CEE 4361
Fluid Mechanics
Flow Through Pipes
TOPICS
• Pipe flow
• Reynold’s number
• Types of flow
• Characteristics of different flow types
• Velocity distribution in different types of flow
• Head loss
• Frictional head loss determination
PIPE FLOW
• Pipes are commonly understood to be conduits of circular cross-section
which flow full and the flow is under pressure.
• Pipe flow can be of two types.
• In some pipe flows, the pipe flows full and the flow remains under
pressure.
Ex. city water and gas main which flow occurs under pressure.
• In some other pipe flows, the pipe does not flow full and the flow is not
under pressure.
Ex. sewer pipes, drainage tiles and culverts.
• Viscous flow pervades the entire flow.
PIPE FLOW
PIPE FLOW
REYNOLDS EXPERIMENT
• Depending on viscosity, flow can be of two types: laminar and turbulent.
• The limiting conditions to determine laminar or turbulent flow were first
investigated by Osborne Reynolds in 1883. Reynolds apparatus consists of two
tanks containing water and dye. A horizontal glass tube is fitted to the tank through
which water can flow. The flow through the glass tube can be regulated by
adjusting the regulating valve.
REYNOLDS EXPERIMENT
REYNOLDS NUMBER
After many experiments, Reynolds showed that, there is a dimensionless ratio of
flow parameters which can predict the change in flow types. This dimensionless
number is known as Reynolds Number. It is represented as Re.
Re =
ρ𝑉𝐷
𝑉𝐷
𝑉𝐷
=
=
µ
µ/ρ
ν
In a circular pipe –
•
•
•
the flow will be laminar, if Re < 2000
The flow will be turbulent, if Re > 4000
The flow will be transitional, if 2000 < Re < 4000
▪ Re (converging pipe) > Re (straight pipe) > Re (diverging pipe)
▪ Re (straight pipe) > Re (curved pipe)
TYPES OF FLOW
• If a fine thread of dye carried by the flowing water is
observed, then it is a laminar flow.
• The velocity of flow at which the dye thread starts
becoming irregular is known as the lower critical
velocity (true critical Reynolds number).
• The velocity at which the whole dye thread is
diffused, is known as the upper critical velocity.
• Beyond the upper critical velocity the dye will fully
mix up with water and show violent mixing. Such a
how is known as turbulent flow.
LAMINAR AND TURBULENT FLOW
Laminar flow
Turbulent
flow
LAMINAR AND TURBULENT FLOW
Part of a water body
can be both laminar
or turbulent.
CHARACTERISTICS OF LAMINAR FLOW
•
Velocity of flow is small and viscous forces are predominant.
•
It is smooth and regular, and thus also known as stream line flow.
•
There is practically no influence of fluid particles of one layer over those of
the adjacent layers. Diffusion or mixing at molecular level may occur, but
macroscopic movement of fluid elements from one layer to another does not
occur.
•
Velocity at any point remains nearly constant in magnitude and direction.
•
Frictional resistance is proportional to the mean velocity of flow.
•
Stagnant rivers and canals, blood streams inside body etc. are a prominent
example of laminar flow.
CHARACTERISTICS OF TURBULENT FLOW
•
The fluid particles no longer move in layers or laminas.
•
Violent mixing of fluid particles takes place due to which they move in
chaotic or random manner. As a result, velocity at any point varies both in
magnitude and direction.
•
Diffusion occur both in molecular level and from instant to instant.
•
Fluctuations occur in wide spectrum.
•
Frictional resistance is proportional to the square of the mean velocity of
flow.
•
All cases of engineering importance are in the turbulent flow region. But if
the fluid is a viscous oil, then laminar flow is often encountered.
Hydraulic Grade Line (HGL):
A hydraulic grade line (HGL) can be drawn to show the variation of the piezometric head (Pressure head +datum
head). The distance from the centerline of the pipe to the HGL is the pressure head.
Energy Grade Line (EGL):
An energy grade line (EGL) shows the variation of the total head. Since the difference between the total head and
the piezometric head is the velocity head, the distance between the EGL and HGL is also the velocity head.
Hydraulic Radius (Rh)
For conduits having non-circular cross section, some value other than
the diameter must be used for the linear dimension in the Reynolds
𝐴
number. Such a characteristic is the hydraulic radius, defined as Rh =
𝑃
Here, A = cross-sectional area of the flowing fluid, P = wetted perimeter
π 2
𝐷
4
𝐴
𝐷 𝑟
For a circular pipe flowing full, Rh = =
= =
𝑃
π𝐷
4
2
• Rh is not the radius of the pipe.
• It is used to express the shape as well as size of the conduit.
• For evaluating Re in a non-circular conduit, it is customary to
substitute 4Rh for D.
HEAD LOSS FOR
TURBULENT FLOW
(DARCY-WEISBACH
AND FANNING’S
EQUATION)
Fanning’s Equation
Applicable for determining
head loss for turbulent flow in
a circular conduit
LIMITATION OF DARCY - WEISBACH EQUATION
1. The loss of head with turbulent flow varies not only as the square of the
mean velocity, but as some power varying from 1.7 to 2 or more
depending on the roughness of pipe. This discrepancy must be taken
care of by varying the value of f. For laminar flow, the loss of head
varies as the first power of the mean velocity.
2. Since V = Q/A = Q/(πd2/4), for a given Q, f and L, the loss of head by
the Darcy-Weisbach formula varies inversely as the fifth power of the
diameter. Tests have shown the actual variation is closer to the 5.25
power and that the exponent of the d of the formula should be close to
1.25. Again, the discrepancy is taken care of by varying the value of f.
HEAD LOSS FOR LAMINAR FLOW (HAZEN-POISEUILLE EQUATION)
FRICTIONAL LOSS IN TURBULENT FLOW
VELOCITY PROFILE IN LAMINAR FLOW
Consider a fluid entering a circular pipe at a uniform velocity. Because of the no-slip condition, the fluid
particles in the layer in contact with the surface of the pipe come to a complete stop. This layer also causes
the fluid particles in the adjacent layers to slow down gradually as a result of friction. To make up for this
velocity reduction, the velocity of the fluid at the midsection of the pipe has to increase to keep the mass flow
rate through the pipe constant. As a result, a velocity gradient develops along the pipe.
The region of the flow in which the effects of the viscous shearing forces caused by fluid viscosity are felt is
called the boundary layer. The hypothetical boundary surface divides the flow in a pipe into two regions: the
boundary layer region, in which the viscous effects and the velocity changes are significant, and the
irrotational (core) flow region, in which the frictional effects are negligible and the velocity remains
essentially constant.
The thickness of this boundary layer increases in the flow direction until the boundary layer reaches the pipe
center and thus fills the entire pipe. The region from the pipe inlet to the point at which the boundary layer
merges at the centerline is called the hydrodynamic entrance region, and the length of this region is called
the hydrodynamic entry length. Flow in the entrance region is called hydrodynamically developing flow
since this is the region where the velocity profile develops. The region beyond the entrance region in which
the velocity profile is fully developed and remains unchanged is called the hydrodynamically fully
developed region.
VELOCITY PROFILE IN TURBULENT FLOW
VELOCITY PROFILE IN LAMINAR AND TURBULENT FLOW
Ratio of the average to
maximum velocity
(V/Vc) in a pipe of
circular cross-section in
turbulent flow
FRICTIONAL
HEAD LOSS
IN LAMINAR
FLOW
FRICTIONAL HEAD LOSS IN LAMINAR FLOW
THICKNESS OF LAMINAR SUBLAYER
Even in turbulent flow there exists next to the wall of the pipe a very thin layer in which the flow is
laminar. This layer is known as the laminar or viscous sublayer. The thickness of this layer is given by
62.4 lb/ft3
HYDRAULICALLY SMOOTH AND ROUGH WALL
HYDRAULICALLY SMOOTH AND ROUGH WALL
A pipe is said to be hydraulically smooth if the height of the roughness elements is less than
the thickness of the laminar sublayer (k < ), i.e. the roughness elements are well covered by
the laminar sublayer. Here, the effect of irregularities or projections extend beyond the
sublayer.
 > k
If the height of the roughness elements are greater than the thickness of the laminar sublayer
(k > ), their presence affects the amount of turbulence and the pipe is said to be hydraulically
rough.
<k
In between these values, the pipe will behave in a transitional mode, that is, neither
hydraulically smooth nor wholly rough.
k><k
As the height of the roughness elements k increases or the thickness of the laminar sublayer
decreases with increasing Reynolds number, the turbulence increases to a maxima level at
which it is said to be "fully developed".
HYDRAULICALLY SMOOTH AND ROUGH WALL
FRICTION FACTOR
For laminar flow using Darcy Weisbach form
For turbulent flow Blasius said equation valid up to Re = 105
For turbulent flow and smooth pipes
Prandtl and
von Karman
For turbulent flow and rough pipes
For turbulent flow in neither smooth nor rough pipes using
White and Colebrook formula for use with commercial pipes
Moody
Diagram
OTHER PIPE FORMULAS
Nafisa Islam, Lecturer, CEE, IUT
LOSS OF HEAD
Head loss is caused by –
i)
Pipe friction along the straight sections of pipe of uniform diameter and
uniform roughness,
ii) Changes in velocity or direction of flow.
Losses of these types are ordinarily referred to as Major loss and minor loss.
Major loss: This is a continuous loss of head, hf, assumed to occur at a uniform
rate along the pipe as long as the size and quality of pipe remains constant, and
is commonly referred to as the loss of head due to pipe friction.
LOSS OF HEAD
Minor Losses: These consist of
1. A loss of head, hc, due to contraction of cross-section. This loss is caused by a
reduction in cross-sectional area of the stream and resulting increase in
velocity.
2. A loss of head, he, due to enlargement of cross-section. This loss is caused by
an increase in cross-sectional area of the stream and resulting decrease in
velocity.
3. A loss of head, hg, caused by an obstruction such as gates or valves which
produces a change in cross-sectional area in the pipe or in the direction of
flow.
4. A loss of head, hb, caused by bends or curves in pipes.
LOSS OF HEAD
Fig. Minor losses in pipe systems
Head loss due to sudden contraction
𝑉22
hc = kc
(value of kc is given in Table 8.2)
2𝑔
Head loss due to at entrance
Here, D2/D1 = 0
𝑉22
hc = kc
2𝑔
Head loss due to gradual contraction
𝑉22
hc = kc
(generally, for 20°- 40° angle, kc = 0.1)
2𝑔
Nozzle at pipe end is a special case of gradual contraction.
Here, kc = 0.04 - 0.2.
In all of these cases, loss occurs after the fluid enters the pipe.
Therefore, V = V2.
Head loss due to sudden expansion
𝑉12 − 𝑉22
he = ke
(ke = 1)
2𝑔
Head loss due to at submerged discharge
It occurs after the fluid leaves the pipe. Here, V2 = 0
𝑉12
hc = ke
2𝑔
Head loss due to gradual expansion
𝑉12 − 𝑉22
he = ke
(value of ke is given in Figure 8.2)
2𝑔
Head loss due to Obstructions
𝑉2
hg = kg
2𝑔
Head loss due to bends and elbows
𝑉2
hb = kb
2𝑔
for 90° bend, kb = k90 = 0.15, when R/D = 2
kb = k90 = 0.10, when R/D = 10
for 22.5° bend, kb = 0.40 k90
for 45° bend, kb = 0.80 k90
PIPELINE WITH PUMP OR TURBINE
Pump: adds energy to the system.
𝑃1
𝑉12
𝑃2
𝑉22
+
+ z1 + hp = +
+ z2 + hf + hm
𝛾
2𝑔
𝛾
2𝑔
Pump efficiency, ƞ =
𝑂𝑢𝑡𝑝𝑢𝑡
𝛾𝑄ℎ𝑝
=
𝐼𝑛𝑝𝑢𝑡
𝑃
Turbine: takes energy away from the system.
𝑃1
𝑉12
𝑃2
𝑉22
+
+ z1 - hT = +
+ z2 + hf + hm
𝛾
2𝑔
𝛾
2𝑔
𝑂𝑢𝑡𝑝𝑢𝑡
𝑃
Turbine efficiency, ƞ =
=
𝐼𝑛𝑝𝑢𝑡
𝛾𝑄ℎ𝑇