White-Box Models

The term “White-Box” refers to the fact that the entire model structure is visible which means that every equation used to describe a specific element of the system is based on physical principles. As described by Merker et al [17], the overall process of the internal combustion engine is very complex. In order to describe the overall process with physical equations, it is first of all necessary to break it down into partial problems which are physically describable and mathematically formulatable. Each problem is then solved by applying first principals from e.g. thermodynamics, fluid dynamics, chemical reactions, mechanics and kinematics. The overall process model is finally formed by connecting the individual solutions for each partial problem with each other. A very logical way of breaking down the overall process was presented by Chow and Wyszynski [58] and Souflas [56]. The entire system can be represented as a combination of five elements, which are cylinder, plenum, pipe, restriction, and turbocharger. These elements can be modelled with different modelling techniques, depending on the desired model fidelity, complexity, simulation speed and cost for parameterisation as shown in Figure 2.9.

Figure 2.9: Top level structure of engine system models Adapted from [56]

equations. In the following explanation, the details of each engine element including the different modelling techniques are reviewed.

• Cylinder: The cylinder element is modelled by applying the filling and emptying modelling technique, which can be derived by applying mass and energy balance for an open system to the cylinder. As described by Souflas [56], the complete cylinder model requires sub models for combustion [17], heat transfer [17], [59], [60], thermodynamic properties [59], [61] and kinematics and mechanics [59], [61], [62]. The reader is referred to Souflas [56] and references therein for a more detailed description and examples.

• Plenum: The plenum element represents the major volumes along the engine air path. This includes the intercooler, intake, and exhaust manifold as well as the catalyst and the silencers. Usually, plenums are modelled using filling and emptying dynamics, which means that the volumes are assumed to be zero dimensional. Two first order differential equations allow to model the dynamic behaviour of pressure and temperature inside the volumes. The required sub models are thermodynamics and heat transfer. The reader is referred to Isermann [13] and Guzzella and Onder [16] for more detailed information. Schaal et all [63] gives a detailed derivation as well as a review of different implementation methods.

• Pipe: The pipes which connect the main volumes along the air-path can be modelled in two different ways. The simple approach is to lump the pipes together with the plenums into big volumes [64]. E.g. lumping the intake runners together with the intake manifold into one volume. This approach allows to apply the filling and emptying dynamics. However, in order to capture the pressure wave propagation through the intake and exhaust system, the pipes have to be modelled using one dimensional gas dynamic modelling techniques [57]. One dimensional flow is derived from the Navier-Stokes equation restricted to one dimension which combines the conservation of mass, energy, momentum, and species inside the pipes as shown in [17]. The challenge in finding a numerical solution is that the set of equations depends on space and time. Early numerical solutions by the mesh method of characteristics were presented by Benson [65], [66]. Current state of the art is to combine finite volume discretization with shock-capturing techniques where the pipes are divided

into many small sub volumes. Toro [67] gives detailed information about this method.

• Restriction: The restriction element represents all valves and orifices along the engine air path such as throttle plate, intake and exhaust valve as well as the waste gate. The mass flow through this restrictions is modelled using quasi steady models which are based on the equation for one dimensional isentropic flow. However, to achieve a highly accurate prediction of the mass flow though the valve it is necessary to include a model for the discharge coefficient which compensates for the fact that the flow is not fully isentropic as shown by Anderson [68] and Hendricks et al [64].

• Turbocharger: The turbocharger element can either be modelled using a purely physics based approach as shown by Mueller [69] or using experimental maps as shown in Moraal and Kolmanovsky [70]. However, due to the extremely nonlinear and complex system behaviour, purely physics based models struggle to provide the required accuracy [69]. The most advanced solutions combine the turbine maps with one dimensional gas dynamics as shown by Macek [71]. Each turbocharger manufacturers delivers high accurate test results about the performance of the turbocharger. These test results are also known as turbocharger maps and usually include values for mass flow, pressure ratio, efficiency and rotational speed. For the turbine and the compressor model a regression model is fitted to the experimental data. For this reason, the turbocharger model could be regarded as a Black-Box model. However, since the data for each turbocharger are available from the manufacturer, no additional experimental testing is required. Recent studies by Wurzenberger [72] show that the maps need to be slightly edited if the pipes are modelled using filling and emptying dynamics instead of gas dynamics. Since this modelling technique does not predict the pressure wave propagation, the efficiency maps of the compressor and the turbine need to be modified slightly to compensate for the loss in information. A common problem is that the turbine and compressor maps do not cover the entire operating range of the turbocharger. Therefore, extrapolation methods are needed

Crank angle resolved simulation of the intake and exhaust events in the cylinder combined with the 1D pressure wave propagation in the intake and exhaust system allow a highly accurate physics based modelling of the air mass flow along the air-path [57]. This results in highly accurate cylinder air charge predictions. The crank angle resolved combustion models combined with the highly accurate air charge prediction allows accurate estimation of trend wise engine torque and exhaust gas temperature without the need of any engine test results. The disadvantage is that even the latest numerical solution methods are too time consuming to run in real time [14]. Only significant simplifications of the wave propagation models allow the models to be run in real time but with a loss in accuracy. 1D crank angle resolved engine models are usually used during a very early stage of engine development where no test results from a prototype engine are available. A common example is the design optimisation of the intake and exhaust system. To test the effect of intake runner lengths and diameters, valve sizes, camshaft profiles and many more geometrical aspects on the test bed is extremely time consuming. Models which simulate the air mass flow through the engine help the designers to find the optimum solution within the given design limits or tell the designers how to tune the components to achieve specific performance requirements. Specially to optimize full load performance engine simulation models are indispensable and routinely used by the industry. Hamilton et al [76], Piscaglia et al [77] and Yarsam et al [78] give some good examples.

In case the pipes are modelled with filling and emptying dynamics instead of 1D gas dynamics, the accuracy of air charge prediction is significantly reduced [57]. On the other hand, 0D model allow a significant increase in simulation speed compared to 1D models which enables the model to run in real time since no time consuming iterative numerical solutions are necessarily required. This significantly expands the application range of the model. Innovative solutions for real time running models were published by Wurzenberger et al [72], [79]. See Alix et al [80] and Chalet et al [81] for detailed information and a comparison with the 1D approach. 0D crank angle resolved models are mainly used during the early stages of engine development and/or for controls development. A common application is HIL (hardware in the loop) testing. One of the key parts in the engine development process is the control strategy development for transient engine control. A real ECU is connected in a loop with a real time running engine model.

The dynamic response of the model to changes in actuator settings allow tuning and testing of the control strategy. An example of this is shown by Pacitti et al [82], Corti et al [83], Wu et al [84]and Chen et al [85].