Lecture 8 Flow in Pipes - LATEST

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Flow in Pipes:

Flow in Pipes

Objectives:

2 Objectives Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow. Calculate the major and minor losses associated with pipe flow in piping networks and determine the pumping power requirements.

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28 August 2011 3 Distribution of liquids

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28 August 2011 4 Oil and natural gas pipelines

Blood flow through arteries and veins:

Blood flow through arteries and veins 28 August 2011 5

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6 In general, flow sections of circular cross section are referred to as pipes (especially when the fluid is a liquid) flow sections of noncircular cross section are referred to as ducts (especially when the fluid is a gas). Smaller diameter pipes are usually referred to as tubes .

Introduction:

7 Introduction Average velocity in a pipe Recall: because of the no-slip condition, the velocity at the walls of a pipe or duct is zero. We are often interested only in V avg or V m which we usually call just V. Friction force of wall on fluid

Laminar and Turbulent Flows:

Laminar and Turbulent Flows 28 August 2011 8

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9 Critical Reynolds number ( Re cr ): Re at which the flow becomes turbulent. For internal flow in a round pipe, Re < 2300  laminar 2300 ≤ Re ≤ 4000  transitional Re > 4000  turbulent Re cr depends upon Pipe roughness Pipe vibrations Upstream fluctuations, disturbances (valves, elbows, etc. that may disturb the flow) Transition from laminar to turbulent flow depends on a dimensionless quantity : Reynolds number, Re .

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10 For non-circular pipes, Re is based on hydraulic diameter defined as: D h = 4A c /P A c = cross-section area of the pipe P = wetted perimeter Example: open channel Ac = 0.15 * 0.4 = 0.06m 2 P = 0.15 + 0.15 + 0.4 = 0.7m Don’t count free surface, since it does not contribute to friction along pipe walls! D h = 4A c /P = 4*0.06/0.7 = 0.34m What does it mean? This channel flow is nearly eqvt to a round pipe of dia 0.34m. Hydraulic Diameter

The Entrance Region:

11 The Entrance Region Consider a round pipe of diameter D. The flow can be laminar or turbulent. In either case, the profile develops downstream over several diameters called the entry length L h . L h

Fully Developed Pipe Flow:

12 Fully Developed Pipe Flow Laminar Can be solved exactly Flow is steady Velocity profile is parabolic Pipe roughness not so important V avg = ½ U max u(r)= 2V avg (1 - r 2 /R 2 )

Fully Developed Pipe Flow:

13 Fully Developed Pipe Flow Turbulent Cannot be solved exactly (too complex) Flow is unsteady (3D swirling eddies) Mean velocity profile is fuller (shape more like a top-hat profile, with very sharp slope at the wall) Pipe roughness is very important V avg = 85% of U max (depends on Re) No analytical solution Instantaneous profiles

Fully Developed Pipe Flow Pressure drop:

14 Fully Developed Pipe Flow Pressure drop There is a direct connection between the pressure drop in a pipe and the shear stress at the wall Consider a horizontal pipe, fully developed, and incompressible flow 1 2 L  w P 1 P 2 V Take CV inside the pipe wall

Pressure Drop:

15 Pressure Drop where f = Darcy Weisbach friction factor For laminar flow: f = 64/Re For turbulent flow: Use Moody chart (famous plot of f vs. Re and  /D) or empirical equations ( Haaland equation or Colebrook equation) to determine f.

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16 For turbulent flow : Moody chart was developed for circular pipes, but can be used for non-circular pipes as well, using hydraulic diameter Colebrook equation: Haaland equation:

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Head Loss:

20 Head Loss In the analysis of piping systems, pressure losses are commonly expressed in terms of the equivalent fluid column height called head loss h L . It also represents the additional height that the fluid needs to be raised by a pump inorder to overcome the frictional losses in the pipe

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Hagen – Poiseuille’s Law:

23 Hagen – Poiseuille’s Law

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28 August 2011 24

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28 August 2011 25

Minor Losses:

26 Minor Losses Piping systems include fittings, valves, bends, elbows, tees, inlets, exits, enlargements, and contractions. These components interrupt the smooth flow of fluid and cause additional losses because of flow separation and mixing. The minor losses associated with these components: K L is the loss coefficient. Is different for different components. Typically provided by manufacturers.

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Minor Losses:

28 Minor Losses Total head loss in a system is comprised of major losses (in the pipe sections) and the minor losses (in the components) If the entire piping system has a constant diameter, then i pipe sections j components

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Example Problem:

33 Example Problem Ans: 169 kPa

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