Spray model

For the spray model the Langrangian model, implemented n STAR CD, (Dispersed multi-phase flow model) was used. For cases in which the number of droplets is relatively small, mass, momentum and energy conservation equations can be used for each element. However, when the number of droplets is high (like this study), a statistical approach is used. In this approach, elements (droplets) with the same properties are grouped into parcels [107]. The total population is represented by a finite number of parcels.

However, the interfacial forces induced by the droplets motion to the continuous phase, relative to the in-cylinder air, may result in unstable behaviour of the droplets. Therefore, a break up model is required to determine the rate of change of the size of the droplets. During this study, the Reitz Diwakar model was used [155, 156]. In this model, the break-up of the droplets due to the aerodynamic forces affecting them, occurs by one of the following two modes [107, 163, 164]:

1) ‘Bag break-up’ mode, in which the droplet is expanded in the low-pressure wake region due to the influences of the non-uniform pressure field around it, and, eventually, when the surface tension forces are overcome, it integrates.

2) ‘Striping break-up’ mode, in which the liquid is removed, stripped or sheared from the surface of the droplet.

In each of these two cases, theoretical studies have provided a criterion for the onset of break-up and concurrently estimations for the break-break-up process time scale, 𝜏_𝑏 and the stable droplet diameter, 𝐷_𝑏 . This allows the calculation of the break-up rate by [107, 163]:

79

𝑑𝐷_𝑑

𝑑𝑡 = −^{𝐷𝑑−𝐷}_𝜏^{𝑑,𝑠𝑡𝑎𝑏𝑙𝑒}

𝑏 (3.50) where 𝐷_𝑑 is the instantaneous droplet diameter.

Moreover, the time scales and the criteria for each one of the two modes are:

‘Bag break-up’ mode

During this mode, a critical value of the Weber number, 𝑊𝑒, is used for the determination of the instability:

𝑊𝑒 ≡^{𝜌|𝑢−𝑢}_2𝜎^{𝑑|2𝐷𝑑}

𝑑 ≥ 𝐶_𝑏𝐼 (3.51) in which 𝐶_𝑏𝐼 is the empirical coefficient and its value ranges between 3.6 to 8.4 [155, 156].

During this research, a value of 𝐶_𝑏𝐼 =6 was used. Moreover, 𝜎_𝑑 is the coefficient of the surface tension and the stable droplet size , 𝐷_𝑑, is one that satisfies the equality in Equation 3.47.

Furthermore, the characteristic time is calculated by:

𝜏_𝑏= ^{𝐶𝑏2 𝜌}

12𝑑 𝐷_𝑑 3 2

4𝜎_𝑑

12 (3.52) where 𝐶_𝑏2 = 𝜋.

‘Striping break-up’ mode

For ‘‘Striping break-up’ mode, the criterion used for the onset of break-up is given by:

𝑊𝑒

√𝑅𝑒_𝑑 ≥ 𝐶_𝑠𝐼 (3.53) in which 𝑅𝑒_𝑑 is the Reynolds number of the droplet, and Cs1 is the empirical coefficient with a value of 0.5 [155, 156].

For this mode the characteristic time is given by:

𝜏_𝑏= ^𝐶₂^𝑠2(^ρ_ρ^𝑑)

1

2 𝐷_𝑑

|𝑢−𝑢_𝑑| (3.54)

80 in which the empirical coefficient 𝐶_𝑠2 is in the range of 2 to 20, 𝑢 is the instantaneous fluid velocity and 𝑢_𝑑is the instantaneous droplet velocity [155, 156].

Turbulence-controlled eddy brake up model (EBU)

For representation of the mixing turbulent chemical reaction, the eddy break up (EBU) model proposed by Magnussen was used [165]. The model was initially constructed for combustion applications and follows two basic assumptions:

1) A single step irreversible reaction is implemented into the chemical kinetics mechanism which involves the fuel (F), the oxidant (O), the products (P) and possible background inert species.

2) The time scale of the reaction is very small so that the rate-controlling mechanism of the reaction can be controlled by the turbulent macromixing.

The consumption rate of the fuel 𝑅_𝐹 is calculated by:

𝑅_𝐹 = −^𝜌𝜀_𝑘 𝐴_𝑒𝑏𝑢min [𝑌_𝐹,^𝑌_𝑆^𝑂

𝑂, 𝐵_𝑒𝑏𝑢 ^𝑌_𝑆^𝑃

𝑃 ] kg/m³s (3.55) Where R and P are the reactant and product respectively, coefficient 𝑘 takes a value between 1 ≤ 𝑘 ≤ 10. 𝐴_𝑒𝑏𝑢 and 𝐵_𝑒𝑏𝑢 are the empirical coefficients of the model. Moreover, the first two terms in the brackets of Equation 3.55 determine the local rate controlling mass fraction, while the third term is used as a reaction inhibitor when the temperature is very low. The micro-mixing time scale is taken to be 𝑘/𝜀, which is the dissipation time scale [107].

In this study, for the simulations of the pilot-injected diesel spray, the ignition and the turbulent mixing representation, C7H16 chemistry was incorporate in the developed mechanisms by using the global single-step reaction, C7H16 + 11O2 = 7CO2 + 8H2O, based on an eddy breakup (EBU) mixing representation and by specifying the reaction parameters of EBU [107]. As mentioned earlier, by using the single step reaction based on the EBU, the time scale of the reaction is very small (activation energy is zero) and therefore, n-heptane is ignited almost immediately. The ignition of n-heptane leads to the creation of a small zone of very high temperature that is sufficient to ignite the premixed syngas fuel. The modelling of pilot-injection n-heptane spray and ignition by using only the single step reaction based on the EBU, can be used accurately in situations when the injected diesel base fuel is very small. For conditions in which the amount of the injected diesel base fuel is higher, the single reaction based on the EBU has to be coupled with the appropriate chemical kinetics mechanism. The reason for that is because when micro-pilot injection is used, the ROHR profiles do not include

81 any changes due to the pilot diesel fuel combustion and the soot formation level is undetectable [62, 63]. However, when the amount of pilot-injected diesel base fuel is high, the total ROHR is significantly affected and the thermodynamic stability of the combustion changes due to the impurities created from the mixing of the diesel spray with the primary premixed syngas fuel [57].

For the mechanisms developed in Chapters 4 and 5, the amount of injected diesel fuel used was 1.2 mg/cycle, which proved to have a negligible effect on the total ROHR [61-63].

Therefore, only the single global reaction was used for the simulations of the pilot-injected diesel spray, the ignition and the turbulent mixing representation. The reaction is implemented into the mechanisms as R1 and can be found in all of the developed mechanisms in Table 4-1, Chapter 4 for the syngas mechanism, in Table 5-1, Chapter 5 for the syngas/NOx mechanism and in Table 6-4, Chapter 6 for the syngas/NOx/n-heptane mechanism. However, it is important to be mentioned here that the final mechanism proposed in Chapter 6 was validated against experimental results by using a higher amount of injected diesel-base fuel (3.0 mg/cycle). Therefore, in order to take into account the effect of the impurities created by n-heptane ignition and the co-oxidation with the premixed syngas fuel, a combination of both the single-step global reaction based on the EBU mixing representation model and the n-heptane chemistry incorporated into the developed mechanism was used. First, for the n-heptane injection and the initial ignition, the single-step global reaction based on the EBU mixing was used. Then, the low and high temperature oxidation of the remaining amount of n-heptane during the combustion process and the co-oxidation with the premixed syngas fuel were simulated using the developed chemical kinetics mechanism.