Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
those that are the supplied pipes, at any junction. For an n-pipe junction, that is basically the
only addition to the theory below other than that the number of equations increases by the
number of extra junctions. With the mathematical technique of solution by the Newton-Raphson
method for multiple polynomial equations, and the matrix arithmetic handled by the Gaussian
Elimination method, the additional computational complexity is negligible.
The sign convention for particle flow is declared as "positive" toward the branch and Fig.
2.13 inherently stipulates that convention. The pressure loss criterion for flow from one branch
to another is set out by Bingham as,
Ap = CLpsC2 (2.14.1)
where the loss coefficient CL is given by the inter-branch angle 9,
C L = 1-6 " -— if 9 > 167 then CL = 0 (2.14.2)
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In any branch there are supplier pipes and supplied pipes. There are two possibilities in
this regard and these lead to two assumptions for their solution. The more fundamental assumption
in much of the theory is that the gas within the pipes is a mixture of two gases and
in an engine context this is logically exhaust gas and air. Obviously, the theory is capable of
being extended to a mixture of many gases, as indeed air and exhaust gas actually are. Equally,
the theory is capable of being extended relatively easily to branches with any number of pipes
at the junction. The theory set out below details a three-way branch for greater ease of understanding.
(a) One supplier pipe
Here there is one supplier pipe and two are being supplied, in which case the solution
required is for the reflected wave amplitudes, Xr, in all three pipes and for the reference
acoustic velocities, ao, for the gas going toward the two supplied pipes. The word "toward" is
used here precisely. This means there are five unknown values needing five equations. It is
possible to reduce this number of unknowns by one, if we assume that the reference acoustic
state toward the two supplied pipes is common. A negligible loss of accuracy accompanies
this assumption. Using the notation of Fig. 2.13, it is implied that pipe 1 is supplying pipes 2
and 3; that notation will be used here only to "particularize" the solution so as to aid understanding
of the analysis. As the pressure in the face of pipes 2 and 3 will be normally very
close, the difference between TQ2 and T03 should be small and, as the reference acoustic
velocity is related to the square root of these numbers, the error is potentially even smaller.
Irrespective of that assumption, it follows absolutely that the basic properties of the gas entering
the supplied pipes is that of the supplier pipe.
(b) Two supplier pipes
Here there is one supplied pipe and two are suppliers, in which case the solution required
is for the reflected wave amplitudes, Xr, in all three pipes and for the reference acoustic
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