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

MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 9: FLOWS IN PIPE. Instructor: Professor C. T. HSU. 9.1 General Concept of Flows in Pipe.

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

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  1. MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 9: FLOWS IN PIPE Instructor: Professor C. T. HSU

  2. 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.

  3. 9.1 General Concept of Flows in Pipe • For fully developed flows (x>>L), flows become parallel, , the mean pressure remains constant over the pipe cross-section

  4. 9.1 General Concept of Flows in Pipe • Flows in a long pipe (far away from pipe entrance and exit region, x>>L) are the limit results of boundary layer flows. There are two types of pipe flows: laminar and turbulent

  5. 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 Re=2300

  6. 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.

  7. 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.

  8. b 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

  9. 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

  10. 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, u=0 at y=R=D/2, there fore C=R2/4 and finally:

  11. 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:

  12. 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,

  13. 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,

  14. 9.2 Poiseuille Flow • At the wall, • So, • As a result, the wall friction coefficient is:

  15. 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.

  16. 9.3 Head Loss in Pipe • So the total head loss between section 1 and 2 of pipes is: • hl=head loss due to frictional effects in fully developed flow in constant area conduits • hlm=minor losses due to entrances, fittings, area changes, etcs.

  17. 9.3 Head Loss in Pipe • So, for a fully developed flow through a constant-area pipe, • And if y1=y2,

  18. 9.3 Head Loss in Pipe • For laminar flow, • Hence

  19. 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,

  20. 9.4 Turbulent Pipe Flow • Therefore, • Dimensional analysis, • Experiments show that the non-dimensional head loss is directly proportional to L/D, hence

  21. 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.

  22. 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.

  23. 9.4 Turbulent Pipe Flow • Miller suggested to use for the initial estimate, • That produces results within 1% in a single iteration

  24. 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

  25. 9.5 Minor Loss • Source of minor loss: 1. Inlets & Outlets 2. Enlargements & Contractions 3. Valves & Fittings 4. Pipe Bends

  26. 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

  27. b a Dh=a Ar=1 a=b b a Dh2a Ar b b a a 9.6 Non-Circular Ducts • Effect of Aspect Ratio (b/a): • For square ducts: • For wide rectangular ducts with b>>a: Thus, flows behave like channel flows • However, pipe flow results can be used with good accuracy only when: 1/3<Ar<3

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