ME 6604 - GAS DYNAMICS AND JET PROPULSION

UNIT – I

BASIC CONCEPTS AND FUNDAMENTALS OF COMPRESSIBLE FLOW

T.SURESHASSISTANT PROFESSOR

DEPT OF MECHANICAL ENGGKAMARAJ COLLEGE OF ENGINEERING

PART - A• FUNDAMENTALS OF COMPRESSIBLE FLOW

• Energy and momentum equations for compressible fluid flows, various regions of flows, reference velocities, stagnation state, velocity of sound, critical states, Mach number, critical Mach number, types of waves, Mach cone, Mach angle, effect of Mach number on compressibility.

•PART – B

• Flow through variable area duct • Isentropic flow through variable area ducts, T-

s and h-s diagrams for nozzle and diffuser flows, area ratio as a function of Mach number, mass flow rate through nozzles and diffusers, effect of friction in flow through nozzles

FLOW THROUGH VARIABLE AREA DUCTS

FLOW THROUGH VARIABLE AREA DUCTS

• As a gas is forced through a tube, the gas molecules are deflected by the walls of the tube. If the speed of the gas is much less than the speed of sound of the gas, the density of the gas remains constant and the velocity of the flow increases. However, as the speed of the flow approaches the speed of sound we must consider compressibility effects on the gas. The density of the gas varies from one location to the next. Considering flow through a tube, as shown in the figure, if the flow is very gradually compressed (area decreases) and then gradually expanded (area increases), the flow conditions return to their original values. We say that such a process is reversible. From a consideration of the second law of thermodynamics, a reversible flow maintains a constant value of entropy. Engineers call this type of flow an isentropicflow; a combination of the Greek word "iso" (same) and entropy.

FLOW THROUGH VARIABLE AREA DUCTS

FLOW THROUGH VARIABLE AREA DUCTS

• The conservation of mass is  a  fundamental  concept  of physics. Within some problem domain, the amount of mass remains  constant;  mass  is  neither  created  or  destroyed. The mass of any object is simply the volume that the object occupies times the density of the object. For a fluid (a liquid or a gas)  the density, volume, and shape of  the object can all change within the domain with time and mass can move through the domain.

• The conservation of mass (continuity) tells us that the mass flow  rate mdot through  a  tube  is  a  constant  and  equal  to the product of the density r, velocity V, and flow area A:

Conservation of mass

Conservation of mass• Solid Mechanics• The conservation of mass is a fundamental concept of physics along with the conservation of energy and

theconservation of momentum. Within some problem domain, the amount of mass remains constant--mass is neither created nor destroyed. This seems quite obvious, as long as we are not talking about black holes or very exotic physics problems. The mass of any object can be determined by multiplying the volume of the object by the density of the object. When we move a solid object, as shown at the top of the slide, the object retains its shape, density, and volume. The mass of the object, therefore, remains a constant between state "a" and state "b."

• Fluid Statics• In the center of the figure, we consider an amount of a static fluid , liquid or gas. If we change the fluid

from some state "a" to another state "b" and allow it to come to rest, we find that, unlike a solid, a fluid may change its shape. The amount of fluid, however, remains the same. We can calculate the amount of fluid by multiplying the density times the volume. Since the mass remains constant, the product of the density and volume also remains constant. (If the density remains constant, the volume also remains constant.) The shape can change, but the mass remains the same.

• Fluid Dynamics• Finally, at the bottom of the slide, we consider the changes for a fluid that is moving through our domain.

There is no accumulation or depletion of mass, so mass is conserved within the domain. Since the fluid is moving, defining the amount of mass gets a little tricky. Let's consider an amount of fluid that passes through point "a" of our domain in some amount of time t. If the fluid passes through an area A at velocity V, we can define the volume Vol to be:

• Vol = A * V * t

Conservation Laws for a Real Fluid

0. Vt

wqVete

.

gVVtV

ij ˆ..

iiij pij '

gpVVtV

ij ˆ.. '

Conservation of Mass Applied to 1 D Steady Flow

0. Vt

Conservation of Mass:

Conservation of Mass for Stead Flow:

0. V

Integrate from inlet to exit :

onstant. CVdVV

One Dimensional Stead Flow

A,V

A+dA,V+dV d

dl

onstant.. CdxdAVV

onstant. CdxdxVAd

0VAd

0AdA

VdVd

Conservation of Momentum For A Real Fluid Flow

pVVij '..

VdpVdVdVVVVV

ij '..

No body forces

One Dimensional Steady flow

A,V

A+dA,V+dV d

dl

dAdxpdxdAdAdxVVV

wVV

ij '.

dxdxpAddx

dxAddx

dxAVd ww

2

pAdAdAVd ww 2

Conservation of Energy Applied to 1 D Steady Flow

wqVete

.

Steady flow with negligible Body Forces and no heat transfer is adiabatic real flow

wVe

.

For a real fluid the rate of work transfer is  due to viscous stress and pressure. Neglecting the the effect of viscous dissipation.

VdAnpVe

.ˆ.

For a total change from inlet to exit :

AV

VdAnpVdVe

.ˆ.

Using gauss divergence theorem:

One dimensional flow 

VV

VdVpVdVe

..

VV

dAdxVpdAdxVe

..

dx

dxpAVddx

dxeVAd

pAVdeVAd

2

2Vue

AVpdVuVAd

2

2

02

2

VpuVAd

02

2

VhVAd

Summary of Real Fluid Analysis

0AdA

VdVd

pAdAdAVd ww 2

02

2

VhVAd

Further Analysis of Momentum equation

pAdAdVAdVVAVd ww

pAdAdVAdV ww

pAdPxddVm w

pAdAddVm ww

pdAAdpPdxxdPPxddVm www

Frictional Flow in A Constant Area Duct

0VdVd

AdpPdxPxddVm ww

02

2

VhVd

Frictional Flow in A Constant Area Duct

AdpPdxdVm w

w

The shear stress is defined as and average viscous stress which is always opposite to the direction of flow for the entire length dx.

AdpPdxPxddVm ww

AdpPdxAVdV w

AdpPdxAVdV w

Divide by AV2

22 Vdpdx

AP

VVdV w

0VdVd

002

2

VdVdTCVhd p

One dimensional Frictional Flow of A Perfect Gas 

0VdVd

0VdVdTC p

2Vdpdx

APf

VdV

TdT

VdV

pdp

TdTd

pdp

Sonic Equation

2

22

2

22 22

RTdTV

RTVdVMdM

RTV

cVM

Differential form of above equation:

TdT

VdV

MdM

2

TdT

VdV

pdp

TdT

MdM

pdp

2

M

dM

M

MTdT

2

2

211

1

Energy equation can be modified as:

TdT

MdM

pdp

2

M

dM

M

MMdM

pdp

2

2

211

121

1D steady real flow through constant area duct : momentum equation

022 Vdpdx

AP

VVdV w

022 pdp

Vpdx

AP

VVdV w

022 pdp

Vpdx

AP

VVdV w

022 pdp

Vpdx

AP

VVdV w

022 pdp

V

pdx

AP

VVdV w

0122 pdp

Mdx

AP

VVdV w

0122 pdp

Mdx

AP

VVdV w

M

dM

M

MTdT

2

2

211

1

M

dM

M

MMdM

pdp

2

2

211

121

TdT

VdV

MdM

2

Differential Equations for Frictional Flow Through Constant Area Duct

TdT

MdM

pdp

2

0122 pdp

Mdx

AP

VTdT

MdM w

0

211

1211

211

12

2

222

2

MdM

M

MMdM

Mdx

AP

VMdM

M

MMdM w

dxAP

VM

MM

MdM w

22

22

12

11

dxAP

VMM

TdT w

22

4

11

dxAP

VMMM

pdp w

22

22

111

dxAP

VM

MM

MdM w

22

22

12

11

Second Law Analysis

vdpdTCTds p

dpTv

TdTCds p

pdpR

TdTCds p

pdpR

TdTCds p

pdp

TdT

Cds

p 1

211

VTC

TdT

TdT

Cds p

p

TTT

TdT

TdT

Cds

p 0211

TTdT

TdT

Cds

p

02

11

T

T

T

T

s

s p iiiTT

dTTdT

Cds

0211

2

1

0

0

/1

lniip

i

TTTT

TT

Css

dxAP

VMM

TdT w

22

4

11

Fanno Line

Adiabatic flow in a constant area with friction is termed as Fanno flow.

Isentropic Nozzle and Adiabatic Duct

C Nozzle Discharge Curve

CD Nozzle + Discharge Curve

Nature of Real FlowEntropy of an irreversible adiabatic system should always increase!

dxAP

VMCds w

p 221

dxAP

VM

MM

MdM w

22

22

12

11

dxAP

VMM

TdT w

22

4

11

dxAP

VMMM

pdp w

22

22

111

M dM dp dT dV<1 +ve -ve -ve +ve>1 -ve +ve +ve -ve

Compressible Real Flow

),(Re, Mdkfunctionf

Effect of Mach number is negligible….

)(Re,dkfunctionf

1Re

n

TT

2

1

2

1

Pressure drop in Compressible Flow

Laminar Flow 

Turbulent Flows

22

2

211

1

MM

MMdMdx

APf

Re16

f

2

9.0Re74.5

7.3log

0625.0

hDk

f

Moody Chart

Compressible Flow Through Finite Length Duct

Integrate over a length l

22

2

211

14

MM

MMdM

Dfdx

h

MdM

MM

MDfdx e

i

M

M

l

h

22

2

0

211

14

22

2

211

14

MM

MMdM

Dfdx

h

22

22

22

211

211

ln2

11114

ie

ei

eih MM

MM

MMl

Df

is a Mean friction factor over a length l . f

Maximum Allowable Length

• The length of the duct required to give a Mach number of 1 with an initial Mach number Mi

Similarly

2

2

2max

211

12

11ln

211114

i

i

ih M

M

Ml

Df

1

2

2

*

211

21

1*

iM

p

p MdM

M

M

pdp

pp

1

2

2

211

1*

ii M

T

T MdM

M

MTdT

2/1

2*

211

21

1

i

i MMpp

2

*

211

21

iMTT

2

5.15 .

1111.384

293103.3

msN

TT

1/2

211

i

o Mpp

*0

*0

*0

0

pp

pp

pp

12

**0

0

21

211

iM

pp

pp

122

*0

0

21

2111

i

i

M

Mpp

Compressible Frictional Flow through Constant Area Duct

HDfL*4

0

*0

pp

pp*

TT *

VV *

M

Frictional Flow in A Variable Area Duct

0AdA

VdVd

A,V

A+dA,V+dV d

042

22

VdVMdx

DfM

pdp

h

TdTd

pdp

0AdA

VdVd

AdA

VdV

TdT

pdp

042

22

VdVMdx

DfM

AdA

VdV

TdT

h

TdT

MdM

VdV

TdT

VdV

MdM

22

02

422

22

TdT

MdMMdx

DfM

AdA

TdT

MdM

TdT

h

M

dM

M

MTdT

2

2

211

1

042

211

1 2

2

2

dx

DfM

MdM

M

MAdA

h

dxDfM

M

M

AdA

M

M

MdM

h

421

211

12

11 2

2

2

2

2

Constant Mach number frictional flow

hDfM

dxdA 22

dxDfMM

AdAM

MdMM

h

422

112

1112

222

Sonic Point : M=1

0422

112

11

dx

Df

AdA

h

0422

12

1

dx

Df

AdA

h

dxDf

AdA

h

42

54

Stagnation Properties

Consider a fluid flowing into a diffuser at a velocity   , temperature T, pressure P, and enthalpy h, etc. Here the ordinary properties T, P, h, etc. are called the static properties; that is, they are measured relative to the flow at the flow velocity.   The diffuser is sufficiently long and the exit area is sufficiently large that the fluid is brought to rest (zero velocity) at the diffuser exit while no work or heat transfer is done.  The resulting state is called the stagnation state. 

V

We apply the first law per unit mass for one entrance, one exit, and neglect the potential energies.  Let the inlet state be unsubscripted and the exit or stagnation state have the subscript o.

q h V w h Vnet net o

o

2 2

2 2

55

Since the exit velocity, work, and heat transfer are zero, 

h hV

o

2

2

The term ho is called the stagnation enthalpy (some authors call this the total enthalpy).  It is the enthalpy the fluid attains when brought to rest adiabatically while no work is done.

If, in addition, the process is also reversible, the process is isentropic, and the inlet and exit entropies are equal.

s so

The stagnation enthalpy and entropy define the stagnation state and the isentropic stagnation pressure, Po.  The actual stagnation pressure for irreversible flows will be somewhat less than the isentropic stagnation pressure as shown below.

56