where Rj is the average gas constant for the zone, mass-averaged across the different species, and Rkj is the gas constant for the kth species in zone j. The mass-averaged values of the various 'bar' terms are from, for example,

While eqns (6.113) and (6.114) appear complex all of the terms in them can be evaluated. Further, they can be manipulated into a form suitable for rapid computer evaluation of P and the J zone values of 7}. The manipulation is straightforward but lengthy, and is not given here. Basically, eqns (6.114), are multiplied through by V} and rearranged to give Vj which is substituted back into eqns (6.113), further rearrangement giving J equations of the form:

where a, b, and c are combinations of calculable terms in (6.113) and (6.114). Again multiplying the J equations (6.114) by Pj and adding them together gives a single equation of the form:

(XjP - Z (PjTj) = Yj - P T, Vj = Yj -PV = e j say, as Z Vj = Vcv where again a, ¡3, y and e are combinations of calculable terms from (6.113) and (6.114). Thus using (6.116)

dP/dt can thus be evaluated from (6.117), and substitution back into (6.116) yields J values of dTj/dt, one for each zone. The equations require (as well as all of the data discussed below) values of individual zone volumes V, and these can be assumed proportional to the zone masses if better knowledge is not available.

To solve the above equations requires values of u, h and R, and their partial derivatives with respect to P, 7 and beyond those in, for example, Figure 6.2. These are given by the empirical methods or property routines, mentioned in section 6.3 due to Kreiger and Borman (KB) and Martin and Heywood (MH), or can be generated from an equilibrium model based on extension of the reactions eqn (6.24). Any enhanced model will have to use some form of property routine, and while the empirical KB and MH methods are sufficient for a thermodynamic model it is necessary to use an equilibrium model, with other extensions, if emissions are to be calculated as the KB and MH methods do not represent individual molecular species.

For high temperatures where dissociation occurs, eqn (6.24) can be extended to represent dissociation species. Normally 10 combustion products are dealt with and (6.24) becomes

There are now 10 unknown ¿>, values and to determine these one uses the four atom balances used with (6.24) together with six dissociation reaction equations. The latter relate reaction equilibrium constants to the concentrations of the different molecules. As the equilibrium constants can be obtained from tables and expressed as polynomials in temperature, these equations can be used with the atom balance to determine the concentrations of the different molecules3'7. Solving 10 simultaneous equations requires quite complex matrix manipulation but it does give the concentrations of the 10 species in an equilibrium mixture for any given P, T, <p set. With the concentrations known, the mixture molecular weight, gas constant R, enthalpy/internal energy, and derivatives of R and u can be found by the same methods as outlined in section 6.3 for the simpler non-dissociated gas mixture. Such an equilibrium model is slow running compared to the empirical methods, but is necessary if emission calculations are to be carried out.

While an equilibrium model is satisfactory for hydrocarbon products of combustion, and hence for their contributions for emissions, it is not suitable for NOv as the NOA reactions are too slow to reach equilibrium. Accordingly for calculation of NO, emissions, the equilibrium model is supplemented with a non-equilibrium NOj calculation, usually based on the extended Zeldovich reaction equations1,3. While the CO reactions are faster than those for NOx, the CO concentration can also differ somewhat from equilibrium and similar techniques have sometimes been used to deal with this rather than assume CO to be at equilibrium concentration.

Before leaving gas properties it is worth mentioning air, unburnt fuel and burnt gas mixtures. For such a mixture the average gas constant is given by

where Ru, Rb are gas constants for air, unburnt fuel and burnt gas respectively, while the x values are corresponding mass fractions. Precisely analogous expressions apply to mixture average enthalpy and internal energy (with mass based hb, ub, etc.). Derivatives for the mixture are then, for example, dT ~ ' ( dT J a dT u dT b dT

with corresponding equations for derivatives w.r.t. P and <j> and for u. For air and unburnt fuel, derivatives with respect to P and <j> are zero, while the temperature derivatives can be found similarly to those for combustion products (but much simpler).

The need for at least two zones in the cylinder has been mentioned as being required to carry out emission calculations, and it is appropriate to briefly introduce more complex submodels of combustion. At least two zones are required because of the very different composition and temperature of the burnt and unburnt parts of the cylinder contents at any instant during combustion. The empirical methods of section 6.7 can in fact be used for a simple two-zone model, as they give the masses of burnt/unburnt gases, but because of the complex mechanism of mixing and burning in a diesel engine in particular, a larger number of zones are really required. Models exist with several hundred zones, though in practice, for run-time reasons these are normally used in a single-cylinder combustion model rather than a complete cycle simulation. It may be noted that eqns (6.113) to (6.117) can be used to determine cylinder pressure and temperatures in any number of zones provided information is available on the contents and properties of each zone.

The normal modelling procedure1'1213 is to model the time behaviour of each injector spray and its surrounding air/residual gas. The spray, including mixing regions at the edges, is divided into a number of zones, new zones being formed as the spray develops and existing zones expanding as the spray moves away from the nozzle and entrains the cylinder air. The fuel starts as a liquid, forms droplets and eventually evaporates, these processes being modelled. Empirical relationships for spray penetration provided the velocity of zones on the spray centreline at any instant, assumptions about jet mixing giving the relative velocities of off-centre zones. With zone velocities known, zone mass and hence the amount of air entrained in a timestep can be calculated by assuming conservation of momentum. Ignition delay for each zone can be calculated by an equation similar to (6.74) Combustion calculations can be based either on achieving a stoichiometric mixture in the zone or on a reaction rate equation of the standard Arrhenius type, rab = ap2[Fuel vapour] [Oxygen]b exp (- E/RT) (6.121)

The burn rate in the zone is thus dependent on concentrations of fuel vapour and oxygen, temperature, density (p) and on empirical reaction activation energy (E); a and b are empirical constants. As well as mass transfer between zones, from the entrainment model, heat transfer also has to be allowed for. With all of this information, the energy equation leads to cylinder pressure, and temperatures of the individual zones. Use of an equilibrium model, with additional calculations for NOx and solids, will then give the relative quantities of species in each zone which can be summed over the cylinder to give total emission products at each timestep. The calculations need to be modified somewhat later in the combustion process both as the spray impinges on the cylinder walls and when all the air/residual gas in entrained, but the principles are similar.

Such models are obviously quite complex but do give a reasonable representation of combustion and allow estimation of emissions, but the complexity of the in-cylinder processes are such that small changes in, for example, cylinder geometry can give quite large changes in emissions. There are unfortunately not many simpler intermediate sub-models currently available lying between two-zone and many-zone sub-models, as these may be more suitable for complete engine simulations.

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