High Speed Ducts for Global Needs of Nation…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Compressible Viscous Fluid Flows in Finite Length Ducts

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

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

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

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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.