Case 2: Turbulent Flow

1
Case 2 Turbulent Flow
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Turbulent Flow
When the flow is turbulent the velocity and
pressure fluctuate very rapidly. The velocity
components at a point in a turbulent flow field
fluctuate about a mean value.
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Turbulent Flow Velocity Profile
For turbulent flow in tubes the time-averaged
velocity profile can be expressed in terms of the
power law equation. n 7 is usually a good
approximation.
where Vc is the velocity at The centerline
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Turbulent Flow Shear stresses

• There are several theoretical models available
  for the prediction of shear stresses in turbulent
  flow. However, there is no general, useful model
  that can accurately predict the shear stress for
  turbulent flow.
• We estimate shear stress by using experimental
  data, semiempirical formulas and dimensional
  analysis

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Losses due to Friction/The Friction Factor
For turbulent flow there is no rigorous
theoretical treatment available. In order to
determine an expression for the losses due to
friction we must resort to experimentation.
where Llength of the pipe, Ddiameter of the
pipe, Vvelocity,
By introducing the friction factor, f
where
(6.16)
(6.15)
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The Friction Factor
The mechanical energy equation (5.1) can be
written
(6.17a)
Or in terms of heads (equation 5.2)
(6.17b)
Knowledge of the friction factor allows us to
estimate the loss term in the energy equation
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Friction factor The Moody Chart
The Moody Chart (Figure 6.10 textbook) provides a
convenient representation of the functional
dependence f f(Re, e/D)

• For laminar flow
• f 16 / Re
• For turbulent flow

(6.18)
Colebrook formula
(6.19a )
(6.19b )

• For turbulent flow, with Relt105 and for
  hydraulically smooth surfaces

(6.20)
Blasius formula
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Summary
For flow of fluids in pipes it is convenient to
use the Mechanical Energy equation in the form
given in equation (6.17a or b)

• To find f
• Calculate Re
• If Relt2100 (laminar flow) find f from eq. (6.18)
• If Regt2100 (turbulent flow)
• Estimate roughness, e (i.e. from Table 6.2)
• Find f from Moody chart (fig. 6.10) or equations
  (6.19), (6.20).

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Special Case Non uniform flows

• The energy equation, Bernoulli equation and
  Mechanical Energy equations have been derived by
  considering uniform flow (ie. as in turbulent
  flow).
• Generalize

where
is a correction factor
(6.21)
For uniform flows a1 For non-uniform flows
agt1. Usually for laminar flow a2
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Example Flow of oil inside a pipe

• Redo the example of page 6.25, using the concept
  of the friction factor

Inclined pipe
Horizontal pipe
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Example Flow of air through tubing

• Air under standard conditions flows through a
  4.0 mm diameter drawn tubing with an average
  velocity of V50 m/s. Determine the pressure drop
  in a 0.1-m section of the tube.
• Assume that for this small section of the tube,
  the density of air does not vary significantly,
  therefore the incompressibility assumption holds

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Minor Losses

• Major losses Associated with the friction in the
  straight portions of the pipes
• Minor losses Due to additional components (pipe
  fittings, valves, bends, tees etc.) and to
  changes in flow area (contractions or expansions)
• Method 1 We try to express the head loss due to
  minor losses in terms of a loss coefficient, KL

Values of KL can be found in the literature (for
example Table 8.2 Munson et al., for losses due
to pipe components, Figure 6.16 textbook for
losses due to change in pipe diameter)
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Minor Losses
Source Munson et al. (1998)
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Minor Losses
The mechanical energy equation can be written
(6.22)
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Example Including Minor Losses

• Water at 60F flows from the basement to the
  second floor through a 0.75-in. (0.0625 ft)
  diameter copper pipe (a drawn tubing) at a rate
  of Q12 gpm (gal/min) 0.0267 ft3/s and exits
  through a faucet of diameter 0.5 in, as shown in
  the figure. Determine the pressure at point (1)
  if
• all losses are neglected
• the only losses included are major losses
• all losses are included

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Calculation of Friction Factor
17
Minor Losses

• Method 2
• Using the concept of equivalent length, which is
  the equivalent length of pipe which would have
  the same friction effect as the fitting.

(6.23)
Values of equivalent length (L/D) can be found in
the literature (for example Table 6.7 or 6.3 in
2nd Ed. textbook )
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Example Including Minor Losses

• Redo the example shown in page 6.39, but using
  the method of equivalent lengths.

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Other considerations Non circular conduits
Define hydraulic diameter, Dh
Then use Reynolds number based on hydraulic
diameter, Dh
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Non circular conduits

• Pipe of circular cross-section

• Annulus (inside diameter D1, outside D2)

• Rectangular conduit (area ab)

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Example Flow in non-circular conduits

• Water (r1000 kg/m3, m 10-3 Pa.s) flows at a
  rate of 0.025 m3/s through a horizontal channel
  of square cross section, having an area of 0.0123
  m2. The channel is 1000 m long and made of cast
  iron. Determine the pressure drop for this flow.

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Multiple Pipe Systems

• Pipes in series

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Multiple Pipe Systems

• Parallel pipes

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Fluid machines

• Pumping machines add energy to a fluid
• Examples Pumps (for liquids), fans, blowers and
  compressors (for gases)
• Positive-displacement machines a finite volume
  of liquid is drawn into a chamber and then forced
  out under high pressure.
• Turbomachines Energy is added to the fluid by
  means of a rotating impeller. A common example is
  the centrifugal pump.
• Turbines extract energy from the fluid.

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Example Pumping of Water

• A house is located near a freshwater lake. The
  homeowner decides to install a pump near the lake
  to deliver 25 gpm of water to a tank adjacent to
  the house. The water can then be used for
  lavatory facilities or sprinkling the lawn. For
  the system sketched below, determine the pump
  power required.

1 exit fitting
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Summary of Topic 6

• Laminar vs. Turbulent Flow
• Reynolds Number
• Pressure driven flow in pipes
• Definition of stress Shear stress profile
• Case 1 Laminar Flow
• Newtons law Definition of viscosity
• Parabolic velocity profile
• Hagen-Poiseuille law
• Losses due to friction

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Summary of Topic 6

• Pressure driven flow in pipes
• Case 2 Turbulent flow
• Flat velocity profile
• Losses due to friction The friction factor
• Mechanical energy equation for pipe flow
• Minor losses
• Non-uniform flows
• Non-circular conduits
• Multiple pipe systems
• Fluid machines

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