5.1 Effects of modeling parameters and discretization on the HCCI-SRM
5.1.2 Effects of modeling parameters that affect mixing and heat transfer
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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
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• average duration of the combustion, time of ignition to the CAD of maximum pressure
• average amount of NO at EVO
• average amount of hydrocarbons at EVO Values chosen for the parameters in the investigation were
• Tau=[0.001, 0.015, 0.025, 0.035]
• Stocon=[10, 20, 30, 40]
The time step size was set to 0.5 CAD and the number of particles to 200.
Definitions of the studied result parameters
Variation in maximum pressure is defined as the difference in pressure between the cycle with the highest maximum pressure and the cycle with the lowest maximum pressure, see Figure 5.6.
This measure is in principle a measure for the cyclic variations, discussed in the previous chapter.
Figure 5.6 Illustration of the difference in pressure between two cycles.
Average maximum derivative of the pressure is the average of the maximum pressure deriva-tive in each cycle, see Figure 5.7. The derivaderiva-tive of the pressure is calculated using the three point formula.
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Figure 5.7 Comparison between the pressure (dashed) and the derivative of the pressure (solid) with the important points and intervals marked.
Average CAD of maximum pressure is calculated by first finding the CAD which gives the maximum pressure for each cycle then taking the average.
Average CAD of maximum pressure derivative is found by the same method as the average CAD of maximum pressure.
Time of ignition is defined as the point when the pressure starts increasing due to the combus-tion. This point is found by studying the derivative of the pressure and finding the minimum before the maximum pressure, see Figure 5.7. The CAD of this minimum becomes the time of ignition. This method will not work for cases where the ignition occur so early that the pressure derivate do not change sign.
Duration of combustion is defined as the difference in CAD between the CAD of maximum pressure and the time of ignition.
Amount NO and hydrocarbons at EVO is taken as the value of the specific species at the final time step.
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Figure 5.8 Left: The amount of NO (solid) with the pressure (dashed). Center: The amount of hydrocarbons (solid) and the pressure (dashed). Right: The mass fraction burnt (solid) with the pressure (dashed).
Statistical analysis methods
To characterize the influence from Tau and Stocon respectively on the different results the Pear-son correlation coefficient was used.
Pearson correlation coefficient gives the linear dependency on the results from the in-parameters.
The correlation coefficient can have a value between -1 and 1 where -1 is a perfect negative li-near relation and 1 is perfect positive relation. For a correlation value larger than 0.5 (or smaller than -0.5) there is a large positive (negative) correlation. For values between 0.3 and 0.5 there is an average correlation, for values between 0.1 and 0.3 there is a small correlation and for values smaller than 0.1 there is no linear correlation. The coefficients are calculated by fixing one of the parameters and varying the other, giving four different coefficients for each parameter and result.
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Results
Variation in maximum pressure
The variations in maximum pressure (Figure 5.9) are in general of the same magnitude, except for cycles with low tau where they become 3-4 times as large. The correlation coefficients for stocon are inconclusive due to the small change in the variation. For tau they are large negative.
Figure 5.9 The variations of the maximum pressure for the different values of tau and stocon.
Studying the ratio between the difference in maximum pressure and the average maximum pres-sure, a clear relation appears (Figure 5.10). The correlation coefficients are large negative for tau and large positive for stocon.
Figure 5.10 The variations of the maximum pressure relative to the average maximum pressure.
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CAD of maximum pressure
The CAD of maximum pressure gets clearly delayed with decreasing tau and increasing stocon (Figure 5.11). The correlation coefficients are large positive for tau and large negative for stocon.
Figure 5.11 The CAD for the maximum pressure for the different tau and stocon.
Maximum derivative of the pressure
Clear trends are seen in Figure 5.12. Large stocon and small tau gives a high derivative in the pressure. In the other end, small stocon and large tau, gives small pressure derivative. These trends are also seen in the correlation coefficients. For tau the correlation is large negative and for stocon it is large positive.
Figure 5.12 The maximum derivative of the pressure for the different tau and stocon.
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CAD of maximum derivative of the pressure
Studying the CAD of the maximum derivative of the pressure in Figure 5.13 the dependence on stocon is stronger than on tau. For larger stocon the CAD for maximum derivative of the pres-sure becomes later. The correlation is large positive for stocon and large negative for tau.
Figure 5.13 The CAD for the maximum derivative of the pressure for the different tau and stocon.
Time of ignition
From the time of ignition in Figure 5.14 it is clear that it becomes later for homogenous mixing.
The correlation coefficients are large positive for stocon and large negative for tau.
Figure 5.14 The time of ignition for the different tau and stocon.
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Duration of combustion
The dependencies for the duration of combustion are not clear. It is quite chaotic as seen in Figure 5.15. The correlation coefficient for tau is average positive and inconclusive for stocon.
Figure 5.15 The duration of combustion for the different tau and stocon.
Amount of NO at EVO
With increasing stocon and decreasing tau the amount of NO in the exhaust decreases. The effect from stocon is stronger. For the correlation coefficients it is large positive for tau and large negative for stocon.
Figure 5.16 The amount of NO for different tau and stocon
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Amount hydrocarbons at EVO
The amount of hydrocarbons at EVO is basically insensitive to changes in tau and stocon.
Conclusions
With low tau the in-cylinder conditions become homogeneous faster. High stocon spreads the heat transfer over a larger portion of the in-cylinder gas, leading to less inhomogeneous condi-tions. The results indicate that inhomogeneties promote earlier ignition and thus a more stable operating condition and less cyclic variations (defined as pressure variations). The pressure deriv-ative is in general terms decreased with inhomogeneous conditions. The value of tau or stocon does not have any major impact on the duration of combustion or the amount of hydrocarbons and NO at EVO.
The reason to do several cycles for each value of tau and stocon, is to be able to get the average of the stochastic variations and also to be able to study range of the stochastic variations. Here one assumption was made: 10 cycles will give the statistical distribution of the variations. With the combinations of tau and stocon that give large variations 10 cycles may not be enough to give the statistical distribution. More cycles may be needed to give the necessary statistics.
In the cases where there was an impact on the result from varying the parameters the correlation was large, for stocon often larger than 0.9. For tau it was more in the range of 0.7. The sign of the coefficients in all cases were opposite to each other. One interesting note is that even for results with little dependence on the value of the parameter, the coefficients become quite large.
In these cases the variations between each coefficient is also large. If the simulations would have been done with more closely spaced values for the parameters, thus giving a larger amount of data to work with, the correlations in these cases would probably have been lower and more coherent.
The main disadvantage of the Pearson correlation coefficient is that it only gives information on how linear the dependency between the variables is.
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Correlation coefficientsTable 5.4 Correlation coefficients.
Stocon 10 Stocon 20 Stocon 30 Stocon 40
Tau 0.035 Tau 0.025 Tau 0.015 Tau 0.001
Variations of maximum pressure
0.67 -0.55 0.10 0.82
-0.94 -0.93 -0.69 -0.84
Variations of maximum pressure relative the average pressure
0.97 1.00 0.96 0.98
-0.78 -0.80 -0.80 -0.85
CAD of maximum pressure
0.81 0.95 0.89 0.98
-0.93 -0.95 -0.81 -0.91
Maximum derivative of the pressure
0.87 0.99 0.91 0.97
-0.76 -0.76 -0.80 -0.84
CAD of the maximum derivative of the pressure
0.84 0.98 0.92 0.97
-0.91 -0.94 -0.80 -0.91
Time of ignition
0.91 0.99 0.95 0.97
-0.83 -0.89 -0.83 -0.90
Duration of combustion
-0.69 -0.79 0.45 0.97
-0.17 0.11 0.71 0.89
Amount NO at EVO
-0.73 -0.95 -0.78 -0.97
0.93 0.80 0.74 0.72
Amount HC at EVO
0.80 -0.77 0.90 -0.77
0.24 -0.92 -0.82 -0.91
How to read the table:
For each upper row stocon is fixed at a value and tau is varied. The first column gives the coeffi-cient when stocon is set to 10, second column when stocon is 20 and so forth. For each lower row tau is fixed and stocon is varied. Each point represents the correlation coefficient of 40 cal-culations.