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.
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• 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
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.