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

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Pressure Driven Flow in pipes
P1
P2
L
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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.

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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)
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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.

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Shear Stress Profile
1
2
ro
r
Force balance on cylindrical fluid element
(6.2)
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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
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Case 1 Laminar Flow
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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

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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
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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.

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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
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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

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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

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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
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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)
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Laminar flow Velocity and Shear stress profiles
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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

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Hagen-Poiseuille Law
The volumetric flowrate through the pipe is
(6.10)
Average velocity
(6.11)
And because of (6.9)
(6.12)
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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)
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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.

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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