146
147
Reference case 1
Table 7.2 Engine operating conditions and model parameters for reference case 1.
Engine speed [rpm] 3250
BMEP [bar] 13.2
Cylinder pressure at IVC [N/m^2] 3.00·106
Number of particles 200
Mixing time [s] 0.0001
Stochastic heat transfer constant Ch 15 Time step size [CAD] 1.0
Reference case 1 is a medium-to-high speed and medium-to-high load case. The fuel injection mass profile was fed to the 1-D-EST which transferred it to a fuel vaporization mass profile, which could be used by the DI-SRM (Figure 7.9). Please note that the rates are normalized. The vaporization profile is quite much delayed and extends much further than the injection profile and does not show almost any trace of the pilot injection.
Figure 7.9 Normalized injection rates for reference case 1.
The calculated pressure has the same maximum pressure as the experimental pressure trace and a reasonable timing. The pressure rate during combustion is clearly overpredicted and the expan-sion to early for the calculated case (Figure 7.10).
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-20 -10 0 10 20 30 40
Normalized Injected mass Normalized Vaporized mass
CAD
148
The heat release rate is very noisy and fluctuating compared to the experimental heat release and the early part of the experimental heat release, CAD -10 to -5, has not been captured at all (Fig-ure 7.10). The general correlation is fairly good though. The reason for the noisy heat release is mainly the coarse discretization with only 200 particles.
If we study the vaporization profile, the heat release rate and the pressure simultaneously it is clear that the experimental heat release starts before there is any major amount of fuel in the model according to the vaporization rate profile. This leads to the author’s belief that the differ-ence in the experimental and calculated pressure is a consequdiffer-ence of a not perfectly described vaporization rate profile.
Figure 7.10 Comparison of experimental and calculated heat release rate (left) and pres-sure (right) for reference case 1.
-50 0 50 100 150 200
-20 -10 0 10 20 30 40
Heat Release Rate
Experiment Calculation
J/deg
CAD
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Pressure
N/m¨2
CAD
149
Reference case 2
Table 7.3 Engine operating conditions and model parameters for reference case 2.
Engine speed [rpm] 1500
BMEP [bar] 14.3
Cylinder pressure at IVC [N/m^2] 2.98·106
Number of particles 200
Mixing time [s] 0.0002
Stochastic heat transfer constant Ch 15
Time step size [CAD] 1.0
Reference case 2 is a low speed and high load case. Just as for reference case 1 the fuel injection mass profile was fed to the 1-D-EST which transferred it to a fuel vaporization mass profile that could be used by the DI-SRM (Figure 7.11). Also in this case is the vaporization profile has the same basic profile and does not show any trace of the pilot injection.
Figure 7.11 Normalized injection rates for reference case 2.
The calculated pressure show the same maximum pressure as the experimental pressure trace and a good timing and an overall good agreement (Figure 7.12). The heat release rate is also for this case very noisy, for the same reason of coarse discretization, and the early part of the experimen-tal heat release, CAD -8 to -3, has not been captured at all (Figure 7.12). The overall correlation is better than for reference case 1, with more of the heat release centered in time.
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-20 -10 0 10 20 30 40
Normalized Injected mass Normalized Vaporized mass
CAD
150
Once again it is clear that experimental heat release starts before there is any major amount of fuel in the model according to the vaporization rate profile.
Figure 7.12 Comparison of experimental and calculated heat release rate (left) and pres-sure (right) for reference case 2.
Parametric studies
All the parametric studies were performed with reference case 2 as the baseline. For clarity all the Figures 7.13-7.18 presents the experimental results as well.
Cycle-to-cycle Variation
Figure 7.13 demonstrates that the calculated engine conditions were stable from cycle to cycle, meaning that combined flow convergence and combustion convergence were maintained and that steady state operation was reached. Stochastic models themselves inherently show increasing cyclic variations when the discretization is getting coarser, but for the calculations presented in this study that has not been an issue. NOx levels are also stable confirming the small cyclic varia-tions.
Number of particles variation
There is a tradeoff for stochastic models between accuracy and calculation time as a function of the discretization. For the DI-SRM the discretization is a function of the number of particles used to describe the in-cylinder conditions and also a function of the time step length. The as-sumption of statistical modeling is based on infinite discretization, which in our case would
-50 0 50 100 150 200
-20 -10 0 10 20 30 40
Heat Release Rate
J/deg
CAD
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Pressure
N/m¨2
CAD
151
translate to an infinite number of particles. Now for those of us who do not have infinite time to solve engineering problems the modeling approach is to have a finite number of particles.
Figure 7.13 Comparison of calculated pressures and NOx levels for cycles 60-65.
With a smaller number of particles the calculation times becomes shorter and one of the impor-tant objectives with the DI-SRM is to have a fast and efficient tool. Of course with a smaller number of particles the process gets less precise eventually boarding to a homogenous process which for this investigation would be completely useless.
Figure 7.14 shows the effects on the calculations with different number of particles. Especially obvious is how the heat release rate evolution gets smoother and more realistic with an increasing number of particles. The penalty in calculation time for the increasing number of particles is more or less a linear factor, so that the difference between the 200 particles and 2000 particles is a factor 10. The pressure and other general parameters are still with only 200 particles quite well captured. Investigations with fewer than 100 particles gave very erratic results, which are contri-buted to the fuel injection process not being modeled reasonable with that small number of particles. The NOx levels are increasing with the number of particles, which cannot be explained by the mean cylinder temperature. Possibly the individual particle temperatures may have a larger spread. Since no experimental value for NOx was given the values with the higher number of particles should be regarded as more correct. The values are typical for this kind of engine.
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Cycle to cycle variation
Experiment Cycle 60 Cycle 61 Cycle 62 Cycle 63 Cycle 64 Cycle 65
Pressure [N/m¨2]
CAD
0 100 200 300 400 500
60 61 62 63 64 65
NOx (ppm)
Cycle
152
Figure 7.14 Comparison of calculated heat release rates (left) and pressures (right) for cases with a number of particles ranging from 100 to 2000.
Figure 7.15 Comparison of calculated NOx levels (left) and temperatures (right) for cases with a number of particles ranging from 100 to 2000.
Time step size variation
With different time step sizes there were not much difference in the results. Combined with the small amount of particles used, 200, the most noticeable difference is in the peak pressure where
-50 0 50 100 150 200
-20 -10 0 10 20 30 40
Heat Release Rate
Experiment 200 2000
J/deg
CAD
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Pressure
Experiment 100 200 300 500 1000 2000
N/m¨2
CAD
0 100 200 300 400 500
100 200 500 500 1000 2000
NOx (ppm)
n-Particles
800 1000 1200 1400 1600 1800 2000 2200
-20 -10 0 10 20 30 40
Temperature
200 2000
K
CAD
153
with a time step size of 0.2 CAD the peak pressure has dip. This particular case showed a less good combination of events with fuel injection and the stochastic mixing process. The NOx levels move in an opposite direction compared with the number of particles variation.
Figure 7.16 Comparison of calculated pressures for three cases with different time step lengths (left) and for four cases with different mixing times (right).
Figure 7.17 Comparison of calculated pressures for three cases with different time step lengths (left) and for four cases with different mixing times (right).
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Time step size variation
Experiment 1.0 CAD 0.5 CAD 0.2 CAD
Pressure [N/m¨2]
CAD
0 50 100 150 200 250 300
1 0.5 0.2
NOx (ppm)
Time Step Size (CAD)
4 106 6 106 8 106 1 107 1.2 107 1.4 107 1.6 107
-20 -10 0 10 20 30 40
Mixing time variation
Experiment 0.001 0.0002 0.0001 0.00007
Pressure [N/m¨2]
CAD
0 100 200 300 400 500
0.001 0.0002 1 10-4 7 10-5
NOx (ppm)
Tau
154
Mixing time variation
One of the parameter for the DI-SRM is the mixing time that either can be given from expe-riments or CFD calculations. If this is not obtainable tau needs to be fitted. The variations of in Figure 7.17 shows the expected behavior for the heterogeneous conditions of a DI engine, where more intense mixing leads to earlier combustion timing and heat release. With a slow mixing process, 0.001 s, the mixing of fuel and air gets so delayed that favorable conditions for combustion are not reached. The levels of NOx have a similar, realistic behavior.
Figure 7.18 Comparison of calculated pressures for three cases with different time step lengths (left) and for four cases with different mixing times (right).
Mixing time variation
The simulations were performed on one of the cores of a 1.8 GHz dual core having 1 GB of RAM. Calculation time for a coupled cycle with a 1-D-EST and the DI-SRM ranges from 17 minutes for 100 particles up to 5 hours and 40 minutes (340 minutes) for 2000 particles (Figure 7.18).