fluid_mechanics

11.4 Isentropic Flow of an Ideal Gas 597
Choked flow occurs
when the Mach
number is 1.0 at
the minimum crosssectional
area.
A very useful means of keeping track of the states of an isentropic flow of an ideal gas involves
a temperature–entropy (T–s) diagram, as is shown in Fig. 11.7. Experience has shown
1see, for example, Refs. 2 and 32 that lines of constant pressure are generally as are sketched in
Fig. 11.7. An isentropic flow is confined to a vertical line on a T–sdiagram. The vertical line in
Fig. 11.7 is representative of flow between the stagnation state and any state within the converging–
diverging nozzle. Equation 11.56 shows that fluid temperature decreases with an increase in Mach
number. Thus, the lower temperature levels on a T–s diagram correspond to higher Mach numbers.
Equation 11.59 suggests that fluid pressure also decreases with an increase in Mach number.
Thus, lower fluid temperatures and pressures are associated with higher Mach numbers in
our isentropic converging–diverging duct example.
One way to produce flow through a converging–diverging duct like the one in Fig. 11.6a is
to connect the downstream end of the duct to a vacuum pump. When the pressure at the downstream
end of the duct 1the back pressure2 is decreased slightly, air will flow from the atmosphere
through the duct and vacuum pump. Neglecting friction and heat transfer and considering the air
to act as an ideal gas, Eqs. 11.56, 11.59, and 11.60 and a T–s diagram can be used to describe
steady flow through the converging–diverging duct.
If the pressure in the duct is only slightly less than atmospheric pressure, we predict with
Eq. 11.59 that the Mach number levels in the duct will be low. Thus, with Eq. 11.60 we conclude
that the variation of fluid density in the duct is also small. The continuity equation 1Eq. 11.402 leads
us to state that there is a small amount of fluid flow acceleration in the converging portion of the
duct followed by flow deceleration in the diverging portion of the duct. We considered this type
of flow when we discussed the Venturi meter in Section 3.6.3. The T–s diagram for this flow is
sketched in Fig. 11.8.
We next consider what happens when the back pressure is lowered further. Since the flow
starts from rest upstream of the converging portion of the duct of Fig. 11.6a, Eqs. 11.48 and
11.50 reveal to us that flow up to the nozzle throat can be accelerated to a maximum allowable
Mach number of 1 at the throat. Thus, when the duct back pressure is lowered sufficiently, the
Mach number at the throat of the duct will be 1. Any further decrease of the back pressure will
not affect the flow in the converging portion of the duct because, as is discussed in Section
11.3, information about pressure cannot move upstream when Ma 1. When Ma 1 at the throat
of the converging–diverging duct, we have a condition called choked flow. Some useful equations
for choked flow are developed below.
We have already used the stagnation state for which Ma 0 as a reference condition. It will
prove helpful to us to use the state associated with Ma 1 and the same entropy level as the flowing
fluid as another reference condition we shall call the critical state, denoted 1 2*.
The ratio of pressure at the converging–diverging duct throat for choked flow, p*, to stagnation
pressure, p 0 , is referred to as the critical pressure ratio. By substituting Ma 1 into Eq. 11.59
we obtain
k1k12
p* 2
a
p 0 k 1 b
(11.61)
T
p 0
p
T
F I G U R E 11.7 The (T–s)
diagram relating stagnation and static states.
s
T 0
F I G U R E 11.8 The T – s diagram
T
p 0
p 1
p 2
T 0
T 1
T 2
(1) (2)
for Venturi meter flow.
s