Design and Simulation of Two Stroke Engines

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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 122
Chapter 2 - Gas Flow through Two-Stroke Engines fact that it requires only two iterations to close a worst-case error of 0.05% on any variable reveals the worth of the Benson "constant pressure" theory for the provision of a very highquality first guess at the "unknowns." No. 1 2 3 4 Pi1 1.4 1.4 1.4 1.4 Table 2.14.1 Input data for the calculations Pi1 1.0 1.0 1.0 1.0 Pi1 1.0 1.0 0.8 1.1 di 25 25 25 25 Table 2.14.2 Output data from the calculations using "constant pressure" theory No. 1 2 3 4 Pr1 0.891 0.841 0.767 0.950 Prt 1.254 1.188 1.086 1.333 Pr1 1.254 1.188 1.345 1.215 d2 25 25 25 25 d3 25 35 25 25 C1 112.6 126.3 148.5 97.0 012 30 30 30 30 c2 56.3 42.7 20.4 72.0 613 180 180 180 180 C3 T01 20 20 20 20 56.3 42.7 128.1 25.0 Table 2.14.3 Output data from the calculations using complex theory No. 1 2 3 4 Pr1 0.905 0.848 0.769 0.974 Pr1 1.228 1.171 1.084 1.291 Pr1 1.273 1.197 1.350 1.245 C1 108.7 124.3 147.7 90.9 c2 51.4 39.5 19.9 64.4 C3 60.8 45.2 129.7 Table 2.14.4 Output on mass flow from constant pressure and complex theory No. 1 2 3 4 Constant Pressure Theory m. 78.3 84.5 93.2 70.4 rhn 39.1 28.5 12.8 52.3 m3 39.1 55.9 80.4 18.1 123 mi 76.4 83.6 92.9 67.2 Complex Theory rh2 31.3 m3 34.6 42.0 25.5 58.1 12.3 80.6 44.8 22.7
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Design and Simulation of Two-Stroke
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A Second Mulled Toast When as a stu
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Acknowledgments As explained in the
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Table of Contents Nomenclature xv C
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Table of Contents 2.10.1 Flow at pi
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Table of Contents 3.5.3.1 The use o
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Table of Contents 6.1.3 The effect
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Nomenclature NAME SYMBOL UNIT (SI)
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speed of rotation speed of rotation
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attenuation or transmission loss wa
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Design and Simulation ofTwo-Stroke
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Design and Simulation ofTwo'Stroke
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Chapter 4 Combustion in Two-Stroke
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Chapter 4 - Combustion in Two-Strok
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a stratilpl o tions to ivior. If m
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CD 30 -, W 20 - LU CO iS _J LU OC 1
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181 also x3 = — = 0.905 x6 = 2.17
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to be cur .3.20) .3.21) .3.22) for
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3.23) com- :on is hhas has a /eng-
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stroke .3.29) mean me to 3land ssio
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a LU z: en 3 CO a LU z EC D m O ho
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CD 1.2 -i 1.0 - di „„ LU 0.8 cc
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CO —> 111" cc LU < LU _J LU CC £
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Combustion in compression-ignition
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stroke ari more often lseful lental
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lat the ssi ;tribu- ELEVATION VIEWS
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CYLINDER HEAD TYPE IS CENTRAL BORE,
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CO E 50 40 - O O _i Hi > I CO 30 Z)
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CURRENT INPUT DATA FOR 2-STROKE 'SP
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Cylin- PP £ Pa- tics of Paper Chap
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vfitro- ;titi >tants 1970. i(In- Ch
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£2 = Pi P2 I Pi J Chapter 4 - Comb
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CHAINSAW AT 9600 rpm. Chapter 4- Co
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LLI •z. O N -z. 0.21 -, 0.20 DC 0
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Chapter 7 Reduction of Fuel Consump
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Chapter 7 • Reduction of Fuel Con
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Chapter 7 - Reduction of Fuel Consu
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^ s C£ Q 2^ E Q. Q_ z' o ISSI HCEM
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o > z~ Q w CO UJ w g x O z o z o m
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0.35 • 0.34 LU ° 0.33 - c 03 DC
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§ 2000
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y, ,6 Chapter 7 - Reduction of Fuel
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Chapter 7- Reduction of Fuel Consum
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INLET FOR r AIR AND FUEL Chapter 7
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Chapter 8 Reduction of Noise Emissi
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Chapter 8 - Reduction of Noise Emis
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• • / . • Appendix Listing of
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Active radical (AR) combustion and
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initial conditions, assumptions reg
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in two-stroke engine (schematic dia
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exhaust timing control valve, effec
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I Exhaust emissions/exhaust gas ana
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Exhaust systems (continued) untuned
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gas constant/universal gas constant
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in crankcase heat transfer analysis
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Mercury Marine air-assisted fuel in
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I PISTON POSITION (computer program
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Pressure wave reflection (in pipes)
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local acoustic velocity, 60 local M
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Roots blower in turbocharged/superc
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Simulation, engine See Computer mod
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temperature vs. crankshaft angle (c
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work per thermodynamic (Otto) cycle
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Design and Simulation of Two Stroke Engines

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