Document em6aLEb7LwGYK3wGJZO8ZgaY4
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CHAPTER 4
1954 Guide
It is convenient in the analysis of compressible flow to introduce the velocity of propagation of pressure impulses or, more familiarly, the
sonic velocity, a. For perfect gases this is given by the equation:
o' = kgp/p = kgRT__
(22)
Accordingly, Equation 21 may be written
KW - VS) +
= 0 (23)
or, by rearrangement,
P 1 Pi
(24)
which permits the calculation of the ratio of pressures at entrance and
exit of the steady flow device--pipe, orifice, or nozzle. From .Equations
19 and 22 it follows that
aS = T, = (pA^ as Ti \m)
(25)
so that
---- (V,* - V,2) + aS - a,2 = 0 2
(26)
ind k- 1 VS
aS _ 1 + 2 aS
a,8 k -- 1 VS
(27)
The ratio of flow velocity to sonic velocity is known as the Mach number,
M = V/a
This parameter is particularly useful in compressible flow analysis. In general, if M g 0.1 the flow may be considered to be incompressible. This is generally true in heating and ventilating air ducts.
In terms of the Mach number
1 + ---- MS
of = Ti =
2
(28)
and The quantity,
(i+--m'Y1
P
Pi k -- 1 ,
K1 + -Tm*I
P = P
(29) (30)
Fluid Flow
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is called the stagnation pressure, and gives a measure of pressure energy. For incompressible flow
P = p+hpVt P + 9
(31)
where
<? = ^pV2 2g
.. (32)
and is the dynamic pressure. A total head tube measures stagnation pres sure directly.
From Equation 29 it follows that for frictionless, adiabatic flow
PS - Pi
(33)
This represents another extension of the Bernoulli equation to compressible flow. Friction will cause a loss in pressure energy.
Ideal Flow through Nozzle or Orifice
The majority of low measuring systems- depend upon a correlation between pressure drop, area, and quantity of flow. The basic formulas may be stated on the assumption that the flow is frictionless and adiabatic. Designating the main stream by station 1, and flow at some measuring restriction by station 2, the flow in pounds per second is
w = pubFi = ptAtFi
(34)
or, in terms of Mach number,
id -- AiMt \/ktypzPr
According to Equation 29
k--1 1 + --w
fc - 1
1 + --"**
from which
(35) (36)
1 (37)
so that
1 + k-----1 MS ah/
1 - MS/MS y k - 1 l_\p,/
J
(38)
If the initial velocity is sufficiently small, Mi will be negligible so that
If this is computed and the figures are plotted, the curved line (partly