Design and Simulation of Two Stroke Engines

Chapter 2 - Gas Flow through Two-Stroke Engines
and replacing the distance from the origin as y, i.e., replacing (x+L), then Eq. A2.1.5 becomes:
a o ( dy V Y_1 3 2 y 9 2 y
3x. dx z at"
(A2.1.6)
This is the fundamental thermodynamic equation and the remainder of the solution is
merely mathematical "juggling" to effect a solution. This is carried out quite normally by
making logical substitutions until a solution emerges. Let _2_ — f _£. I, in which case the
following are the results of this substitution: °t ^" x '
Thus, by transposition:
^ r f ^ i / V ) where f ( ^
3t ydx J dt ydx J ydx)
df .dx,
"3y
.3x,
2.. /^.A r> fr\..\ f^..\ ^ ( C^,S\
at 2 lax J ax vat J vaxjaxl ydx))
Hence ^y = f(^V(—1—
at 2 ydx) ydx) dx 2
or
( r*y>tf a 2 y
r ydxjj
v
dx'
This relationship is substituted into Eq. A2.1.6 which produces:
4 ^yV7 " 1 9 2 y (
dx. 3x'
-Y-l
'dy\
V3x>
f
dy X\ 2 d 2 y
ydx) j dx'
= ±i-f(^
a
0 ydx J
Integrating this expression introduces an integration constant, k:
l-Y
'dy_y
.dx;
2a0 ydx)
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