Page 741 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 2 2. 1 Exterior noise: Assessment and control

Now as so
gP
2
vf v u r ¼ a 2 (22.1.40)
¼ (22.1.33)
vt vxvt
Substituting equation (22.1.40) into equation (22.1.36)
most workers omit that term so that the 1-D, non-con- and thus assuming isentropic expansion of the gas as it
servative, inviscid, irrotational, homentropic momentum enters the cylinder:
equation, along a streamline (!) becomes:

1=g
gp 01 p 2
r ¼ (22.1.41)
2
g 1 2 a 2 p 01
2
2
a ¼ a þ 2 u (22.1.34) 01
0
Re-writing equation (22.1.34) with the appropriate
Benson (1982) famously calls this an energy equation subscripts
(although this is probably a misnomer).
From the derivation of equation (22.1.34) a simple 2 2 g 1 2
and very well-known intake valve flow model can be a 01 ¼ a þ 2 u 2 (22.1.42)
2
constructed that assumes a flow from a large reservoir
(representing the manifold) into one of the cylinders via Re-arranging equation (22.1.42)
a single orifice of negligible length. This was first derived
by Tsu in 1947. Stagnation conditions (subscript 0,1 u ¼ða 2 a Þ 2 (22.1.43)
2
2
meaning zero flow velocity in zone 1) are assumed in the 2 01 2 g 1
cylinder.
For inflow to the cylinder via the intake valve, the Substituting equation (22.1.39) into equation (22.1.37)
intake manifold is assumed to constitute a sufficiently gives, with the appropriate subscripts,
large volume for constant pressure conditions to occur, 2 g 1=g
and the conditions in the manifold are given subscript ‘2’. T 2 ¼ a 01 p 2 (22.1.44)
For outflow (reverse flow through the intake valve) the gR p 01
subscripts are reversed.
From a principle of continuity of mass, assuming quasi Use of equation (22.1.39) once more to replace T 2
yields:
steady flow it can be written that:
g 1=g
2
_ m ¼ r u 2 A m (22.1.35) a ¼ a 2 01 p 2 (22.1.45)
2
2
p 01
where Substituting equation (22.1.45) into equation (22.1.43):
1
_ m ¼ mass flow rate (kg s )
A m ¼ open flow area of the valve (given in Appendix p 2 g 1=g 2
2
22.1A at the end of this chapter, after Heywood u ¼ a 2 01 a 2 01
2
[1988]) p 01 g 1
The following isentropic relationships apply for an 2 2a 2 01 p 2 g 1=g
u ¼ 1 (22.1.46)
2
ideal gas: g 1 p 01

p 2 1=g Now substituting both equation (22.1.46) and equation
r ¼ r 1 (22.1.36) (22.1.41) into (22.1.35) gives:
2
p 1

p 2 g 1=g 1=g 2a 2 1=2 g 1=g 1=2
T 2 ¼ T 1 (22.1.37) gp 01 p 2 01 p 2
p 1 _ m ¼ 2 1 A m
a p 01 g 1 p 01
01
Now also for an ideal gas, where R is the specific gas 2 2=g g 1=g 1=2
constant _ m ¼ p 01 A m 2g p 2 1 p 2
a 01 g 1 p 01 p 01
p ¼ rRT (22.1.38) (22.1.47)
and Equation (22.1.47) presents the mass flow rate as
p ffiffiffiffiffiffiffiffiffiffi a function of the open valve area and the pressure ratio
a ¼ gRT (22.1.39) across the valve for subsonic flow through the orifice

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