Design and Simulation of Two Stroke Engines

whence the particle movement, dx, is given by:
Chapter 2 - Gas Flow through Two-Stroke Engines
dx = csdt = 99.1 x 0.333 x 10~ 3 = 33 x 10~ 3 m
From the above superposition time element it is seen that this computes to a numerical
value of 0.333 ms. Assuming a friction factor Cf of 0.004, the above equations in the section
show that the loss of superposition pressure occurs over a distance dx of 33 mm within the
duct and has a magnitude of 122 Pa. The pressure ratio of the rightward wave drops from 1.2
to 1.198 and that of the leftward wave rises from 0.8 to 0.8023. The rightward propagation
velocity of the compression wave during superposition drops from 440.5 to 439.5 m/s while
that of the leftward expansion wave rises from 242.3 to 243.4 m/s. The superposition particle
velocity drops from 99.1 m/s to 98.05 m/s.
It is evident that the pressure loss due to friction reduces the amplitude of compression
waves and slows them down. The opposite effect applies to an expansion wave: it raises its
absolute pressure, i.e., weakens the wave, and thereby moves its propagation velocity from a
subsonic value toward sonic velocity.
While the likelihood of a single traverse of a pressure wave in a duct of an engine is
remote, nevertheless it should be considered theoretically. By definition, friction opposes the
motion of a pressure wave and does so continuously. This means that a train of pressure
waves is sent off in the opposite direction to the propagation of the wave train and with a
magnitude which can be calculated from the above equations. Use the data above, but with
the exception that the single wave, p^ traveling rightward, i.e., in the positive direction as far
as the sign convention is concerned, has a pressure ratio of 1.2. All other data remain the same
and a fixed friction factor of 0.004 is employed. All of the above equations can be used with
the value of p2 inserted as being identical to po, i.e., with a pressure ratio of 1.0. The results
show that the ongoing wave pressure ratio, Pif, is reduced to 1.1994 and the reflected wave,
P2f, is 1.0004. If the calculation is repeated to find the effect of friction on a single traverse of
an expansion wave, i.e., by inserting Pi as 0.8 and P2 as 1.0, then Pjf and P2f become 0.8006
and 0.9995, respectively.
In Sec. 2.19 the traverse of a single pressure wave in a duct is described as both theory
and experiment.
2.3.1 Friction factor during pressure wave propagation
It is possible to predict the value of friction factor more closely by considering further
information available in the literature of experimental and theoretical fluid mechanics. The
properties of air for thermal conductivity, Ck, and viscosity, u, are a function of absolute
temperature, T, and are required for the calculation of the shearing forces in air from which
friction factor can be assessed. The interconnection between friction factor and shear stress
has been set out in Eq. 2.3.1. The thermal conductivity and viscosity of air can be found from
data tables and curve fitted to provide high accuracy from values of T from 300 to 2000 K as:
Ck = 6.1944 x 10- 3 + 7.3814 x 10~ 5 T - 1.2491 x lO" 8 T 2 W/mK (2.3.10)
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