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With the fuel fraction burnt, x, known at any instant the total fuel burnt can be obtained simply by multiplication by the cylinder's fuel charge per cycle (Mh). To solve eqn (6.6) the rate of burning fuel (dM,/dt) is required and this can be obtained by either differencing (xMft) values at successive timesteps or by differentiating eqns (6.76) and (6.77):

.¿P = 5000Ci2"-' (1 - Q" )5000-1 0 (6.78)

where 6 is the (non-dimensional) rate of change of crank angle (6N/BDUR for N rpm, BDUR in deg. CA). The value of BDUR in the equation is not critical as long as it is at least as long as actual combustion duration and a value of 125 degrees was used originally.

To a reasonable approximation for much work (dM/dt) in eqn (6.6), the rate of change of total cylinder mass at any instant, can be taken to be the same as (dMf/dt) and the actual injection pattern ignored; this is equivalent to the assumption that the fuel is injected as in is burnt. Alternatively an injection time profile can be input, either giving the variation in rate of fuel per unit time or the injector open area; in the latter case fuel flow rate at any instant can be calculated from the hydraulic eqn (6.44), using an appropriate discharge coefficient and an instantaneous pressure difference across the injector taken as fuel rail pressure less current cylinder pressure. If injection is represented explicitly then the fuelling rate is included in the (dM/dt) term of (6.6); (dMf/dt) is still the fuel burning rate as the form of model described uses a burnt-fuel based equivalence ratio. Whatever method is used to represent injection/combustion it will be necessary to provide the model with data on start of injection time for each cylinder, normally speed dependent. For steady state modelling it is usually sufficient to specify input fuel per cycle per cylinder as an input variable but alternatives such as specifying fuel-air ratio and letting the model determine fuelling are fairly straightforward. For transient modelling (section 6.4) it will be necessary to in corporate a fuel pump/governor sub-model to determine fuelling as engine speed and load change; there will usually also need to be an aneroid control override to limit fuelling at low boost pressure. For more modern fuelling systems fuel control 'maps' (look-up tables) will be required

(again generally specified in terms of engine current speed and load) which can be interpolated to obtain current injection timing, pattern, fuel quantity, etc.

Before leaving combustion it is worth giving some further information on heat release. First, the approximate heat release up to any instant is given by

where x is the current fraction burnt from (6.75,6.76,6.77), Mfl is the fuel charge and <2i.cv- the fuel's lower calorific value. The heat released of course is split approximately between raising the charge temperature, doing work on the piston and heat transfer through the chamber walls. It is also worth mentioning that when a fuel-air mix bums, there is no change in specific enthalpy, that is ha = hb or tiu + RTU = ub + RTb (6.80)

with obvious subscripts. If the specific internal energies of the unburnt mix and burnt gas are known, with the unburnt mix temperature immediately before combustion, the equation gives the adiabatic flame temperature. Finally, while it should be apparent that by deriving fuel burn data from eqns (6.74)—(6.78) and combining this with the energy equation it is possible to compute the time variation of P and Tin the cylinder, it may not be apparent that one can do the reverse. That is, using a measured cylinder pressure diagram one can reverse the procedure to calculate burnt fuel fraction variation and, by repeating the exercise for alternative engine speeds and loads, go further back and generate values of the various coefficients in eqns (6.74)-(6.78) for the specific engine of interest. However, this is quite a lengthy exercise analytically, and it is important to have very accurate cylinder pressure measurements as any small ripple in the pressure curve (real or otherwise) will cause a large bump in the resulting burn rate curve. Normally, empirical adjustment of the coefficients in the equations is adequate and quicker.

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