Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
an engine simulation value, i.e., the "actual" coefficient of discharge, Cda, is then determined
as:
u _ Aeff n At
k eff - —— - C da -— " C da k (A2.3.4)
A2 A2 v
The algebraic solution to this iterative procedure (for it requires a solution for the several
unknowns from an equal number of simultaneous polynomial equations) is not trivial. The
number of unknowns depends on whether the flow regime is inflow or outflow, and it can be
subsonic or sonic flow for either flow direction; the number of unknowns can vary from two
to five, depending on the particular flow regime encountered. The iterative procedure is completed
until a satisfactory error band has been achieved, usually 0.01% for any one unknown
variable.
Then, and only then, with the incorporation of the actual discharge coefficient, Cda,at the
same cylinder-to-pipe pressure ratio, P, and geometric area ratios, k, into the simulation will
the correct value of mass flow rate and pressure wave reflection and formation be found in the
replay mode during an unsteady gas-dynamic and thermodynamic engine computer simulation.
Apart from some discussion in a thesis by Bingham [2.64], I am unaware of this approach
to the determination of the actual coefficient of discharge, Cda, being presented in the literature
until recent times [5.25]. I have published a considerable volume of Cd data relating to
both two- and four-stroke engines, but all of it is in the format of Cdi and all of it is in the
traditional format whereby the ideal mass flow rate was determined by an isentropic analysis.
Where the original measured data exist in a numeric format, that data can be re-examined and
the required Cda determined. Where it does not, and the majority of it no longer exists as
written records, then that which I have presented is well nigh useless for simulation purposes.
Furthermore, it is the complete digitized map, such as in Fig. 2.27, that is needed for each and
every pipe discontinuity to provide accurate simulation of unsteady gas flow through engines.
Some measurements ofCj at the exhaust port of an engine
In Figs. A2.4 to A2.7 are the measured discharge coefficients for both inflow and outflow
at the exhaust port of a 125 cm 3 Grand Prix racing motorcycle engine, as shown in Fig. A2.3
[5.25]. These figures plot the discharge coefficients with respect to pressure ratio from the
cylinder to the pipe, P, and for geometrical area ratios, k, of 0.127, 0.437, 0.716 and 0.824,
respectively. On each figure is shown both the actual discharge coefficient, Cda, and the ideal
coefficient, Cdi, as given by Eq. A2.3.3. It should be noted that any difference between these
two numbers, if replayed back into an engine simulation at identical k and P values, will give
precisely that ratio of mass flow rate difference.
At a low area ratio, k, there is almost no difference between Cdi and Cda- The traditional
analysis would be equally effective here. As the area ratio increases, and where the mass flow
rate is, almost by definition, increasing, then the mass flow rate error through the application
of a Cdi value also rises. The worst case is probably inflow at high area ratios, where it is seen
to be 20%.
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