The starting point of the present work is a global model of NOx formation for stoichiometric and lean combustion of hydrocarbons developed on the basis of a single non-linear algebraic equation. The latter is the exact solution of a system of differential equations describing the main kinetic reaction schemes of NOx formation, because it’s been analytically derived. The NOx sub-model incorporates the well-established thermal (extended Zeldovich) and the N2O reaction paths, which are considered to be the most relevant NOx production paths under certain operating conditions in arbitrary engine application. Furthermore, the NOx sub-model proposed here relies on well-established and adopted mechanisms like the GRI-Mech 3.0 [25] and consequently requires no parameter adjustment.
The single equation NOx sub-model has been developed by the authors in a previous study [14] and shown satisfactory results when validated against test bench data of two different engines operated under stoichiometric and lean burn combustion conditions respectively. Therefore, there is a strong evidence, that its implementation on embedded systems for "in-situ" and "in memory" analysis of engine process data, or even its application as a virtual sensor, is of great importance. Unfortunately, the previous developed NOx sub-model requires a few seconds running time per engine cycle. This long running time makes the model though less attractive for a real-time application. In the current study the main goal is to drastically reduce the computational times without compromising robustness and accuracy. The requirements for the time resolution on a dSpace Microautobox (MAB) is set to at least 1 kHz, meaning running times of the NOx sub-model of 1 ms per engine cycle, while its accuracy needs to be ensured at the levels of the detailed NOx sub-model as validated in [14].
The computationally most expensive steps have been identified and concern on the one hand the chemical equilibrium calculations based on the minimization of the free Gibbs energy and on the other hand the iterative solution method of the non-linear algebraic equation for the determination of the actual NOx concentration. Approaches for both steps have been developed and tested on the dSpace MAB II leading to an average computational time of 20 μs per point. Main focus of the present work is on the NOx sub-model and not the associated thermodynamic sub-models needed to describe the whole process. In order to cope with any uncertainties in NOx sub-model’s input parameters (p, T and mixture composition) only one calibration factor has been introduced and hence leads to low calibration effort. The final NOx sub-model results are compared to the detailed model and show a very good agreement.