CH.8 Viscous Flow in Pipes

1
CH.8 Viscous Flow in Pipes
object of this chapter application of the
basic principles to a specific, important
topic-the flow of viscous, incompressible fluids
in pipes and ducts closed conduit pipe
commonly called a pipe if it is of round cross
section duct if it is not round basic
components of a typical pipe system see p460
Fig. 8.1 solving method
2
(No Transcript)
3
8.1 General Characteristics of Pipe Flow
1. Pipe and Duct Flow pipe flow considerable
pressure difference
duct flow small
pressure difference 2. Difference between
open-channel flow and the pipe flow






(see p460 Fig. 8.2) - driving force
open channel flow gravity alone pipe flow
pressure gradient along the pipe(main)
gravity If the pipe is not full, it is
not possible to maintain this pressure
difference,
4
(No Transcript)
5
8.1.1 Laminar or Turbulent Flow
1. Laminar, transitional and turbulent flow (p461
Fig.8.3) 2. Time dependence of fluid velocity at
a point (p461 Fig. 8.4) 3. Most important
dimensionless parameter for pipe flow
Reynolds number 4. Laminar Flow(?? ??) and
Turbulent Flow(?? ??) - For a sufficiently low
Reynolds number (in a pipe flow and in a wide
channel) a laminar flow results. - At
sufficiently high Reynolds number a turbulent
flow occurs. 5. Critical Reynolds Number(?? ????
?) 1 Role delineating the regimes of laminar
and turbulent flow 2 lower and upper critical
Reynolds number
6

• 1gt upper critical(???) Reynolds number
• It is indefinite because the upper limit of
  laminar flow is indefinite.
• 2gt lower critical(???) Reynolds number
  
• - a limit below which laminar flow will always
  occur
• - Below all turbulence entering
  the flow from any source will eventually be
  damped out by viscosity.
• In general it is called .
• 3
  Critical Reynolds number is very much a
  function of boundary geometry.
• 4 Example of
• - for flow in a circular pipes(mean velocity and
  diameter) 2,100
• - for flow between parallel walls(mean velocity
  and spacing)
• 1,000

7
- for flow in a wide open channel(mean velocity
and water depth) 5,000 - for flow
about sphere(approach velocity and diameter)
1 5 Determination of must be
determined experimentally because of the complex
origins of turbulence. 6. Comparison of Laminar
with Turbulent Flow 1 Agitation of Fluid
Particle 1gt Laminar Flow - irregular
molecular motionlength scale of the order of
the mean free path(??????) of the molecules -
macroscopically a well-ordered flow (fluid
particles are constrained to motion in
essentially parallel paths by the action of
viscosity)
8
2gt Turbulent Flow - rapid and continuous
macroscopic mixing of the flowing fluid with
length scales which are very much greater than
the molecular scales in laminar flow -
macroscopically a disordered flow (fluid
particles do not remain in layers, but move in a
heterogeneous fashion through the flow, sliding
past other particles and colliding with some in
an entirely haphazard manner) 2 Stability
against Disturbances(??, ??) 1gt Laminar Flow
stable against disturbance If the laminar
flow is disturbed by wall roughness(????) or some
other obstacles, the disturbances are rapidly
damped by viscous action, and downstream the flow
is smooth again. 2gt Turbulent Flow unstable
against disturbances 3 Dominant Force
9
(No Transcript)
10
(No Transcript)
11
Osborne Reynolds was a mathematics graduate of
Cambridge in 1867 and became the first professor
of engineering in Manchester in 1868. He held
this post until he retired in 1905. He became a
Fellow of the Royal Society in 1877 and, 11 years
later, won their Royal Medal. His early work was
on magnetism and electricity but he soon
concentrated on hydraulics and hydrodynamics. In
1886 he formulated a theory of lubrication and
three years later he developed the standard
mathematical framework used in the study of
turbulence. The 'Reynolds number' used in
modelling fluid flow is named after him.
Born 23 Aug 1842 in Belfast, IrelandDied 21
Feb 1912 in Watchet, Somerset, England
12
1gt Laminar Flow viscous forces are
dominant ---gt low Reynolds number -
low velocity flow --gt - small length
scale(e.g. small pipe diameter) -
fluid with high kinematic viscosity 2gt Turbulent
Flow - Inertia forces(associated with the
accelerations during the motion) are dominant
---gt high Reynolds number - high
velocity flow --gt - large length scale
- fluid with low kinematic viscosity
13
4 Steady and Unsteady 1gt Laminar Flow in
general steady flow 2gt Turbulent Flow in
general unsteady flow 5 Shearing Stress 1gt
Laminar Flow 2gt Turbulent Flow where
eddy viscosity(?????)
14
Example 8.1 ltSol.gt a)
15
(No Transcript)
16
8.1.2 Entrance Region and Fully Developed Flow
Entrance Region, Entrance Length 1 Definition
flow region over which the velocity profile
changes in the flow direction region of flow
near where the fluid enters the pipe 2 Extent
3 Dimensional Analysis 4 Velocity Profile
5 entrance length(????) - for laminar flow
17
(No Transcript)
18
- for turbulent flow for many practical
engineering problems 2.
Boundary Layer and Inviscid Core (p463 Fig. 8.5)
3. Fully Developed Flow(??? ??? ??) Region 1
Definition flow region after which the velocity
profile(????) ceases to change in the flow
direction 2 Extent
3 Downstream of the velocity
profile is constant, the wall shear is constant,
and the pressure drops linearly with x for
either laminar or turbulent flow.
19
4. Developing Flow
20
8.1.3 Pressure and Shear Stress

• 1. Driving Forces in Fully Developed Flow in a
  Constant Diameter Pipe
• - inclined pipe flow gravity force, pressure
  force
• - horizontal pipe flow pressure force
• i.e. pressure difference
• 2. Viscous Effects
• - Viscous effects provide the restraining force
  that exactly balances the pressure force, thereby
  allowing the fluid to flow through the pipe with
  no acceleration.
• If viscous effects were absent in such flows, the
  pressure would be constant throughout the pipe,
  except for the hydrostatic variation which is
  negligible.
• - The fact that there is a nonzero pressure
  gradient along the horizontal pipe is a result of
  viscous effects. If the viscosity were zero, the
  pressure would not vary with x.

21
3. Pressure Distribution along a Horizontal
Pipe(see p465 Fig. 8.6) In non-fully developed
flow region Such as the entrance region of a
pipe, the fluid accelerates or decelerates as it
flows (the velocity profile changes from a
uniform profile at the entrance of the pipe to
its fully developed profile at the end of the
entrance region). Thus, in the entrance region
there is a balance between pressure, viscous, and
inertia(acceleration) forces. in the
entrance region gt in the fully developed
region 4. Need for the Pressure Drop viewed
from 2 Different Standpoints 1 Force Balance
In terms of force balance, the pressure force
is needed to overcome the viscous forces
generated.
22

• 2 Energy Balance
• In terms of an energy balance, the work done by
  the pressure force is needed to overcome the
  viscous dissipation of energy throughout the
  fluid.
• 5. Nature of the Pipe Flow Laminar Flow,
  Turbulent Flow
• - The nature of the pipe flow is strongly
  dependent on whether the flow is laminar or
  turbulent.
• - This is a direct consequence of the
  differences in the nature of shear stress in
  laminar and turbulent flows.
• The shear stress in laminar flow is a direct
  result of momentum transfer among the randomly
  moving molecules (a microscopic phenomenon).
• The shear stress in turbulent flow is largely a
  result of momentum transfer among the randomly
  moving, finite-sized bundles of fluid particles
  (a macroscopic phenomenon).

23
(No Transcript)
24
8.2.1 From Applied to a Fluid Element
8.2 Fully Developed Laminar Flow 1. Control
Volume( Fluid Element) (see p466 Fig. 8.7)
circular cylinder of fluid of length and
radius centered on the axis of a horizontal
pipe of diameter 2. Fully Developed, Steady
Flow - local accelerationo steady flow
- convective accelerationo fully
developed flow 3. Free-body diagram (see p466
Fig. 8.8)
25
(No Transcript)
26
(No Transcript)
27

• 4. Newtons Second Law of Motion
• - First Governing Law for Fully Developed
  Laminar Flow of a Newtonian Fluid within a
  Horizontal Pipe Eq. (8.3)
• 
  
  
  
• If the viscosity is zero there would be no
  shear stress, and the pressure would be constant
  throughout the horizontal pipe.
• - Eq. (8.3) represents the basic balance in
  forces needed to drive each fluid particle along
  the pipe with constant velocity.
• -
• shear stress distribution(see p467 Fig. 8.9)

28
- pressure drop wall shear stress
29

• A small shear stress can produce a large
  pressure difference if the pipe is relatively
  long( ).
• Equations (8.3), (8.4), (8.5) are valid for both
  laminar and turbulent flow.
• 5. Definition of Newtonian Fluid
• 6. Velocity Distribution , Average
  Velocity , Maximum Velocity , Volume
  Flow Rate
• - velocity distribution (Fig. 8.9)
  

30

• average velocity and maximum velocity
• 
  
• - volume flow rate
• 
  
• 7. Equations for a Nonhorizontal Pipe(see p469
  Fig. 8.10)

31
(No Transcript)
32
Example 8.2 ltSol.gt
33
(No Transcript)
34
8.2.3 From Dimensional Analysis
1. Functional Relationship of Pressure Drop 2.
Result of Dimensional Analysis 3. Other
Important Results
35
8.2.4 Energy Considerations

• Energy equation for Incompressible, Steady Flow
  (Eq. 5.89)
• - where
• kinetic energy coefficient
  (compensate for the fact that the velocity
  profile across the pipe is not uniform)
• - For uniform velocity profile , whereas
  for any nonuniform profile, .
• - The head loss term accounts for any energy
  loss associated with the flow. This loss is a
  direct consequence of the viscous dissipation
  that occurs throughout the fluid in the pipe.
• - For the ideal (inviscid) cases,
  and the energy equation reduces to the
  familiar Bernoulli equation (Eq. 3.7).

36
2. Head Loss for Fully Developed Flow - Fully
developed flow, the velocity profile does not
change, so , energy
equation becomes The energy dissipated by
the viscous forces within the fluid is supplied
by the excess work done by the pressure and
gravity forces. - from
It is the shear stress at the wall (which is
directly related to the viscosity and the shear
stress throughout the fluid) that is responsible
for the head loss.
37

• It is valid for both laminar and turbulent flow.
• Example 8.3
• ltsol.gt
• If the flow is laminar, from Eq. 8.9
• b)

38
(No Transcript)
39
(No Transcript)
40
8.3.1 Transition from Laminar to Turbulent Flow
8.3 Fully Developed Turbulent Flow 1.
Characteristics of Turbulent Flow 1
Irregularity (or randomness in time and space)
disordered (chaotic) motion, unsteady motion,
unstable motion 2 Diffusivity (or rapid mixing)
1gt The diffusivity of turbulence causes rapid
mixing and implies high rates (compared to
laminar flow) of momentum and heat transfer
through the flow. 2gt If a flow pattern looks
random but does not exhibit spreading of velocity
fluctuation through the surrounding fluid, it is
surely not turbulent.
41
3 High Reynolds Number 1gt Turbulence often
originates as an instability of laminar flows if
the becomes too large. 2gt The instability
are related to the interaction of viscous terms
and nonlinear inertia terms in the equation of
motion. 4 3-Dimensional Vorticity Fluctuation
1gt Turbulence is rotational and 3-dimensional.
2gt Turbulence is characterized by high levels of
fluctuating vorticity, so vorticity dynamics
plays an essential role in the description of
turbulent flows. 5 Dissipation of the kinetic
energy of the turbulence by viscous shear
stresses. 1gt Turbulent flows are always
dissipative. 2gt Viscous shear stresses perform
deformation work which
42
increases the internal energy of the fluid at the
expense of kinetic energy of turbulence. 3gt
Turbulence needs a continuous supply of energy to
make up for these viscous losses. If no energy is
supplied, turbulence decays rapidly. 6
Turbulence is a continuum phenomenon even at the
smallest scale. 1gt Turbulence is a continuum
phenomenon, governed by the equations of fluid
mechanics. 2gt Even the smallest scales occurring
in a turbulent flow are ordinarily far larger
than any molecular length scale. 7 Turbulence is
a feature of fluid flows, not a property of the
fluids themselves. Most of the dynamics of
turbulence is the same in all fluids, whether
they are liquids or gases.
43
2. Transition from laminar to turbulent flow in a
pipe (see p477 Fig. 8.11) 1. Occurrence of
Turbulent Flow Most flows encountered in
practical applications are turbulent flows.
Turbulent flow is assumed to occur whenever the
Reynolds number in a pipe exceeds 4,000. 3.
Description of Velocity for Turbulent Flow(see
p478 Fig. 8.12) 4. Mixing Process in Turbulent
and Laminar Flow(p478) 5. Benefit of Turbulence
and Laminar Flow 6. Critical Reynolds Number for
Flow along a Flat Plate
44
(No Transcript)
45
(No Transcript)
46
8.3.2 Turbulent Shear Stress
1. Instantaneous velocity where time
mean(or time average) fluctuating
part(fluctuation) 2. Turbulence intensity
(level of the turbulence) (see p480, Fig.
8.13) 3. Mechanism of change of momentum in
laminar and turbulent flow (see p481, Fig.
8.14) 4. Apparent stress, Reynolds stress (
)
47
(No Transcript)
48
(No Transcript)
49
5. Structure of Turbulent Flow (see p482, Fig.
8.15) viscous sublayer, outerlayer, overlap
layer 6. Eddy Viscosity , Mixing Length
50
(No Transcript)
51
Prandtl was born in Freising, Germany, in
1875. He studied mechanical engineering in
Munich. For his doctoral thesis he worked on a
problem on elasticity under August Foppl, who
himself did pioneering work in bringing together
applied and theoretical mechanics. Later Prandtl
became Foppl's son-in-law, following the good
German academic tradition in those days. In 1901
he became professor of mechanics at the
University of Hanover, where he continued his
earlier efforts to provide a sound theoretical
basis for fluid mechanics. The famous
mathematician Felix Klein, who stressed the use
of mathematics in engineering education, became
interested in Prandtl and enticed him to come to
the University
Ludwig Prandtl (1875-1953)
52
of Gottingen. He served as a professor of applied
mechanics at Gottingen from 1904 to 1953 the
quiet university town of Gottingen became an
international center of aerodynamic research..
In 1904 Prandtl conceived the idea of boundary
layer, which adjoins the surface of a body moving
through a fluid. It is perhaps the greatest
single discovery in the history of fluid
mechanics. He showed that frictional effects in a
slightly viscous fluid are confined to a thin
layer near the surface of the body the rest of
the flow can be considered inviscid. The idea led
to a rational way of simplifying the equations of
motions in the different regions of the flow
field. Since then the boundary layer technique
has been generalized and has become a most useful
tool in many branches of science. He made
notable innovations in the design of wind tunnels
and other aerodynamics equipment. His advocasy of
monoplanes greatly advanced the heavier-than-air
aviation. In experimental fluid mechanics he
designed the pitot-static tube for measuring
velocity. In turbulence theory he contributed the
mixing length theory. Toward to end of his career
Prandtl became interested in dynamic meteorology
and published a paper generalizing the Ekman
spiral for turbulent flows.
53
Prandtl was endowed with rare vision for
understanding physical phenomena. His mastery of
of mathematical tricks was limited indeed many
of his collaborators were better mathematician.
However, Prandtl had an unusual ability in
putting ideas in simple mathematical forms. In
1948 Prandtl published a simple and popular
textbook on fluid mechanics, which has been
referred to in several place here. His varied
interest and simplicity of analysis is evident
throughout this book. Prandtl died in Gottingen
in 1953.
54
8.3.3 Turbulent Velocity Profile
1. Viscous Sublayer (see p484 Fig. 8.16) -
Within this region the viscous shear stress is
dominant compared with the turbulent (or
Reynolds) stress, and the random, eddying nature
of the flow is essentially absent. - In this
region the fluid viscosity is an important
parameter the density is unimportant. -
velocity profile (law of the wall) where
friction velocity(is not an actual velocity
of the fluid- it is merely a quantity that has
dimensions of velocity) - Eq. (8.29) is valid
very near the smooth wall, for 2. Velocity
Profile in Overlap Region(see p484 Fig. 8.16)
55
Velocity should vary as the logarithm of y.

3. Velocity
Profile in the Outer Turbulent Layer -
Logarithmic law - Power-Law Velocity Profile

where
(see p485, Fig. 8.17) - The one-seventh power
law velocity profile (n7) is often used as a
reasonable approximation for many practical
flows.(see p485, Fig. (8.18) 4. Remarks
56

• Note from Fig. 8.18 that the turbulent profiles
  are much flatter than the laminar profile and
  that this flatness increases with Reynolds number
  (i.e., with n).
• - fictitious uniform velocity profile turbulent
  velocity profile

57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
Boussinesq, Valentin Joseph (1842-1929)
French physicist and mathematician who received
his Ph.D. in 1867. He was professor of
differential and integral calculus at the Faculty
of Sciences of Lille (1872-86), and professor of
physics and mechanics at Sorbonne, Paris (1886).
He was a member of the French Académie des
Sciences (1886), the teacher of mathematics at
Agde, Le Vigan, and Gap (1866-1872), and retired
in 1918.
Boussinesq made important contributions to all
branches of mathematical physics, except that of
electromagnetism. His work on hydraulics was
considerable. He studied whirlpools, liquid
waves, the flow of fluids, the mechanics of
pulverulent masses, the resistance of a fluid
against a solid body, and the cooling effect of a
liquid flow. His contributions to the study of
turbulence were praised by Saint Venant, and
those on theory of elasticity by Love.
61
Although he approached mathematics only in order
to apply it practically, he still made important
contributions. Notably, in 1880 he came upon
nonanalytic integrals of hydrodynamic equations.
Boussinesq left a considerable body of work,
carefully quoted in Poggendorff volumes III to
VI.
62
Example 8.4
63
8.4.1 The Moody Chart

• 8.4 Dimensional Analysis of Pipe Flow
• application We will consider only typical
  constant diameter pipes with relative roughness
  in the range
• Functional Form of Pressure Drop for Steady,
  Incompressible Turbulent Flow in a Horizontal
  Round Pipe
• where
• mean (average) velocity,
  length of pipe
• pipe diameter viscosity,
  density of fluid
• measure of the roughness of the pipe
  wall
• - A fundamental difference between laminar and
  turbulent flow is
• that the shear stress for turbulent flow is a
  function of the
• density of the fluid .

64

• pressure drop for laminar pipe flow
• 2. Result of Dimensional Analysis
• - difference between laminar case and turbulent
  case (see p490)
• pressure drop
• so
• For laminar fully developed flow
• For turbulent flow
  (obtain from Moody chart)

65
3. Darcy-Weisbach Equation Henri Philibert
Gaspard Darcy (1803-1858) Performed extensive
tests on filtration and pipe resistance
initiated open-channel studies carried out by
Bazin
is valid
for any fully developed, steady, incompressible
pipe flow whether the pipe is horizontal or on a
hill. 4. Energy Equation for a Fully Developed
Flow Part of the pressure change is due to
the elevation change and part is due to the head
loss associated with frictional effects, which
are given in terms of the friction factor, .
66

• 5. Moody Diagram
• Lewis Ferry Moody (1880-1953) Provided many
  innovations in the field of hydraulic machinery
  proposed a method of correlating pipe resistance
  data which is widely used
• For laminar flow, , which is
  independent of relative roughness.
• For very large Reynolds numbers, ,
  which is independent of the Reynolds number. For
  such flows, commonly termed completely turbulent
  flow (or wholly turbulent flow), the laminar
  sublayer is so thin (its thickness decreases with
  increasing Re) that the surface roughness
  completely dominate the character of the flow
  near the wall.
• For flows with moderate values of Re, the
  friction factor is indeed dependent on both the
  Reynolds number and relative roughness

67
- The gap in the figure for which no values of
are given ( ) is
a result of the fact that the flow in this
transition range may be laminar or turbulent (or
an unsteady mix for both) depending on the
specific circumstance involved. - The Moody
chart is universally valid for all steady, fully
developed, incompressible pipe flows.
68
Henri-Philibert-Gaspard Darcy Born 10 June
1803 died 3 Jan 1858. French hydraulic
engineer who first derived the equation (now
known as Darcys law) that governs the laminar
(nonturbulent) flow of fluids in homogeneous,
porous media. In 1856, modern studies of
groundwater began when Darcy was commissioned to
develop a water-purification system for the city
of Dijon, France. He constructed the first
experimental apparatus to study the flow
characteristics of water through the earth. From
his experiments, he derived the Darcy's Law
equation, describing the flow of water in nature,
which is fundamental to understanding groundwater
systems. He performed extensive tests on
filtration and pipe resistance. He initiated the
open-channel studies carried out by Bazin 
69
6. Hydraulically Smooth Wall (pipe) and Rough
Wall (see p492) smooth wall rough
wall where viscous sublayer
thickness average wall
roughness height 7. Empirical Equation for -
Colebrook formula Eq. (8.35) valid for the
entire nonlaminar range of Moody chart -
Blasius formula for turbulent flow in smooth
pipes with
70
(No Transcript)
71
(No Transcript)
72
(No Transcript)
73
Example 8.5 ltSolgt
74
(No Transcript)
75
8.4.2 Minor Losses
1. Definition Most pipe systems consist of
considerably more than straight pipes. These
additional components (valves, bends, tees, and
the like) add to the overall head loss of the
system. Such losses are generally termed minor
losses, with the apparent implication being that
the majority of the system loss is associated
with the friction in the straight portions of the
pipes, the major losses. 2. Loss Coefficient
- -
76

• 3. Equivalent Length (????)
• 4. Losses due to Various Transition Sections
• - Many pipe systems contain various transition
  sections in which the pipe diameter changes from
  one size to another.
• Any change in flow area contributes losses that
  are not accounted for in the fully developed head
  loss calculation (the friction factor).
• 1 Entrance Flow Conditions and Loss coefficient
• - flow into a pipe from a reservoir
• - Fig. 8.22
• - typical flow pattern for flow entering a pipe
  through a square-edge entrance Fig. 8.23 ,
  Vena Contracta
• - entrance head loss (pressure drop)

77
(No Transcript)
78
(No Transcript)
79
(No Transcript)
80

• Although a fluid may be accelerated very
  efficiently, it is very difficult to slow down
  (decelerate) a fluid efficiently.
• typical values for the loss coefficient for
  entrances with various amounts of rounding of the
  lip Fig. 8.24
• 2 Exit Head Loss flow from a pipe into a tank
  
• in any cases
• 3 Loss Coefficient for a Sudden Contraction
• Fig. 8.26
• 4 Loss Coefficient for a Sudden Expansion see
  Fig. 8.27, 28
• - It is the dissipation of kinetic energy
  (another type of viscous effect) as the fluid
  decelerates inefficiently.

81
(No Transcript)
82
(No Transcript)
83
(No Transcript)
84
(No Transcript)
85
(No Transcript)
86
5 Loss Coefficient for a Typical Conical
Diffuser Fig. 8.29 - diffuser a device
shaped to decelerate a fluid 6 Flow in a Conical
Contraction - Typical loss coefficient based on
the downstream (high-speed) velocity can be quiet
small, ranging from for
, to for 7 Loss due to
90 Smooth Bend Fig. 8.30 8 Loss due to Miter
Bend Fig. 8.31 9 Loss Coefficients for Pipe
Components such as Elbows, Tees, Valves,
Return Bends Table 8.2 - The values of
for such components depend strongly on the shape
of the component and only very weakly on the
Reynolds number for typical large Re flows. -
When the valve is closed, the value of is
infinite and no fluid flows. Opening of the valve
reduces , producing the
87
desired flow rate. - As with many system
components, the head loss in valves is mainly a
result of the dissipation of kinetic energy of
high-speed portion of the flow.
88
(No Transcript)
89
(No Transcript)
90
(No Transcript)
91
(No Transcript)
92
(No Transcript)
93
(No Transcript)
94
Example 8.6 ltSolgt
95
(No Transcript)
96
(No Transcript)
97
(No Transcript)
98
8.4.3 Noncircular Conduits

• 1. Hydraulic Diameter (Fig. 8.34)
• where perimeter
• 2. Head Loss
• - Laminar flow where
  
• value of (Table
  8.3)
• - Turbulent flow
• 3. Accuracy Such calculations are usually
  accurate to within about 15.

99
(No Transcript)
100
(No Transcript)
101
Example 8.7 ltSolgt
102
(No Transcript)
103
8.5.1 Single Pipes
8.5 Pipe Flow Examples 2 classes of problems
- single pipe (whose length may be interrupted by
various components) - multiple pipes in
parallel, series, or network configurations -
The nature of the solution process for pipe flow
problems can depend strongly on which of the
various parameters are independent parameters
(the given) and which is the dependent
parameter (the determine). - Pipe Flow Types
(Type I, II, III) (see p511 Table 8.4)
104
(No Transcript)
105
Example 8.8 Type 1, determine pressure
drop ltSolgt
106
(No Transcript)
107
(No Transcript)
108
(No Transcript)
109
(No Transcript)
110
(No Transcript)
111
Example 8.9 Type 1 determine head loss
112
(No Transcript)
113
Example 8.11 Type II determine flowrate
114
ltSol.gt
115
Example 8.13 Type III with minor losses determine
diameter
116
ltSolgt
117
(No Transcript)
118
(No Transcript)
119
(No Transcript)