Modern Engineering Thermodynamics

668 CHAPTER 16: Compressible Fluid Flow
Diameter ratio D/D*
8
7
6
5
4
3
2
1
0
0 1 2 3 4
Mach number M
FIGURE 16.17
Diameter ratio variation in a converging-diverging isentropic nozzle for air ðk = 1:4Þ, from Eq. (16.23b).
For a circular cross-section, A/A = ðD/D Þ 2 . Figure 16.17 shows how this diameter ratio varies with Mach
number for air according to Eq. (16.23b). If a converging-diverging nozzle does not have the exact shape dictated
by Eq. (16.23a), it may still produce supersonic flow; however, the flow will not be isentropic.
EXAMPLE 16.8
To deflate an automobile tire, the valve core must be removed from the tire’s valve stem. When the core is removed, the
valve stem approximates an isentropic converging nozzle with an internal diameter of 0.0938 in. If the tire is initially at
50.0 psia (35.3 psig) and 70.0°F,
■
■
■
Is the flow in the open valve stem initially choked?
If so, at what tire pressure does it unchoke?
How long does it take to unchoke, if the tire is assumed to have a constant volume of 1.00 ft 3 and a constant internal
temperature ðT os Þ of 70°F?
Solution
First, draw a sketch of the system (Figure 16.18).
a. From Eq. (16.19) for air (k = 1.40), the flow is choked if
k/ðk−1Þ 1:40/ð1:4−1Þ
p exit /p os < p /p os =
2
=
2
= 0:528
k + 1 1 + 1:40
Here, p exit /p os = 14:7/50:0 = 0:294, which is