T
rad
=
T + T
(F=1:1)
2
: (32)
The temperature of combustion products at F = 1:1 is computed from a correlation of instantaneous air
temperature and pressure as follows [23]
T
F=1:1
= (1 + :0002317(T � 950))(2726:3+ :906P � :003233P
2
) (33)
for 800K < T < 1200K, and
T
F=1:1
= (1 + :000249(T � 650))(2487:3+ 4:7521P � :11065P
2
+ :000898P
3
) (34)
for 450K < T < 800K
The apparent emissivity is assumed to vary linearly between its maximum value :9 to zero over the
expansion stroke. The cylinder wall temperature is updated at each step by using the one-dimensional
heat conduction model, and an electrical analogy model, shown in Figure 2:
Q
0
ht1
+Q
0
ht2
=
(T
w
� T
coolant
)
R
wc
+R
w
(35)
where R
wc
=
1
h
coolant
A
is the thermal resistance from wall to coolant. The thermal resistance for the
conduction through the wall is R
w
=
t
KA
, where t is the wall thickness and K is the thermal conductivity.
Then the fourth order polynomial equation in T
m
as an independent variable is obtained and the solution
gives the wall temperature.
(T � T
wall
)
R
gw
+ "�A(T
4
rad
� T
4
wall
) =
(T
w
� T
coolant
)
R
wc
+R
w
(36)
where R
gw
=
1
h
con
A
is the thermal resistance from gas to wall.
R
gw
R
w
R
wc
T
T
r
Radiation resistance.
T
w2
T
coo
T
w1
Figure 2: Thermal resistance for the cylinder
Then the cylinder heat release rate can be calculated from Equations 29 and 31:
Q
0
ht
= Q
0
ht1
+Q
0
ht2
(37)
7
T
m
T
air
T
ex
R
ew
R
ma
Radiation resistance.
Figure 3: Thermal resistance for the exhaust manifold
2.5 Equations for Exhaust Manifold Heat Transfer
The exhaust gas ow is considered turbulent. The forced heat convection between the exhaust gas and
the manifold wall, the natural convection, and radiation between the manifold and ambient air are taken
into account. The conductive transfer through the wall thickness is neglected. An electrical analogy
model is shown in Figure 3. The manifold is assumed to have a cylindrical shape, homogeneous gas and
wall temperatures. The heat transfer rate from exhaust gas to manifold is given by
Q
0
ht
= h
f
A
m
(T � T
m
) (38)
The heat transfer rate from manifold to air is given by
Q
0
ma
= h
n
A
m
(T
m
� T
air
) (39)
The heat transfer rate due to radiation is given by
Q
0
r
= "�A
m
(T
4
m
� T
4
air
) (40)
where h
f
is the forced heat convective coe�cient given by [24]
h
f
= :023
K
D
i
R
:8
e
P
:3
r
(41)
where R
e
is the Reynolds number, P
r
is the Prandtl number:
R
e
=
$D
i
�
(42)
P
r
=
C
p
�
K
(43)
� is the kinematic viscosity, D
i
is the inner manifold diameter and C
p
is the speci�c heat. The dynamic
viscosity � can be calculated from [21]:
� =
3:3� 10
�7
T
:7
1 + :027F
: (44)
The average ow rate is given by [24]:
$ =
NV
120(
�D
2
i
4
)
: (45)
The natural heat convective coe�cient h
n
is given by [24]:
h
f
= :53
K
D
o
(G
r
P
r
)
:25
(46)
8
where G
r
is the Grashof number [24]:
G
r
=
g&(T
m
� T
air
)(:9D
o
)
3
�
2
(47)
where g is the gravitational acceleration, D
o
is the outer manifold diameter and & is the volumetric
coe�cient and it is taken equal
1
T
air
[25]. Applying the energy conservation principle to the volume,
a non-linear algebraic equation in T
m
is obtained. The solution to this equation gives the manifold
temperature which can be substituted into Equation 38 to calculate the manifold heat transfer rate.
T
m
� T +
D
o
D
i
h
n
h
f
(T
m
� T
air
) +
D
o
D
i
�"
h
f
(T
4
m
� T
4
air
) = 0 (48)
2.6 Cylinder, Intake and Exhaust Manifolds Modelling
The solution of Equations 9, 10 and 14 are considered for cylinder, inlet and exhaust manifolds in turn.
In all cases it is assumed that T;m; F , and P are known at the beginning of each step. For the �rst step,
estimate values are used; for the remaining the values steps are known from the previous step.
2.6.1 Cylinder
i) Compression stroke, from close of the intake valve to open of the exhaust valve: During this period
F
0
= 0; m
0
= 0; h
0
inorout
= 0; m
0
fb
= 0
Equation 9 becomes
T
0
=
2
4
X
j
Q
0
htj
1
m
�
RT
V
V
0
3
5
�
@u
@T
: (49)
ii) Combustion and expansion: During this period m
0
= m
0
f
; h
0
inorout
= 0
Equations 9 and 14 become
T
0
=
2
4
0
@
X
j
Q
0
htj
+m
0
f
h
for
� um
0
1
A
1
m
�
RT
V
V
0
�
@u
@F
F
0
3
5
�
@u
@T
: (50)
F
0
=
�
1 + f
s
F
m
� �
1 + f
s
F
f
s
m
0
bf
� Fm
0
f
�
(51)
iii) Exhaust stroke: During this period, combustion is assumed to be completed and the equivalence
ratio is constant. i.e F
0
= 0, m
0
in
= 0, h
0
0in
= 0, m
0
f
= 0 and m
0
= m
0
out
: Equations 9 becomes
T
0
=
2
4
0
@
X
j
Q
0
htj
+
X
out
h
0out
m
0
out
� um
0
1
A
1
m
�
RT
V
V
0
3
5
�
@u
@T
: (52)
iv) Intake stroke: During this period, m
0
out
= 0, m
0
f
= 0, m
0
fb
= 0 and m
0
= m
0
in
: Equations 9 and 14
become
T
0
=
2
4
0
@
X
j
Q
0
htj
+
X
in
h
oin
m
0
in
� um
0
1
A
1
m
�
RT
V
V
0
�
@u
@F
F
0
3
5
�
@u
@T
: (53)
F
0
= �
�
1 + f
s
F
m
�
[Fm
0
in
] (54)
9
2.6.2 Inlet Manifold
The air ow is supplied, hence the equivalence ratio is zero, so F
0
= 0. In addition m
0
fb
= 0, Q
0
ht
= 0,
V
0
= 0 and m
0
= m
0
in
�m
0
out
: Equation 9 becomes
T
0
=
"
X
in
h
oin
m
0
in
�
X
iout
h
0out
m
0
out
� um
0
!
1
m
#
�
@u
@T
: (55)
2.6.3 Exhaust manifold
Here, m
0
fb
= 0, V
0
= 0 and m
0
= m
0
in
�m
0
out
: Equations 9 and 14 become
T
0
=
2
4
0
@
X
j
Q
0
htj
+
X
in
h
oin
m
0
in
�
X
out
h
0out
m
0
out
� um
0
1
A
1
m
�
@u
@F
F
0
3
5
�
@u
@T
: (56)
F
0
= �
�
1 + f
s
F
m
�
[Fm
0
] (57)
2.7 Engine dynamic model
Figure 4 shows a model of an engine coupled to a dynamometer. The model is discussed in greater detail
in [26, 19].
J
J
1
S
D
Engine Coupling Dyn.
I
�
1
P
T
Lj
II
�
T
b
Figure 4: An engine coupled to a dynamometer
The following two equations, derived from Lagrangian principles, describe the dynamics of the system:
T
i
�
5
X
k=1
T
fk
� T
S
� T
D
� T
r
= J(�)
�
� +
1
2
@J(�)
@�
_
�
2
; (58)
T
D
+ T
S
�
N
X
j=1
T
Lj
= J
1
�
�
1
: (59)
The indicated engine torque, T
i
, is generated by the conversion of energy from chemical to thermal
to mechanical, during the combustion process. The relationship between the indicated gas pressure, P
i
,
10
and the indicated torque, T
i
, is deterministic and is a function of engine geometry. This relationship is
given by
T
i
= (P
i
� P
atm
)ArG(�); (60)
where
G(�) =
sin(� + �)
cos�
= sin � +
r
1� �
�
cos �; (61)
� = 1�
�
� + r sin(� � �)
L
�
2
; (62)
where � is the crank shaft angular position, � is the angle of connecting rod, r is the crank radius (equal
to half of the stroke), L is the connecting rod length, � is the piston pin o�set, and � is the connecting
rod angle when the piston is at the top dead centre position.
From the piston-crank geometry, the piston displacement, y, is given by
y =
p
(r + L)
2
� �
2
� [L cos� + r cos(� � �)]; (63)
where the angles � and � are given by
� = sin
�1
�
r + L
and � = sin
�1
� + r sin(� � �)
L
: (64)
The friction torque terms include piston assembly friction, T
f1
; bearing friction torque, T
f2
; valve train
friction torque, T
f3
; pumping losses torque, T
f4
; and pumps losses torque, T
f5
[20]. These are subtracted
from the instantaneous indicated torque value to produce the brake torque at the shaft. Subsequently,
the resistance torque,
P
n
j=1
T
Lj
, which is the result of external loading imposed on the engine by the
dynamometer, is subtracted from the brake torque and the net value is passed on to the engine dynamic
model.
The reciprocating torque, T
r
, is produced due to the motion of the piston assembly and the small end
of the connecting rod and is given by:
T
r
=MrG(�)�y =MrG(�)[G
1
(�)
_
�
2
+G
2
(�)
�
�]; (65)
where G
1
(�) and G
2
(�) are functions of the engine geometry:
G
1
(�) = r
"
cos(� � �)
"
1 +
�
r
L
�
cos(� � �)
�
3
2
#
�
r
1� �
�
sin(� � �)
#
; (66)
G
2
(�) = r
"
sin(� � �) +
r
1� �
�
cos(� � �)
#
; (67)
M is the mass of piston, rings, pin and the small end of the connecting rod, and �y is the acceleration of
the reciprocating components.
The reciprocating mechanisms have a variable inertia function due their changing geometry through
a crank revolution. This is due to the piston and connecting rod masses changing their position relative
to the crankshaft axis and hence changing the e�ective inertia about this axis. The same path is followed
by these parts during each revolution of the crankshaft and thus the inertia is a smoothly varying as a
periodic function. A variable inertia (with respect to the crankshaft rotational axis) as a function of the
crankshaft position is used. For a given crank-angle, the inertia with respect to the centre line of the
crankshaft is de�ned by considering the equivalent inertia of the piston, connecting rod, and crankshaft
assembly [27, 28, 29, 30]. In this study the piston pin o�set is taken into consideration during the analysis
of the variable inertia.
The crank-angle-varying inertia of an engine crankshaft assembly is given by:
J(�) = J
c
+m
c
(gr)
2
+ J
R
�
�
r
L
�
2
1
�
cos
2
�
�
+m
p
r
2
r
1� �
�
cos � + sin �
!
2
+m
R
r
2
(1� })
2
cos
2
� +m
R
r
2
}
r
1� �
�
cos � + sin �
!
2
(68)
11
and
@J(�)
@�
= 2J
R
�
�
r
L
�
3
p
1� �
�
2
cos
3
� �
�
r
L
�
2
1
�
cos � sin �
�
� 2m
R
(1� })
2
r
2
cos � sin �
�m
R
r
2
(}
r
1� �
�
cos � � sin �)
"
�
�
r}
L
p
�
3
�
cos
2
� � cos � + }
r
1� �
�
sin �
#
�m
p
r
2
(
r
1� �
�
cos � + sin �)
"
�
r
L
p
�
3
�
cos
2
� � cos � +
r
1� �
�
sin �
#
(69)
where J
c
is the moment of inertia of crankshaft, J
R
is the moment of inertia of connecting rod, m
c
is the
mass of crankshaft, m
R
is the mass of connecting rod, m
p
is the mass of piston, } is the length ratio of
CB to CA, and g is the length ratio of ED to DC as shown in Figure 1.
The torsional sti�ness torque, T
S
, and damping torque, T
D
, at the coupling between the engine and
the dynamometer, are given by
T
S
= S(� � �
1
); (70)
and
T
D
= D(
_
� �
_
�
1
); (71)
where S and D are sti�ness and damping coe�cients respectively of the coupling between the engine and
dynamometer.
To develop an engine dynamic model in the crank angle domain, consider the crankshaft angle � as
an independent variable. Then the independent variable transformation can be obtained using the chain
rule as follows:
_
� =
d�
dt
= !(�) (72)
�
� =
d!
dt
d�
d�
= !
d!
d�
(73)
_
�
1
=
d�
1
dt
=
d�
1
dt
d�
d�
=
d�
1
d�
! = !
1
(�) (74)
�
�
1
=
d!
1
dt
d�
d�
= !
d!
1
d�
(75)
From Equation 74, the relationship between the dynamometer angular position �
1
and the independent
variable � is given by:
Z
�
1
0
d�
1
=
Z
�
0
!
1
(�)
!(�)
d� ) �
1
=
Z
�
0
!
1
(�)
!(�)
d� (76)
By substituting Equations 72 to 76 into Equations 58 to 71 as well as friction components equations,
then the engine dynamic model is obtained in the crank-angle domain.
2.8 Cylinder Volume and Area
From the piston-crank mechanism, the cylinder volume is
V =
V
d
c� 1
+
�d
2
4
h
p
(r + L)
2
� �
2
�
�
p
L
2
� (� + r sin(� � �))
2
+ r cos(� � �)
�i
: (77)
The cylinder heat transfer area is represented by
A = �
�d
2
4
+ �d
h
p
(r + L)
2
� �
2
�
�
p
L
2
� (� + r sin(� � �))
2
+ r cos(� � �)
�i
: (78)
12
Where � > 2 for a non- at piston and cylinder head, and � = 2 for a at piston top and cylinder head
bottom. The derivative of Equation 77 is
V
0
=
�d
2
4
r
"
sin(� � �) +
cos(� � �)(� + r sin(� � �))
p
L
2
� (� + r sin(� � �))
2
#
: (79)
3 Model implementation
The model is implemented using MATLAB/SIMULINK [17, 18]. The main advantage of SIMULINK is
its capability for representing the entire engine model by an assemblage of interconnected blocks. Also
it has eight variable-step solvers and six �xed-step solvers for the integration of di�erential equations,
and hence the most suitable integration method can be chosen. Input design parameters are passed on
to the blocks from an input �le, but all of the operating parameters come from the block (functions) for
the other components of the system. Fourth order Runge-Kutta numerical integration is used to solve
the di�erential equations for the detailed model. The integration step for the program is chosen to be
one crank-angle degree. The indicated torque from the cylinder is updated at each step, and then used
in the non-linear dynamic model to update the instantaneous crankshaft angular velocity. The current
simulation program is executed on a (400 MHZ) Pentium II PC, and the simulation takes about 8 seconds
for a 720 degree engine cycle at an engine speed of 950 rpm.
4 Model behaviour and validation
In order to validate the behaviour of the engine model with experimental results, simulations were per-
formed for a F1L 210 DEUTZ MAG DI single cylinder diesel engine. The geometrical speci�cations for
this Engine are shown in Table 1.
Bore (d) 95 [mm]
Stroke 95 [mm]
Crank radius (r) 47.5 [mm]
Connecting rod length (L) 160 [mm]
Displacement (V
d
) 673 [cm
3
]
Compression ratio (c) 17
Piston pin o�set (�) 1.65 [mm]
Mass of piston and rings (M) 1.18 [kg]
Mass of piston (m
p
) .98 [kg]
Mass of connecting rod (m
R
) .65 [kg]
Injection timing (BTDC) 23 [deg]
Intake valve opens (BTDC) 21 [deg]
Intake valve close (ABDC) 62 [deg]
Exhaust valve opens (BBDC) 62 [deg]
Exhaust valve close (ATDC) 21 [deg]
Crankshaft moment of inertia (J
c
) .3825 [kgm
2
]
Dynamometer moment of inertia (J
1
) .37 [kgm
2
]
Table 1: Engine geometrical speci�cations
A comparison between predicted and measured values of indicated pressure and crankshaft instanta-
neous angular velocity during engine transient behaviour is illustrated in Figure 5 and Figure 6 respec-
tively. The measured values are taken from [31]. Figure 5 shows a comparison between the predicted
and measured cylinder gas pressure with �ring at an average engine speed of 920 rpm and 25 mm
3
of
fuel per cycle. The overall agreement between the measured and predicted curves is good. Owing to the
measured data of the gas pressure from the engine without an exhaust manifold, the gas pressure during
the exhaust stroke is overestimated. The slight underestimation during the combustion stroke may be
13
due to the fact that a single zone combustion model is used here; a slight di�erence in engine timing
between the practical and simulated one and a minor change between the real heat transfer, heat release
and the simulated one.
The slight underestimation of the gas pressure during combustion has resulted in an underestimation
of the instantaneous engine speed near TDC. Also the underestimation of instantaneous engine speed
during the exhaust stroke is due to the gas pressure overestimation as shown in Figure 6. In addition,
there are another reasons like, inaccuracies in the values of the engine model parameters and changes in
friction behaviour. The overall agreement between the measured and predicted curves is good.
The normalized rate of heat release is shown in Figure 7. The high left peak in the curve represents
the premixed phase. Combustion of fuel which has mixed with air to within the ammability limits
during the ignition delay period occurs rapidly in a few crank angle degrees. When this burning mixture
is added to the fuel it burns during this phase, and a high heat release rate is characteristic of this phase.
The low right peak is the di�usion phase; once the fuel and air premixed during the ignition delay have
been consumed, the burning rate is controlled by the mixture available for burning. The heat released
decreases as this phase progresses as shown in Figure 7.
The instantaneous torque produced by the engine at 40 mm
3
of fuel per cycle for four engine cycles is
shown in Figure 8. The maximum torque value represents the maximum pressure in the cylinder during
the combustion stroke as illustrated in the same �gure. As a consequence of the uctuations in engine
torque during the cycle, variations of the instantaneous crankshaft rotational speed occur.
The convection and radiation heat transfer from the cylinder is shown in Figure 9. In the vicinity
of ignition at top dead centre, the magnitude of the heat transfer is high. This is due to a high gas
temperature inside the combustion chamber at that moment, as shown in Figure 10.
Figure 11 shows a pressure-volume diagram. The compression curve behaviour is similar to the
expansion curve. Since both the compression of the unburnt mixture prior to combustion and expansion
of the burnt gases following the end of combustion are close to an adiabatic isentropic process, the
observed behaviour is as expected; the compression and expansion processes �t well with the polytropic
relation PV
1:3
=constant.
The inlet ow using the one dimensional compressible ow equation is shown in Figure 12. The rate
of the equivalence ratio is presented in Figure 13. Since air exists only during the compression stroke,
there is a zero or small equivalence ratio. Then the injection starts suddenly, and a constant rate of fuel
is injected at a constant equivalence ratio rate. After the completion of the injection the equivalence ratio
become constant with a zero rate of change.
The exhaust gas temperature is shown in Figure 14. Initially, the manifold temperature is from the
previous cycle. Then when the gas enters in from the combustion chamber, heat transfer by convection
and radiation takes place leading to a decrease in the exhaust gas temperature.
The rate of engine inertia is shown in Figure 15.
@J(�)
@�
is a bounded periodic function with an average
value of zero. Therefore, although there are periods during crank rotation when the velocity term has
a positive damping e�ect and other periods where it represents negative damping, the net e�ect for a
complete rotation is zero. The physical reason for this term is that, as the inertia decreases and less work
is required for motion, it appears as if energy is added to the system. Similarly, as the inertia increases
and more work is required, energy appears to be removed from the system. In the same �gure, the inertia
of the crank assembly is not constant, but it is a function of crank-angle. Due to the attachment of the
big end of the connecting rod with the crankshaft, the maximum crank assembly inertia is near
�
2
and
3�
2
of the crankshaft angular position and the minimum is near 0 and �.
5 Conclusions
This paper presents a non-linear thermodynamic and dynamic based model for single cylinder diesel
engine. The model is used to predict the in-cycle variation of engine parameters for both transient
and steady-state operating conditions. The model has been implemented in SIMULINK. Validation has
been performed for a F1L 210 DEUTZ MAG DI single cylinder diesel engine. Predicted pro�les of the
cylinder gas pressure and the instantaneous crankshaft angular velocity through the transient are in
good agreement with measurements. The crank-angle-varying inertia with respect to the centre line of
the crankshaft is taken in consideration. The model also includes consideration of the piston pin o�set.
14
A dynamic dynamometer model is also included, which enables a variety of engine tests to be carried-
out. In addition, the model is based in the crankshaft angle domain. This type of model is a better
candidate for model-based diagnostics since oscillatory engine states are intrinsic in the model and can
be related to engine failures. Also, inclusion of the inertia variation e�ects, especially at high engine
speeds, signi�cantly decrease the occurrence of false diagnostic alarms due to more accurate pressure
estimation. Future advances in microprocessor speed will allow the use of such those non-linear models
for estimation and control in real time.
The model has been developed with the aim of investigating di�erent strategies for transient fuel
control. The model is being extended for multicylinder engines. Work is also ongoing to include modelling
of engine operation from cold start.
Acknowledgments
The authors would like to thank Professor M. Yianneskis for his suggestions and help. This research is
mainly �nanced by the Hashemite Kingdom of Jordan by Mu'tah University. The �rst author is most
grateful to Professor E. Dahiyat for his encouragement and help.
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16
150 200 250 300 350 400 450 500 550 600 6500
1
2
3
4
5
6
7x 10
6
Crankshaft angle [deg]
Cyl
inde
r pr
essu
re [
Pa]
Figure 5: Comparison between predicted (|{) and measured (- - -) cylinder pressure under �ring
0 100 200 300 400 500 600 700
880
900
920
940
960
980
1000
Crankshaft angle [deg]
Eng
ine
spee
d [r
pm]
Figure 6: Comparison between predicted (|{) and measured (- - -) instantaneous crankshaft angular
velocity under �ring
17
300 320 340 360 380 400 4200
5
10
15
20
25
30
35
40
Crankshaft angle [deg]
Rat
e of
hea
t rel
ease
[1/
Ca]
Figure 7: Normalized of heat release at fuel per cycle 25 mm
3
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
x 106
Crankshaft cycle number
Cyl
inde
r pr
essu
re [
Pa]
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4−500
0
500
1000
Crankshaft cycle number
Indi
cate
d to
rque
[N
m]
(b)
Figure 8: (a) Cylinder pressure at fuel per cycle 40 mm
3
, (b) Indicated pressure at fuel per cycle 40 mm
3
18
340 350 360 370 380 390 400 410 4200
1
2
3
4
5
6
7
8
9x 10
4
Crankshaft angle [deg]
Hea
t tra
nsfe
r ra
te [
W]
Figure 9: Wall heat transfer rate as a function of the crankshaft angle at fuel per cycle 40 mm
3
350 360 370 380 390 400 410600
700
800
900
1000
1100
1200
1300
1400
1500
Crankshaft angle [deg]
Cyl
inde
r te
mpe
ratu
re [
K]
Figure 10: Gas temperature pro�le as a function of the crankshaft angle at fuel per cycle 40 mm
3
19
0 1 2 3 4 5
x 10−4
0
2
4
6
8
10
12x 10
6
Cylinder volume [m3]
Cyl
inde
r pr
essu
re [P
a]
Figure 11: Pressure-volume diagram
20 40 60 80 100 120 140 1600
5
10
15
20
25
30
35
40
45
50
Crankshaft angle [deg]
Inle
t mas
s flo
w [
g/C
a]
Figure 12: Flow rate through the inlet valve
20
335 340 345 350 355 360 365 3700
1
2
3
4
5
6
Crankshaft angle [deg]
Rat
e of
equ
ival
ence
rat
io [
1/C
a]
Figure 13: Rate of the equivalence ratio for the mixture inside the cylinder
480 500 520 540 560 580 600300
400
500
600
700
800
900
Crankshaft angle [deg]
Exh
aust
man
ifold
tem
pera
ture
[K
]
Figure 14: Temperature pro�le of the exhaust gas
21
0 50 100 150 200 250 300 350
0.3835
0.384
0.3845
0.385
0.3855
0.386
0.3865
Crankshaft angle [deg]
Iner
tia o
f cra
nk A
s. [
kg m
2 ]
(a)
0 50 100 150 200 250 300 350
−3
−2
−1
0
1
2
3
x 10−3
Crankshaft angle [deg]
Rat
e of
iner
tia [
kg m
2 /rad
]
(b)
Figure 15: (a) Inertia of the crankshaft assembly, (b) Rate of the inertia for the crankshaft assembly
22
