As compared with the HCCI-SRM model described in chapter 4 and the formal features of the SRM itself, the DI-SRM has additional features, direct injection into the cylinder during the closed cycle being one of these. The operator splitting loop is also expanded to include the direct injection and an additional pressure correction step (Figure 7.1).
The DI-SRM
The DI-SRM, or direct injection stochastic reactor model, is based on the partially stirred reac-tor model (PaSPFR) described in chapter 4. It also contains the same PDF based Monte Carlo type simulation assumption, involving use of an MDF and an operator splitting loop. The oper-ator split loop has been extended with one additional step for fuel injection. The DI-SRM mod-el does not include any modmod-el for ignition assistance but rmod-elies entirmod-ely on chemical kinetics.
The fuel injection model employed can model multiple direct injections in which the injection profiles can be of any shape. To simplify implementation, the fuel is assumed to be vaporized instantaneously at the moment of injection, and no vaporization model is employed. There are two implications that follow from this.
1. Rather than a fuel mass injection curve, a vaporized mass injection curve needs to be used as input to the model.
2. The energy needed for vaporization of the injected fuel needs to be taken into account in the model.
During the combustion calculations a control is made at each time step to determine whether there is any fuel to be injected. If such is the case, the vaporized amount to be injected is deter-mined by a linear interpolation of the fuel injection profile provided.
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Figure 7.1 Operator split loop in the DI-SRM involving an added fuel injection step.
The energy needed to vaporize the fuel is determined by lowering the temperature on a suffi-ciently large mass in the cylinder to the level of the fuel vaporization temperature. This mass, referred to here as the mixing mass, is taken from the already existing particles and is combined with the mass of the injected fuel that forms new particles (Figure 7.2).
138
Figure 7.2 Particle handling during injection.
When fuel is added to the cylinder, the mass fractions and the temperatures of the current set of particles is changed, in line with the composition of the fuel and its liquid injection tem-perature. Thus, for each particle the source terms need to be supplemented by extra terms that are added (superscript ):
, , 1, … ,
, , 1, … , 1
∑ , ,
where is the index of the temperature, if first soot moments are included in calcula-tions. The total mass of the cylinder gas also changes in accordance with the fuel that is add-ed. The mixing mass is taken from the existing cylinder gas particles. Figure 7.3 presents a
sche-the mixing mass injected
fuel
a) Before injection c) After injection Cylinder gas
(air particles, rest gas particles)
fuel mixed with air, distributed
in new added particles
b) Gas mixing mass, as portions of the cylinder air, taken over from cylinder gas for mixing
(7.1)
(7.2)
(7.3)
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matic for redistribution of the mass and species in a particle after a certain portion of it, represented by a dashed line, is removed and is used for mixing with the fuel.
Figure 7.3 Schematic of an existing particle and of the ratios of masses transferred from it.
A given species , represented as gray in the figure, is proportionally distributed between the remaining particles and the mass collected for mixing with the fuel. For each individual particle used in mixing, account is taken of the distribution of the particles species.
If , denotes the mass of species taken from particle for contributing to the mixing mass, for the old particles the mass fraction of the updated species in particle is:
∆ , , 1, … ,
whereas the new and added particles all have the same species mass fractions, where , is the mean mass of species for particles present:
∆ , , , 1, … ,
The updated temperatures of the old particles and the new and added ones respectively are:
∆ , 1, … ,
∆ , 1, … ,
mp(t)
mmp
mip
(t)
mi,mp
i
(7.4)
(7.5)
(7.6)
(7.7)
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The gas mass needed for vaporizing the fuel is calculated, account being taken of the pressure at which the process occurs being constant, and of the final temperature of both the injected fuel and the mixing mass after completion of the process being equal to the vaporization tempera-ture, which corresponds to that specific pressure:
, ,
where the index denotes the injected fuel, and the index vaporization.
In the case in which the fuel is a mixture of several different species, each species is dealt with separately in terms of Equation 7.8, since each has a specific vaporization temperature.
The process of fuel injection is considered in itself as being a distinct event in the series of opera-tor splits. The final step in the fuel injection process is the re-initialization of the PDF particles when the new fuel particles are introduced, the masses of all existing particles being adjusted accordingly.
Figure 7.4 Basic steps in the fuel injection model.
The algorithm for the fuel injection event at each time step in the calculations, shown in Figure 7.4, is as follows:
1. Check if current time is in the range of any of the injection events. If true, follow steps 2…5, otherwise step out.
(7.8)
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2. Calculate the amount of vaporized fuel (for each fuel species) from the externally pre-scribed vaporization rate that corresponds to the time step.
3. Determine the vaporization temperature that corresponds to the current pressure.
4. Calculate the amount of mixing mass that corresponds to the vaporization energy.
5. Reset the particles properties, when new particles are added, in accordance with Equa-tions (7.5) – (7.8).
Figure 7.5 provides a picture of what occurs when one investigates the evolution of the particle masses during a set of calculations. The x-axis is evolution over time (CAD), y-axis is the particle mass and z-axis is the number of particles. Only a small number of particles are represented. At the start, only equal particles are present.
Figure 7.5 Representation of a selection of the particle masses during fuel injection.
In the course of time, fuel particles are added consuming mass from all the existing particles, which leads to the mass of the particles decreasing, but since it is assumed that all the particles are of the same weight each fuel particle is only allowed to grow until it reaches the mass of all of the other particles. When this happens, any fuel mass remaining at a given time step leads to the creation of new fuel particles, all of them obeying this condition, except for the single fuel par-ticle created last which at any given time step can have a smaller mass than any of the others. At the next time step, this last particle is filled up, this process being repeated until all the fuel has been injected. Figure 7.5 shows the smooth evolution of the particle masses.