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Friction in noncircular conduits of fluid.pptx
1.
2. Mostly the conduits used in engineering practice
are of circular cross sections
Equations studied in the previous topic can
therefore be modified for application to non-
circular cross sections using the concept of
hydraulic radius.
As circular pipes are not that common, most of the
equations use diameter D instead of hydraulic
radius Rh as linear dimension
3. So for a circular pipe flowing full,
(19)
This can also be called the equivalent diameter.
Substituting this in (13)
(20)
Then the relation for Reynold’s number will take the
form
(21)
h
h R
D
D
R 4
4
/
g
V
R
L
f
h
h
f
2
4
2
h
R
V
R
4
4.
5. Consider steady uniform and laminar flow of
fluid in a pipe of constant diameter.
6. As per Newton’s law of viscosity,
(22)
Where u is the velocity at a distance y from the
boundary.
If an element of fluid is annular in shape at a
distance r from the centre line is considered, with
thickness dr then
(23)
Where negative sign shows that u decreases as r
increases
dy
du
dr
du
7. The coefficient of viscosity is normally constant for
any particular fluid at constant temperature
the wallSo if shear stress varies zero at the centre of the
pipe to maximum at the wall, the slope of the velocity
profile will be zero at the centre and will be having a
steeper gradient approaching .
For laminar flow in a circular pipe, substituting (23) into
(16)
(24)
r
L
dr
du
hf
2
8. (25)
Integrating (25) and determining the constant of
integration from the fact that u = umax, when r = 0,
Then
(26)
So it can be concluded that velocity profile is a
parabola. Also
2
max
2
max
4
kr
u
r
L
h
u
u
f
L
h
k
f
4
rdr
L
h
du
f
2
9. At the wall where r = r0, u = 0 and umax = Vc
(centerline velocity) then from (26)
(26) can then be written as
(27)
2
0
/ r
V
k c
2
0
2
2
2
0
1
r
r
V
r
r
V
V
u c
c
c
11. 1. Derive the relation (Hagen-Poiseulle Law) for the
head loss due to friction in Laminar flow case in
pipes. Also prove that mean velocity V is half of the
maximum velocity for laminar flow case.
Hint: consider a fluid element at a distance ‘r’ from
the center of the pipe. Let ‘dr’ be its thickness and
‘dA’ be its area.
12. 2. Prove that the frictional factor ‘f’ for laminar flow
under pressure in a circular pipe is given by
3. For the case of laminar flow in pipe, at what
distance from the centerline (in terms of pipe
radius) does the mean velocity occur.
4. An oil with kinematic viscosity 0.004ft2/sec
weighs 62 lb/ft2. Determine its flow rate and head
loss in a 2750-ft length pipe of 3inch diameter
when Reynolds no. is 950.
R
f /
64
13.
14. Consider a pipe connected to a reservoir with
rounded entrance
Consider section AA’ where fluid is just
entering into the pipe
So because of minimum friction, velocity
distribution will be uniform at this stage.
At section BB’, some sort of friction is exerted
by the walls of the pipe, thereby resulting in
negligible velocity at the pipe wall.
18. It is observed that as the distance from the
pipe wall increases, the frictional effect
reduces up till certain depth ‘y’ from the
wall.
This will consequently result in the increase
in the velocity up to certain depth ‘y’
beyond which the velocity distribution will
again be uniform up to the centre of pipe.
Thus the flow at section BB’ can be divided
into two components
19. ◦ Central frictionless cone
◦ Boundary layer
Boundary layer is the region of the fluid
extended from the pipe wall in which
velocity variation is taking place. So in this
range
Thickness of the boundary layer is normally
zero at the pipe entrance and increases as
the distance from the entrance increases
0
;
0
dy
du
20. Developing or establishing length
◦ It is that distance from the pipe entrance to a
section CC’ where a parabolic velocity profile first
becomes fully consistent
◦ Also, it is the distance from the pipe entrance at
which thickness of the boundary layer becomes r0.
◦ Beyond section CC’ for same straight pipe the
velocity profile does not change, and hence the flow
is known as Established Flow or Fully Developed
Flow.
◦ Given as
D
R
L e
e 058
.
0
21. For critical value of Reynold’s number, the
entrance length Le equals 116 times pipe
diameter.
In other cases of laminar flow with Re less
than 2000, Le will correspondingly be low.
22.
23. Point velocity in the flow field normally
fluctuates both in magnitude and direction
The fluctuation is normally due to the
formation of small eddies created by
viscous shear between adjacent particles
With the passage of time, the eddies grow
in size and then disappear because of their
mixing with adjacent eddy particles, thereby
resulting in a mixture of all the particles in
the fluid flow.
24. In fluid dynamics, an eddy is the swirling of
a fluid and the reverse current created when
the fluid is in a turbulent flow regime.
25.
26. In a manner similar to laminar flow, if the
molecules in the flow are replaced by
eddies, then the shear stress due to
turbulence will be directly proportional to
the velocity gradient
or
dy
du
tur
dy
du
tur
27. where is the eddy viscosity which is not normally
constant for a given temperature as it depends on the
flow turbulence.
can also be called coefficient of momentum transfer
expressing the momentum transfer from points of low
velocity to higher ones or vice versa.
Total shear stress for turbulent flow case is usually taken
as the summation of laminar shear stress and the
turbulent shear stress given by
29. kinematic viscosity
kinematic eddy viscosity
For any flow case for a smooth walled pipe,
shear stress is normally due to laminar flow
alone and therefore, for these cases
dy
du
30. For the case of turbulent flow, at some
distance “0.2r” from the pipe wall, the value
of du/dy becomes small.
This consequently results in a decrease of
viscous shear i.e. as compared to
turbulent shear.
The later can be large even though even
though du/dy is small. This normally happens
due to a relatively higher value of because
of higher turbulence.
viscous
lamor
31. If a fluid of mass m, below ab,
moving with temporal mean axial
velocity u moves upward where the
temporal mean axial velocity is
u+Δu, the momentum in axial
direction increases by mΔu and vice
versa.
The back and forth transfer of
momentum results in shear along
the plane ab proportional to Δu.
If distance Δy is chosen in such a
way that average value of +u′ in
upper zone over a time period (long
enough to include many velocity
fluctuations) is equal to Δu i.e.
u
u
32. The distance between the two streams will be
called mixing length ‘l’, equal to Δy.
If a mass is moving upward from below ab with a
velocity v′ it will transport into upper zone where
the velocity is u+u′, the momentum per unit time
will be given by
(Kg/m3)(m/s)(m2)(m/s) = Mass x velocity = Momentum/unit time
Slower moving mass below ‘ab’ will tend to retard
the upper moving mass, resulting in shear along
the plane ab. This shear force will be given by
=(kg/m3)(m3/s)(m/s) = (kg.m/s2)
Newton = (kg.m/s2)
)
)(
( u
dA
v
dA
v
u
u
u
u
dA
v
V
Q
dA
F
)
)(
(
)
(
)
( V
Q
dA
F
33. In case of a larger number of fluctuations,
In modern turbulence theory, is known
as Reynold’s stress.
-ve sign shows that the product is always
negative on average.
v
u
dA
F
/
v
u
v
u
34. ‘l’ is the distance
transverse to the flow
direction such that
From figure
So
u
u
dy
ldu
u /
dy
ldu
u /
35. If
And if l accounts for the constant of proportionality,
according to Prandtl,
varies as
Thus
v
u
v
u
2
2
/ dy
du
l
2
2
dy
du
l
v
u
36. In any experiment that determines the pipe
frictional loss, one can determine and
at any radius.
and at any radius
Mixing length l as a function of pipe radius.
0
u dy
du /
37. The mixing length is a
distance that a fluid
parcel will keep its
original characteristics
before dispersing them
into the surrounding
fluid. Here, the bar on
the left side of the figure
is the mixing length.
38.
39. If Reynold’s number is greater than the critical
value, up to some point, the entrance condition is
same as Laminar flow
Due to the increase in the thickness of the laminar
boundary layer, a certain point of transition occurs
and the boundary layer becomes turbulent
Turbulent boundary layer increases in thickness
much more rapidly than a laminar layer
Transition normally occurs where the length xc of
the laminar portion of the boundary layer is about
or
U
/
500000 500000
/
Ux
Rx
42. Turbulent boundary layer increases in thickness
much more quickly than the laminar layer therefore
the length of the inviscid core is slightly shorter
than the one for laminar flow case.
Velocity profile is generally fully developed within
20-40 diameters.
43.
44.
45. Even for the case of turbulent flow, the flow next to
the wall is essentially laminar due to
◦ Smaller velocity of flow near the pipe wall results in smaller
Reynold’s number so flow becomes laminar (No Slip
Condition)
◦ Lateral component of velocity is not possible so the flow is
laminar
◦ Some transient effects are still induced in the flow due the
movement of adjacent turbulent layer resulting in some
disruption in the sublayer.
◦ Since this layer is not a true laminar layer and shear in the
layer is due to viscosity alone, it is called viscous sublayer.
46. Even though the viscous sub layer is extremely thin, it
still has a great effect because of the very steep velocity
gradient within it.
For studying the velocity profiles, shear stress velocity
or frictional velocity is quite important and is given by
(A)
Although its dimensions are of velocity, scientists have
named it as shear stress velocity or friction velocity u*
although it is actually not flow velocity.
/
0
*
u
47. When there is an extremely thin layer of fluid
next to the wall where viscous shear
dominates, its velocity profile can be scarcely
distinguished from a straight line, so for
linear velocity profile
or (B)
y
u
0 y
u
o /
/
48. Squaring eq. (A) and equating to eq. (B)
(C)
y
u
u
2
*
*
*
yu
u
u
This is known as law of the wall
49. The linear relation for u(y) approximates experimental
data well in the range of
From this the thickness for the viscous sub-layer comes
out to be
5
/
0 *
yu
*
/
5 u
L
50. Since we know from previous = fV2ρ/8 or
= fV2Ƴ/8g (8.19 of the book)
Simplifying the above equation we get
/ρ = fV2/8
Hence (a)
Also (b)
Substituting equation (a) in equation (b)
8
0
*
f
V
u
0
0
0
*
/
5 u
L
f
R
D
f
V
L
14
.
14
14
.
14
51. It can be seen from the expression that
higher the velocity or lower the kinematic
viscosity, thinner is the viscous sub-layer.
For a given constant pipe diameter, thickness
of the viscous sub-layer decreases as the
Reynold’s number increases.
53. If the irregularities on a surface are small enough that
the effects of the projections do not pierce through the
viscous sublayer, the surface is hydraulically smooth
If the effects of the projections extend beyond the
sublayer, the laminar layer is broken up and the surface
is no longer hydraulically smooth.
If the roughness projections are large enough to
protrude through the transition layer, it is totally broken
up, the resulting flow is fully rough (pipe) flow.
54. if e is the equivalent height of roughness projection,
then for viscous sublayer completely
buries the surface roughness, the roughness has
no effect on friction and the pipe is hydraulically
smooth.
5
/
*
eu
55. If the pipe will behave as fully
rough.
70
/
*
eu
56. If the pipe will behave in a transition
mode, neither hydraulically smooth nor fully rough.
70
/
5 *
eu
57.
58. Turbulent flow in a pipe is strongly influenced by
the flow phenomena near the wall (Prandtl).
at some point P very close to the wall.
Mixing length l near the wall is proportional to the
distance from the wall
from experiments K=0.4.
y is the distance of point P from wall (outside
viscous sublayer).
0
Ky
l
59. r0 r
y
Particle at Point
‘P’ near the
edge of the
Pipe wall
y=r
Umax
Viscous Sublayer
0
60. If at P
and we know that
for
From Prandtl’s mixing length theory
0
tur
vis
0
vis
0
tur
2
2
0
dy
du
l
tur
62. Integrating both sides to obtain expression for u with
respect to y.
--(A) {Since ∫ (1/y) dy = ln y + c}
“C” is a constant which can be determined from
boundary conditions
C
y
u
ln
5
.
2 0
max
0 u
u
r
y
C
r
u
0
0
max ln
5
.
2
63. -- (B)
So
or
The rearrangement of the above equation is also
known as “Velocity Defect Law”
0
0
max ln
5
.
2 r
u
C
0
max
0
ln
5
.
2
ln
5
.
2 r
u
y
u o
r
r
r
u
u
o
o 0
max ln
5
.
2
r
r
y
0
64.
65. r0
r
y=r
Umax
r
dr
Consider turbulent flow of an incompressible fluid in
a pipe of constant diameter D (radius r0).
Consider an annular shape element of fluid at a
distance “r” having thickness “dr”
Area of the fluid element will be given by
rdr
dA
2
66. and moves with velocity ‘U’ (m/s) given as
r
r
r
u
u
o
o 0
max ln
5
.
2
67. Since discharge is product of area and velocity
dQ = U x dA -------(X)
Putting values of dA and U in expression (X) and
integrating
Dividing by the pipe area and simplifying the
above expression, the mean velocity is given by
0
0 0
0
0
max ln
5
.
2
2
r
rdr
r
r
r
rdr
u
Q
0
0
0
2
0
0
0
max ln
2
ln
5
.
2
r
dr
r
r
r
r
r
u
V
68. After integration we will get
As we know that
Then
or
8
*
0 f
V
u
f
V
u
f
u
V 326
.
1
8
5
.
2
)
5
.
1
( max
max
f
V
V
u 326
.
1
max
f
V
u 326
.
1
1
max
0
max 75
.
3
u
V
69. The rearrangement of the previous expression as is
also called as pipe factor
f
u
V
326
.
1
1
1
max
