Description of the process that involves the one-dimensional transient simulation of intake ducts in reciprocating SI Engines

1. End of career project summary
Politecnica de Madrid university
1D Simulation of intake manifolds in single-cylinder
reciprocating engine
Juan Manzanero*, Juan Ramo´n Arias1, A´ ngel Vela´zquez1
Abstract
In the scenario of the Motostudent championship, there was the need of the 1D gas dynamics code development, in order to
achieve enough detailed understanding of the complex processes taking part in a complete thermodynamic cycle inside a
reciprocating engine.
1D gas-dynamics simulations in intake ducts and manifolds interior was required to feed more complex 3D simulations
employing professional software such as ANSYS FLUENT. Due to equations simplicity and low computational requirements, a
parametric design was possible allowing the engine performance optimization.
For that purpose, MATLAB language was used in combination with Godunov-Roe based finite volume theory, and thus
introducing the unidimensional flow in ducts. This combined with a Wiebe law-governed cylinder, completes the whole engine
model.
Results were compared with professional 1D gas-dynamics software to check the model’s performance, concluding with an
excellent wave phenomena approach despite the theory engaged simplicity.
Keywords
Fluid dynamics — Reciprocating engines — 1D simulation — Matlab
1Applied thermo-fluid dynamics and propulsion department, Madrid.
*Corresponding author: j.manzanero@alumnos.upm.es – j.manzanero1992@gmail.com
Contents
Introduction 1
1 Methods 1
1.1 Intake runner model . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Cylinder model . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Valve model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Results and Discussion 4
2.1 Analysis of a complete cycle . . . . . . . . . . . . . . . . . 4
2.2 Comparison with GT-Power . . . . . . . . . . . . . . . . . . 4
2.3 Performance with crankshaft speed . . . . . . . . . . . . 4
2.4 Performance with runner length . . . . . . . . . . . . . . . 4
Introduction
The target of this project is the enforcement of the engine
model illustrated in figure 1. As can be seen in the mentioned
figure, the model comprises the intake runner duct, the com-bustion
chamber, a quasi-stationary valve model, and finally a
0D exhaust model.
The intake runner will be simulated using an unidimensional,
adiabatic, and frictionless model, and this yields to Euler equa-tions
of fluid dynamics.
The combustion chamber as well, will be simulated as a tank
which is feeded with air flowing through intake runner. Its
volume changes according to the piston kinematic law, and
once the valves are closed, a Wiebe heat model will be em-ployed
to include a combustion phase.
The valve model will act as a boundary condition for the in-take
runner, and thus connecing it with the cylinder.
Once the whole model is assembled, several cycles simula-tions
are performed until solution converges and does not
change from one cycle to another (with some error tolerance).
When this cyclic convergence is reached, it will be possible
to obtain all engine performance parameters such as power,
volumetric efficiency, work or fuel consumption.
1. Methods
1.1 Intake runner model
As worked out in the introduction, the intake manifold
model will be governed with the Euler equations of fluid
dynamics, which are formulated as follow:

2. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 2/8
ri(tn);ui(tn); pi(tn) ro(tn);uo(tn); po(tn)
x1 x2 xM
valve combustion chamber
r(tn;xk);u(tn;xk); p(tn;xk)
rg(tn);ug(tn); pg(tn)
rc; pc;Tc;mc;Vc(q)
pamb
ramb
pamb
ramb
Figure 1. Variables and components scheme.
¶r
¶t
+
¶ (ru)
¶ x
= 0 (1)
¶ (ru)
¶t
+
¶ (ru2+ p)
¶ x
= 0 (2)
¶E
¶t
+
¶ (u[E + p])
¶ x
= 0 (3)
due to their characteristics, they are written in conservative
form in variables ~U = fr;ru;Eg, allowing a finite volume
scheme implementation aiding to include discontinuous solu-tions
such as shock waves.
This yields to a scheme known as the Godunov scheme and
can be summarized with the following relation
~Un+1
i = ~Un
i +
Dtn
Dxi
~Fi1=2~Fi+1=2
(4)
where the intercell fluxes are computed using the approximated-state
Roe flux, obtaining the Riemann problem solution with
enough accuracy.
This method is implemented in MATLAB language, obtaining
the following function that obtains the flow solution in the
next timestep.
1 function [rhotubosig,utubosig,ptubosig]=
2 GodunovRoe(rhotuboprev,utuboprev,ptuboprev,
3 gamma,WM0,WM05,dt,dx)
4 M=length(rhotuboprev);
5 %Variable allocation
6 Flux=zeros(3,M+1);
7 tildeu=zeros(1,M-1);
8 tildeH=zeros(1,M-1);
9 tildea=zeros(1,M-1);
10 tildelambda1=zeros(1,M-1);
11 tildelambda2=zeros(1,M-1);
12 tildelambda3=zeros(1,M-1);
13 tildeK1=zeros(3,M-1);
14 tildeK2=zeros(3,M-1);
15 tildeK3=zeros(3,M-1);
16 tildealpha1=zeros(1,M-1);
17 tildealpha2=zeros(1,M-1);
18 tildealpha3=zeros(1,M-1);
19
20 %Step 1: Boundary conditions
21 Flux(:,1)=fixF(swapWU(WM0,gamma,+1)',gamma);
22 Flux(:,M+1)=fixF(swapWU(WM05,gamma,+1)',gamma);
23
24 % Roe method starts -------
25 auxFlux(:,1)=fixF(swapWU([rhotuboprev(1),
26 utuboprev(1),ptuboprev(1)],gamma,+1),gamma);
27 %Step 2: Roe intercell coefficients
28 %calculations
29 H=gamma*ptuboprev(1)/((gamma-1)*rhotuboprev(1))
30 +0.5*utuboprev(1)ˆ2;
31
32 for i=1:M-1
33 tildeu(i)=(sqrt(rhotuboprev(i))*utuboprev(i)
34 +sqrt(rhotuboprev(i+1))*utuboprev(i+1))/
35 (sqrt(rhotuboprev(i))+
36 sqrt(rhotuboprev(i+1)));
37
38 tildeH(i)=sqrt(rhotuboprev(i))*H;
39 H=gamma*ptuboprev(i+1)/((gamma-1)*
40 rhotuboprev(i+1))+0.5*utuboprev(i+1)ˆ2;
41 tildeH(i)=(tildeH(i)+sqrt(rhotuboprev(i+1))
42 *H)/ (sqrt(rhotuboprev(i))+
43 sqrt(rhotuboprev(i+1)));
44 tildea(i)=sqrt((gamma-1)*
45 (tildeH(i)-0.5*tildeu(i)ˆ2));
46
47 %Eigenvalues calculation
48 tildelambda1(i)=tildeu(i)-tildea(i);
49 tildelambda2(i)=tildeu(i);
50 tildelambda3(i)=tildeu(i)+tildea(i);
51
52 %Eigenvectors calculation
53 tildeK1(:,i)=[1;tildelambda1(i);
54 tildeH(i)-tildeu(i)*tildea(i)];
55 tildeK2(:,i)=[1;tildeu(i);
56 0.5*tildeu(i)ˆ2];
57 tildeK3(:,i)=[1;tildelambda3(i);
58 tildeH(i)+tildeu(i)*tildea(i)];
59
60 %Alpha coefficient calculation
61 Du1=rhotuboprev(i+1)-rhotuboprev(i);
62 Du2=rhotuboprev(i+1)*utuboprev(i+1)
63 -rhotuboprev(i)*utuboprev(i);
64 Du3=(0.5*utuboprev(i+1)ˆ2*
65 rhotuboprev(i+1)+ptuboprev(i+1)/

3. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 3/8
66 (gamma-1))-(0.5*utuboprev(i)ˆ2*
67 rhotuboprev(i)+ptuboprev(i)/
68 (gamma-1));
69
70 tildealpha2(i)=(gamma-1)/(tildea(i)ˆ2)*(Du1
71 *(tildeH(i)-tildeu(i)ˆ2)+
72 tildeu(i)*Du2-Du3);
73 tildealpha1(i)=1/(2*tildea(i))*(Du1*
74 (tildeu(i)+tildea(i))-
75 Du2-tildea(i)*tildealpha2(i));
76 tildealpha3(i)=Du1-(tildealpha1(i)+
77 tildealpha2(i));
78
79 %Step 3: Roe flux assembly
80 Flux(:,i+1)=0.5*auxFlux-0.5*(tildealpha1(i)
81 *abs(tildelambda1(i))*tildeK1(:,i)+
82 tildealpha2(i)*abs(tildelambda2(i))*
83 tildeK2(:,i)+
84 tildealpha3(i)*abs(tildelambda3(i))*
85 tildeK3(:,i));
86 auxFlux(:,1)=fixF(swapWU([rhotuboprev(i+1),
87 utuboprev(i+1),ptuboprev(i+1)],gamma,+1),
88 gamma);
89 Flux(:,i+1)=Flux(:,i+1)+auxFlux*0.5;
90 end
91
92 %Step 4: Performing a step in Godunov scheme
93 Wsig=Godunovcommander([rhotuboprev(1,:);
94 utuboprev(1,:);ptuboprev(1,:)],
95 Flux,dt,dx,gamma);
96 rhotubosig(1,:)=Wsig(1,:);
97 utubosig(1,:)=Wsig(2,:);
98 ptubosig(1,:)=Wsig(3,:);
99
100 end
Listing 1. Next solution calculation code according to
Godunov-Roe algorithm
1.2 Cylinder model
In order to simulate the processes taking part inside the
cylinder, a tank model will be employed, and thus, the conser-vation
equations of fluid dynamics will be applied to calculate
the changes in its variables.
rc; pc;Tc;mc;Vc(q)
Figure 2. 0D cylinder model scheme
dmc
dt
= å
i2enters
m˙ i å
i2exits
m˙ i (5)
dEc
dt
= å
i2enters
m˙ iHi å
i2exits
m˙ iHi pc
dVc
dt
+
dQ
dt
(6)
pc = rcRTc (7)
Where the heat deposition law is given by the Wiebe expres-sion:
Q(q) = QTOTAL
Z q
qs
1ea
qqs
qd
n
dq (8)
This yields to an algebraic equation using Euler method for
differential equations.
1.3 Valve model
Implementing a boundary condition for the Euler equa-tions
needs the discussion of the flow characteristics patterns,
to choose the appropriate equations in every conditions.
Depending if flow enters/exits the pipe, or if exceeds or not
the sound speed delimitates the four different cases imple-mented
in the code. This flow patterns are shown in figure 3,
in which can be seen how the duct either uses the information
contained in some characteristics or not.
l1
l3
l2
x
t
(a) Subsonic outlet
l1
l3
l2
t
x
(b) Supersonic outlet
l2
l3
l1
x
t
(c) Subsonic inlet
l2
l1 l3
x
t
(d) Supersonic inlet
Figure 3. Characteristics waves patterns. l1 = ua,
l2 = u, l3 = u+a

4. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 4/8
2. Results and Discussion
2.1 Analysis of a complete cycle
The complete simulation of a 250cc single-cylinder en-gine
yields to the following results during a complete cycle
are shown in figure 4.
In figure 4(a), the wave phenomena can be appreciated. When
the valve opens, flow enters the cylinder, and thus pressure
diminishes. Also in the first opening phases, as pressure in the
pipe exceeds pressure in cylinder, flow enters into the chamber
even if the piston goes upwards.
The pressure minimuns bounces in the pipe’s open end chang-ing
into a overpressure. That overpressure helps as well avoid-ing
the flow exiting the cylinder when the piston starts going
upwards again before the intake valve closes.
After that, when the intake valve is closed, the pipe becomes
a closed tube with the other end opened to ambient, and thus
pressure oscillates in its interior with its natural frequency
( f = a=4L).
2.2 Comparison with GT-Power
The mass flow profile obtained with this MATLAB simu-lation
was compared with the one obtained with professional
software for 1D engine calculation such as GT-Power.
As this software contemplates heat transfer inside the runners
and also friction, the mass flow curves obtained with MATLAB
exceed in values the ones obtained with GT.
2.3 Performance with crankshaft speed
The engine studied consists in a MOTO3 class one, so that
is called for high performance at high revs, maintaining a
good behavior at lows an thus, allowing a high corner exit
speed.
That requirement forces the use of a short length runner pipe
in order to take advantage of wave phenomena in a high speed
range. That assures that the wave travelled as cylinder goes
downwards has time enough to bounce in the open end and
thus returning to the valve transformed into a high pressure
pulse.
In figure 7 results are shown and from then, the engine perfor-mance
characteristics are worked out, summarised in
table 1
(a) Intake Mass flow curves
(b) Pressure curves
Figure 5. Results obtained with GT-Power
Parameter Value
Maximun volumetric efficiency 1.195 at 7600 rpm
Maximun Torque 39.6 Nm at 7600 rpm
Maximun Power 59.8 CV at 13400 rpm
Table 1. Engine performance characteristics using a 300mm
length runner.
2.4 Performance with runner length
As runner length is changed, the behavior of the engine is
switched. Longer runners lowers maximum power and its cor-responding
speed, while shorter ones allow obtaining higher
values sacrificing power at low speeds. The best solution is
for sure an intermediate state that maintains high level both at
high and low speeds.
Results are shown in figure 8 , confirming previous statements.

5. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 5/8
0.12
0.1
0.08
0.06
0.04
0.02
0
−0.02
0 90 180 270 360 450 540 630 720
3 deg
m˙ kg/s
Gasto masico de entrada al cilindro
(a) Intake Mass flow curves
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0 90 180 270 360 450 540 630 720
3 deg
p(3) bar
Presiones en el tubo
valvula punto medio exremo abierto
(b) Pressure curves
Figure 6. Results obtained with MATLAB

6. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 6/8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0 90 180 270 360 450 540 630 720
3 deg
p(3) bar
Presiones en el tubo
valvula punto medio exremo abierto
(a) Intake runner pressure distribution (referred to an atmospheric
gauge pressure)
150
100
50
0
−50
−100
−150
0 90 180 270 360 450 540 630 720
3 deg
u(3) m/s
Velocidades en el tubo
valvula punto medio exremo abierto
(b) Intake runner velocity distribution
9x 106
8
7
6
5
4
3
2
1
0
0 90 180 270 360 450 540 630 720
3 deg
pc(3) Pa
Presion en el cilindro
pc
mass in
mass out
(c) Cylinder pressure and intake/exhaust mass flow
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0 90 180 270 360 450 540 630 720
3 deg
m˙ kg/s
Gasto masico de entrada al cilindro
(d) Intake mass flow
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 90 180 270 360 450 540 630 720
3 deg
Mg
Numero de Mach en la garganta
(e) Mach number in the valve throat
Figure 4. Simulation results accross a complete thermodynamic cycle. M=50 (Number of volumes), Dq = 0:05deg, L = 0:3m
(Pipe length), n = 10:000rpm

7. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 7/8
1.3
1.2
1.1
1
0.9
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
x 104
0.8
n, rpm
2v
(a) Volumetric efficiency
40
38
36
34
32
30
0.6 0.8 1 1.2 1.4
x 104
28
n, rpm
T (Nm)
(b) Torque
60
55
50
45
40
0.6 0.8 1 1.2 1.4
x 104
35
n, rpm
, CV
˙W
(c) Power
185
184
ce, g/kWh (d) Fuel consumption
183
182
181
180
0.6 0.8 1 1.2 1.4
x 104
179
n, rpm
20
19
18
17
16
15
0.6 0.8 1 1.2 1.4
x 104
14
n, rpm
pme, bar
(e) Mean effective pressure
0.6
0.58
0.56
0.54
0.52
0.5
0.48
0.6 0.8 1 1.2 1.4
x 104
0.46
n, rpm
2i
(f) Thermodynamic efficiency (dashed line refers to Otto cycle)
Figure 7. Simulation results changing crankshaft speed. M = 20, Dq = 0:05deg, L = 300mm

8. 1D Simulation of intake manifolds in single-cylinder reciprocating engine — 8/8
1.4
1.3
1.2
1.1
1
0.9
0.8
0.6 0.8 1 1.2 1.4
x 104
0.7
n, rpm
2v
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(a) Volumetric efficiency
40
35
30
25
0.6 0.8 1 1.2 1.4
x 104
20
n, rpm
T , Nm
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(b) Torque
65
60
55
50
45
40
35
0.6 0.8 1 1.2 1.4
x 104
30
n, rpm
, CV
˙W
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(c) Power
186
185
184
183
182
181
180
0.6 0.8 1 1.2 1.4
x 104
179
n, rpm
ce (g/kWh)
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(d) Fuel consumption
20
18
16
14
0.6 0.8 1 1.2 1.4
x 104
12
n, rpm
pme, bar
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(e) Mean effective pressure
0.55
0.5
0.6 0.8 1 1.2 1.4
x 104
0.45
n, rpm
2i
L=0.2
L=0.25
L=0.3
L=0.35
L=0.4
(f) Thermodynamic efficiency (dashed line refers to Otto cycle)
Figure 8. Simulation results changing crankshaft speed and intake horn length. M = 20, Dq = 0:05deg, L = 300mm