I. Introduction of two-zone models

1.1 Introduction of two-zone models

  During last simulation of one-zone models, we have established an ideal model for a closed, non-reacting system and assume the enclosed fluid is homogenous. However this kind of system doesn’t exchange mass. So on this basis, then we developed two-zone models which contain gas-exchange models for backflow, crossflow and intake, exhaust mass flow, combustion models and heat loss models. A two-zone model is defined to split the total cylinder volume in two distinct zones: and unburnt zone and a burnt zone which separated by a flame front. This flame front propagates into the unburnt zone and thereby converts fresh mixture into burnt gasses. This process is described by a conservation law for mass, the first law of thermodynamics and the equation of state. And then, we can get a system of equations to be solved as follows:
$\underbrace{ \begin{pmatrix} 0 & {m_uC_{v,u}(T_u)} & 0& -p \\ 0 & 0 & {m_bC_{v,b}(T_b) }& p\\ V_u & -m_u(\frac{\overline{R}}{M_u}) & 0 & -p\\ V_b & 0 & -m_b(\frac{\overline{R}}{m_b} & p) \end{pmatrix}}_{\text{M}} {\begin{pmatrix} \frac{\partial p}{\partial t}\\\\ \frac{\partial T_u}{\partial t}\\\\ \frac{\partial T_b}{\partial t}\\\\ \frac{\partial V_b}{\partial t}\end{pmatrix}}=\begin{pmatrix}RH_1\\RH_2\\RH_3\\RH_4 \end{pmatrix}\\ RH_1\ =\ -e_u\frac{\partial{m_u}}{\partial t}+\frac{\partial Q_u}{\partial t}-\frac{\partial V}{\partial t}+\sum^{port}{\bigg[h_p \frac{\partial m_u}{\partial t}\bigg]}_{gas\ exch.}-\bigg[h_u \frac{\partial m_b}{\partial t} \bigg]_{reac} \\RH_2 \ =\ -e_b\frac{\partial{m_b}}{\partial t}+\frac{\partial Q_b}{\partial t}+\sum^{port}{\bigg[h_p \frac{\partial m_b}{\partial t}\bigg]}_{gas\ exch.}+\bigg[h_u \frac{\partial m_b}{\partial t} \bigg]_{reac} \\ RH_3\ = \ -p\frac{\partial V}{\partial t}+T_u \bigg(\frac{\overline R}{\widetilde M_u}\bigg)\frac{\partial m_u}{\partial t} \\RH_4 \ =\ T_b\bigg(\frac{\overline R}{\widetilde M_b}\bigg)\frac{\partial m_b}{\partial t}$M⎝⎜⎜⎜⎛​00Vu​Vb​​mu​Cv,u​(Tu​)0−mu​(Mu​R​)0​0mb​Cv,b​(Tb​)0−mb​(mb​R​​−pp−pp)​⎠⎟⎟⎟⎞​​​⎝⎜⎜⎜⎜⎜⎜⎜⎜⎛​∂t∂p​∂t∂Tu​​∂t∂Tb​​∂t∂Vb​​​⎠⎟⎟⎟⎟⎟⎟⎟⎟⎞​=⎝⎜⎜⎛​RH1​RH2​RH3​RH4​​⎠⎟⎟⎞​RH1​ = −eu​∂t∂mu​​+∂t∂Qu​​−∂t∂V​+∑port​[hp​∂t∂mu​​]gas exch.​−[hu​∂t∂mb​​]reac​RH2​ = −eb​∂t∂mb​​+∂t∂Qb​​+∑port​[hp​∂t∂mb​​]gas exch.​+[hu​∂t∂mb​​]reac​RH3​ = −p∂t∂V​+Tu​(Mu​R​)∂t∂mu​​RH4​ = Tb​(Mb​R​)∂t∂mb​​
  In order to close this system, coupled ODE’s sub-models have to be supplied. The important point is in the RHS(right hand side). Here we construct kinematic model of the crankshaft con-rod movement; combustion models; models for mass transfer through valves; heat loss models.

  In our models, combustion model is one of the most well-known phenomenological models for SI engines----the Wiebe Law. For gas exchange through valves, we apply models for isentropic ducted flow in which the flow is assumed to be steady the conservation of energy. And we can take into account the backflow and crossflow and choke. Also, the discharge coefficient can be defined to indicate a real intake or exhaust runner friction and aerodynamic losses due to the curvature of the runner occur. And we should know the effective minimum cross-sectional surface.

  After that, we could add models for knock predictions and supercharging or turbocharging. The whole structure of our system is as figure 1.

Figure 1. Structure of the whole system

1.2 Results and analysis of the basic models

  After running the main program with all the functions, we can get the results as follows:

Figure 2. Results of the two-zone model   From figure 2, we can know the pressure, temperature, volume and mass flow of unburnt, burnt gas and mixture variation with crank angle. The first iteration is the initial condition where we set the burnt gas as the initial and after that we can change it to be the real one by the result of the first calculation. And from the shape of curve, we can figure out which points are ignition, EVO, IVO, EVC, IVC. Here we can know the clapping of intake and exhaust opening and closing. Also we can know which segments are backflow and crossflow.

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Figure 3. Diagram of pressure iteration

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Figure 4. P-V diagram of the 3rd cycle

  We can expand the range of CA, that means do different quantities of iteration and we can know when it will converge. As we can see in figure 3, actually, from the second cycle our pressure has converged, which indicates our choosing 3rd cycle is reasonable.
  Finally we can plot the diagram of pressure-volume of the 3rd cycle in figure 4, it is like a typical Otto cycle in real circumstance. If we change different engine parameters, the cycle can be different.

Figure 5. Knock index in the normal condition

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Figure 6. Instantaneous torque of 4 cylinders

  In order to know the knock issue, we can know from the figure 5, where we plot the KI(knock index) with CA, so we can say that if KI value at the end of combustion is less than 1, the engine knock won’t occur else it will. In our condition here, KI is 0.2<1, so there is no knock.
  In figure 6, we can know the instantaneous torque, and total torque of our 4-cylinder engine. We can also calculate the ratio between total torque with mean torque to know our engines performance.

II. Influence of design parameters on performance

  In this part, we want to perform some tests on engine performance when changing the design parameters by using our models to know how it works.

2.1 Influence of compression ratio

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Figure 7. P-V diagram for different Compression Ratio

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Figure 8. Pressure and temperature variation in different Compression Ratios

  We do a testing loop to change the value of compression ratio from 2 to 30 bar. From figure 7 and figure 8, P-V diagram and the pressure and temperature variation with CA in typical compression ratios, we can know the trend pressure and temperature will increase with higher compression ratio. That’s because compression ratio can determine the maximum pressure in compression stroke and then maximum temperature can increase with the pressure increasing. And there will be more heat loss so the curve of expansion stroke at the same volume is lower, the end volume of the expansion stroke can be less.

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Figure 9. Engine performance variation in different Compression Ratios

  From the figure 9, we compute global engine performance indicators with different compression ratio. The cycle efficiency will increase sharply and then slowly, which indicates the sensibility of the efficiency with CR. As for KI, we can estimate if CR>11 the KI will more than 1, and there can be knock issue. Engine power and IMEP will increase slowly after CR=20, that means increasing CR when CR>20 has little influence with engine output.

2.2 Influence of spark advance

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Figure 10. Pressure and temperature variation in different Compression Ratios

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Figure 11. Efficiency & P-V diagram for different Compression Ratio

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Figure 12. Engine performance variation in different Compression Ratios

2.3 Influence of combustion duration

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Figure 13. Pressure variation w/ Combustion Duration

Figure 14. P-V diagram for different Combustion Duration

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Figure 15. Cycle efficiency related to SA   In this part, we want to know the influence of combustion duration. From the figure, with the combustion duration increasing, the Tmax, Pmax will decrease, because the shorter combustion, the higher heat release ratio, so that the maximum pressure and temperature will increase. ,  Figure 16. Engine performance variation in different Combustion Duration   As for the engine output and engine performance, the engine power and IMEP will decrease slowly and then sharply, that means the engine can have bad performance when the combustion takes too much time. And the maximum pressure will always decrease with the combustion duration increasing. The maximum temperature will first increase and then decrease, 35CAD is the best one.

  For the cycle efficiency, we plot the contour with spark advance and combustion duration in figure 17, then we can know the trend of changing the two variation. For example, if we fix the combustion duration to 10 cad, and just increase the increase the spark advance, then we know the efficiency increases first and then decreases.

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Figure 17. Contour of engine efficiency with spark advance and combustion duration

2.4 Influence of valve overlapping

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Figure 18. P-V diagram for different valve overlapping   We can observe on these graphs the influence of valve overlap on the general perfor-mances of the engine. The – domain means less overlap the + domain means more overlap.

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Figure 19. Engine performance variation in different valve overlapping   We reach peak torque at peak volumetric efficiency, around -20°. Which mean we get the best volumetric efficiency while cams profiles are fully separated. In our case, we don’t take into account the inertia of gases, and as exhaust pressure is higher than intake pressure, all the overlap time is lost time to flow fresh air into the cylinder. Positive overlap is crossflow into the intake and reduces overall efficiency as well as rated torque. Too much negative overlap will place the lift profile in a bad position during the cycle hence all major values decreases.

2.5 Influence of valve lift

  We can now observe the influence of the total valve lift on the cycle. We can see that very low lift (lower than 0.3x regular value) our engine efficiency is dropping quiet a lot. This is explained by the sonic behavior of the flow through the valves as entering and expelling air has difficulties to go past valves. We can observe this phenomenon on the pumping loop from the cycle, giving high exhaust pressure and low intake pressure. We can also see that for our engine specs, even 0.5x factor in the lift allows for good engine working properties. Cycle efficiency is the highest at the highest lift, but volumetric efficiency prefers the intermediate values ( 0.5-0.8 coefficient ). That means we get better torque at this working point, but the efficiency/fuel consumption is not the best it can be.

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Figure 20. P-V diagram for different valve lift

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Figure 21. Engine performance in different valve lift ## 2.6 Influence of addition of a super-charger

  This first study of a supercharger considers a stable Pintake at 1atm. Fresh gas loop is without intercooler. And we simulate the effect of a supercharger at different ambient pressure, as if we go up and down in altitude. As we go up in altitude, pressure drops, and we drive the super charger at a higher compression ratio to compensate. As a result, temperature out of the supercharger increases and we get less air into the cylinder (in mass).
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Figure 22. P-V diagram for different valve lift

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Figure 23. Pressure variation of the supercharger   Second test we made is with a fixed ambient pressure of 1 bar, and we modified the intake pressure linearly, still without intercooler:

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Figure 24. Engine performance in different supercharger   Without considering the engine knock, we can see that supercharger brings a significant performance increase (>50%) thanks to the higher amount of fresh gases pushed into the cylinder. On the other hand, we get a sharp increase in mechanical stress, because of higher maximum pressure in the cylinder and burned gas temperature.   Super charger allows to increase the thermodynamic efficiency of the cycle thanks to the higher intake pressure, reducing the pumping loop. But it draws energy from the crank-shaft to compress air, and we deduced this energy from the useful work delivered by the cycle. Overall trend is a decrease in the engine efficiency as the intake pressure goes up.

III. Knock predictions in SI IC-engines

3.1 Influence of compression ratio on engine knock

RPM	Cr_limit
1000	7.2
2000	9.7
3000	11.7
4000	13.5
5000	15.0

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Figure 25. The KI in different backpressure w/o intercooler   In this part we go a bit more in detail about when knock is occurring and how does it influences engine behavior. We consider here that combustion parameters are fixed (-10 ignition, 60 duration) and we look at the knock index through the operating range. We can see that the higher the rpm is the less it is prone to knock. This is due to the fact that auto ignition delay barely changes at similar loads, and higher rpm means shorter cycle dura-tion and then less time to auto-ignite.   We observe that a compression ratio of already 10 already bring knocking issue at 2000rpm. When we design the engine, we need to decide whether we put a high com-pression ratio and then needs to deteriorate the combustion to avoid knock or build an engine with lower Cr and better timed combustion. This is all about compromise. We aim for 12 in a NA engine, and closer to 10 in a boosted one.

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Figure 26. Efficiency of the ideal cycle at 4000RPM   We can see that choosing a lower Cr deteriorates the overall cycle efficiency through compression and expansion, and choosing a higher compression ratio leads to worse combustion phasing at low rpm and reduced efficiency because of that.

3.2 Influence of supercharging on engine knock occurrence

  Using different fuels from a knock stand point is like studying three different factors.
  First fuels have different energy density, and stoichiometric ratios. In that regards a fuel with lower AFR will be injected in higher quantities during intake stroke. Every fuels also have its own latent vaporization energy. If you combine both we get the cooling effect of the fuel when injected into the port (for PFI engine) or into the combustion chamber (for GDI).The lower the AFR the more fuel you need to vaporize. The higher the latent vaporization heat the more heat the fuel absorbs. This is very useful for knocking issue as more heat is absorbed, engine is less likely to knock.
Alcohol based fuel is exactly in this case with about 850kJ/kg of LHV, it is more than twice the value of heptane (320kJ/kg), combined with a AFR of 9, it is taking way more heat out of the combustion chamber. Furthermore the Octane number of the fuel is very important value. Gasoline would be rated between 87-93 while pure ethanol reaches 110. This means that auto ignition delays for a mixture of air and ethanol are longer than the gasoline ones, leading to less knock issue at low rpm. We are able to run higher compression ratio with the better fuels described above, with the same combustion characteristics, hence increasing overall engine efficiency.

3.3 Influence of supercharging on engine knock occurrence

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Figure 27. The KI in different backpressure w/o intercooler

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Figure 28. The KI in different backpressure w/ intercooler In the figure 20 and 21, the curves are plot to show the Knock Index changes with the pressure and the compression ratio with or without intercooler. In general, when the pressure increase the whole curve will go up, that means for the same compression ratio, the KI will increase. And for K=1 which illustrates the critic KI for knock issue at the specific compression ratio. When Pb >1.1, which means we apply supercharger to our engine. The influence is the compression ratio decreases to prevent from knock. As for intercooler, w/o intercooler, when Pb=1.1bar, the critic CR is 12 compared with 12.3 with intercooler. So with intercooler, we can have more CR in the same Pb without producing knock.