Pipes for Global Energy Needs of Nation…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Gas Flows & Maximum Capacity of A Pipe

World's Longest Natural Gas Pipelines

• West-East Pipeline :Length: 5,410 miles, Start: Xinjiang, China -- Finish: Shanghai.

• GASUN Pipeline :Length: 3,100 miles, Places: Starts in Bolivia, ends in Brazil.

• Yamal-Europe Pipeline: Length: 2,608 miles, Places: Starts in Siberia, ends in Germany.

• Trans-Saharan Pipeline: Length: 2,565 miles, Places: Starts in Nigeria, ends in Algeria.

• TransCanada Pipeline: Length: 2,005 miles, Places: Starts in Alberta, ends in Quebec.

• Rockies Express Pipeline (REX): Length: 1,678 miles, Places: Starts in Colorado, ends in Ohio.

• Transcontinental Pipeline: Length: 1,671 miles, Places: Starts in Texas, ends in New York.

• Trans-Mediterranean Pipeline: Length: 1,610 miles, Places: Starts in Algeria, ends in Italy

• Northern Border Pipeline: Length: 1,391 miles, Places: Starts in Canada, ends in Chicago.

• Nord Stream Pipeline: Length: 759 miles, Places: Starts in Russia, ends in Germany

Indian Pipe Flow for Better economy & Ecology

• Hajira-Bijapur-Jagdishpur (HBJ) Gas Pipeline:

• This is 1,750 km long and connects Hazira in Maharashtra to Bijapur in M.P. and Jagdishpur in U.P.

• This is the world s largest underground pipeline. tal city.

• Jamnagar-Loni LPG Pipeline:

• This 1,269 km long pipeline has been constructed by Gas Authority of India Limited (GAIL).

• This is the longest LPG pipeline of the world.

• It is like transporting 5.0 lakh cylinder per day.

• It will result in net saving of Rs. 500 crore per year by eliminating road tanker movement and lead to reduction of about 10,000 tonnes of pollutant emission per year.

Experimental Analysis of Pipe Flows

The Capacity of A Pipe

tioncross

udAQsec

Rr

rrudrQ

02

tioncross

udAmsec

Rr

rrudrm

02

• The mean velocity is defined by

A

Qu

A

mu

Friction Factor

w is proportional to mean velocity.

• It is customary, to nondimensionalize wall shear with the pipe dynamic pressure.

2

2uf wall

This is called as standard Fanning friction factor, or skin-friction coefficient.

22

2uf

dx

dpRwall

Evolution of Mean Velocity of Compressible Frictional Flow in A Constant Area Duct

0V

dVd

02

2

Vhd

w

Self similar compressible fully developed flow through ducts

Adiabatic flow through Pipe:

0VdVdTC p

Energy Conservation Equation in terms of Mach Number

0VdVdTC p

01

VdVdTR

01

2 V

dVV

T

dTRT

01

1 2 V

dVM

T

dT

Sonic Equation

RT

V

c

VM

2

2

22

Differential form of above equation:

T

dT

V

dV

M

dM

2

2

222

RT

dTV

RT

VdVMdM

021

1 2

T

dT

M

dMM

T

dT

Sonic Equation into Energy Equation 01

1 2 V

dVM

T

dT

M

dM

M

M

T

dT

2

2

21

1

1

Energy equation can be modified as:

0V

dVd

T

dTd

p

dp

T

dT

V

dV

p

dp

T

dT

V

dV

M

dM

2

T

dT

M

dM

p

dp

2

M

dM

M

M

T

dT

2

2

21

1

1

Energy equation can be used to replace temperature terms in pressure equation:

T

dT

M

dM

p

dp

2

M

dM

M

M

M

dM

p

dp

2

2

21

1

1

2

1

Frictional Flow in A Constant Area Duct

AdpPdxdVm w

w

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

AdpPdxAVdV w

Divide by V2

22 V

dpdx

A

P

VV

dV w

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

02 2

V

dpdx

A

Pf

V

dV

02 2

p

dp

V

pdx

A

Pf

V

dV

02 2

p

dp

V

pdx

A

Pf

V

dV

22 V

dpdx

A

P

VV

dV w

02 2

p

dp

V

pdx

A

Pf

V

dV

02 2

p

dp

V

p

dxA

Pf

V

dV

01

2 2

p

dp

Mdx

A

Pf

V

dV

01

2 2

p

dp

Mdx

A

Pf

V

dV

M

dM

M

M

T

dT

2

2

21

1

1

M

dM

M

M

M

dM

p

dp

2

2

21

1

1

2

1

T

dT

V

dV

M

dM

2

Differential Equations for Frictional Flow Through Constant Area Duct

01

2 2

p

dp

Mdx

A

Pf

T

dT

M

dM

0

2

11

1

2

11

22

11

1

2

2

22

2

M

dM

M

M

M

dM

Mdx

A

Pf

M

dM

M

M

M

dM

dxA

Pf

M

MM

M

dM

212

11

2

22

Differential Equations for Frictional Flow Through Constant Area Duct

dxA

Pf

M

M

T

dT

21

12

4

dxA

Pf

M

MM

p

dp

21

112

22

dxA

Pf

M

MM

M

dM

212

11

2

22

Differential Equations for Frictional Flow Through Constant Area Duct

M dM dp dT dV

<1 +ve -ve -ve +ve

>1 -ve +ve +ve -ve

Compressible Flow Through Finite Length Duct

22

2

21

1

12

MM

M

M

dM

D

fdx

h

dxA

Pf

M

MM

M

dM

212

11

2

22

Integrate over a length l

M

dM

MM

M

D

fdx e

i

M

M

l

h

22

2

0

21

1

12

Maximum Length of A Pipe

22

22

22

21

1

21

1ln

2

11114

ie

ei

eih MM

MM

MMl

D

f

Using a Mean friction factor over a length l .

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

2

2

2max

21

1

12

11

ln2

11

114

i

i

ih M

M

Ml

D

f

Compressible Frictional Flow through Constant Area Duct

Fanno Line

p

dpR

T

dTCds p

p

dp

T

dT

C

ds

p 1

2

11

V

TC

T

dT

T

dT

C

ds p

p

TT

T

T

dT

T

dT

C

ds

p 0

11

TT

dT

T

dT

C

ds

p

02

11

T

T

T

T

s

s p iiiTT

dT

T

dT

C

ds

02

11

2

1

0

0

/1

lniip

i

TT

TT

T

T

C

ss

Fanno Line

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

Degree of Creeping

• How deep the presence of a boundary can propagate into the flow field?

• An almost imperceptible flow field (creeping flow field) completely respects the presence of a solid boundary.

• How to define the degree of creeping?

• What if the fluid particle can move much faster than the speed at which the effect of solid boundary propagates into the flow field?

• No effect of Wall at all or something else?

An Ingenious Lecture

• A29 year old professor in Hanover, Germany delivered in a 10 minutes address in 1904 on this topic.

• This concept is a classic example of an applied science greatly influencing the development of mathematical methods of wide applicability.

• Prof. Ludwig Prandtl.

• Prandtl had done experiments in the flow of water over bodies, and sought to understand the effect of the small viscosity on the flow.

• Realizing that the no-slip condition had to apply at the surface of the body, his observations led him to the conclusion that the flow was brought to rest in a thin layer adjacent to the rigid surface.

• The boundary layer.

The Boundary Layer Effect : The Leader of Asymptoticity

• Prandtl reasoning suggested that the Navier-Stokes equations should have a somewhat simpler form owing to the thinness of this layer.

• This led to the equations of the viscous boundary layer.

• Boundary-layer methods now occupy a fundamental place in many asymptotic problems for partial differential equations.

• Ludwig Prandtl, with his fundamental contributions to hydrodynamics, aerodynamics, and gas dynamics, greatly influenced the development of fluid mechanics as a whole.

• His pioneering research in the first half of the 20th century that founded modern fluid mechanics.

Publications by Ludwig Prandtl

• 1913 , The doctrine of the fluid and gas movement.

• 1931, Demolition the Str¨omungslehre.

• 1942, Essentials of Fluid Mechanics

• An indication of Prandtl 's intentions to guide the reader on a care carefully thought-out path through the different areas of fluid mechanics .

• On his way, Prandtl advances intuitively to the core of the physical problem, without extensive mathematical derivations.

• After Prandtl’s death, his students Klaus Oswatitsch and Karl Wieghardt undertook to continue his work, and to add new findings in fluid mechanics in the same clear manner of presentation.