Closed Conduit Hydraulics

1. Closed Conduit Hydraulics CE154 - Hydraulic Design Lecture 6 CE154

2. Hydraulics of Closed Conduit Flow • Synonyms- closed conduit flow- pipe flow- pressurized flow • Objectives – to introduce- basic concepts of closed conduit flow, - its hydraulics, and - design method CE154

3. Concepts • Closed Conduit vs. Open Channel CE154

4. Concepts – Reynolds Number • Reynolds Number (ratio of inertia force to viscous force)V = velocity (ft/sec)D = pipe diameter (ft) = density of fluid (lbm/ft3) = dynamic viscosity of fluid (lbm/ftsec or lbfsec/ft2) = kinematic viscosity (ft2/sec) CE154

5. Concepts – Froude Number • Froud Number (ratio of inertia force to gravitational force) • V = velocityg = gravitational accelerationh = depth of water CE154

6. Concepts - Turbulence • Turbulent vs. laminar flow CE154

7. Concepts – turbulent flow • Turbulent flow - Critical Re (laminar to turbulent) in the order of 1000 CE154

8. Concepts – laminar flow • Turbulent and Laminar flows CE154

9. Concepts – uniform & steady flow • Uniform flow – constant characteristics with respect to space • Steady flow – constant characteristics with respect to time. Often adopted when establishing pipe system design parameters (pressure & flow at certain locations). Consider unsteady (transient) phenomena to refine design (pipe pressure class and thickness) CE154

10. Conservation of Mass 1 Control Volume 2 CE154

11. Conservation of Mass • Consider the control volume CE154

12. Conservation of Mass • For steady & incompressible flow, dS/dt = 0I = OV1A1 = V2A2 ViAi =  VoAo CE154

13. Conservation of Mass • Apply to a pipe junction, Q1+Q2 = Q3+Q4 CE154

14. Conservation of Momentum • Newton’s 2nd law – the resultant of all external forces on a system is equal to the time rate of change of momentum of this system CE154

15. Conservation of Momentum • Consider this control volume (CV) of fluid in a pipe elbow x1=v1t 1 1’ 2 2’ x2=v2t CE154

16. Conservation of Momentum • In a time t the fluid originally at Section 1 moves to 1’, and that at Section 2 moves to 2’ • The control volume lost momentum equal to that of the fluid contained between 1 and 1’(A1x1)V1 = A1V12t = (QV1)tAt the same time it gained momentum (QV2)t CE154

17. Conservation of Momentum • The time rate of change of momentum is (QV2)- (QV1) • Hence, the 2nd Law becomes • This is the momentum equation for steady flow. Use this convention: • QVx1 Fx = QVx2 • QVy1 Fy = QVy2 • Where  depends on the direction of the force w.r.t. the coordinate system CE154

18. Application of Momentum Eq. • Forces on a pipe elbow:Taking momentum balance in the x direction,QV1 + (PA)1 – Fx = Q(0)Fx = (PA)1 + QV1 CE154

19. Application of Momentum Eq. • Taking momentum balance in the y direction,External y force = (PA)2 - FyRate of change of momentum = QV2 (where V2 is in the negative direction) (PA)2 - Fy = QV2 Fy = (PA)2 - QV2 = (PA)2 + QV2 CE154

20. Conservation of Energy • In pipeline design, most often consider steady state – flow not varying with time - first • Steady state (SS) Bernoulli Equation along a streamline: CE154

21. Conservation of Energy • Pressure head p/ • Elevation head z • Velocity head V2/2g • Piezometric head p/ + z (hydraulic grade line) • Total head p/ + z + V2/2g (energy grade line) • Head Loss h CE154

22. Uniform Flow: CE154

23. CE154

24. Example 3-1 • A plane jet of unit discharge q0 strikes a boundary at an angle of 45, what will be the ratio of q1/q2 for the divided flow? CE154

25. Head Losses • Include mostly 2 types of losses: • Friction Loss- resulting from friction between the fluid and pipe wall • Minor Loss- resulting from pipe entrance, transition, exit, valve and other in-line structures CE154

26. Friction Loss • Most useful head loss equation for closed-conduit flow – Darcy-Weisbach equation Pipe length Friction head loss Pipe velocity Dimensionless Friction coefficient Gravitational acceleration Pipe diameter CE154

27. Friction Loss • Darcy-Weisbach equation- derived from basic relationships of physics -  dimensionless, app. to all unit systems-  determined from experimental data • Other friction loss relationships – Hazen-Wiliams, Manning, Chezy, etc. – are also used in the industry, but are less accurate and will not be discussed here CE154

28. Darcy-Weisbach  • Laminar flow (Re<2000)Turbulent flow in smooth pipes (Re>4000) CE154

29. Darcy-Weisbach  (cont’d) • Turbulent flow in rough pipesTransition between turbulent smooth and rough pipes CE154

30. Friction Loss CE154

31. CE154

32. Darcy-Weisbach  • Most recent development of Darcy Weisbach coefficient - Explicit equation [Swamee and Jain, 1976] applicable to entire turbulent flow regime (smooth, transition and rough pipes): CE154

33. Minor Loss • Use minor loss coefficient (k) in this form CE154

34. Minor Loss CE154

35. Minor Loss • For abrupt expansion, from D1 to D2, the loss coefficient may be estimated by CE154

36. Minor Loss • American Water Works Association – Steel Pipe, A guide for design and installation, Manual of Water Supply Practices, M11, 4th Edition, 2004 CE154

37. Minor Loss CE154

38. Minor Loss • Valve manufacturer has loss curves typically presented in terms of Cv vs. valve opening degrees. Cv is defined as the flow rate in gallons per minute of 60 water that flows through the valve under 1 psi of head loss. CE154

39. CE154

40. Globe Valve CE154

41. Angle Valve CE154

42. Example – using HGL & EGL CE154

43. Example 3-2 • p. 2.24 of Mays’ Hydraulic Design Handbook – Calculate f and e/D from given discharge V2/2g=1.21 m Atmospheric Pressure P=3MPa L=2500 m El. 200 m D=27 in El. 100 m Q=1.8 cms CE154

44. Example 3-3 • Same problem but now we have an 20” in-line ball valve with a 20” bore opened at 70 from closed position, a contraction and expansion section each connected to the valve, and 2 90 elbows with r/D=2. What is the f now? CE154