Decomposition Analysis of an Automotive Powertrain Design ...

the surface-to-volume ratio at a volume 1/3 of swept volume. About 50% of the heat loss in the
Otto cycle occurs by that time in the thermodynamic cycle.
The breathing characteristics of a naturally aspirated engine depend on the ability of the
fluid motion to follow the piston motion. The piston motion effects a rarefaction wave that
propagates through the induction system to the throttle and results in an attendant compression
wave which arrives at the intake valve. For engines with fixed geometry induction systems, the
single torque peak occurs where the compression wave is 'tuned' to the runner lengths, cam
events, and piston speed. For variable geometry induction systems, multiple torque peaks can be
obtained. This tuning behavior is described with the Mach index, Z, the volumetric efficiency, η v
and the tuning pressure, ∆p t . These are, respectively, a normalized speed of sound, a normalized
mass flow, and a scalar approximation to the effects of the compression wave.
The Mach index, defined as (Taylor, op cit.),
Z = 2 s N e (b/d i ) 2 /(60 a o C s n v ) (46)
is the ratio of an idealized flow velocity past the valve to the speed of sound. The discharge
coefficient, C s , accounts for flow losses through the induction system and is expressed as
C s = C stiff C m C p (i vc - i vo )/180 (C m
2 + C p
2 - C m
2 C p
2 ) -0.5 . (47)
The volumetric efficiency is the volume of air flow (referenced to standard conditions)
normalized against the volume displaced by the piston. The tuning behavior is characterized by
the volumetric efficiency curve as a function of Mach index and engine variables. It can be
represented at various levels of detail. In the simplest approach, a baseline is determined and
represented as function of Z,
η v = f (Z). (48a)
As bore and stroke are changed the engine speed of the tuning peak changes accordingly. Such an
approach only describes the relationship between piston speed and compressibility.