Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
(b) two supplier pipes, i.e., pipes 1 and 2 supplying pipe 3,
CL12=0 CU3=1.6-^3. (21419)
ID/
Xs? 1 - XS G 2 72 = 0 Pofx^ 71 - Xg 7e3 ) = CL13ps3cs 2 3 (2.14.20)
In the analysis for the First Law of Thermodynamics and the pressure loss terms it should
be noted that the relationships for Xs and cs for each pipe are written in full in the continuity
equation.
These functions can be solved by a standard iterative method for such problems. I have
determined that the Newton-Raphson method for the solution of multiple polynomial equations
is stable, accurate and rapid in execution. The arithmetic solution on a computer is
conducted by a Gaussian Elimination method.
Note that these are the three reflected wave pressures and T02 and/or T03, depending on
the thermodynamic assumptions debated earlier. In practice it has been found that the "constant
pressure" criterion provides excellent initial guesses for the unknown variables. This
permits the numerical solution to arrive at the final answers for them in two or three iterations
only to a maximum error for the worst case of just 0.05% of its value.
This more sophisticated branched pipe boundary condition can be incorporated into an
unsteady gas-dynamic code for implementation on a digital computer, papers on which have
already been presented [2.40, 2.59].
2.14.1 The accuracy of simple and more complex branched pipe theories
The assertion is made above that the "constant pressure" theory of Benson is reasonably
accurate. The following computations put numbers into that statement. A branch consists of
three pipes numbered 1 to 3 where the initial reference temperature of the gas is 20°C and the
gas in each pipe is air. The angle 612 between pipe 1 and pipe 2 is 30° and the angle ©13
between pipe 1 and pipe 3 is 180°, i.e., it is lying straight through from pipe 1. The input data
are shown in Table 2.14.1. The pipes are all of equal diameter, d, in tests numbered 1, 3 and 4
at 25 mm diameter. In test number 2 the pipe 3 diameter, d3, is 35 mm. In tests numbered 1 and
2 the incident pulse in pipe 1 has a pressure ratio, PJI, of 1.4 and the other pipes have undisturbed
wave conditions. In tests numbered 3 and 4 the incident pulse in pipe 1 has a pressure
ratio, Pji, of 1.4 and the incident pulse in pipe 3 has a pressure ratio, PJ3, of 0.8 and 1.1,
respectively.
The output data for the calculations are shown in Table 2.14.2 where the "constant pressure"
theory is used. The symbols are Pri, Pr2, and Pr3 for the pressure ratios of the three
reflected pressure waves at the branch, for superposition particle velocities ci, C2, and C3. The
output data when the more complex theory is employed are shown in Table 2.14.3. The computed
mass flow rates (in g/s units) riij, m2, and m3 are shown in Table 2.14.4. In Table
2.14.5 are the "errors" on the computed mass flow between the "constant pressure" theory
and the more complex theory. The number of iterations is also shown on the final table; the
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