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Fundamentals of Thermodynamics
Chapter 15
Compressible Flow
Thermal Engineering Lab.
reversible adiabatic deceleration
V = 0
- isentropic stagnation state
흐르는유체가 을
통해 가 되었을때의상태
2
02
V- h h stagnation enthalpy
15.1 Stagnation properties
2
Chapter 15. Compressible Flow
Thermal Engineering Lab.
Ts
h
vdPdhTds
P
3
Chapter 15. Compressible Flow
Thermal Engineering Lab. 4
Ex. 15.1 Air flows in a duct at a pressure of 150 kPa with a velocity of 200 m/s.
The temperature of the air is 300 K. Determine the isentropic stagnation
pressure and temperature.
Chapter 15. Compressible Flow
Thermal Engineering Lab.
0
x
C.V.
x e ex i ix
d mVSSSF
dt
F m V mV
에서
C.V.에 가해지는 힘
volumecontrol : -)(
mass fixed : )(
..
exeixix
vc
x
xx
VmVmFdt
mVd
Fdt
mVd
15.2 The momentum equation for a control volume
Chapter 15. Compressible Flow
5
Thermal Engineering Lab.
- C.V.(유체)에가해지는 힘
x ex ixF m V V
Chapter 15. Compressible Flow
6
Thermal Engineering Lab. 7
Ex. 15.2 On a level floor, a man is pushing a wheelbarrow (Fig. 15.3) into which
Sand is falling at the rate of 1 kg/s. The man is walking at the rate of 1
m/s, and the sand has a velocity of 10 m/s as it falls into the wheelbarrow.
Determine the force the man must exert on the wheelbarrow and the
force the floor exerts on the wheelbarrow due to the falling sand.
Chapter 15. Compressible Flow
Thermal Engineering Lab.
R: external force C.V. 에의해 에가해지는 힘
0
0
0
0
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) 0
x x
x x
x x x
x i i e e t i e xx
i i e e i e x x
i i o i e e e x x
t x
F P A P A P A A A R
P A P A P A A R
P A P A P A P A R
P A
15.3 The momentum equation for a control volume
8
Chapter 15. Compressible Flow
Thermal Engineering Lab. 9
Ex. 15.3 A jet engine is being tested on a test stand (Fig. 15.5). The inlet area to
the compressor is 0.2 m2, and air enters the compressor at 95 kPa, 100
m/s. The pressure of the atmosphere is 100 kPa. The exit area of the
engine is 0.1 m2, and the products of combustion leave the exit plane at a
pressure of 125 kPa and a velocity of 450 m/s. The air-fuel ratio is 50
kilograms of air to one kilograms of fuel, and the fuel enters with a low
velocity. The rate of air flow entering the engine is 20 kg/s. Determine
the thrust, Rx, on the engine.
Chapter 15. Compressible Flow
Thermal Engineering Lab.
e i i i e e
i i e e
- m = m = AV = AV
AV AV
2 2
2 2
02 2
02 2
i ei i e e
e ie i e i
- first law
V Vq w h gz h gz
V Vh h g z z
10
Chapter 15. Compressible Flow
15.4 Adiabatic, one-dimensional, steady-state flow of an incompressible fluid through a nozzle
Thermal Engineering Lab.
.
e i
e i
e
e i e ii
- second law
s s
rev adiabatic s s
Tds dh vdP dh vdP
h h vdP v P P
Bernoulli eq.
2 2
02 2
e ie i e i
V Vv P P g z z
11
Chapter 15. Compressible Flow
Thermal Engineering Lab. 12
Ex. 15.4 Water enters the diffuser in a pump casing with a velocity of 30 m/s, a
pressure of 350 kPa, and a temperature of 25℃. It leaves the diffuser
with a velocity of 7 m/s and a pressure of 600 kPa. Determine the exit
pressure for a reversible diffuser with these inlet conditions and exit
velocity. Determine the increase in enthalpy, internal energy, and entropy
for the actual diffuser.
Chapter 15. Compressible Flow
Thermal Engineering Lab.
17.4 Ex 보충설명
P
ieieie
i
ePie
iseie
ieie
eiie
ieiePie
i
ePie
P
P
P
P
c
PPvhhTT
T
Tcssleirreversib
PPvhh
ssTTisentropic
VVhh
PPvTTchh
vdPTdsdh
T
Tcss
dTT
cds
T
vibleincompress
dPT
vdT
T
cds
)()(
ln:
)(
,:
22
)()(
ln
,0
,
22
Ex. 15.4
13
Chapter 15. Compressible Flow
Thermal Engineering Lab.
- pressure disturbance : sonic velocity
- sound wave : very small pressure disturbance
의 전파속도
14
Chapter 15. Compressible Flow
15.5 Velocity of sound in an ideal gas
Thermal Engineering Lab. 15
Chapter 15. Compressible Flow
2
2 2
0
0
0, 0
,
2
s
s
i) cA= +d c dV A= m
cd dV
c dVcii) h+ = h+dh +
2 2
dh cdV
dP dPiii) isentropic Tds = dh dh =
dP dP cdcdV c
dP Pc c
d
Pc
과정에서 이므로
1
2
Continuity와 열역학 1, 2법칙을 적용
Thermal Engineering Lab. 16
Chapter 15. Compressible Flow
2
( ) ( )
0
( ) ( ) ( )
s
Ac d A c dV
cd dV
PA P dP A m c dV c Ac c dV c
dP cdV
cddP c
dP Pc c
d
Continuity와 momentum eq.을 적용
Thermal Engineering Lab. 17
Chapter 15. Compressible Flow
Ideal gas sonic velocity의
kPv const
ln lnP k v C
2
1 1
0
dP dv+k = 0
P v
v dv d
dP dk
P
미분
이므로
2
dP kP PP = RT = RT
d
dPkRT c kRT c kRT
d
에서
Thermal Engineering Lab. 18
Ex. 15.5 Determine the velocity of sound in an air at 300 K and at 1000 K.
Chapter 15. Compressible Flow
Thermal Engineering Lab. 19
Chapter 15. Compressible Flow
15.6 Reversible, adiabatic, one-dimensional flow of an ideal gas through a nozzle
0
0
0
0
2Vh+ = const
2
dh+VdV = 0
continuity AV m const
d AV dA V dV A
d dA dV
A V
property relation
Tds = dh vdP rev. adiabatic isentropic
dh vdP
dPdh
미분하면
에서
1
2
3
Thermal Engineering Lab. 20
Chapter 15. Compressible Flow
dPdh VdV
dPdV
V
에서
2
2
2 2
1 1
1 1
dA d dV
A V
d dP dP
dP V
dP
VdP
d
dP
c V
에서
2 3
1
와
Thermal Engineering Lab. 21
Chapter 15. Compressible Flow
2
2 2
2
2
1
1
:
dP V VM
V c c
dPM
V
VMach number M
c
M 1 : super sonic
M 1 : sub sonic
M 1 : sonic
Thermal Engineering Lab. 22
Chapter 15. Compressible Flow
- : 0
; 1 0
nozzle dP
subsonic M dA
; 1 0supersonic M dA
- : 0
; 1 0
diffuser dP
subsonic M dA
supersonic ; M 1 dA 0
1 0M dA throat
Nozzle
Diffuser
2
21
dA dPM
A V
Thermal Engineering Lab. 23
Chapter 15. Compressible Flow
0
0
0
0 0
0
0
1
P
P
P
P v
P
v
CC R
k
k C kR
C C R
Ck
C
0
2
0
2 00 0
22 2 0
22 0
2
20
2
2 2 2 11
21
1
21
1
11
2
P
Vh h
TkRT- V h h C T T
k T
Tc- c kRT V
k T
TVM
c k T
kTM
T
이므로
Thermal Engineering Lab. 24
Chapter 15. Compressible Flow
11 1
0 0 0 0
11 1
2 20 0
*
0
1* *1 1
0 0
,
1 11 1
2 2
1
2
1
2 2
1 1
kk k
kk k
kk k
T P T-
T P T
k kP- M M
P
- throat M
T
T k
P
P k k
이므로
에서 인 경우
Thermal Engineering Lab. 25
Chapter 15. Compressible Flow
15.7 Mass flow rate of an ideal gas through an isentropic nozzle
0
0
2
0
0
1
0 2 12
0
1
0 2 12
1
11
2
11
2
11
2
k
k
k
k
m PV- V
A RT
TPV k PV k
c R T TkRT RT
PM k kM
RT
P k M
RT kM
P k M
RT kM
0
121
12
kk
PP
kM
Thermal Engineering Lab. 26
Chapter 15. Compressible Flow
*
0
1*
0 2 1
1
2 12
*
1
2
, 1,
1
1 2 11
1 2
k
k
k
k
k
- At throat M A A
Pm k
A RT
A k- M
A M k
Thermal Engineering Lab. 27
Chapter 15. Compressible Flow
( )- Effect of back pressure converging nozzle
Thermal Engineering Lab. 28
Chapter 15. Compressible Flow
( )- Effect of back pressure converging diverging nozzle
Thermal Engineering Lab. 29
Ex. 15.6 A convergent nozzle has an exit area of 500 mm2. Air enters the nozzle
with a stagnation pressure of 1000 kPa and a stagnation temperature of
360 K. Determine the mass rate of flow for back pressures of 800 kPa,
528 kPa, and 300 kPa, assuming isentropic flow.
Chapter 15. Compressible Flow
Thermal Engineering Lab. 30
Ex. 15.7 A converging-diverging nozzle has an exit area to throat area ratio of 2.
Air enters this nozzle with a stagnation pressure of 1000 kPa and a
stagnation temperature of 360 K. The throat area is 500 mm2. Determine
the mass rate of flow, exit pressure, exit temperature, exit Mach number,
and exit velocity for the following conditions:
a. Sonic velocity at the throat, diverging section acting as a nozzle.
(corresponds to point d in Fig. 15.13.)
b. Sonic velocity at the throat, diverging section acting as a diffuser.
(correspints to point c in Fig. 15.13.)
Chapter 15. Compressible Flow
Thermal Engineering Lab. 31
Chapter 15. Compressible Flow
15.8 Normal shock in an ideal gas flowing through a nozzle
: &shock wave extermely rapid abrupt change of state
22
;2 2
;
2 : 0
yxx y ox oy
x x y y
x y y x
y x
VVh h h h
mV V
A
A P P m V V
nd law S S
v
v
v
v
Fanno line
Rayleigh line
Rankine Hugoniot line
Thermal Engineering Lab. 32
Chapter 15. Compressible Flow
Thermal Engineering Lab. 33
Chapter 15. Compressible Flow
22
2 2
1 1 1
2
0
1 1
oy y y x
ox x x y
x x x y y y
y x y x
x y
y x
Fanno line
h h V
h h V
Rayleigh line
P V P V
Rankine Hugoniot line
h h P P
- s s x y
M M
이므로 로 일어남
로 일어남
v
v
v
Thermal Engineering Lab. 34
Chapter 15. Compressible Flow
Normal shock governing equation 의 Fanno - line
2 2
2
2
1 11 , 1
2 2
11
2 - 17.491
12
ox oy ox oy P
oyoxx y
x y
xy
xy
x x y y
yxx y
x y
- T T h h assume const C
TT k k- M M
T T
kMT
-kT
M
- V V
PP
RT RT
에서
그런데
또한 이므로
5
Thermal Engineering Lab. 35
Chapter 15. Compressible Flow
2 2
- 17.50
y y yy y y y y y yx
x y x x x x x x x x x
y y y y y y
x x x x xx
P M TT P P V P M c
T P P V P M c P M T
T P M T P M
P M T P MT
2
2
11
2: - 17.51
11
2
x xy
x
y y
kM M
P- Fanno - line
P kM M
5
5
Thermal Engineering Lab. 36
Chapter 15. Compressible Flow
2 2
2 2
2 2 2 2
x y y x y y x x
x x x y y y
x x x x y y y y
xx
Rayleigh - line
m- P P V V V V
A
- P V P V
- P M c P M c
P- P
R
xT
2
xM k R xT y
y
PP
R
yT
2
yM k R yT
2 2
2
2
1 1
1; - 17.52
1
x x y y
y x
x y
P kM P kM
P kMRayleigh - line
P kM
2
2
2
17.51 , 17.52
2
1 ,2
11
x
y y x
x
MkM M f M k
kM
k
에서
5
Thermal Engineering Lab. 37
Ex. 15.8 Consider the convergent-divergent nozzle of Example 15.7, in which the
diverging section acts as a supersonic nozzle (Fig. 15.16). Assume that a
normal shock stands in the exit plane of the nozzle. Determine the static
pressure and temperature and the stagnation pressure just downstream of
the normal shock.
Chapter 15. Compressible Flow
Thermal Engineering Lab. 38
Thermal Engineering Lab. 39
Ex. 15.9 Consider the convergent-divergent nozzle of Examples 15.7 and 15.8.
Assume that there is a normal shock wave standing at the point where
M = 1.5. Determine the exit-plane pressure, temperature, and Mach
number. Assume isentropic flow except for the normal shock (Fig. 15.18).
Chapter 15. Compressible Flow
Thermal Engineering Lab. 40
Chapter 15. Compressible Flow
:
17.9 Nozzle and Diffuser Coefficients
N
nozzle efficiency
- actual flow velocity coefficient
discharge coefficient
- nozzle efficiency
actual kinetic energy at nozzle exit
kinetic energy at nozzle exit with
isentropic f
low to same exit pressure
15.9 Nozzle and diffuser coefficients
Thermal Engineering Lab. 41
Chapter 15. Compressible Flow
V
V
- Velocity coefficient : C
Actual velocity at nozzle exit C =
Velocity at nozzle exit with isentropic flow to same exit pressure
ev N
s
VC
V
Thermal Engineering Lab. 42
Chapter 15. Compressible Flow
D
D
: 1
- Discharge coefficient : C
Actual mass rate of flowC =
Mass rate of flow with isentropic flow
s
a
s
choked flow throat M mm
m 에서 기준
Thermal Engineering Lab. 43
Chapter 15. Compressible Flow
3 1 3 1
2
1 1 1 2 1
:
2
sD
o o
h h h h hDiffuser efficiency
V h h h h
Thermal Engineering Lab. 44
Chapter 15. Compressible Flow
0
0
3 1
1
3 1 1
2
12 1
12
2 2 2 3 211 1 1 1
1 1
12
12
1
2
, , ,1
1
D
o
P
kk
oP
kk
o
D
T TT
T T T
VT T
C
T PckRC T V M c
k kR T P
cP
P
kR
2 2
1 1M c
2kR
1
2
1
2
1
1
1
2
1
kk
oP
P
k M
k
Thermal Engineering Lab. 45
Chapter 15. Compressible Flow
1
2 21
1
2
1
11 1
2
1
2
kk
o
o
D
PkM
P
kM
11 1
2 1 2
1 1 1
kk kkk k
o o o
o
P P P
P P P
1
211
1
11
2
k kk
oP kM
P
1k1k
k2
1
11
2
kM
Thermal Engineering Lab. 46
Chapter 15. Compressible Flow
15.9 Nozzle and orifices as flow-measuring devices
2
2 222 2 2 2
12 11 2 1 2
1 2
2 2
2
1
,
2 2
2
1
incompressible reversible
AV V
AV Vv P P v P P
v P PV
AA
Thermal Engineering Lab. 47
Chapter 15. Compressible Flow
ΔP ideal gas: Incompressible 가작은경우 로계산 가능
0
2 2
2 2
1 1
1
2
2 2
2
1 1
1 11 1
2
i ei e
e ii e P i e
k kk k
e e i i
i i i i
kk
i i
i i
V V- h h
V Vh h C T T
T P T T P P
T P T P
T P
T P
T k P k
T k P k
0 0 0
2
2
2 2
1
11
2
i
i i
e i i i iP i P i P i
i i i
P
P
T k P
T k P
V V T T T k PC T C T C T
T T k P
k
1
R
k
1k
ki
i
i
PT
P
v P
Thermal Engineering Lab. 48
Chapter 15. Compressible Flow
Pitot tube
Thermal Engineering Lab. 49
Chapter 15. Compressible Flow
2
0 0
0
2
2
incompressible
Vh h v P P
V v P P
인 경우
0 0
0
2
00 0
1
0
12120
2
12
1
1 11 1
2 2
P P
kk
P
kk kk
compressible
TVh h C T T C T
T
PC T
P
P k k VM
P c
인 경우
그런데
Thermal Engineering Lab. 50
Chapter 15. Compressible Flow
0 0
2
0
2
0
22 2
0
22
0
2 2
2 22200
2 2 2
2
2 1 1
2 1 1
221
1 1
2 11
2 2
P P
VC T C T
V kR kRT T
k k
cV c
k k
cc
V k V k
c V kcc k
V V V
Thermal Engineering Lab. 51
Chapter 15. Compressible Flow
1
22
0
2
0
)1(2
2
2
)1(1
kk
kVc
Vk
P
P
12
0
1
2
0
22
0
1
222
0
22
0
1
22
0
2
0
2
)1(1
4
)}1(2{2
2)1()}1(2{2
)}1(2{2
)1(2
2
2
)1(1
1
kk
kk
kk
kk
c
Vk
c
kVc
VkkVc
kVc
kVc
VkP
P
Thermal Engineering Lab. 52
Chapter 15. Compressible Flow
2 1
0 0
2 4
0 0 0
2 4
00 0 0
0 0
11
2
12 8
2 8
kk
P k V
P c
P k V k V
P c c
kP kV k VP P P P
c c
2
0
2
P V
k 0
k
RT
22
0
08
P V V
c k
2
0
2
0
0
22 2
0 00
0
22 2
0 00
0
2
0
0
0
2
2
2 8
1
2 2 4
11
4
1 for incompressible flow
V
V
RT
V V VP
c
V V VP P
c
P P V
c
P P
Thermal Engineering Lab. 53
Chapter 15. Compressible Flow
:참고자료 Fanno, Rayleigh, Rankine-Hugoniot Line 작도
22
222
2
222
2
2
)
2 2
1
2 2
1
2 2
2
yxx y
x x y y
y y yx x x xy x x x x
y x y y x x y
yx xP y x x
x y
yx xP x y x
x y
x
i Fanno - line
VVh h
V V
RT T TP P PV V V V V
RT P P T T P
TV PC T T V
T P
TV PC T T V
T P
VC
2
2 2 2
12 2
2 2
2 2
1
2
2 2
ln
xP x y y x y
x
x xx y x y
x xy
x P x y x P x y
y P y x x
y y
y x P
x x
PT T P V T
T
P PV T V T
T TP
V C T T V C T T
h C T T h
T PS S C R
T P
Thermal Engineering Lab. 54
Chapter 15. Compressible Flow
2 2
)
yx
x x x y y y x y
x y
x
xx x y y y x
y
ii Rayleigh
PPP V P V P P
RT RT
P
RV V V V
x
y
T
P
R
22
2 2
22 2
2
2
2
2 2
1 1
x y
x x
x y
y
y yx xx x y x
x y x y
x xx y x x y
x x y
x y xy x y x
x x x
P TV V
T P
T
P TP PP V P V
RT RT T P
P PP P V V T
RT R T P
RT P PT P P V
P V RT
Thermal Engineering Lab. 55
Chapter 15. Compressible Flow
2 2
2 2
)
2 2
1 1
2 2
1 1
2 2
1 1 1
2
2
12
x xx y
y x x y x y x y
x y y x
y x
y x x y y x x y
y x
x y
yxP y x y x
x y
P y x
y
iii Rankine - Hugoniot line
V Vh h
h h V V V V V V
P P A m V V
P P A A Ah h V V P P V V
m m m
P P
TTC T T R P P
P P
C R P PP
에서
2
2
12
xy P x y x
x
xP x y x
xy
P y x
y
TT C T R P P
P
TC T R P P
PT
C R P PP
Thermal Engineering Lab. 56
Chapter 15. Compressible Flow
Ex. 17.8 작도 계산예
2
2
93.9 : 1.4, 0.287 /
183.2 1.005 /1
2.197
596.1 /
271.3 /
1
2
1
2
x
x P
x
x
x
x xy
x
y y
Fanno - line
P kPa air k R kJ kg K
kRT K C kJ kg K
k
M
V m s
c m s
kM M
P
P kM M
512.7
339.7
?
0.547
y
y
y
y
P kPa
T K
V
M
Ex. 15.8
Thermal Engineering Lab. 57
Chapter 15. Compressible Flow
22
22
2
2
1.4 0.287 1000 200 283.5 /
2 2
2 2 1005 183.2 200 596.1 567.1 /
567.12.000
283.5
1.4 12.197 2.197
2.197 0.982521.2067
2.000 0.89441.4 12.000 2.000
2
y
y y
yxx y
y p x y x
y
y
x
T
c kRT m s
VVh h
V C T T V m s
M
P
P
S
i) 가정(200K)
200
1.005ln 0.287 ln 1.2067 0.03425 /183.2
y xS kJ kg K
Thermal Engineering Lab. 58
Chapter 15. Compressible Flow
2
2
1.4 12.197 2.197
2.197 0.982521.2067
2.000 0.89441.4 12.000 2.000
2
2001.005ln 0.287 ln 1.2067
183.2
0.03425 /
y
x
y x
P
P
s s
kJ kg K
Thermal Engineering Lab. 59
Chapter 15. Compressible Flow
150
160
170
180
183.2
339.7
340
350
400
x x x yP T V T
Thermal Engineering Lab. 60
Chapter 15. Compressible Flow
22
22
)
1 1.4 2.19711.1754
1 1 1.4 2.000
2001.005ln 0.287 ln 1.1754
183.2
0.04180 /
y x
x y
y x
ii Rayleigh line
P kM
P kM
s s
kJ kg K
Thermal Engineering Lab. 61
Chapter 15. Compressible Flow
40000
60000
80000
93900
100000
120000
500000
512700
520000
yP
Thermal Engineering Lab. 62
Chapter 15. Compressible Flow
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1ln
P
TP TP
T
air table
reversible process
dPdh vdP C dT RT
P
C sdP dT P dTC
P R T P R T R
2 2
1 1
22 1
1
r
r
r
r
P P
P P
PP P
P
:r
P relative pressure