**7. Conclusions**

This article reported on the application and comparisons of two di fferent chemistry solvers for in-cylinder combustion simulations of compression and spark-ignited engines using a zero-dimensional PDF-based framework. A well-stirred reactor-based online chemistry solver was compared to the developed progress-variable-based solver noted as CPV. Latent enthalpy is chosen as the only parameter for the formulation of the reaction progress variable while a dedicated source term-based method is applied to thermal NO formation. Verification of the newly introduced CPV solver was first assessed under homogeneous constant pressure reactor conditions in order to optimize the table density/interpolation accuracy trade-o ff. Subsequently, a stochastic reactor model was used to validate the interaction of chemistry and flow during engine combustion processes. Model performance with respect to experimental data and the two solvers was assessed under heavy-duty Diesel engine as well as passenger-car SI engine conditions.

The SRM was shown to be capable of predicting the mixing controlled as well as flame propagation driven combustion processes independently of the chemistry solver. Main engine-out emissions (CO, CO2, NO and uHC) as well as combustion phasing parameters (CA50, PCP location) are in good agreemen<sup>t</sup> with experimental data. With respect to the results obtained with the online and tabulated chemistry solvers, minor di fferences have been noticed for the start of combustion location and CO emissions. Although still limited to a magnitude of 2.0 CAD, more noticeable discrepancies between the two solvers, in terms of combustion onset, were seen at low-load low-speed in both the Diesel and the gasoline engine simulation campaigns. Under these conditions, the fuel undergoes the NTC behavior where the interpolation error becomes more evident due to highly non-linear reactivity trajectories of the reacting mixture. Despite the thorough table grid optimization study performed to minimize interpolation error, when low temperature combustion is the dominant phenomenon, a tighter, if not adaptive, tabulation grid point distribution may be needed to match even better the start of combustion location predicted by the online chemistry solver. Nevertheless, given the accuracy level shown in the present work, it was concluded that the predictive capabilities of the 0-D SRM is well within commonly noted uncertainty ranges caused by, for instance, sensors inaccuracy or cycle-to-cycle variability.

From the computational cost standpoint, the CPV solver was found to be at least three orders of magnitude faster than the online chemistry solver while keeping the same order of chemical and physical models. The proposed approach is therefore a competitive tool, in terms of CPU time, to lower order methods (i.e., multizone Vibe models) widely used in 0-D/1-D engine performance studies. Generally, CPU cost is one of the main burdens when deployment of detailed chemical mechanisms in 0-D and 3-D CFD simulations is concerned. In particular, if simulations aim to an accurate prediction of exhaust emissions, it often comes a point where a trade-o ff has to be made between computational performances and size of the chemical mechanisms. Employing a tabulated chemistry solver has the potential to break this tread-o ff, by using the large mechanism only during table generation (a one-time process) while keeping the high-fidelity combustion and emission predictive capability. In conclusion, it can be stated that the present validation of CPV tabulated chemistry solver allows the SRM to be a useful CAE tool, which holds the accuracy of the overall model tool chain.

**Author Contributions:** Conceptualization, A.M. and T.F.; methodology, A.M., T.F. and L.C.G.M. Gonzales Mestre.; software, A.M., A.B. and T.F.; validation, A.M., T.F. and L.C.G.M. Gonzales Mestre.; formal analysis, A.M., T.F. and F.M.; investigation, A.M. and T.F.; resources, A.M.; data curation, A.M. and T.F.; writing—original draft preparation, A.M.; writing—review and editing, A.M. and T.F.; visualization, A.M.; supervision, A.B. and F.M.; project administration, A.M.; funding acquisition, A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/under REA gran<sup>t</sup> agreemen<sup>t</sup> n◦ 607214. Funding from the Swedish Energy Agency (P39368-2-F-Flex2) is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.
