gas dynamics relations
TRANSCRIPT
Bulk Modulus of a gas /vapour
k=(increase in pressure)/(relative change in volume)
( )÷ö
çæ-=÷
öçæ D-=÷
÷ö
ççæ
-D+=
dppppp
-ve sign because Volume decreases with Pressure increase
( )÷øö
çèæ-=÷
øö
çèæDD
-=
÷÷÷
øççç
è÷øö
çèæ D-
-D+= ®D dv
dpv
Vp
V
VV
pppk oplim Eq. 1
Bulk Modulus of a gas /vapour
k=(increase in pressure)/(relative change in volume)
÷÷ö
ççæ
=÷÷÷ö
çççæ
÷÷ö
ççæ-=Þ÷÷
öççæ
= r dpdpkv
11Eq. 2÷÷
ø
öççè
æ=
÷÷÷÷
øçççç
è÷÷ø
öççè
æ÷÷ø
öççè
æ-=Þ÷÷
ø
öççè
æ=
rr
rrr d
dp
d
dpkv
111
÷÷ø
öççè
æ=
rr
ddp
k
¢ For Isothermal Process
¢ For Isentropic Process
¢ From Eqs. 1 & 3, Isothermal Bulk Modulus
÷øö
çèæ-=÷
øö
çèæÞ=
vp
dvdp
constpv .
÷øö
çèæ-=÷
øö
çèæÞ=+Þ= -
vp
dvdp
dpvdvvpconstpvgg ggg 0)(. 1 Eq. 4
Eq. 3
pkk ==
¢ From Eqs. 1 & 4, Isentropic Bulk Modulus
¢ Summary
pkk T ==
pkk s g==
Ts kk g= Eq. 5
Coeff of Compressibilitykc=(relative change in volume/(increase in pressure)
÷ö
çæ÷ö
çæ-=÷
öçæ D÷ö
çæ-=÷
÷ö
ççæ
÷øö
çèæ D-
=dvVV
V11
-ve sign because Volume decreases with Pressure increase
÷÷ø
öççè
æ÷øö
çèæ-=÷÷
ø
öççè
æDD
÷øö
çèæ-=
÷÷÷
øççç
èD
øè=dpdv
vpV
VpVkc
11Eq. 6
¢ For Isothermal Process
¢ For Isentropic Process
¢ From Eqs. 6 & 7, Coeff. of Isothermal compressibility
÷÷ø
öççè
æ-=÷÷
ø
öççè
æÞ=
pv
dpdv
constpv .
÷÷ø
öççè
æ-=÷÷
ø
öççè
æÞ=+Þ= -
pv
dpdv
dpvdvvpconstpvg
g ggg 0)(. 1 Eq. 8
Eq. 7
÷÷ö
ççæ
=÷÷ö
ççæ
== kk11
¢ From Eqs. 6 & 8, Coeff. of Isentropic compressibility
¢ Summary
÷÷ø
öççè
æ=÷÷
ø
öççè
æ==
TcTc kp
kk11
cscT kk g= Eq. 9
÷÷ø
öççè
æ=÷÷
ø
öççè
æ==
scsc kp
kk11
g
¢ Important parameter in compressible flow is the speed of sound (a)— Speed at which infinitesimally
small pressure wave travels
¢ Consider a duct with a moving piston (Velocity c)
adc
Speed of Sound (a) and MachNo (M/ Ma)
piston (Velocity c)— Creates a sonic wave moving to
the right with Velocity a.— Fluid to left of wave front
experiences incremental change in properties
— Fluid to right of wave front maintains original properties
dc
SPEED OF SOUND AND MACH NUMBER
¢ Construct CV that encloses wave front and moves with it
¢ Mass balance
Wavefront is made Stationary by imposing Opp Velocity a
)()( dcaAdAa -+= rrr
rr addc = Eq. 21
a-dc a )( dcdaddcaAAa rrrrr -+-=0=- dcad rr
SPEED OF SOUND AND MACH NUMBER
¢ Momentum Equation givesWavefront is made Stationary by imposing Opp Velocity C
[ ] [ ]))(()(.
dpppAadcam +-=--
[ ] [ ])()(.
dpAdcm =[ ] [ ])()( dpAdcm =
Acm r=.
adcdp r=Eq. 22
a-dc a
SPEED OF SOUND AND MACH NUMBER
¢ Momentum Equation gives
adcdp r= Eq. 22
rr addc = Eq. 21
= r2
÷÷ø
öççè
æ==
=
r
r
ddp
aSoundVel
dadp 2aa-dc
Eq. 23
SPEED OF SOUND AND MACH NUMBER
TMR
TRpk
ddp
aw
UG
s ggrg
rr=====
Sound Velocity is related to T,Ks,KC,& Mw
Eq. 24
SPEED OF SOUND AND MACH NUMBER
¢ Since — RG is constant for a given
gas
TMR
TRaw
UG gg ==
gas— g is only a function of T— Hence, Speed of sound is
only a function of temperature for a given gas.
SPEED OF SOUND AND MACH NUMBER
¢Second important parameter is the Mach number Ma
¢Ratio of fluid velocity to the speed of sound
C=320
C=320
M < 0.33 : Low Speed Aerodynamics
0.33<M < 1 : Subsonic
M » 1 : Transonic ( 0.8<M<1.2)
M = 1 : Sonic
M > 1 : Supersonic
M > 4 : Hypersonic
M=c/aFlow regimes Classification based on Mach No
Eq. 24
Mach Angle/ Mach Cone¢ A source of disturbance is moving from right to
left with a velocity u in the fluid¢ Pt. S represents present location of source while
1,2 &3 show its location before 1,2 &3 seconds 1,2 &3 show its location before 1,2 &3 seconds respectively.
¢ Distance travelled by sound is a,2a,3a meters in 1,2 ,3 seconds respectively.
¢ Four cases considered areMach No =( 0,1/2,1 & 2)
Energy Conservation Equation for Compressible flows CV
CS
Outlet 2
Outlet 4
Inlet 1
C1,z1,h1,T1,u1
Unsteady Flow Energy Equation (2-2)
·
Q
·
CVETime Rate of Energy Change in CV
=
For flow Process
Outlet 4Inlet 3
þýü
îíì
úû
ùêë
é+++ú
û
ùêë
é+++-
þýü
îíì
úû
ùêë
é+++ú
û
ùêë
é+++=
·······
44
24
422
22
233
23
311
21
1 2)(
2)(
2)(
2)(
hgzc
mhgzc
mWhgzc
mhgzc
mQECV
·
WTime Rate of Energy Inflowsto CV
Time Rate of Energy Outflows from CV
-
Eq. 31
Energy Conservation Equation for Compressible flows CV
CSUnsteady Flow Energy Equation (0-0)For non- flow Process
·
Q
·
CVETime Rate of Energy Change in CV
=
þýü
îíì-
þýü
îíì=
···
WQECV
·
WTime Rate of Energy Inflowsto CV
Time Rate of Energy Outflows from CV
+
þýü
îíì+
þýü
îíì=
þýü
îíì ···
CVEWQ
þýü
îíìÑ+
þýü
îíì=
þýü
îíì ···
CVUWQEq. 32
Energy Conservation Equation for Compressible flows CV
CS
Outlet 2
Outlet 4
Inlet 1
Steady Flow Energy Equation (2-2)C1,z1,h1,T1,u1
Time Rate of Energy Change in CV
= 0
For flow Process
·
Q
Outlet 4Inlet 3
úû
ùêë
é+++ú
û
ùêë
é+++=ú
û
ùêë
é+++ú
û
ùêë
é+++
······
44
24
422
22
233
23
311
21
1 2)(
2)(
2)(
2)(
hgzc
mhgzc
mWhgzc
mhgzc
mQ
4231
····
+=+ mmmm
·
W
Eq. 33
Energy Conservation Equation for Compressible flows CV
CS
Outlet 2Inlet 1
Steady Flow Energy Equation (2-1)
·
Q
C1,z1,h1,T1,u1
Inlet 3
úû
ùêë
é+++=ú
û
ùêë
é+++ú
û
ùêë
é+++
·····
22
22
233
23
311
21
1 2)(
2)(
2)(
hgzc
mWhgzc
mhgzc
mQ
···
=+ 231 mmm
·
W
Energy Conservation Equation for Compressible flows CV
CS
Outlet 2
Outlet 4
Inlet 1
Steady Flow Energy Equation (1-2)
·
Q
C1,z1,h1,T1,u1
Outlet 4
úû
ùêë
é+++ú
û
ùêë
é+++=ú
û
ùêë
é+++
·····
44
24
422
22
211
21
1 2)(
2)(
2)(
hgzc
mhgzc
mWhgzc
mQ
421
···
+= mmm
·
W
Energy Conservation Equation for Compressible flows CV
CS
Outlet 2Inlet 1
úû
ùêë
é+++=ú
û
ùêë
é+++
····
22
22
211
21
1 2)(
2)(
hgzc
mWhgzc
mQ
Steady Flow Energy Equation (1-1)
·
Q
C1,z1,h1,T1,u1
úû
ùêë
é++-ú
û
ùêë
é+++=
····
22
22
211
21
1 2)(
2)(
hgzc
mhgzc
mQW
·
··
þýü
îíì
úû
ùêë
é++-ú
û
ùêë
é++= 22
22
11
21
2)(
2)(
hgzc
hgzc
mW
But Mass Conservation gives that ···
== mmm 21
And for Adiabatic flow,
·
W
Adiabatic Energy Equation
Eq. 34
Application of Steady Flow Energy Conservation (SFEE) Equation
úû
ùêë
é+++=ú
û
ùêë
é+++
····
22
22
211
21
1 2)(
2)(
hgzc
mWhgzc
mQ
Steady Flow Energy Equation (1-1)
úû
ùêë
é++-ú
û
ùêë
é+++=
····
22
22
211
21
1 2)(
2)(
hgzc
mhgzc
mQW
···
== mmm 21
For Insulated Devices·
··
þýü
îíì
úû
ùêë
é++-ú
û
ùêë
é++= 22
22
11
21
2)(
2)(
hgzc
hgzc
mW
Adiabatic Energy Equation
Application of Steady Flow Energy Conservation (SFEE) Equation
Steady Flow Energy Equation (1-1)
·
··
þýü
îíì
úû
ùêë
é++-ú
û
ùêë
é++= 22
22
11
21
2)(
2)(
hgzc
hgzc
mW
Adiabatic Energy Equation
For Insulated Devices
Applicable Equation for Turbines & Turbo compressors:
úû
ùêë
é++=ú
û
ùêë
é++ 22
22
11
21
2)(
2)(
hgzc
hgzc
Applicable Equation for Nozzles & Diffusers (W=0):
úû
ùêë
é+=ú
û
ùêë
é+ 2
22
1
21
2)(
2)(
hc
hc ú
û
ùêë
é+=ú
û
ùêë
é+ 2
22
1
21
2)(
2)(
hc
hc
Note: dh=d(u+pv)=Cp(dT) Eq. 35
STAGNATION STATE
úû
ùêë
é+=ú
û
ùêë
é+ 2
22
1
21
2)(
2)(
hc
hc
When Flow is isentropically decelerated to final Zero velocity, We get Stagnation State
[ ] úû
ùêë
é+= 1
21
0 2)(
hc
h
0=+ cdcdh
[ ] úû
ùêë
é+= 1
21
0 2)(
TCc
TC pppC
cTT
2)( 2
110 +=
Eq. 36
ûë 2 ûë 2 pC2
1
21
1
0
2)(
1TC
cTT
p
+=úû
ùêë
é
1
21
1
0
12
)(1
TR
cTT
G÷÷ø
öççè
æ-
+=úû
ùêë
é
gg
1
21
1
0 )(21
1TR
cTT
Ggg
÷øö
çèæ -
+=úû
ùêë
é
÷÷ø
öççè
æ-
=÷÷ø
öççè
æ-
=1
&1 g
gg
Gp
Gv
RC
RC
Eq. 37
STAGNATION & STATIC STATES
1
21
1
0 )(21
1TR
cTT
Ggg
÷øö
çèæ -
+=úû
ùêë
é2
1
21
1
0
)()(
21
1ac
TT
÷øö
çèæ -
+=úû
ùêë
é g
( )21T öæ -+=
ùé g( ) ( )1
0
1
0
1
00
---÷÷ø
öççè
æ -
úù
êé
=úù
êé
=úù
êé
=úù
êé
gggg
rvpTBut( )21
1
0
21
1 MTT
÷øö
çèæ -
+=úû
ùêë
é g1
0
1
0
1
0
1
0øè
úû
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
érr
vv
pp
TT
But
÷÷ø
öççè
æ-÷÷
ø
öççè
æ-
úû
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
é 11
1
0
1
01
1
0
1
0 &gg
g
rr
TT
TT
pp
Eq. 40 Eq. 41
RELATIONSHIP BETWEENSTAGNATION & STATIC PROPERTIES
MTT 2
11
0
21
1 ÷øö
çèæ -
+=úû
ùêë
é g ÷÷ø
öççè
æ-÷÷
ø
öççè
æ-
úû
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
é=ú
û
ùêë
é 11
1
0
1
01
1
0
1
0 &gg
g
rr
TT
TT
pp
÷÷ø
öççè
æ-
úù
êé
÷ö
çæ -
+=úù
êé 12
10
21
1gg
gMp
pEq. 42ú
ûêë
÷ø
çè
+=úû
êë
11 2
1 Mp
÷÷ø
öççè
æ-
úû
ùêë
é÷øö
çèæ -
+=úû
ùêë
é 11
2
11
0
21
1gg
rr
M
Eq. 42
Eq. 43
Static Velocity of Sound at state x
xGx TRa g=
Stagnation Velocity of Sound
( ) ( ) oopoGo hTCTRa 11 -=-== ggg
Stagnation & Static Enthalpies
÷÷ø
öççè
æ+=÷÷
ø
öççè
æ+
22
21
1
2 ch
ch oo ÷÷
ø
öççè
æ+=
2
21
1
chho
Co=0
( ) ( )1212 TTChh p -=- ( ) ( )refxprefx TTChh -=-
Eq. 44
Stagnation & Static Enthalpies
00 == refref forTh
( ) ( )refxprefx TTChh -=-
For Gases, Defining
xpx TCh == 00 TCh p==
RELATIONS FOR STAGNATION ENTHALPY
÷÷ø
öççè
æ+=
2
2chho
TRTCh Gp ÷÷ø
öççè
æ-
==1g
gTRp Gr=
TRa Gg=2
÷÷ø
öççè
æ÷÷ø
öççè
æ-
=÷÷ø
öççè
æ-
==rg
gg
paTCh p 11
2
TRa Gg=
÷÷ø
öççè
æ+÷÷ø
öççè
æ÷÷ø
öççè
æ-
=÷÷ø
öççè
æ+÷÷ø
öççè
æ-
=÷÷ø
öççè
æ+=÷÷
ø
öççè
æ+=
212122
22222
0
cpcacTC
chh p rg
gg
÷÷ø
öççè
æ=÷÷
ø
öççè
æ
-=÷÷
ø
öççè
æ+÷÷ø
öççè
æ-
=÷÷ø
öççè
æ+=
21212
2max
20
222
0
cacacTCh p gg
RELATIONS FOR STAGNATION ENTHALPY
÷÷ø
öççè
æ=÷÷ø
öççè
æ
-=÷÷ø
öççè
æ+÷÷ø
öççè
æ÷÷ø
öççè
æ-
=÷÷ø
öççè
æ+÷÷ø
öççè
æ-
=÷÷ø
öççè
æ+=÷÷
ø
öççè
æ+=
21212122
2max
20
22222
0
cacpcacTC
chh p grg
gg
00max 12
2 ahc ÷÷ø
öççè
æ
-==
g
÷ö
çæöæöæ 222 cca 2 22
÷ö
çæ
÷ö
çæ ca
Eq. 45
÷÷ø
öççè
æ=÷÷ø
öççè
æ+÷÷ø
öççè
æ- 221
2max
22 ccag ( )
11
22
max
2
2max
2
=÷÷ø
öççè
æ+÷÷ø
öççè
æ
- c
c
c
a
g
12max
2
20
2
=÷÷ø
öççè
æ+÷÷ø
öççè
æ
c
c
a
aEq. 46
STEADY FLOW ELLIPSE& DIFFERENT REGIMES OF FLOW
12max
2
20
2
=÷÷ø
öççè
æ+÷÷ø
öççè
æ
c
c
a
aGoverning Equation Eq. 46
GAS DYNAMICS RELATIONS…
¢ Max Flow Velocity ( cmax)
00max 12
2 ahc ÷÷ø
öççè
æ
-==
g
airforc
..24.22max =÷
öçæ
=÷ö
çæ
Eq. 47
Eq. 48airforac
..24.21
2
0
max =÷÷ø
öççè
æ
-=÷÷
ø
öççè
æg
Eq. 48
GAS DYNAMICS RELATIONS…
¢ Critical Flow Velocity ( ccritical / C*)
It is the flow velocity when M=1
*** == TRaC Gg Eq. 49
( )211
0
21
1 MTT
÷øö
çèæ -
+=úû
ùêë
é g
( ) ÷øö
çèæ +
=÷øö
çèæ -
+=úûù
êëéÞ÷
øö
çèæ -
+=úûù
êëé
**
* 21
21
121
1 020 gggTT
MTT
Eq. 50
GAS DYNAMICS RELATIONS…
¢ Critical Flow Velocity ( ccritical / C*)*** == TRaC Gg ÷
øö
çèæ +
=úûù
êëé
* 210 g
TT
00
12
12 a
TRaC G ÷
÷
ø
ö
çç
è
æ÷÷ø
öççè
æ+
=÷÷ø
öççè
æ÷÷ø
öççè
æ+
== **
ggg
Eq. 51
011G ÷ø
çè
÷ø
çè +÷
øçè
÷ø
çè + gg
**
÷÷ø
öççè
æ
-+
=÷÷ø
öççè
æ
-+
=÷÷ø
öççè
æ
-== caahc
11
11
12
2 00max gg
gg
g
airforac
cc
..45.211
**maxmax =÷÷
ø
öççè
æ
-+
=÷øö
çèæ=÷
øö
çèæ
gg
Eq. 56
TEMPERATURE RATIOS
( )20
21
1 MTT
÷øö
çèæ -
+=úûù
êëé g
÷øö
çèæ +
=úûù
êëé
* 210 g
TT
&
2* 12
MT
÷÷ö
ççæ -
+÷÷ö
ççæ
=úù
êé g 2
11
12
MTT
÷÷ø
öççè
æ+-
+÷÷ø
öççè
æ+
=úû
ùêë
égg
g Eq. 57