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The range of cycle-to-cycle variations in HCCI-SRM

5.1 Effects of modeling parameters and discretization on the HCCI-SRM

5.1.1 The range of cycle-to-cycle variations in HCCI-SRM

Cycle-to-cycle variation is a phenomenon present in all engines to a greater or lesser degree. The cycle-to cycle variation has its origins in the variations concerning the in-cylinder gas motion and heat transfer interaction, the mixture preparation, the pressure fluctuations in the gas ex-change processes. The cyclic variations are more pronounced in SI and HCCI engines than in DICI engines that do not have mixture preparation and is not so sensitive to in-cylinder gas motion and heat transfer since they only have basically air in the cylinder during most of these processes.

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One of the major assets of the SRMs is their inherit cycle-to-cycle variation. This is a feature of the stochastic heat transfer model, with its random temporal variation and random wall surface interaction variation (if several wall temperatures are assigned) which are real physical origins for real cyclic variations in engines. It is also a feature of the stochastic mixing model which dis-perses the variation in accordance with the real stochastic turbulent gas motions in the cylinder.

The ability to model the cyclic variations is very useful to assess the stability of real engines, especially in operating ranges close to misfire or close to pressure rates that may be destructive for the engine.

Consequently with the SRMs, one single calculation will be somewhere in the range of the cyclic variations. Where that single calculation is in that range, is not known, unless a number of calcu-lations is done to establish the borders of the cyclic variations.

One issue with stochastic models is that when the discretization is getting too coarse due to a too small number of particles or too long time steps, the stochastic processes get too violent and the range of the cyclic variation too big. This is not a physically correct phenomenon but a conse-quence of insufficient discretization. What is sufficient discretization depends on the case, where in general terms homogeneous cases as SI and HCCI are less demanding, while heterogeneous cases as DICI are more demanding. To control that, it is advisable to routinely check the current ranges of the cyclic variations to check that the discretization is sufficient. Often this control is done by running 10 calculations to define the range of the cyclic variations and to define an average cycle.

The present investigation answers the question how well 10 cycles covers the range of cyclic variations compared to 100 cycles, and if 10 cycles are enough to ensure that average values from 10 cycles are useful.

A number of cases with different settings for stocon and tau were set up. For each of the cases, 10 calculations were performed with the same initial conditions. The number of particles was set to 100 and ∆CAD to 0.5. For this investigation one case with large cyclic variations (tau=0.001 and stocon=10) and one case with small cyclic variations were selected (tau=0.035 and sto-con=40), (Figure 5.3).

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Figure 5.3 The two selected cases exhibiting large respectively small cycle-to-cycle

varia-tions, from 10 calculations with the same initial conditions each.

The case with large variations was quite uncharacteristic compared to all other investigated cases that were rather similar to the case with small cyclic variations. Nevertheless it was used since tau is within its physical bound and such setting may be applied. Stocon which has an empirical connection is also within its normal bounds.

For the two selected cases, a total of 100 calculations each were performed so that the cycle-to-cycle variation ranges and averages could be compared with those of the initial 10 calculations.

All calculations for each case used the same initial conditions.

The following result parameters were compared for the calculations with 10 and 100 calcula-tions:

– Maximum pressure – CAD of maximum pressure – CAD of Ignition

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Figure 5.4 Left: The pressure curve for the case with the highest maximum pressure and the case with the lowest maximum pressure for simulations with 100 cycles (solid) and 10 cycles (dashed). Right: The average pressure curve for simu-lation with 100 cycles (solid) and 10 cycles (dashed).

As can be seen in the left graph in Figure 5.4, 100 calculations gives a larger spread between maximum and minimum pressure than for 10 calculations. This is the case for the other result parameters as well and it is as expected. The averages for the 100 calculations and the 10 calcula-tions are approximately the same.

Figure 5.5 Left: The pressure curve for the case with the highest maximum pressure and the case with the lowest maximum pressure for simulations with 100 cycles (solid) and 10 cycles (dashed). Right: The average pressure curve for simu-lation with 100 cycles (solid) and 10 cycles (dashed).

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The results for tau = 0.035 and stocon = 40 are similar to the previous, although the averages are even more closely fit (Figure 5.5).

Table 5.2 Results for 100 and 10 calculations for case with Tau = 0.001 and stocon 10.

Maximum pressure

[106 N/m2]

CAD of maximum pressure [CAD]

CAD of Ignition [CAD]

n Calculations 100 10 100 10 100 10

Max 10.0 9.89 19.5 13.0 9.5 6.0

Min 5.92 7.91 5.5 6.0 -1.5 -0.5

Variation 4.09 1.98 14.0 7.0 11.0 6.5

Average 8.79 8.89 9.87 9.50 3.51 3.20

Std deviation 0.71 0.54 2.34 1.90 1.62 1.70

Table 5.3 Results for 100 and 10 calculations for case with Tau = 0.035 and stocon 40.

Maximum Pressure

[106 N/m2]

CAD of maximum pressure [CAD]

CAD of Ignition [CAD]

n Calculations 100 10 100 10 100 10

Max 9.66 9.50 12.0 10.5 4.0 3.0

Min 8.15 8.55 7.0 7.5 0.0 0.5

Variation 1.51 0.94 5.0 3.0 4.0 2.5

Average 8.99 9.03 9.07 8.95 1.99 1.90

Std deviation 0.30 0.28 0.99 0.86 0.76 0.67

Conclusions

This study demonstrates that with 100 particles and a timestep of CAD 0.5 the discretization is coarse enough to give quite substantial cyclic variations for certain conditions. For the more extreme case it is obvious that one single calculation may differ from another in predicting the time of ignition with more than 11 CAD. For a study with a single calculations and using an unfortunate combination of tau and stocon the result would be easily misinterpreted.

However, most combinations of stocon-tau do not create such large cyclic variations. Also aver-age values from as few as 10 calculations are useful to establish the soundness of the results.

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The engine case simulated in this study is according to my experience, from a physical point of view, a medium stable case. Engine operating conditions that is more unstable, verging to mis-fire, may themselves demonstrate large cyclic variations. In such circumstances the model would also experience larger cyclic variations. However the ability to predict cyclic variations is certainly useful while studying engine operating regimes. But for the SRM one should remember that this is an effect that may origin from the incorrect use of discretization, and thus not a physically correct feature.

The remedy for this is to use sufficiently many particles, and to check the cyclic variations from calculations on a regular basis.

5.1.2 Effects of modeling parameters that affect mixing and heat transfer