Zero-dimensional Approach

Zero dimensional model are generally conceived from empirical and semi-empirical approaches. In general, they are considered as simple modeling approach to study different systems and processes.

The results obtained from more complex simulations (for example CFD) or from the experimental tests are used for the validation of 0D models. The outcomes of 0D models has to be in good agreement with the experimental or CFD ones. 0D models can present different complexity levels, depending on the modeled system and on the desired outputs. The complexity of the 0D models

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may vary from the simple mathematical models to phenomenological models. 0D models are currently employed for the simulation of systems and sub-systems for automobile industry. As an example, 0D models can be adopted for a fist satisfactory evaluation of engine performance, consumption, noise or thermal fluxes [29] but they do not take into account the flow modeling i.e.

geometric/dimensional features of the flow motion. For the latter reason they are called zero-dimensional. The most important advantage of 0D modelling approach is the very low computational time and it is comparable with the one of the real time application. The latter characteristic makes these models very attractive for automobile industry. Another relevant advantage of this modeling approach is that it can be extended to different sectors, allowing to simulate several types of system, including mechanical, thermal, electrical etc. In the case of internal combustion engines simulation, 0D modeling approach is widely adopted in order to reproduce the in-cylinder phenomena. It is also employed to reproduce the intake and exhaust flow when the wave propagation effects within pipes can be neglected. In many cases a 0D/1D modeling approach is preferred for the simulation of ICEs: 0D model for volume elements (i.e. cylinders) and 1D model for pipes (i.e. intake/exhaust systems). In particular, as said, 0D modeling approach has been widely adopted in the past and it is still used for the simulation of in-cylinder phenomena, including the turbulent combustion process both for Spark-Ignition and Diesel engines and 0D models. Indeed, various 0D phenomenological models have been developed and oriented to primarily reproduce the turbulent combustion phenomenon and then the performance and emissions. As known, with the adoption of a zero-dimensional approach the variables which characterize the thermodynamic state of fluid within the engine cylinder are considered to be uniform in the cylinder volume while they are time-dependent. For the engine cylinder a 0D dimensional model provides the resolution of the mass and energy conservation equations. These equations, for the volume of an engine cylinder, are generally written in terms of crank angle:

mass conservation equation: ∑ ∑ (2.1)

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exchanged through the walls of the combustion chamber and ∑ ^̇ is the energy term due to the flux of mass through the control surface.

Normally, the enthalpy term and the system mass variation, due to leakage, are neglected. In this case, ∑ ^̇ and by considering the idel gas law, the energy conservation equation (2.2) can

Once the engine system is assigned, the heat release rate is known, the in-cylinder heat exchange is modeled as reported in the literature [64], [65] and by take into account the ideal gas law, the energy equation allows to obtain the in-cylinder pressure trace as a function of the crank angle.

Referring to the in-cylinder models, the zero-dimensional ones are the simplest and most suitable to evaluate the effects of variations in the engine operating parameters on the overall heat release rates and in-cylinder pressure evolutions.

Zero dimensional models are further sub-divided into:

1. Single zone models;

2. Two zone models;

3. Multi-zone models.

In single zone models, the working fluid in the engine is assumed to be a thermodynamic system, which undergoes energy and/or mass exchange with the surroundings and the energy released during the combustion process is obtained by applying the first law of thermodynamics to the system. In two zone models, the working fluid consists of two zones, an unburned zone and a burned zone. These zones are actually two distinct thermodynamic systems with energy and mass interactions between themselves and their common surroundings, the cylinder walls. The mass-burning rate (or the in-cylinder pressure), as a function of crank angle, is then numerically computed by solving the simplified equations resulting from applying the first law to the two zones.

Single zone and to zone models can be employed in two different directions:

1. In one way, both these models can predict the in-cylinder pressure as a function of crank angle from an assumed energy release or mass burned profile (Wiebe equation);

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2. Another use of these models lies in determining the energy release/mass burning rate as a function of crank angle from experimental in-cylinder pressure data.

Multi-zone models consider energy and mass balances over several zones, thus obtaining results that are closer to reality. Concerning the modeling of the combustion process, many combustion models have been developed to date and they essentially allow to predict the rate of fuel burning. In the case of Spark Ignition ICEs, various approaches for combustion modeling have been proposed and used trying to reproduce the flame development inside a turbulent flow field [23]. Among the different approaches, the “fractal” combustion model appears to be the one which better describes the turbulent flame propagation phenomenon inside an internal combustion engine [6], [7], [66], [67]. It is based on the hypothesis that aerodynamic turbulence enhances the burn rate by increasing the surface of the flame front with respect the laminar extent and does not alter significantly the engines because it allows for the description of the entire engine system with satisfactory accuracy, lower computational effort and reduced computational time. Indeed, this approach is usually employed during the engine development phase to study the global engine system. The 1D modeling of the entire engine, thanks to the above discussed advantages, allows to analyze (in addition to steady state engine operation) more complex processes such as the transient operation of the engine, unsteady behavior of engine and its sub-systems, including turbocharger group, cooling system behavior, etc. Furthermore, the 1D approach allows to analyze different engine architectures (including VVA/VVT systems, EGR circuits, water injection systems, turbocharging systems, components for the abatement of pollutant emissions, exhaust and muffler systems) and to rapidly test the effects of the modified engine configurations on its overall performances (torque/power, fuel consumption, noxious emissions, gas-dynamic noise, etc.). 1D model also furnishes the possibility to investigate a wide range of engine operating conditions such as engine operations which differ in the air-to-fuel ratio, speed, fuel injection strategies, etc. In particular, as said before,