Modern Engineering Thermodynamics

16.6 Choked Flow 665
T a
a
p a = constant
T
T b
T a′
b
a ′
p exit = p b = constant
FIGURE 16.14
Isentropic expansion of a vapor near its saturation at state a through a nozzle to a nonequilibrium supersaturated state at a’. This is
followed by an irreversible nucleation and condensation process to equilibrium state b.
s
16.6 CHOKED FLOW
The mass flow rate per unit area in an isentropic nozzle can be determined for an ideal gas from Eqs. (16.11),
(16.12), and (16.13) and the ideal gas formula, ρ = p/RT, as
_m /A = ρV = ðp/RTÞðMcÞ
pffiffiffiffiffiffiffiffiffiffiffiffi
= pMð kg c RTÞðT os /T os Þ 1/2 ðp os /p os Þ/RT
= p os Mðkg c /RT os Þ 1/2 ðT os /TÞ 1/2 ðp/p os Þ
(16.21)
= p os M kg 1/2
c
1 + k − 1 ðk+1Þ/2ð1−kÞ
RT os 2 M2
Figure 16.15 is a schematic of the variation in _m /A with the ratio of the back pressure p B to the upstream stagnation
pressure p os .WhenM throat = M = 1:0, the nozzle is passing the maximum possible flow _m max :
Since we always have M throat ≤ 1:0, clearly the maximum nozzle mass flow rate occurs when the throat velocity
is sonic. Then _m = _m max ,M throat = M = 1:0, A throat = A , and Eq. (16.21) becomes
1/2
_m max /A kg
= p c k + 1 ðk+1Þ/2ð1−kÞ
os
(16.22a)
2
RT os
Note that the value of _m max /A depends only on the
upstream isentropic stagnation properties and is completely
independent of the downstream conditions. For air,
k = 1.40 and, in Engineering English units, this equation
reduces to
ð _m max /A Þ air = 0:532 lbm pffiffiffi
. R
os
pffiffiffiffiffiffi
(16.22b)
lbf .s T os
where _m max must be in lbm/s, A must be in in 2 , p os
must be in lbf/in 2 ,andT os must be in R. This is called
Fliegner’s (or sometimes Zeuner’s) formula and was
experimentally discovered in the 1870s. In metric
SI units, Eq. (16.22a) reduces to
ð _m max /A Þ air
= 0:0404 kg pffiffiffi
. K
os
pffiffiffiffiffiffi
(16.22c)
N.s T os
where _m max must be in kg/s, A must be in m 2 , p os must
be in N/m 2 ,andT os must be in K.
m/A
m max /A*
Choked flow
region
M throat = 1
0 p B /p os
M throat < 1
FIGURE 16.15
The relative variation in dimensionless isentropic converging nozzle
air mass flow rate with increasing back pressure to upstream
stagnation pressure ratio. Since the Mach number in a converging
nozzle cannot exceed 1.0, the maximum flow rate through the
nozzle occurs when the Mach number at the throat is 1.0.
1.0