### A DAE Formulation for Multi-Zone Thermodynamic Models and its Application to CVCP Engines

In the automotive area there are ever increasing demands from legislators and customers on low emissions and good fuel economy. In the process of developing and investigating new technologies, that can meet these demands, modeling and simulation have become important as standard engineering tools. To improve the modeling process new concepts and tools are also being developed.
A formulation of a differential algebraic equation (DAE) that can be used for simulation of multi-zone in-cylinder models is extended and analyzed. Special emphasis is placed on the separation between thermodynamic state equations and the thermodynamic properties. This enables implementations with easy reuse of model components and analysis of simulation results in a structured manner which gives the possibility to use the formulation in a large number of applications. The introduction and depletion of zones are handled and it is shown that the DAE formulation has a unique solution as long as the gas model fulfills a number of basic criteria. Further, an example setup is used to validate that energy, mass, and volume are preserved when using the formulation in computer simulations. In other words, the numerical solution obeys the thermodynamic state equation and the first law of thermodynamics, and the results are consistent and converge as tolerances are tightened. As example applications, the DAE formulation is used to simulate spark ignited SI and Diesel engines as well as simple control volumes and 1-dimensional pipes. It is thus shown that the DAE formulation is able to adapt to the different requirements of the SI and Diesel engine models.
An interesting application is the SI engine with continuously variable cam phasing (CVCP), which is a technology that reduces the fuel consumption. It influences the amount of air and residual gases in the engine in a non trivial manner and this SI application is used to evaluate three control oriented models for cylinder air charge and residual mass fraction for a CVCP-engine both for static and transient conditions. The models are: a simple generalized flow restriction model created with physical insight and two variants of a model that is based on an energy balance at intake valve closing (IVC). The two latter models require measurement of cylinder pressure and one also requires an air mass flow measurement. Using the SI model as reference it is shown that transients in cam positions have a large impact on air charge and residual mass fraction, and the ability of the models to capture these effects is evaluated. The main advantages of the generalized flow restriction model are that it is simple and does not require measurement of the cylinder pressure but it is also the model with the largest errors for static operating points and highest sensitivity in transients. The two models that use an energy balance at IVC both handle the transient cycles well. They are, however, sensitive to the temperature at IVC. For static cycles it is therefore advantageous to use the model with air mass flow measurement since it is less sensitive to input data. During transients however, if the external measurement is delayed, it is better to use the model that does not require the air mass flow.
The conclusion is that the DAE formulation is a flexible, robust, tool, and that it is well suited for multi-zone in-cylinder models as well as models for manifolds and pipes outside the cylinder.

*Per Öberg*

**2009**

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Senast uppdaterad: 2021-11-10