Modeling Ignition Delay in a Diesel Engine

Transcription

1 Modeling Ignition Delay in a Diesel Engine Ivonna D. Ploma Introduction The object of this analysis is to develop a model for the ignition delay in a diesel engine as a function of four experimental variables: 1. Speed 2. Load 3. Percent of alcohol (by mass in fuel) 4. Ignition timing. Ignition delay is defined as the amount of time between the start of injection and the start of combustion in a diesel engine. For this study, ignition delay was assessed using the average of approximately 40 values from a pressure trace and is claimed to be reproducible within 0.01 crank angle degrees. The data set consists of 47 runs (see Table 1). Note that no data was collected for the combination of injection timing 30 Ca 0 and speed 2000 rpm. The researcher claims that the data for this combination is consistent with the trends exhibited by the other three combinations of injection timing and speed. Plots of the ignition delay versus percent alcohol for each load/injection timing combination are shown in Appendix 1. With the exception of the data for load 70, the trends are very similar. For this reason, we first focus on a model for loads 20, 40, and 60 only. Modeling ignition delay for loads 20, 40, and 60 The plots (see Appendix 1) suggest that increasing load decreases the ignition delay. How much it decreases, however, appears to depend on the speed and/or injection timing. Similarly, the ignition delay appears to be related to the square of alcohol but the degree of curvature appears to depend on the combination of speed, load, and injection timing. In all model fitting, proportion instead of percent alcohol was used. Furthermore residuals were plotted against the variables and predicted responses for all models. These plots will not be discussed further unless noteworthy. Finally all predictor variables, except load, were centered and standardized prior to fitting the models. Table 2 summarizes the full model fits to each timing and speed combination. 95% confidence intervals for each of the model terms are listed in the last 6 columns. The last term in the table assesses if the coefficient of the quadratic term should vary according to load. Since 0 is in each of these confident intervals, this suggests that the degree of curvature does not depend on load but rather the combination of speed and injection timing. Almost all other confidence intervals do not contain 0, suggesting that the remaining terms are needed in the final models.

2 Table 1: Data on Diesel Engine run speed(rpm) load (pounds force) alcohol (mass %) Injection timing (Ca o ) ignition delay (Cao) normal normal normal normal normal normal normal normal normal normal normal normal normal retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard retard normal normal normal

3 Table 2: Full model fits for each load and timing combination Timing Speed R 2 Constant Load Alcohol Load*Alcohol Alcohol 2 Load* Alcohol (1.27, 1.38) ( , ) (0.211, 0.464) (0.0004, ) (0.249, 1.188) (-0.011, 0.016) (1.19, 1.33) ( , ) (0.437,0.868) ( , ) (0.114, 1.808) (-0.038, 0.010) (0.91, 1.06) ( , ) (0.457, 0.868) ( , ) (0.407, 2.001) (-0.050, 0.003) I then combined these separate models into one that also included speed and injection timing. This model included all main effects, some two-factor interactions, and one three-factor interaction. In total there were 13 parameters in this model. The results are shown below. Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept <.0001 alc <.0001 load <.0001 injtiming speed <.0001 load_speed alc_speed alc_injtiming alc_load alcsq <.0001 load_injtiming alcsq_injtiming alc_injtiming_load

4 The predicted values of this model are shown in Appendix 2 for all loads. While the model did a good job fitting the data to loads 20, 40, and 60 (compare to figures in Appendix 1), it suffers to predict 70, especially for the injection timing 24 data. Modeling ignition delay for loads 20, 40, 60, and 70 This reversal of trend in ignition delay at high loads has been noted in other research. Therefore, it cannot be assumed that the results for load 70 were due to chance variation. From the fits in Appendix 2, we clearly cannot use the previous model for loads beyond 60. In order to fit all the data well, we need a model that modifies the quadratic alcohol term for load 70. Because we do not have data for loads between 60 and 70, we cannot determine at what load this modification should occur. Thus, the best I can do is to only treat load on a continuum between 20 and 60 and then let load 70 behave differently. This model involves 21 parameters and has an R 2 of 99.6%. Plots of the fits are shown in Appendix 3. Notice how this model now accommodates the load 70 data but has not really changed the previous fits. Summary The general modeling approach has been to first fit small sections of data and then use these models to build a more general model. For all loads but 70, simple models were found for each injection timin and speed combination. These three models were then incorporated into one single general model by the addition of shifts in parameters based on injection timing, speed, and their interaction. The resultant model was rather complicated (13 parameters) but fit the data in the range for load well and can be used to predict the response for any load within this range. Modeling the data to include load=70 was more complex. Because of the change in trend relative to alcohol, we ended up basically combining the previous model with one that modeled load 70 separately. The final model had an extremely good R2 (also see Appendix 3 compared to Appendix 1) but this model does not allow one to predict the response for any load greater than 60 except for load=70. Further research that investigated loads in this range would be needed to build one overall unifying model. Conclusions Through perseverance and ingenuity perhaps an overall unifying model could have been found for the complete set of load values. However, given that there was only one load (load 70) above load 60, I believe any model would be suspect in terms of prediction for any load between 60 and 70. Further data in this range is needed to build one unifying model that can

5 be used to predict for any combination of load, speed, injection timing, and alcohol percent within the range of data. Appendix 1: Scatterplots of the Data

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7 Appendix 2: Scatterplots of the predicted values from load model

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9 Appendix 3: Scatterplots of the predicted values from combined model

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