gas dynamics relations

Gas Dynamics

BULK MODULUS (K) & COEFF. OF COMPRESSIBILITY(KC)

V.Uma Maheshwar,

Faculty,MED, OUCE

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

¢Speed of Sound (a)

¢ 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


¢ Momentum Equation givesWavefront is made Stationary by imposing Opp Velocity C

[ ] [ ]))(()(.

dpppAadcam +-=--

[ ] [ ])()(.

dpAdcm =[ ] [ ])()( dpAdcm =

Acm r=.

adcdp r=Eq. 22

a-dc a


¢ Momentum Equation gives

adcdp r= Eq. 22

rr addc = Eq. 21

= r2

÷÷ø

öççè

æ==

=

r

r

ddp

aSoundVel

dadp 2aa-dc

Eq. 23


TMR

TRpk

ddp

aw

UG

s ggrg

rr=====

Sound Velocity is related to T,Ks,KC,& Mw

Eq. 24


¢ 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.


¢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)

Incompressible flow (M~0,u/a=0) Subsonic flow (M<1, u=a/2)

Sonic flow (M=1, a=u)

Supersonic flow (M=2, a=u/2)

÷øö

çèæ=Þ== -

MMutatSin

1sin

1/ 1aa

Semi Cone Angle=Mach Angle

Eq. 24

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


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


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


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


CS

Outlet 2

Outlet 4

Inlet 1


·

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


CS

Outlet 2Inlet 1

úû

ùêë

é+++=ú

û

ùêë

é+++

····

22

22

211

21

1 2)(

2)(

hgzc

mWhgzc

mQ


·

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


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


Application of Steady Flow Energy Conservation (SFEE) Equation


·

··

þýü

îíì

úû

ùêë

é++-ú

û

ùêë

é++= 22

22

11

21

2)(

2)(

hgzc

hgzc

mW


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


¢ 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


¢ 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

CROCCO NUMBER

÷÷ø

öççè

æ÷øö

çèæ=÷÷

ø

öççè

æ=

max

*

*max c

ccc

cc

Cr Eq. 58

÷÷ö

ççæ +

=÷ö

çæ 1max gc

But÷÷ø

ççè -

=÷ø

çè 1*

max

gcBut

÷÷ø

öççè

æ

+-

÷øö

çèæ=÷÷

ø

öççè

æ=

11

*max g

gcc

cc

Cr

gas dynamics relations

Engineering