*4.3. Results*

In this section, the in-cylinder pressure trace simulated by the analytic model under four di fferent operating conditions are compared with the measured ones to validate its correctness as shown in Figure 20. It can be observed from this figure that the simulation results agree well with the measured results and can capture the variation trend of in-cylinder pressure trace with crank angle during each working phase. During the gas exchange phase (blow-down, scavenging and post-exhaust), the relative errors of the simulation results are relatively higher than the other phases, which is mainly caused by the simplifications and assumptions made for these phases, i.e., two simple linear functions are adopted to approximate the in-cylinder pressure trace; however, it should be noted that the absolute errors during the gas exchange phase are less than 0.15 bar, which is still satisfactory with respect to the variation range of in-cylinder pressure during the whole working cycle. During the combustion phase, the relative errors of most of the simulation results are less than 5%, which is mainly benefiting from the e ffective calibration of Wiebe function model parameters as introduced in Section 4.2.3. It should be noted that by superposing more single Wiebe functions, the simulation accuracy during the combustion phase will improve, but this will make the Wiebe function model parameters calibration process laborious. Actually, the current simulation accuracy is already satisfactory to some extent. At the start of compression phase, the relative errors of part of the simulation results approach 20%, which is maybe attributed to two reasons: (1) the absolute value of in-cylinder pressure is relatively lower and small absolute errors will lead to obvious relative errors; (2) the actual compression process is approximated by a polytropic process. Note that as part of the input variables of the analytic model are from the MVEM, thus the predictive errors of MVEM will transform to the analytic model and lead to predictive errors.

**Figure 20.** *Cont*.

**Figure 20.** Prediction result and error distribution of the cylinder pressure analytic model: (**a**) predictive results; (**b**) error distribution.

For practical engineering practice, the marine engineers normally judge the engine's operation status by checking the compression pressure *pcomp*, maximum pressure *p*max and its crank angle position <sup>θ</sup>*p*,max. Table 5 compares the predicted and measured results of *pcomp*, *p*max and <sup>θ</sup>*p*,max. As can be observed from this table, the cylinder pressure analytic model is capable of predicting *pcomp* and *p*max satisfactorily with relative errors less than 1%. Although the relative errors of <sup>θ</sup>*p*,max are relatively higher, the absolute errors are generally less than one crank angle, indicating that the predictive accuracy can be accepted to some extent.


**Table 5.** Prediction error of the cylinder pressure analytic model.

Predicted value; 2 Measured value; 3 Relative error.

1

The inputs of the analytic model, such as scavenging air pressure and temperature and engine speed, are completely from the MVEM and the calculation process does not influence the MVEM; however, it should be noted that the MVEM and the cylinder pressure analytic model have different simulation speed, which must be handled when they are coupled together and applied in the engine room simulator. Aiming at this problem, two threads are adopted to depart the analytic model from the MVEM as shown in Figure 21. At steady state conditions, the simulation results of the MVEM are the same in every engine cycle, meaning that the engine performance parameters transformed to the analytic model also remain unchanged, therefore, the in-cylinder pressure trace simulated by the analytic model will also be the same in every engine cycle; on the other hand, the in-cylinder pressure trace is not required to be updated in real-time in the engine room simulator, and the "Pressure-Crank Angle" diagram or the "Pressure-Volume" diagram need to be displayed only when the user switches to the corresponding simulation interfaces. Based on the two factors, at steady state conditions, the in-cylinder pressure trace only needs to be calculated for one engine cycle by the analytic model, therefore, the running speed of the whole model can be as fast as the MVEM.

**Figure 21.** Synchronization approach for the MVEM and the cylinder pressure analytic model: (**a**) *X* = 0; (**b**) *X* = 1; (**c**) the calculation frequency of the MVEM is many times the analytic model.

During the transient process, the engine performance parameters and the in-cylinder pressure trace are different in every engine cycle; on the other hand, the MVEM and the cylinder pressure analytic model have different simulation speed. Therefore, in order to keep the simulation results in sync, the MVEM has to wait for some time before analytic model completes the calculation of an engine cycle as shown in Figure 21a, as a result, the simulation speed of the whole engine model is

determined by the cylinder pressure analytic model, which is actually slower than the MVEM. To accelerate the simulation speed of the whole model, the idea of abandoning engine cycles is adopted. Figure 21b presents the case where the calculation frequency of the MVEM is two times the analytic model, meaning that the MVEM has already completed the calculation of two engine cycles when the analytic model only completes the calculation of one engine cycle. Figure 21c presents the case where the calculation frequency of the MVEM is many times the analytic model. By adopting this method, we can keep the MVEM and the cylinder pressure analytic model in sync, furthermore, the simulation speed of the whole model will ge<sup>t</sup> faster if more engine cycles are abandoned by the analytic model. The relationship between the number of abandoned cycles *X* and the reduced time *Y* can be expressed as *Y*=*X*/*(X*+*1)*. It should be noted that the synchronization mentioned above is a fake synchronization during the transient process and this is because that the current simulated in-cylinder pressure trace simulated by the analytic model does not correspond to the current engine operating conditions outputted by the MVEM but falls behind to some extent.

To assess the improvement in simulation speed of the extended MVEM developed in this paper, it is compared with the merged model developed by Tang et al., where the 0-D model was adopted to calculate the in-cylinder pressure and the MVEM was used to simulate other engine performance parameters with the similar synchronization approach as shown in Figure 21 [5]. The comparison of actual execution time for a 100 s simulation time is presented in Table 6.


**Table 6.** Comparison of simulation speed.

As can be found from Table 6, the engine model developed in this paper is able to achieve similar actual execution time of about one second but only four cycles are abandoned, whereas it is 49 for the merged model developed by Tang et al. [5]. This is due to the fact that the cylinder pressure analytic model runs much faster than the 0-D model and therefore similar actual execution time can be obtained by abandoning less number of engine cycles. As a result, the model developed in this paper can capture the transient response of the in-cylinder pressure trace more accurately.
