One dimensional modelling
of an internal combustion engine.
Clive Lewis 680113-2157
[email protected]
Mekanik fördjupning 10p
5C1010
2007-06-08
Preface ................................................................................................................... 3
Abbreviations, symbols and values ....................................................................... 4
Model Dimensions: ............................................................................................... 4
1. Background ....................................................................................................... 5
2. Discription of model.......................................................................................... 6
3. Sub-models ........................................................................................................ 7
3.1 Piston velocity ......................................................................................... 7
3.2 Intake conditions...................................................................................... 8
3.2.1 Incompressible flow ......................................................................... 8
3.2.2 Compressible flow ............................................................................ 9
3.3 Exhaust flow .......................................................................................... 11
3.4 Flow passed the mouthpeice .................................................................. 12
4. Comparison of compressible and incompressible flow .................................. 13
5. Conclusions ..................................................................................................... 14
6. References ....................................................................................................... 14
7. Appendixes ...................................................................................................... 15
7.1. Matlab code ............................................................................................... 15
2
Preface
The story of the internal combustion engine started back in the 1870’s when the german
inventor Niklaus Otto (1832-1891) came upon the idea of an engine which worked by
converting the chemical energy in a fuel to mechanical power via combustion which took
place inside the engine. In 1876 Otto invented the four cycle engine which became known
worldwide as the “Otto-cycle” motor. Eight years later in 1884 the magneto was built which
meant that the source of ignition could be transported with the engine thus making the internal
combustion engine a perfect means of propulsion.
The internal combustion engine is at first glance a relatively simple machine. A mixture
of air and fuel is ignited in the combustion chamber and the following expansion of the gas
pushes the piston down creating torque. Even though the engine has been in use for more 230
years nobody yet fully understands what happens during the combustion process making the
internal combustion engine in reality a highly complex system, which leads to problems when
modelling. To overcome these difficulties a number of problems need to be simplified, for
example it is assumed that there is no roughness on the walls of the inlet and inlet manifold
which can lead to changes in the flow due to friction losses.
Flow can change backwards and forwards between incompressible and compressible
under its time in the engine. The flow around the valves are at much higher velocities which
lead to compressible flow, this changes the character of the gas. At high flow velocity the
Mach number can be as high as unity which means choking, at this point the engine has
reached maximum speed.
Matematical methods can find the limits of engine speed and the surronding conditions,
several of these are shown in the included graphs. To conclude the rapport a short section is
included which compares the results of the different metods used to find the conditions during
compressible and incompressible flow.
I am greatful for the help I reccieved from both Tony Burden and Arne Karlsson from
the Institute for mechanical engineering, KTH.
3
Abbreviations, symbols and values
BDC
TDC
i.c.e.
S.I.
C.I.
rpm
ρ
B
di
dim
div
dev
de
dem
S
c
Le
Li
Tatm
Patm
θ
vp
M
vk
Bottom dead center
Top dead center
Internal combustion engine
Spark ignition
Combustion ignition
Revolutions per minute
Density
Bore
Diameter of inlet
Diameter of inlet manifold
Diameter of inlet valve
Diameter of exhaust valve
Diameter of exhaust
Diameter of exhaust manifold
Length of stroke
Length of con-rod
Length of exhaust
Length of inlet
Atmosphere temperature
Atmosphere pressure
Crank angle
piston velocity
mach number
kinematic velocity
Model Dimensions:
Bore
Stroke
Inlet Ø
Inlet port Ø
Exhaust port Ø
Exhaust Ø
Conrod length
Inlet length
Exhaust length
84*10-3 m
90*10-3 m
54*10-3 m
27*10-3 m
27*10-3 m
54*10-3 m
0.250 m
0.10 m
2.00 m
B
S
di
dip
dep
de
c
Li
Le
General values used:
Density of air
Temp of air
Atmospheric pressure
Gamma (ideal gas)
Gas constant
ρatm
Tatm
Patm
γ
R
1.2 kg*m-3
293 K
101*103 Pa
1.4
287
4
1. Background
The aim of this project is to build a one dimensional model of an i.c.e. so that it is
possible to calculate the speed, density, pressure and temperature of the flow in various places
in the engine. One dimensional flow is defined as flow which has small changes in the
conditions normal to the streamlines compared to changes along the streamlines. Because the
shapes and sizes of the passages change, the air or air/fuel mixture will have different
velocities so therefore one must decide if the flow is compressible or incompressible.
Incompressible flow is defined as where there is no change in the mass flow rate through a
reference volume which lies across the streamlines. An analys of the conditions around the
inlet valve, with the help of simple mathematical models, give us the conditions when the
flow changes from incompressible to compressible.
Concerning the engine there are two different Otto-cycle engines, one has spark-ignited
combustion (S.I.) and the other has compression ignition (C.I.), these are commonly refered to
as petrol engines and diesel engines. The petrol engine has the fuel and air premixed before
entering the combustion chamber whereas the diesel engine sprays in the fuel directly before
ignition. Because the C.I. engine uses pressure in the ignition process the compression ratio of
a diesel engine is much higher than that of a petrol engine.
The engine draws air from the atmosphere through an inlet followed by an inlet
manifold which depending upon the number of cylinders in the engine, will have a certain
number of sub-inlet ducts (Fig.1). The air, or in the case of the petrol engine, air/fuel mixture
then passes the inlet valve and into the combustion chamber. The air/fuel mixture is first
compressed with a compression stroke, when the piston is at TDC and maximum compression
is reached a spark ignites the mixture and the rapid expansion of the mixture/gas forces the
piston down in the power stroke. When maximum compression is reached in a diesel engine
the higher compression conditions force the mixture to self-ignite thus leading to the
expansion or power stroke. On the power or expansion stroke the piston travels down
increasing the volume of the cylinder and the pressure lowers. When the piston reaches BDC
the exhaust valve opens and the piston pushes out the burnt air/fuel mixture. After the exhaust
stroke follows the intake stroke in which the piston draws fresh mixture into the cylinder past
the inlet valve as it travels down towards BDC. Now the engine is back at the start of the
compresion stroke when both the inlet and exhaust valves are closed.
If it is assumed that the engine is running at a constant rpm then the mass flow rate from
the atmosphere into the intake will also be constant. When the mixture of air and fuel nears
the inlet valve it will start to acclerate and become compressible, it can be shown that the flow
is compressible even with low engine speeds. What happens in the moment of combustion
will be omitted due to the complexity of the process, the aim of this project is to look at the
conditions surronding the flow. The journey out of the cylinder, past the exhaust valve,
through the exhaust and into the atmosphere is princibly the same as the intake procedue.
5
2. Discription of model
The model which is used to calculate the conditions in the engine system is represented
by Fig. 1.
Fig 1. Model of the engine
Air taken from the surronding atmosphere is drawn into the inlet through a mouthpiece. The
lack of carburetor, turbocharger or control system contributes to the simplification of the
model. After the inlet a four branch manifold feeds each of four cylinders, the mixture is then
drawn past one inlet valve per cylinder and into the combustion chamber. Modern engines can
have several inlet and exhaust valves, again to simplify the model only one inlet and one
exhaust valve have been included. The burnt mixture is then pushed out through the exhaust
manifold, down the length of the exhaust and finaly released into the atmosphere. All forms of
exhaust gas control have also been omitted, the model needs to be as simple as possible to
avoid complex calculations.
The model can now be divided into four sub-models to be able to be analysed correctly.
The first sub-model is used to calculate the piston speed when the engine rpm is constant.
Second is the sub-model which includes the intake, intake manifold and the inlet valve, here
the conditions are calculated for the flow into the cylinder. The third sub-model looks at the
exhaust side of the engine system, meaning the flow past the exhaust valve. Sub-model four
looks briefly at the effects the shape of the mouth of the inlet can have on the flow.
Approximations in this model as a whole include the following assumptions.
It is assumed that the walls of all pipes and ducts are smooth so as to eliminate losses due to
friction. Because the optimal fuel air mixture calculates to a 5% mass content of fuel the
mixture is treated as an ideal gas, this allows the use of the general gas law for an ideal gas.
6
3. Sub-models
3.1 Piston velocity
To calculate the piston velocity it is neccessery to observe the piston in the
cylinder from the side and form some geometric equations (Fig. 2.). The speed of the piston is
the deritive of position by time.
2
S
S
⎞
⎛
y (θ ) = * cos θ + ⎜ c 2 − * sin θ ⎟
2
2
⎠
⎝
⎡
⎢
S
S * cos θ
&y (θ ) = * sin θ ⎢− 1 −
⎢
2
S 2 * sin 2 θ
2
2 * Lc −
⎢
4
⎣
v p _ max ⇒ θ =
θ& =
π
2
⇒ v p _ max =
⎤
⎥
⎥θ&
⎥
⎥
⎦
S &
*θ
2
rpm * π
S * π * rpm
⇒ v p _ max =
30
60
Fig. 2 Side view of piston
7
[1]
3.2 Intake conditions
3.2.1
Incompressible flow
Once the maximum speed of the piston has been found as a function of rpm it is
possible to use the mass conservation law for incompressible fluids [2] to calculate the mach
number for the flow as it passes the inlet valve. To simplify the calculations the passage past
the inlet valve will be modelled as a Venturi tube (Fig. 3). Equation [3] leads to the flow
velocity past the valve ( uip) as a function of rpm.
u ip _ max * Aip = v p _ max * A p
u ip _ max
⎛ B
= v p _ max * ⎜
⎜d
⎝ ip
⎞
⎟
⎟
⎠
[2]
2
[3]
Fig. 3 Inlet port
If the limit of incompressible flow is set at a mach number of M=0.1 then the corresponding
engine speed can be computed.
u
M =
a
where a is sonic velocity for air (343 m/s for ideal gas at 293 K). Eq. [1], [2] & [3] give
M * a * 60 ⎛ d ip
* ⎜⎜
rpm =
S *π
⎝ B
⎞
⎟⎟
⎠
2
rpm = 752
This shows that the flow past the inlet valve must be treated as compressible even at low
engine speeds.
8
[4]
3.2.2
Compressible flow
To calculate the flow conditions for the compressible flow at higher engine speeds it is
neccesary to start from the general gas law for an ideal gas.
P *V = m * R * T
[5]
To calculate the mass of gas which is drawn into the cylinder (min) with each intake stroke it
is assumed that the combustion chamber has atmosphere pressure and the gas is at atmosphere
temperature upon entering the cylinder. The volume of the cylinder to be filled Vc will be
π * B2 * S
[6]
Vc =
4
and the mass of the gas drawn in is given with [5] & [6]
Patm * π * B 2 * S
min =
= 5.99 *10 − 4 kg
[7]
4 * R * Tatm
The mass flow rate is found by dividing by the time the valve is open
m
[8]
m& in = in
t open
which can be calculated by the following
−1
t open
30
⎛ rpm ⎞
= (180 ) = ⎜
sec
⎟ * 0.5 =
rpm
⎝ 60 ⎠
0
[9]
putting [7] & [9] into [8]
m& in =
5.99 *10 −4 * rpm
= 2.00 *10 −5 * rpm kg * s −1
30
[10]
The continuity equation can now be writen as a function of temperature, density, pressure and
mach number by seperating varibles.
m& in = ρ * u ip * Aip = ρ * M * a * Aip = ρ * M * γ * R * T * Aip =
=
T
ρ
* ρ0 * M * γ * R *
* T0 * Aip
T0
ρ0
ρ
Q m& in = ρ 0 * γ * R * T0 * Aip *
ρ0
⎛T
* ⎜⎜
⎝ T0
1
⎞2
⎟⎟ * M
⎠
With the insertion of density and temerature ratios in [10]
−1
ρ ⎡ 1
⎤ γ −1 T ⎡ 1
⎤
= ⎢1 + (γ − 1)M 2 ⎥ &
= ⎢1 + (γ − 1)M 2 ⎥
T0 ⎣ 2
ρ0 ⎣ 2
⎦
⎦
−1
−1 1
−
2
⎡ 1
⎤ γ −1
m& in = ρ 0 * γ * R * T0 * Aip * ⎢1 + (γ − 1)M 2 ⎥
⎣ 2
⎦
[10] and [11] become
[
8.484 *10 −5 * rpm = M * 1 + 0.2M 2
]
−3
The final equation [12] can now be plotted with help of MATLAB[1] (Fig.4).
9
[11]
[12]
rpm as a function of mach number
10000
9000
8000
7000
rpm
6000
5000
4000
3000
2000
1000
0
0
0.5
1
1.5
Mach number
2
2.5
3
Fig.4 rpm as function of mach number
Fig. 4 shows how the engine speed increases as the mach number increases up to about
7000rpm, this is to be expected, a top engine speed of 7000 rpm is also quiet realistic. As the
mach number rises over 1 the engine speed decreases, here it is apparent that the model stops
working and values over mach number 1 can be ignored. Using the pressure and density ratios
it is possible to calculate the coniditions in the inlet port.
−1
⎡ 1
⎤ γ −1
ρ ip = ρ 0 ⎢1 + (γ − 1)M 2 ⎥
⎣ 2
⎦
⎤
⎡ 1
Tip = T0 ⎢1 + (γ − 1)M 2 ⎥
⎣ 2
⎦
−1
The results can be seen in Fig. 5
Values for the velocity in the inlet can be calculated with the help of computed temperature,
density and velocity in the inlet port.
Assumed engine speed = 2500 rpm
Matlab gives
Tip = 290.2 K
ρip = 1.174 kg/m3
aip = γ * R * Tip = 341.47 m/s
uip = 0.22*341.47 = 75 m/s
10
M = 0.22
density as a function of mach number
density [kg/m3]
2
1.5
1
0.5
0
0
0.1
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
mach number
temp as a function of mach number
0.8
0.9
1
0.8
0.9
1
temp [K]
300
250
200
0.3
0.4
0.5
0.6
mach number
0.7
Fig. 5 Density and temperature as a function of mach number
Mass conservation equation for compressible fluid
ρ ip * u ip * Aip = ρ i * u i * Ai
2
ρ ip
ui =
ρi
⎛ d ip ⎞
⎟⎟ * u ip
* ⎜⎜
⎝ di ⎠
Density in the inlet is assumed to be ρatm , which gives
ui = 18.3 m/s
[13]
3.3 Exhaust flow
The principle idea for calculation of the coniditions surronding the flow out of the
engine are the same as the intake with the exceptions of the temperature of the gas and the
size of the exhaust valve or port which in reality is always larger than the inlet.
From Heywoods book “Internal combustion engine fundamentals”[2]
Tex = 500 K
gives
a = 448 m/s
Assume that he pressure in the cylinder Pex = Patm at BDC before exhaust stroke starts
P *π * B 2 * S
mout = atm
= 3.5104 *10 − 4 kg
4 * R * Tex
3.5104 *10 −4 * rpm
= 1.1701 *10 −5 * rpm
30
with the same metod as for intake gives the following
m& out =
[
]
−3
1.1701 *10 −5 * rpm = M * 1 + 0.2M 2
This produces Fig. 6 which shows the engine speed as a function of mach number
11
[14]
[15]
rpm as a function of mach number (exhaust)
15000
rpm
10000
5000
0
0
0.5
1
1.5
mach number
2
2.5
3
Fig. 6 rpm as a function of mach number for exhaust flow
Fig. 6 shows how a much higher engine speed is possible before choking starts to affect
engine performance. It is obvious that the inlet side steers the limits of the engine, the larger
valves on the exhaust side give more room for the gas to be pushed out thus insuring that the
combustion chamber is always emptied before the intake stroke begins.
3.4 Flow passed the mouthpeice
There are many factors which affect the flow, most of which have been neglected in the
model. Roughness on the walls create friction losses, so called “wall wetting” which means
fuel molecules attach themselves to the sides of the ducts and swirling where the flow
becomes tubulent are all factors which could contribute to an enormously complex model.
Instead of looking at all of the above only the fact that the entrance to the inlet is “bellmouth”
shaped (see Fig. 7) will be analysed. From the book Blevins- Applied fluid dynamics
handbook[3].
Patm − Pi =
1
f *l ⎞
2⎛
⎟
* ρ atm * u i ⎜⎜1 + k +
2
d i ⎟⎠
⎝
[16]
f * l 64 * l * v k
=
[17]
2
di
ui * d i
u *d
r
64 * l
assuming
Re = i i [20]
= 0.05 ⇒ k = 0.4 [18]
f *l =
[19]
di
Re
vk
putting [20] into [19] then [19] into [17] then [17] & [18] into [16]
Pi = 1.007 * 10 5
The pressure lost due to the geometry of the mouthpiece is thus calculated to 300 pa which is
negligible compared to the pressure of the atmosphere outside the entrance to the inlet.
12
Fig. 7 Bellmouth entrance to the inlet
4. Comparison of compressible and incompressible flow
It has been shown that even for low engine speeds the flow passed the inlet valve must
be considered compressible. How do the two different models used to calculate the flow vary
in the limits of incompressible and compressible flow? Can the flow be calculated using both
models in region between 500 and 1000 rpm? Fig. 8 shows the values for both models in the
same graph
Incompressible and compressible
10000
9000
8000
7000
rpm
6000
5000
4000
3000
2000
1000
0
0
0.5
1
1.5
mach number
2
2.5
3
Fig.8 Incompressible and compressible
The striaght line of the incompressible flow model shows a slightly higher mach number for
the same rpm than the compressible flow.
13
5. Conclusions
It has been shown that with relatively simple mathematical equations based upon some
of the ground laws of physics, together with a collection of assumptions, it is possible to find
the conditions of the flow in a number of places in the models. The limits of the engine speed
are decided by the shapes and dimensions of the passages which the gas has to flow together
with the size of the piston and cylinder in which it moves. It is also intressing to note that att
maximunm engine speed the phenomen “choking” prevents the engine from going any faster,
this takes place at the the critical mach number 1. Verification of the models validity can be
seen in the maximum rpm (7000rpm), this is not unrealistic of todays high performance petrol
engines.
The constant improvments in the field of rescearch and devolpment see more and more
control units included in the engine to steer everything from the temperature of the gas
entering the cylinder to the amount of unburnt fuel which is sent back into the cylinder after
the exhaust stroke. Todays engineers are faced with challenging problems which are based on
a 230 year old invention and the modelling of such a machine demands models of a much
higher complexity than shown in this rapport.
6. References
[1]. MATLAB.: The MathWorks,inc
[2]. Heywood, J.: Internal combustion engine fundamentals McGraw-Hill 1998
[3]. Blevins.: Applied fluid dynamics handbook
[4]. Daneshyar, H.: One dimensional compressible flow Pergamon international library 1976
14
7. Appendixes
7.1. Matlab code
%A program to calculate and show the curves for compressible and
%incompressible flow, temperature and density,flow for exhaust gas
%and to compare the two metods for calculating the mach number.
clear all
close all
clc
M = 0 : 0.01 : 3.0;
%a vector which represents all the mach numbers
rpm = (1/(8.484*10^-5))*M.*(1+0.2*M.^2).^-3;
dens = 1.2*(1+0.2*M.^2).^(-2.5);
temp = 293*(1+0.2*M.^2).^(-1);
rpm_ex = (1/(4.9637*10^-5))*M.*((1+0.2*M.^2).^-3);
Incomp = M*(343*60/(0.09*pi))*(27/84)^2;
figure (1)
plot(M,rpm)
axis([ 0.0 3.0 0.0 10000.0]);
title('rpm as a function of mach number');
xlabel('mach number');
ylabel('rpm');
figure (2)
subplot (2,1,1);
plot(M,dens)
axis([ 0.0 1.0 0.0 2.0]);
title('density as a function of mach number');
xlabel('mach number');
ylabel('density [kg/m^3]');
subplot (2,1,2);
plot(M,temp)
axis([ 0.0 1.0 200.0 300.0]);
title('temp as a function of mach number');
xlabel('mach number');
ylabel('temp [K]');
figure(3)
plot(M,rpm_ex)
axis([ 0.0 3.0 0.0 15000.0]);
title('rpm as a function of mach number (exhaust)');
xlabel('mach number');
ylabel('rpm');
figure (4)
plot(M,Incomp,M,rpm)
axis([ 0.0 3.0 0.0 10000.0]);
title('Incompressible and compressible');
xlabel('mach number');
ylabel('rpm');
15