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Flow in Pipes Fluid Friction

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Title: Flow in Pipes Fluid Friction


1
TOPIC 6
  • Flow in Pipes Fluid Friction

2
The Pressure-Drop Experiment
P1
P2
Q
L
3
Laminar vs. Turbulent Flow
  • Laminar flow Fluid flows in smooth layers
    (lamina) and the shear stress is the result of
    microscopic action of the molecules.
  • Turbulent flow is characterized by large scale,
    observable fluctuations in the fluid and flow
    properties are the result of these fluctuations.

4
Reynolds Number
  • The Reynolds number can be used as a criterion
    to distinguish between laminar and turbulent flow

(6.1)
  • For flow in a pipe
  • Laminar flow if Re lt 2100
  • Turbulent flow if Re gt 4000
  • Transitional flow if 2100lt Re lt 4000
  • For very high Reynolds numbers, viscous forces
    are negligible inviscid flow
  • For very low Reynolds numbers (Reltlt1) viscous
    forces are dominant creeping flow

5
Pressure Driven Flow in pipes
P1
P2
L
6
Forces acting on a fluid
  • The forces acting on a fluid are divided into two
    groups
  • Body forces act without physical contact. They
    act on every mass element of the body and are
    proportional to its total mass. Examples are
    gravity and electromagnetic forces
  • Surface forces require physical contact (i.e.
    surface contact) with surroundings for
    transmission. Pressure and stresses are surface
    forces.

7
Stresses
  • In fluid mechanics it is convenient to define a
    force per unit area (F/A), called a stress (same
    units as pressure).
  • Normal stress acts perpendicular to the surface
    (Fnormal force).

F
F
F
F
A
A
Tensile causes elongation
Compressive causes shrinkage
(Pressure is the most important example of a
compressive stress)
8
Stresses
  • Shear stress acts tangentially to the surface
    (Ftangential or shear force).
  • Recall from Topic 1
  • A fluid is defined as a substance that deforms
    continuously when acted on by a shearing stress
    of any magnitude.

9
Shear Stress Profile
1
2
ro
r
Force balance on cylindrical fluid element
(6.2)
10
Shear Stress profile
From (6.3) shear stress varies linearly with r
(6.3)
At the wall (rro)
(6.4b)
(6.4a)
or
(6.5)
Shear stress is a function of the radial
coordinate
11
Case 1 Laminar Flow
12
Shear Flow
  • NO-SLIP CONDITION The fluid sticks to the
    solid boundaries. The velocity of the fluid
    touching each plate is the same as that of the
    plate (Vo for the top plate, 0 for the bottom
    plate).
  • The velocity profile is a straight line The
    velocity varies uniformly from 0 to Vo

13
Shear Flow
The force, F is proportional to the velocity Vo,
the area in contact with the fluid, A and
inversely proportional to the gap, yo
Recall, shear stress, t F / A
In the limit of small deformations the ratio
Vo/yo can be replaced by the velocity gradient
dV/dy
Rate of shearing strain or shear rate
14
Newtons law of Viscosity
  • Newtons law of viscosity

(6.6)
m N/m2 . sPa . s Viscosity n m/r
Kinematic viscosity m2/s
Newtonian fluids Fluids which obey Newtons law
Shearing stress is linearly related to the rate
of shearing strain.
  • The viscosity of a fluid measures its resistance
    to flow under an applied shear stress.

15
Example Shear stress
  • The space between two plates, as shown in the
    figure, is filled with water. Find the shear
    stress and the force necessary to move the upper
    plate at a constant velocity of 10 m/s. The gap
    width is yo0.1 mm and the area A is 0.2
    m2. The viscosity of water is 0.001 Pa.s.

Vo
F
A
t F/A
yo
Water
16
Effect of temperature on viscosity
  • Viscosity is very sensitive to temperature
  • The viscosity of gases increases with temperature

Power-law
Sutherland equation
  • The viscosity of liquids decreases with
    temperature

17
Non-Newtonian fluids
  • Non-Newtonian fluids Fluids which do not obey
    Newtons law Shearing stress is not linearly
    related to the rate of shearing strain.
  • Bingham plastics
  • Shear thinning
  • Shear thickening
  • The study of these materials is the subject of
    rheology

18
Laminar Flow Velocity profile
Lets consider again the flow of a fluid inside a
pipe. In cylindrical coordinates (6.6) can be
written
(6.7)
By combining (6.3) and (6.7) and integrating
6.8 (a)
6.8 (b)
Velocity profile is parabolic
19
Laminar Flow Velocity profile
  • Minimum velocity, V0 at the pipe wall
  • Maximum velocity Vmax at pipe centerline (located
    at r0)

(6.9)
The velocity profile can be written
6.8 (c)
20
Laminar flow Velocity and Shear stress profiles
21
Fully Developed Flow
  • Flow in the entrance region of a pipe is complex.
  • Once the velocity profile no longer changes, we
    have reached fully developed flow. Mathematically
    dV/dx 0
  • Typical entrance length, 20 D lt Le lt 30 D

22
Hagen-Poiseuille Law
The volumetric flowrate through the pipe is
(6.10)
Average velocity
(6.11)
And because of (6.9)
(6.12)
23
Losses due to Friction
Mechanical energy equation (5.2) between
locations 1 and 2 (page 6.8) in the absence of
shaft work
For flow in a horizontal pipe, under SS
conditions and no diameter change
  • Because of (6.10)
  • Because of (6.4)

The shear stress at the wall is responsible for
the losses due to friction
(6.13)
(6.14)
24
Example 1 Laminar Flow in Pipes
  • A polymer of density r0.80 g/cm3 and viscosity
    m230 cP flows at a rate Q1560 cm3/s in a
    horizontal pipe of diameter 10 cm. Evaluate the
    following
  • The mean (average) velocity
  • The Reynolds number Re. Is the flow laminar?
  • The maximum velocity. Where does the maximum
    occur?
  • The pressure drop per unit length
  • The wall shear stress
  • The frictional dissipation (losses due to
    friction) for 100 cm of pipe.

25
Example 2 Flow of oil inside a pipe
  • An oil with viscosity of m 0.4 N s/m2 and
    density r900 kg/m3 flows in a pipe of diameter
    D0.020 m.
  • What is the pressure difference P1-P2 needed to
    produce a flow rate of Q2 10-5 m3/s if the pipe
    is horizontal and has a length of 10 m?
  • What is the pressure difference if the pipe is
    located on a hill with inclination q 13.34?
  • What would the pressure difference be if the oil
    flowed downwards instead?

Inclined pipe
Horizontal pipe
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