Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
The continuity equation set in Eq. 2.16.4 reduces to:
.G5, vG5
PoiXt CdAtct - Po2(Xi2 + Xr2 - l) u;, A2G5ao2(Xi2 - Xr2) = 0 (2.16.8)
The First Law of Thermodynamics in Eq.2.16.5 reduces to:
G5(aoiX!) 2 - (G5a02(Xi2 - Xr2)) + G5ag2(Xi2 + Xr2 - if (2.16.9)
The First Law of Thermodynamics in Eq. 2.16.6 reduces to:
G< ( a 0lXi) 2 - (a0iXt) 2 - cf = 0
The momentum equation, Eq. 2.16.7, reduces to:
Po[xt G7 - (Xi2 + Xr2 - 1) G7
+[p02(Xi2 + Xr2 - 1) G5 x G5a02(Xi2 - Xr2)] x
[c, - G5a02(Xi2 - Xr2)] = 0
(2.16.10)
(2.16.11) '
The five equations, Eqs.2.16.8 to 2.16.11, cannot be reduced any further as they are polynomial
functions of the four unknown variables, Xr2;, X(, ao2; and ct. 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.
2.16.1 Outflow from a cylinder where sonic particle velocity is encountered
In the above analysis of unsteady gas outflow from a cylinder the particle velocity at the
throat will quite commonly be found to reach, or even attempt to exceed, the local acoustic
velocity. This is not possible in thermodynamic or gas-dynamic terms. The highest particle
velocity permissible is the local acoustic velocity at the throat, i.e., the flow is permitted to
become choked. Therefore, during the mathematical solution of Eqs. 2.16.8 to 2.16.11, the
local Mach number at the throat is monitored and retained at unity if it is found to exceed it.
As, Mt = it
a 01 X t
= 1 then ct = a01Xt (2.16.12)
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