Análisis de la relación entre las variables

For the numerical simulation of ICEs behavior, as discussed before, several modeling approaches can be adopted according to the analysis type, the desired results and their level of detail. During the last years, three-dimensional codes have been more and more used to describe various thermo-fluidynamic processes, allowing for detailed and accurate prediction of fluid-dynamic conditions within the intake and exhaust manifolds and engine cylinders. In particular, 3D analyses are very helpful to study the in-cylinder phenomena, including turbulence and combustion processes, ab-normal combustion (knock phenomenon), fuel spray evolution, fuel evaporation, mixing process and pollutant species generation. Even if 3D CFD models allow to be closer to the actual engine fluid-dynamic behavior, they still require high computational times, which grows with the engine dimensions, making the considered models not compatible with the development phase of ICEs. For this reason, the employment of a sole 3D CFD model for ICEs is limited to few cases and oriented to better understanding the physical phenomena and to find information which cannot be easily obtained with the experimental tests. On the other hand, 0D/1D models allow to describe the whole engine system (including external sub-systems such as turbocharger) with reduced computational times and satisfactory accuracy. With this modeling approach, virtual analyses of new engine configurations can be performed. The main drawbacks of 0D/1D models are usually represented by the description of the in-cylinder phenomena, because an inadequate formulation of in-cylinder processes (especially for combustion process) leads to not correct assessment of engine performance and engine-turbocharger matching. A very promising numerical methodology showing the potentialities to support the engine development phase and to overcome the limits of both 0D/1D and 3D models, is represented by a proper integration of these types of models. The role of each model within the integrated 0D-1D/3D methodology is univocally identified. As an example, a brief description of an integrated procedure for the simulation of SI ICEs performance is here reported and it is adopted within this work. The considered integrated method requires, as the initial step, the engine geometrical data provided by the manufacturer. They are used to build the 1D

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model of the whole engine and to carry out preliminary 1D simulations under motored operations.

Then, multi-cycle motored 3D analyses, applying 1D computed time-varying pressure and temperature traces as boundary conditions, are performed. Detailed information about the in-cylinder mean and turbulent flow fields are extracted from the 3D analyses over a whole engine cycle. The results obtained by the latter analyses are utilized to calibrate a 0D turbulence model coupled to a proper phenomenological combustion model, which will be discussed in detail in the following paragragh. Once the turbulence sub-model is tuned, the 1D engine simulation is carried out under fired operations. The combustion process description is properly linked to the previously tuned in-cylinder turbulence model. The combustion model is tuned in order to fit the in-cylinder pressure cycles at full load operation. Once validated, the 1D model is capable to reproduce the engine perfoemance and it can be employed for further numerical analyses, including the description of knock occurrence and cycle-by-cycle variations. In Figure 2.2, a flowchart of the described integrated 1D/3D methodology is fully summarized.

Figure 2.2 – Flowchart of the integrated 1D/3D hierarchical methodology.

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2.3 0D Combustion Modeling: “Fractal Combustion Model”

Fractal Combustion Model is a phenomenological 0D model, based on the fractal description of the flame front and it is able to sense both the combustion system geometry and the operating parameters, such as air-to-fuel ratio, spark advance, boost level, residual gas content etc. This model is capable to take into account the in-cylinder turbulence evolution thanks to a proper 0D turbulence sub-model. Referring to the 0D combustion modelling, in the wrinkled-flamelet regime of combustion occurring inside an ICE [71], [72], [73], the burning rate can be expressed as:

. / (2.23)

ρu being the unburned gas density. Equation (2.23) puts into evidence that the burning rate is mainly increased by the wrinkling of the flame surface AT determined by the turbulent flow field, with respect to the corresponding smooth surface AL, occurring in a laminar combustion process. Flame propagation indeed locally proceeds at the stretched laminar flame speed, SL, which is a function of the fuel type, air-to-fuel ratio and residual gas fraction. The employed correlation also accounts for flame stretch mainly occurring during the first phase of burning process [74]. Under this schematization, moreover, the burning rate can be easily computed once the increase in flame area has been established. Basing on the concepts of the fractal geometry, the latter can be easily expressed as a function of a minimum and maximum flame wrinkling scales, Lmin-Lmax, and to its fractal dimension D3 as reported in equation (2.24):

.

/ (2.24)

Lmin is assumed equal to the Kolmogorov length scale, which, under the hypothesis of isotropic turbulence, follows the expression (2.25):

 _ (2.25)



u being the kinematic viscosity of the unburned mixture. The turbulent intensity and the integral length scale LT are given by the K-k model described in the following. Lmax, indeed, representing the dimension of the maximum flame front wrinkling, is related to some characteristic dimension the

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flame front, such as the flame radius or the square root of the laminar surface. In the present version of the model, the latter choice is followed, and the Lmax is computed according to equation (2.26):

√ (2.26)

cwrk being a tuning constant. The D3 dimension only depends on the ratio between the turbulence intensity and the laminar flame speed SL, reported in [73].

The above described fractal model is really valid for a fully developed and freely expanding turbulent flame. During both early flame development and combustion completion, a different approach is required. Initial flame development is in fact dominated by the kernel formation phase;

its duration is calculated by an Arrhenius-like formulation, also accounting for residual fraction contents. At the end of the kernel formation time, the computation of the combustion process starts with a stable and spherically-shaped smooth flame, of radius r0. Flame wrinkling process also starts at a rate, ωwr (equation (2.27)), which increases with the normalized elapsed time after the combustion start.

(2.27)

The transition time, ttrans, in the above equation (2.27) is hence considered as a time scale during which the flame front evolves from an initial smooth surface – corresponding to a laminar-like combustion – to a fully developed turbulent wrinkled flame. The transition time is assumed proportional to a characteristic turbulent time scale, calculated as reported in equation (2.28):

(2.28)

ctrans being a tuning constant, while k and Dr are the turbulence kinetic energy and its dissipation rate. Flame wrinkling rate ωwr affects the calculation of the maximum fractal dimension of the flame, according to equation (2.29):

( ) (2.29)

WF being the well-known Wiebe function. In this way, the D3 expression in [73] is redefined as:

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(2.30)

With this formulation, the first phase of the combustion process (w1~ 0) is characterized by a fractal dimension close to 2.00, corresponding to an initial laminar burning process.

A different combustion rate is also specified when the flame front reaches the combustion chamber walls (wall combustion phase). Wall-combustion burning rate is simply described by an exponential decay, as reported in equation (2.31):

. /

(2.31)

η being the characteristic time scale of the above process. The overall burning rate is consequently defined as a weighted mean of the two described combustion rates in (2.32):

. / . /

( ) . /

(2.32)

The switch between the two combustion modes gradually starts when a threshold value, cwc, of the burned gas fraction is reached (transition time, twc). At this time, the characteristic time scale in equation (2.31) is computed assuming that the wall combustion burning rate equals the one derived from the fractal model, equation (2.23), hence:

₍ ^{( )}₎

_{( )}

(2.33)

The above η value is then kept fixed during the subsequent wall combustion process. The weight factor w2 indeed increases with time, depending on the instantaneous unburned mass (m - mb), compared to the one occurring at the transition time, twc. In this way, a smooth transition between the two modes is easily obtained. It is important to note that the above schematization of the wall combustion, based on the analysis of typical heat release data, does not include any additional tuning constant. Finally, two corrections are introduced in the turbulent flame front area calculation, as shown in equation (2.34):

.

/ , ( )- 0 1 (2.34)

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Ka in the above equation is the Karlovitz number which is computed as reported in equation (2.35):

(_ ) ^ (2.35)

f being the flame front thickness. The first correction accounts for the enhanced species mixing in the flame front and is activated only when the Ka > 1 inequality verifies. In this case, in fact, the flame thickness is larger than the Kolmogorov turbulence micro-scale and turbulent eddies are able to enhance the burning speed. This verifies in the case of high turbulence levels (high engine speed) or slow laminar flame speed (large amount of trapped EGR, very lean or very reach mixtures, for instance). Simultaneously, at very high Karlovitz numbers, the flame deformation determined by the turbulent flow field can be so intense to produce a multiple connected flame front, with

“islands” of unburned mixture trapped within the burned gas zone. This is mainly due to the increased convective action of the turbulent field, that can stretch and break the flame, determining pockets of unburned mixture within the burned gas zone. As a consequence, the reaction surface and the burning speed increase.

The second correction term is justified by the possibility of a flame front distortion, that may occur when an intense mean flow is present in the cylinder. Each term is weighted by additional tuning constants, and .