Design and Simulation of Two Stroke Engines

Chapter 2 • Gas Flow through Two-Stroke Engines
ing effect can be calculated as the friction force, F, is available from the combination of Eqs.
2.3.1 and 2.3.2 and the work done by this force acting through distance, dx, appears as heat in
the gas element involved.
F = 7cd
2
r Ps c s cs dt (2.3.15)
Thus the work, dWf, resulting in the heat generated, dQf, can be calculated by:
8Wf = Fdx = Fcsdt = 7ldCfPs ° sdt = 5Qf (2.3.16)
All of the relevant data for the numerical example used in this section are available to
insert into Eq. 2.3.16, from which it is calculated that the internal heating due to friction is
1.974 mJ. While this value may appear miniscule, remember that this is a continuous process
occurring for a pressure wave during its excursion throughout a pipe, and that this heating
effect of 1.974 mJ takes place in a time frame of 0.333 ms. This represents a heating rate of
5.93 W which puts the heating effect due to friction into a physical context which can be more
readily comprehended.
One issue which must be taken into account by those concerned with computation of
wave motion is that all of the above equations use a length term within the calculation for
friction force with respect to the work done or heat generated by opposition to it. This length
term is quite correctly computed from the particle velocity, cs, and the time period, dt, for the
motion of those particles. However, should the computation method purport to represent a
group of gas particles within a pipe by the behavior of those particles at the wave point under
calculation, then the length term in the ensuing calculation for force, F, must be replaced by
that length occupied by the said group of particles. The subsequent calculation to compute the
work, dWf, i.e., the heat quantity dQf generated by friction, is the force due to friction for all
of the group of particles multiplied by the distance moved by any one of the particles in this
group, which distance remains as the "csdt" term. This is discussed at greater length in Sec.
2.18.6.
2.3.2 Friction loss during pressure wave propagation in bends in pipes
This factor is seldom considered of pressing significance in the simulation of engine
ducting because pressure waves travel around quite sharp kinks, bends and radiused corners
in ducting with very little greater pressures-loss than is normally associated with friction, as
has been discussed above. It is not a subject that has been researched to any great extent as
can be seen from the referenced literature of those who have, namely Blair et al. [2.20]. The
basic mechanism of analysis is to compute the pressure loss, dpi,, in the segment of pipe of
length which is under analysis. The procedure is similar to that for friction:
Pressure loss dpb = Cbpsc^ (2.3.17)
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