Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
The First Law of Thermodynamics in Eq. 2.12.6 reduces to:
GsfMXn + J^ - l)) 2 - (a01Xt) 2 } + (GsaoiJX,, - Xrl)) 2 - c 2 = 0 (2.12.11)
The momentum equation, Eq. 2.12.7, reduces to:
PoA2[x? 7 - (Xi2 + Xr2 - l) Gr
+[poi(Xn + Xrl - l^A^aoifXn - Xrl)][ct + G5a01(Xi2 - Xr2)] = 0 (2A2A2)
The five equations, Eqs. 2.12.8 to 2.12.12, cannot be reduced any further as they are
polynomial functions of all five variables, Xri, Xr2>, Xt, ao2i 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. Numerical examples are given in Sec. 2.12.2.
It is not easy to supply initial guesses which are close to the final answers for the iteration
method on a computer. The Benson "constant pressure" assumption is of great assistance in
this matter. Even with this assistance to the numerical solution, this theory must be programmed
with great care to avoid arithmetic instability during its execution.
2.12.1 Flow at pipe restrictions where sonic particle velocity is encountered
In the above analysis of unsteady gas flow at restrictions in pipe area 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. 2.12.8 to 2.12.12,
the local Mach number at the throat is monitored and retained at unity if it is found to exceed
it.
c
As, M t = —— = 1 then c t = amXt (2.12.13)
a 01 A t
This simplifies the entire procedure for this gives a direct relationship between two of the
unknowns and replaces one of the equations employed above. It is probably easier from an
arithmetic standpoint to eliminate the momentum equation, but probably more accurate thermodynamically
to retain it!
The acquisition of all related data for pressure, density, particle velocity and mass flow
rate at both superposition stations and at the throat follows directly from the solution of the
four polynomials for Xri, Xr2, Xt, ao2) and ct.
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