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Problem Set 4
Fa 2009
Solution
1. A large vacuum tank, held at 66,400 Pa, sucks sealevel Standard Air through a con
verging nozzle with a throat diameter of 4 cm. Estimate the following:
(a) The Mach number at the throat.
(b) The temperature of the gas at the throat.
(c) The mass ﬂow rate through the nozzle.
Solution
(a) We are given the pressure at the throat (66.4 kPa) and so we can take advantage
of the relationship for pressure in an isentropic process
P
0
P
=
±
1 +
(
γ

1)
2
M
2
²
γ/
(
γ

1)
Solving this expression for
M
2
yields
M
2
=
2
(
γ

1)
"
³
P
0
P
´
(
γ

1)
/γ

1
#
Substituting numbers (for air setting
γ
= 1
.
4)
M
2
=
2
(0
.
4)
"
³
101
,
325
66
,
400
´
0
.
4
/
1
.
4

1
#
= 0
.
64
Taking the square root yields
∴
M
= 0
.
8
(b) The temperature at the throat can be calculated from the relation
T
0
T
= 1 +
(
γ

1)
2
M
2
= 1 + (0
.
2) (0
.
64) = 1
.
13
Solving for the temperature yields
∴
T
=
288
.
16
1
.
13
= 255
.
4 K
(c) The mass ﬂow rate through the nozzle is by deﬁnition
˙
m
=
ρuA
1
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View Full Documentwhere
ρ
,
u
and
A
are the density, velocity and area at the nozzle throat. Since
we know
P
and
T
at the throat, we can compute the density from the ideal gas
law
∴
ρ
=
P
RT
=
66
,
400
(287)(255
.
4)
= 0
.
91kg
/
m
3
The velocity can be determined from the known Mach number and sound speed
∴
u
=
Ma
=
M
p
γRT
= 0
.
8
p
(1
.
4)(287)(255
.
4) = 256
.
3 m
/
s
The mass ﬂow rate is
∴
˙
m
= (0
.
91)(256
.
3)
π
(0
.
04)
2
4
= 0
.
29 kg
/
s
2. A converging/diverging nozzle has a throat area
A
*
= 0.25 m
2
and is designed to carry
a mass ﬂow rate of 25 kg/s. The temperature at the throat is 195 K and the ﬂow is
choked (i.e., sonic).
(a) Determine the pressure, density and velocity at the throat (
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 Fall '08
 CAUGHEY

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