Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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Exercise
supersonic upstream, for example
0.2
M
, the chocking occurs at ap-
proximately 3.0
max2
L . Further extending the tube length causes a forma-
tion of a normal shock wave upstream and at the exit of the tube, where the
flow reaches 1
2
M . The mass flow rate does not change for a tube
greater than
max2
L . The position of the shock is that subsonic flow behind
the shock accelerates to sonic condition 1
2
M at the exit.
Fig. 5.17 Variation of Mach number
Similarly the integration of Eqs. (11) to (15) yields the following rela-
tionships
U
U
*
*
¿
¾
½
¯
®
2
1
2
21
1
Mk
k
M
u
u
(18)
2
*2*
21
1
¸
¹
·
¨
©
§
a
a
Mk
k
T
T
(19)
2
1
2*
21
11
¿
¾
½
¯
®
Mk
k
M
p
p
(20)
263
5 Compressible Flow
12
1
2
1
211
In
¿
¾
½
¯
®
k
k
k
Mk
MR
ss
*
(21)
and writing Eq. (21) by
TT using Eq. (19), we have
»
»
»
»
¼
º
«
«
«
«
¬
ª
°
¿
°
¾
½
°
¯
°
®
¸
¹
·
¨
©
§
2
1
1
1
1
21
ln
k
T
T
k
T
T
R
ss
k
*
*
(22)
As an alternative of Eq. (22),
*
/ TT can be replaced by the enthalpy, using
the relationship
***
/// TTTcTchh pp . This resultant expression gives
the equivalent form of Eq. (5.3.6)
Exercise 5.3 Rayleigh-line Flow Relations and Chocking
Consider a flow of an ideal gas in a horizontal tube in a constant cross sec-
tion. The flow in the tube is assumed to be frictionless, but there is heat
transfer between the tube wall and the fluid. Such a flow is called the
Rayleigh-line flow. Derive the Rayleigh-line flow relationships and discuss
the possibility of chocking condition.
Ans.
As schematically indicated in Fig. 5.18, we will apply the mass conti-
nuity, the momentum and the energy equations, denoting that q
G
is the
heat transfer to the control volume.
Fig. 5.18 Rayleigh-line flow,
q
G
the heat transfer
264
Exercise
(i) Mass continuity equation
0
u
du
d
U
U
(1)
where the cross-section area A is constant.
(ii) Momentum equation
0
1
2
p
dp
kM
u
du
(2)
Denote that
w
W
in Eq. (3) in Exercise 5.2 is set at zero, and letting
U
kpa
2
.
(iii) Energy equation
For an open system at a steady state, the energy balance equation of the
control volume can be written as
t
dLduupvddeq u
G
(3)
where de is the increase of internal energy,
U
/pdpvd is the in-
crease of flow work,
udu is the increment of kinetic energy, and
t
dL is the
work transfer (including work done due to a frictional effect). Defining en-
thalpy pveh and recognizing Tch
p
for an ideal gas, we can reduce
Eq. (3) for a frictionless flow under a condition of no work transfer, i.e.
0
t
dL , as follows
ududhq
G
ududTc
p
(4)
Both side of Eq. (4) is divided by Tc
p
, and by defining the stagnation
enthalpy
00
Tch
p
, we have (similar to Eq. (7) in Exercise 5.2)
h
dh
T
dT
u
du
Mk
T
dT
h
q
00
2
1
G
(5)
(iv) The entropy change and Mach number.
From the definition of entropy for a reversible process, it follows that
RT
q
R
ds
G
(6)
265
5 Compressible Flow
The increase of a Mach number is written, according to Eq. (10) in Ex-
ercise 5.2, as
T
dT
u
du
M
dM
2
1
(7)
Along the similar manner to derive the Fanno-line flow relationships,
we can now derive the Rayleigh-line flow relationships in terms of the
Mach number as written below
U
U
d
dM
kM
Mu
du
¸
¹
·
¨
©
§
2
1
12
(8)
dM
kM
kM
MT
dT
¿
¾
½
¯
®
2
2
1
121
(9)
h
q
M
kM
dM
kM
k
M
p
dp
G
11
2
2
2
2
¸
¹
·
¨
©
§
(10)
The entropy change is also derived from Eq. (6), to give
dM
kM
M
k
k
MR
ds
»
»
¼
º
«
«
¬
ª
¿
¾
½
¯
®
¸
¹
·
¨
©
§
2
2
1
12
1
1
(11)
and, the change of the total temperature
0
dT in Eq. (5) is derived from
dM
kM
kM
MT
dT
¸
¸
¹
·
¨
¨
©
§
2
2
0
1
12
(12)
Equations (8) to (12) give the change of properties, sTu ,,,
U
M and
0
T
for a state of a Mach number, to which in Table 5.2 the summarized results
are listed in a case of heating, .0
!q
G
It should be kept in mind that in the
case of cooling, ,0
q
G
the trends in Table 5.2 are opposite. From Table
5.2, we see that the heating of an ideal gas flow causes the fluid to tend
toward 1
M
for both initially subsonic and supersonic conditions. There-
fore, if a tube is heated to transfer heat to an ideal gas, the flow is choked;
likewise in the Fanno-line flow (in the case of frictional effect). This phe-
nomenon is sometimes called thermal choking.
,
266
Exercise
Table 5.2 Change of property in Rayleigh-line flow
0!q
G
Property Subsonic flow
1
M
Supersonic flow
1!
M
s
M
u
U
T
kM 1
kM 1!
p
0
T
In order to verify thermal choking, we will consider the total tempera-
ture ratio
*
00
/TT . It is readily confirmed that
*
00
/TT can be directly ob-
tained by integrating Eq. (12) from an arbitrary point to 1
M
, yielding
22
2
1
1
¸
¹
·
¨
©
§
¿
¾
½
¯
®
a
a
kM
Mk
T
T
*
(13)
and using Eq. (5.4.7) together with following two relations for 1
M
2
0
2
1
1
M
k
T
T
(14)
1
2
*
0
*
k
T
T
(15)
By eliminating
T
and
*
T
from Eqs. (13), (14) and (15), we have
^
`
2
2
22
0
0
1
121
kM
MkkM
T
T
(16)
Now consider a case with reference to Fig. 5.19, where a section of a
tube is heated by q (heat transfer to the flow), as is written below
0102
TTcq
p
(17)
where
01
T and
02
T are the temperatures at the points where 1 and 2 are
along the tube respectively. It will be convenient to write Eq. (17) as
267
5 Compressible Flow
0101
02
1
Tc
q
T
T
p
(18)
Thus, as a result, the temperature rises due to heating, as written as Eq.
(18) and now it is desired as an expression in terms of the Mach numbers
at point (1) and point (2), which is given as
°
¿
°
¾
½
°
¯
°
®
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
2
1
2
2
2
2
2
1
2
1
2
01
0
0
02
01
02
12
12
1
1
Mk
Mk
kM
kM
M
M
T
T
T
T
T
T
(19)
Fig. 5.19 Chocking by heating in Rayleigh-line flow
In order to characterize the system, Fig. 5.20 is a plot for
0102
/TT
of Eq.
(19) versus
2
M , while keeping
1
M constant. As seen in the diagram, when
a flow at point (1) is subsonic 1
M
and the total temperature of
01
T , the
total temperature
02
T
at point (2) rises due to heating (according to Eq.
(18)), for example (1)ĺ(2) in 5.0
1
M as indicated in Fig. 5.20, follow-
ing the simultaneous rise of the Mach number of
2
M . Then, at 1
2
M ,
0102
/TT
reaches its maximum for a given
1
M
and
01
T
, where the
Rayleigh-line flow becomes sonic, and the flow is said to be thermally
chocked.
If the cooling is started from the point (2) to down stream, the total
temperature drops and the supersonic flow becomes possible. Further cool-
ing the tube to absolute zero, the Mach number of the flow asymptotically
approaches infinity. In contrast, if the section of the tube is further heated,
02
T
rises, for example with reference to Fig. 5.20
)(2(2)
c
o
, where we
keep the state chocked at 1
2
M , since the only maximum Mach number
possible at
2
M is 1
2
M . With reference to the state at )(2
c
, the Mach
268
Exercise
number at point (1) should be lower, for example, with reference to Fig.
5.20
)(1(1)
c
o
, when followed by the Rayleigh-line flow, while
01
T
is
kept constant. In this situation, the mass flow rate drops, compared to that
of the original state.
Fig. 5.20
0201
TT
, Mach number relation (for
4.1 k
) in subsonic flow
case
In this case, if the Mach number at point (1) is supersonic 1
1
!M and a
section of the tube is heated, the Mach number at the exist of a tube, where
002
TT , becomes the sonic 1
2
M , leading the flow thermally chocked.
Figure 5.21 is a plot of Eq. (16) for air
4.1 k . As the section is heated
from a total temperature of
01
T with supersonic flow
1
M at the point (1),
the Mach number
M
along the tube decreases to point (2). If there is a
normal shock wave that exists in a section of the tube, the Mach number
jumps to a state of
)(2
c
, where the Mach number behind the shock wave is
subsonic. From )(2
c
, the Mach number again increases toward the exit,
where the Mach number is in unity. When the heating is high enough, the
shock wave is formed further upstream, i.e. shifting (2) to further high
Mach numbers and resulting in
)(2
c
to a further lower Mach number to-
ward the point (1), where the heating is originally started in the section of
the tube.
In engineering practice of observing shock wave formation, a higher
heating of the section of the tube may shift the shock wave to the throat of
a Laval nozzle, in which the shock disappears. As it has been verified, it is
269
5 Compressible Flow
interesting to note that the heating (and cooling) can control the shock
wave positions in a tube.
Fig. 5.21
002
TT
, Mach number relation (for air
4.1 k
) in supersonic
flow case
Similar to the Fanno-line flow, the integration of Eqs. (8) to (11) yields
the following relations
U
U
2
2
1
1
kM
Mk
u
u
(20)
2
1
1
kM
k
p
p
(21)
»
»
»
¼
º
«
«
«
¬
ª
¿
¾
½
¯
®
k
k
kM
k
M
k
k
R
ss
1
2
2
1
1
ln
1
(22)
»
»
»
»
¼
º
«
«
«
«
¬
ª
°
¿
°
¾
½
°
¯
°
®
r
¸
¹
·
¨
©
§
k
k
k
k
TTkkk
T
T
k
k
1
2
1
2
411
ln
1
*
/
(23)
It is noted that as an alterative of Eq. (23),
*
/ TT can be also replaced by
the enthalpy, using the relation
TTTcTchh
pp
/// , and the resultant
expression gives the equivalent form of Eq. (5.3.19).
270
Problems
Problems
5-1. A model wing is placed in a uniform air flow of pres-
sure
25
1
N/m10450 u .p and the temperature is
K253
1
T
. When a
measurement of pressure and the Mach number on a wing was carried
out, they were, respectively,
25
2
N/m10350 u .p and
750
2
. M
.
Obtain the speed and the Mach number of the uniform air flow, as-
suming that the flow is isentropic.
Ans.
>@
420 and m/s134
11
. Mu
5-2. A supersonic plane traveling at a speed with a Mach number of
252.
passes m12000 above an observer. Determine how far does the plane
travel from the point beyond the observer, when the acoustic distur-
bance of the plane was first heard. Assume the speed of sound to be
m/s330 , and that it is independent of the altitude.
Ans.
>@
32.5 safter m24174 t
5-3. A compressed gas of a specific heat ratio
k is discharged in the at-
mospheric pressure
a
pp
2
through the Borda’s mouth piece of a
cross-section area
1
A
from a large reservoir tank, where the pressure
is kept
0
p , as schematically displayed in Fig. 5.22. If the area of the
vena contracta is
2
A and the Mach number is
2
M at the vena con-
tracta, prove that the contraction coefficient
1
2
AAC
c
, is given by
the following formula
2
2
2
0
1 kM
p
p
C
c
¸
¸
¹
·
¨
¨
©
§
(1)
and for isentropic flow also prove this is written as
°
¿
°
¾
½
°
¯
°
®
¸
¹
·
¨
©
§
1
2
1
1
1
1
1
2
2
2
2
k
c
M
k
kM
C
(2)
'
271
5 Compressible Flow
Fig. 5.22 Flow through
B
Borda’s mouth piece
sudden expansion as shown in Fig. 5.23. The flow is subsonic and the
conditions at (1) and (2) are respectively
(1)
11111
,,,, MupA
U
(2)
22222
,,,, MupA
U
show that
212212
uuupp
U
2
21
2
1
1
2
1
1
kM
AAkM
p
p
where the flow is assumed to be adiabatic.
Fig. 5.23 Gas flow through sudden expansion
5-4. An incompressible gas of the specific heat ratio k is passing through a
272