Design and Simulation of Two Stroke Engines

Design and Simulation of Two-Stroke Engines
The First Law of Thermodynamics, Eq. 2.10.3, reduces to:
(G5a01(Xu - Xrl)) + G5a 2 Ql(Xn + Xrl - if
(G5a02(Xi2 - Xr2)) 2 + G5a22(Xi2 + Xr2 - 1)' = 0
The momentum equation, Eq. 2.10.4, reduces to:
Po A 2 (Xu + Xrl - 1) G7 - (Xi2 + Xr2 - if 7
Poi( x il + Xrl - lJ^GsaoifXu - Xrl)_
G5a0l{Xn - Xrl) + G5a02(Xi2 - Xr2)] = 0
(2.10.8)
(2.10.9)
The three equations cannot be reduced any further as they are polynomial functions of all
three variables. 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.
As with all numerical methods, the computer time required is heavily dependent on the
number of iterations needed to acquire a solution of the requisite accuracy, in this case for an
error no greater than 0.01% for the solution of any of the variables. The use of the Benson
"constant pressure" criterion, presented in Sec. 2.9, is invaluable in this regard by considerably
reducing the number of iterations required. Numerical methods of this type are also
arithmetically "frail," if the user makes ill-advised initial guesses at the value of any of the
unknowns. It is in this context that the use of the Benson "constant pressure" criterion is
indispensable. Numeric examples are given in Sec. 2.12.2.
2.10.1 Flow at pipe expansions where sonic particle velocity is encountered
In the above analysis of unsteady gas flow at expansions in pipe area the particle velocity
at section 1 will occasionally be found to reach, or even attempt to exceed, the local acoustic
velocity. This is not possible in thermodynamic or gas-dynamic terms as the particles in unsteady
gas flow cannot move faster than the pressure wave signal that is impelling them. The
highest particle velocity permissible is the local acoustic velocity at station 1, i.e., the flow is
permitted to become choked. Therefore, during the mathematical solution of Eqs. 2.10.7,
2.10.8 and 2.10.9, the local Mach number at station 1 is monitored and retained at unity if it is
found to exceed it.
M = ° sl = G 5 a 0l( X il ~ X rl) = G 5( X il ~ X rl)
l
sl a a X
sl 01 sl Xn + Xrl "il rl - 1
104
(2.10.10)