## 1300 1100 900 800 700

Core Temperature at injection (K)

Figure 7.2 Comparison of measured39 and predicted38 ignition delay times for constant volume bomb at different chamber gas temperatures nozzle at the spray edge at about 9 degrees BTDC. This is evidenced by the high temperature contours which are seen to surround the computed spray. The build-up and subsequent rapid consumption of the intermediate, branching and radical species that control the ignition process are also shown as a function of crank angle in the ignition cell in Figure 7.1. After the ignition, the gas temperature variation in the ignition cell reflects the balance between the energy released due to combustion and the energy required to vaporize the liquid fuel.

The Shell model has been modified to account also for fuel composition (Cetane number)40 and for the effect of residual gases41. Since the formation of the branching species is crucial to the ignition process, the approach was to adjust the rate of reaction of reaction (7.31d). The formation rate of Q is f4Kp, where/4 = Am exp (- Ef4/RT) [02 ]14 [RH]U with £f4 = 3.0 x 104, x4 = - 1, v4 = 0.35, and the rate Kp is given by Kong et al.3S. To account for the effect of the residual gas the constant Af04 was adjusted as a function of the cylinder gas temperature at intake valve closure, Tt ivo as

Equation (7.32) shows that the autoignition reaction becomes slower as the initial temperature increases (i.e. as the amount of residual gas increases). This is consistent with expectation since an increase in the amount of residual gas leads to a reduction in the concentration of the reactants. To account for fuel effects the activation energy of reaction (7.3 Id), E{4, was modified by the factor 65l(CN + 25)40. In other words, for a Cetane number, CN, of 40, which is typical for diesel fuels, the factor is unity and for higher values of CN the activation energy decreases, resulting in shorter ignition delays and vice versa, consistent with experimental data5.

In practical combustors, once ignition has occurred, the majority of combustion is thought to be mixing-controlled, and the interactions between turbulence and chemical reactions have to be considered. In recent years several turbulent combustion models have been proposed which include the effects of turbulence on mean reaction rates based on flamelet concepts. The model of Pitsch et al.2 combines a detailed elementary chemical kinetics model with the flamelet model concept, while the models of Dillies et al42 and Musculus and Rutland43 use simpler reduced chemistry models.

A simpler combustion model that has proved to be successful is based on the characteristic-time combustion model of Reitz and Bracco44, which was originally applied to spark-ignition engine combustion by Abraham et al.4 . The diesel combustion model used by Kong et al.2S,3S and Xin et al41 combines the 'Shell' ignition model and the characteristic time combustion model. In the combustion model, the time rate of change of the partial density of species i due to conversion from one chemical species to another, is given by dYi/dt = - (Y¡ - F,*)/tc

where K, is the mass fraction of species i. and the * indicates local and instantaneous thermodynamic equilibrium values. rc = T| + t, is the characteristic time for reaction which is formulated as a sum of timescales such that the longest timescale controls the combustion rate. T, is a laminar Arrhenius (high temperature) chemistry time, and r, =fC2 k/e is the turbulence mixing time. k and e are the turbulence kinetic energy and its dissipation rate, and C2 = 0.1 when k and e are computed using the RNG model of Han and Reitz12./is a 'delay coefficient' that accounts for the fact that turbulence does not influence early flame growth, /= ( 1 - e~r)/0.632, where r is the ratio of the amount of products to that of total reactive species. Its value varies from 0 (no combustion yet) to 1 (complete consumption of fuel and oxygen). Accordingly, the delay coefficient/changes from 0 to 1 depending on local conditions.

Xin et al*1 accounted for the effect of residual gas concentration on combustion through the laminar timescale

by introducing the residual gas concentration in the pre-exponential constant with A = (1 + 3.3xr) x 3.24 x 10~12, where xr is the residual gas mass fraction13. This ensures that the laminar characteristic time increases with increasing residual gas concentration, consistent with experimental data on laminar flame speeds, such as that of Metghalchi and Keck46. This concept of modifying the laminar timescale to account for residual gas effects was also used successfully by Kuo and Reitz47.

With the combustion model in place, the chemical source term in the species eqn (7.1) and the chemical heat release in the energy eqn (7.5) are found from

A Pn

where At is the numerical timestep, and p';„ = Apm/At and Qc = AQ/At. Further details of the combustion model are given by Kong and Reitz33 and Xin et al.*1.

Modelling of engine emissions is of critical importance for the engine industry, which is facing stringent emission regulations. Unburned hydrocarbon emissions (HC) are predicted by the characteristic time combustion model in regions where the temperatures become low enough that combustion times become very large. For NO, the extended Zeldovich mechanism48 has been widely used:

O + N2 <-» NO + N N + 02 <-» NO + O N + OH NO + H

The reaction equations are solved by assuming a steady state population of N and equilibrium for O + OH 02 + H. The resulting NO formation rate is rf[NO] = 2/?i (1 - a2 ) dt ~1 + (ccRl/(R2 + R3))

where a is the ratio of the kinetic [NO] to that which would be in equilibrium and reaction rates, R„ are given by Patterson et al.49. To close the solution, N2, O, 02, and OH are assumed to be in equilibrium at local conditions. The factor fa0 in eqn (7.38) is a constant calibration factor adjusted to allow NOx predicted from the NO model to match the engine-out NOx data. Other more comprehensive NO models have also been proposed (e.g. Miller and Bowman50). However, adequately accurate results have been obtained with the extended Zeldovich model.

Similarly, detailed soot models have been proposed (e.g., Pitsch et al?, and Fusco et al.5. However, good results have been obtained using the soot formation model of Hiroyasu52 with the Nagle-Strickland Constable (NSC) oxidation model53. The Hiroyasu soot formation model uses an Arrhenius rate expression to compute the rate of soot formation, i.e., dM%l dMt.

dMa dt where dt dt

dM, form dt

and Mfv, Mform. Mmii, Afsool are the fuel vapour, formed, oxidized and net soot masses, respectively, P is the pressure, and the Arrhenius pre-exponential and activation terms are Af = 100, £f = 12500 cal/mole, respectively. In the NSC oxidation model, carbon oxidation occurs by two mechanisms whose rates depend on surface chemistry involving more reactive A sites and the less reactive B sites. The net reaction rate is

where x , the proportion of A sites, is given by

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