MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 9: FLOWS IN PIPE

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MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 9
FLOWS IN PIPE

• Instructor Professor C. T. HSU

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9.1 General Concept of Flows in Pipe

• As a uniform flow enters a pipe, the velocity at
  the pipe walls must decrease to zero (no-slip
  boundary condition). Continuity indicates that
  the velocity at the center must increase.
• Thus, the velocity profile is changing
  continuously from the pipe entrance until it
  reaches a fully developed condition. This
  distance, L, is called the entrance length.

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9.1 General Concept of Flows in Pipe

• For fully developed flows (xgtgtL), flows become
  parallel, , the mean pressure
  remains constant over the pipe cross-section

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9.1 General Concept of Flows in Pipe

• Flows in a long pipe (far away from pipe entrance
  and exit region, xgtgtL) are the limit results of
  boundary layer flows. There are two types of pipe
  flows laminar and turbulent

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9.1 General Concept of Flows in Pipe

• Whether the flow is laminar or turbulent depends
  on the Reynolds number, where Um is the
  cross-sectional mean velocity defined by
• Transition from laminar to turbulent for flows in
  circular pipe of diameter D occur at Re2300

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9.1 General Concept of Flows in Pipe

• When pipe flow is turbulent. The velocity is
  unsteadily random (changing randomly with time),
  the flow is characterized by the mean
  (time-averaged) velocity defined as
• Due to turbulent mixing, the velocity profile of
  turbulent pipe flow is more uniform then that of
  laminar flow.

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9.1 General Concept of Flows in Pipe

• Hence, the mean velocity gradient at the wall for
  turbulent flow is larger than laminar flow.
• The wall shear stress, ,is a function of the
  velocity gradient. The greater the change in
  with respect to y at the wall, the higher is the
  wall shear stress. Therefore, the wall shear
  stress and the frictional losses are higher in
  turbulent flow.

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9.2 Poiseuille Flow

• Consider the steady, fully developed laminar flow
  in a straight pipe of circular cross section with
  constant diameter, D.
• The coordinate is chosen such that x is along the
  pipe and y is in the radius direction with the
  origin at the center of the pipe.

y
x
D
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9.2 Poiseuille Flow

• For a control volume of a cylinder near the pipe
  center, the balance of momentum in integral form
  in x-direction requires that the pressure force,
  
• acting on the faces of the
  cylinder be equal to the shear stress
  acting on the circumferential area, hence
• In accordance with the law of friction (Newtonian
  fluid), have

since u decreases with increasing y
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9.2 Poiseuille Flow

• Therefore
• when is constant (negative)
• Upon integration
• The constant of integration, C, is obtained from
  the condition of no-slip at the wall. So, u0 at
  yRD/2, there fore CR2/4 and finally

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9.2 Poiseuille Flow

• The velocity distribution is parabolic over the
  radius, and the maximum velocity on the pipe axis
  becomes
• Therefore,
• The volume flow rate is

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9.2 Poiseuille Flow

• The flow rate is proportional to the first power
  of the pressure gradient and to the fourth power
  of the radius of the pipe.
• Define mean velocity as
• Therefore,
• This solution occurs in practice as long as,

Hence,
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9.2 Poiseuille Flow

• The relation between the negative pressure
  gradient and the mean velocity of the flow is
  represented in engineering application by
  introducing a resistance coefficient of pipe
  flow, f.
• This coefficient is a non-dimensional negative
  pressure gradient using the dynamic head as
  pressure scale and the pipe diameter as length
  scale, i.e.,
• Introducing the above expression for (-dp/dx),
• so,

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9.2 Poiseuille Flow

• At the wall,
• So,
• As a result, the wall friction coefficient is

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9.3 Head Loss in Pipe

• For flows in pipes, the total energy per unit of
  mass is
• given by where the correction
  factor is
• defined as,
• with being the mass flow rate and A
  is the cross sectional area.

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9.3 Head Loss in Pipe

• So the total head loss between section 1 and 2 of
  pipes is
• hlhead loss due to frictional effects in fully
  developed flow in constant area conduits
• hlmminor losses due to entrances, fittings, area
  changes, etcs.

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9.3 Head Loss in Pipe

• So, for a fully developed flow through a
  constant-area pipe,
• And if y1y2,

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9.3 Head Loss in Pipe

• For laminar flow,
• Hence

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9.4 Turbulent Pipe Flow

• For turbulent flows we cannot evaluate the
  pressure drop analytically. We must use
  experimental data and dimensional analysis.
• In fully developed turbulent pipe flow, the
  pressure drop, , due to friction in a
  horizontal constant-area pipe is know to depend
  on
• Pipe diameter, D
• Pipe length, L
• Pipe roughness, e
• Average flow velocity, Um
• Fluid density,
• Fluid viscosity,

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9.4 Turbulent Pipe Flow

• Therefore,
• Dimensional analysis,
• Experiments show that the non-dimensional head
  loss is directly proportional to L/D, hence

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9.4 Turbulent Pipe Flow

• Defining the friction factor as,
  , hence
• where f is determined experimentally.
• The experimental result are usually plotted in a
  chart called Moody Diagram.

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9.4 Turbulent Pipe Flow

• In order to solve the pipe flow problems
  numerically, a mathematical formulation is
  required for the friction factor, f, in terms of
  the Reynolds number and the relative roughness.
• The most widely used formula for the friction
  factor is that due to Colebrook,
• This an implicit equation, so iteration procedure
  is needed to determine.

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9.4 Turbulent Pipe Flow

• Miller suggested to use for the initial estimate,
• That produces results within 1 in a single
  iteration

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9.5 Minor Loss

• The minor head loss may be expressed as,
• where the loss coefficient, K, must be
  determined experimentally for each case.
• Minor head loss may be expressed as
• where Le is an equivalent length of straight
  pipe

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9.5 Minor Loss

• Source of minor loss
• 1. Inlets Outlets
• 2. Enlargements Contractions
• 3. Valves Fittings
• 4. Pipe Bends

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9.6 Non-Circular Ducts

• Pipe flow results sometimes can be used for
  non-circular ducts or open channel flows to
  estimate the head loss
• Use Hydraulic Diameter,
• A - Cross section area P - Wetted perimeter
• For a circular duct,
• For rectangular duct,
• where Ar b/a is the geometric aspect ratio

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9.6 Non-Circular Ducts

• Effect of Aspect Ratio (b/a)
• For square ducts
• For wide rectangular ducts with bgtgta
• Thus, flows behave like channel flows
• However, pipe flow results can be used with good
  accuracy only when

1/3ltArlt3