1. Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Closed Conduit Flow CEE 332

2. Closed Conduit Flow ä Energy equation ä EGL and HGL ä Head loss ä major losses ä minor losses ä Non circular conduits ä Energy equation ä EGL and HGL ä Head loss ä major losses ä minor losses ä Non circular conduits 

3. Conservation of Energy ä Kinetic, potential, and thermal energy hL =hL = hL =hL = h p = ht =ht = ht =ht = head supplied by a pump head given to a turbine head loss between sections 1 and 2 Cross section 2 is ____________ from cross section 1! downstream Point to point or control volume? Why  ?

4. Energy Equation Assumptions hydrostatic density Steady kinetic ä Pressure is _________ in both cross sections ä pressure changes are due to elevation only ä section is drawn perpendicular to the streamlines (otherwise the _______ energy term is incorrect) ä Constant ________at the cross section ä _______ flow ä Pressure is _________ in both cross sections ä pressure changes are due to elevation only ä section is drawn perpendicular to the streamlines (otherwise the _______ energy term is incorrect) ä Constant ________at the cross section ä _______ flow

5. EGL (or TEL) and HGL velocity head velocity head elevation head (w.r.t. datum) elevation head (w.r.t. datum) pressure head (w.r.t. reference pressure) pressure head (w.r.t. reference pressure) downward lower than reference pressure ä The energy grade line must always slope ___________ (in direction of flow) unless energy is added (pump) ä The decrease in total energy represents the head loss or energy dissipation per unit weight ä EGL and HGL are coincident and lie at the free surface for water at rest (reservoir) ä If the HGL falls below the point in the system for which it is plotted, the local pressures are _____ ____ __________ ______ ä The energy grade line must always slope ___________ (in direction of flow) unless energy is added (pump) ä The decrease in total energy represents the head loss or energy dissipation per unit weight ä EGL and HGL are coincident and lie at the free surface for water at rest (reservoir) ä If the HGL falls below the point in the system for which it is plotted, the local pressures are _____ ____ __________ ______

6. Energy equation z = 0 pump Energy Grade Line Hydraulic G L velocity head pressure head elevation datum z static head

7. Bernoulli Equation Assumption density Steady streamline Frictionless ä _________ (viscosity can’t be a significant parameter!) ä Along a __________ ä ______ flow ä Constant ________ ä _________ (viscosity can’t be a significant parameter!) ä Along a __________ ä ______ flow ä Constant ________

8. Pipe Flow: Review dimensional analysis ä We have the control volume energy equation for pipe flow. ä We need to be able to predict the head loss term. ä How do we predict head loss? __________ _______. ä We have the control volume energy equation for pipe flow. ä We need to be able to predict the head loss term. ä How do we predict head loss? __________ _______.

9. Dimensional Analysis ä What are the important forces? ______, ______. Therefore _________ number. ä What are the important geometric parameters? _________________________ ä Create dimensionless geometric groups ______, ______ ä Write the functional relationship ä What are the important forces? ______, ______. Therefore _________ number. ä What are the important geometric parameters? _________________________ ä Create dimensionless geometric groups ______, ______ ä Write the functional relationship Inertial diameter, length, roughness height Reynolds l/D viscous  /D

10. Dimensional Analysis ä How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow? ä If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will C p change? ä How will the results of dimensional analysis guide our experiments to determine the relationships that govern pipe flow? ä If we hold the other two dimensionless parameters constant and increase the length to diameter ratio, how will C p change? C p proportional to l f is friction factor

11. Dimensional Analysis Darcy-Weisbach equation Pressure Coefficient and Head Loss Always true (laminar or turbulent) Assume horizontal flow More general Darcy-Weisbach equation

12. Friction Factor : Major losses ä Laminar flow ä Hagen-Poiseuille ä Turbulent (Smooth, Transition, Rough) ä Colebrook Formula ä Moody diagram ä Swamee-Jain ä Laminar flow ä Hagen-Poiseuille ä Turbulent (Smooth, Transition, Rough) ä Colebrook Formula ä Moody diagram ä Swamee-Jain

13. Hagen-Poiseuille Darcy-Weisbach Laminar Flow Friction Factor Slope of ___ on log-log plot

14. Turbulent Pipe Flow Head Loss ä ___________ to the length of the pipe ä Proportional to the _______ of the velocity (almost) ä ________ with surface roughness ä Is a function of density and viscosity ä Is __________ of pressure ä ___________ to the length of the pipe ä Proportional to the _______ of the velocity (almost) ä ________ with surface roughness ä Is a function of density and viscosity ä Is __________ of pressure Proportional Increases independent square

15. (used to draw the Moody diagram) Smooth, Transition, Rough Turbulent Flow ä Hydraulically smooth pipe law (von Karman, 1930) ä Rough pipe law (von Karman, 1930) ä Transition function for both smooth and rough pipe laws (Colebrook) ä Hydraulically smooth pipe law (von Karman, 1930) ä Rough pipe law (von Karman, 1930) ä Transition function for both smooth and rough pipe laws (Colebrook)

16. Moody Diagram 0.01 0.1 1E+031E+041E+051E+061E+071E+08 Re friction factor laminar 0.05 0.04 0.03 0.02 0.015 0.01 0.008 0.006 0.004 0.002 0.001 0.0008 0.0004 0.0002 0.0001 0.00005 smooth

17. Swamee-Jain ä 1976 ä limitations   /D < 2 x 10 -2 ä Re >3 x 10 3 ä less than 3% deviation from results obtained with Moody diagram ä easy to program for computer or calculator use ä 1976 ä limitations   /D < 2 x 10 -2 ä Re >3 x 10 3 ä less than 3% deviation from results obtained with Moody diagram ä easy to program for computer or calculator use no f Each equation has two terms. Why?

18. Pipe Roughness pipe material pipe roughness   (mm) glass, drawn brass, copper 0.0015 commercial steel or wrought iron 0.045 asphalted cast iron 0.12 galvanized iron 0.15 cast iron 0.26 concrete 0.18-0.6 rivet steel 0.9-9.0 corrugated metal 45 PVC 0.12

19. Solution Techniques ä find head loss given (D, type of pipe, Q) ä find flow rate given (head, D, L, type of pipe) ä find pipe size given (head, type of pipe,L, Q)

20. Exponential Friction Formulas C = Hazen-Williams coefficient range of data ä Commonly used in commercial and industrial settings ä Only applicable over _____ __ ____ collected ä Hazen-Williams exponential friction formula ä Commonly used in commercial and industrial settings ä Only applicable over _____ __ ____ collected ä Hazen-Williams exponential friction formula

21. Head loss: Hazen-Williams Coefficient CCondition 150PVC 140Extremely smooth, straight pipes; asbestos cement 130Very smooth pipes; concrete; new cast iron 120Wood stave; new welded steel 110Vitrified clay; new riveted steel 100Cast iron after years of use 95Riveted steel after years of use 60-80Old pipes in bad condition CCondition 150PVC 140Extremely smooth, straight pipes; asbestos cement 130Very smooth pipes; concrete; new cast iron 120Wood stave; new welded steel 110Vitrified clay; new riveted steel 100Cast iron after years of use 95Riveted steel after years of use 60-80Old pipes in bad condition

22. Hazen-Williams vs Darcy-Weisbach preferred ä Both equations are empirical ä Darcy-Weisbach is dimensionally correct, and ________. ä Hazen-Williams can be considered valid only over the range of gathered data. ä Hazen-Williams can’t be extended to other fluids without further experimentation. ä Both equations are empirical ä Darcy-Weisbach is dimensionally correct, and ________. ä Hazen-Williams can be considered valid only over the range of gathered data. ä Hazen-Williams can’t be extended to other fluids without further experimentation.

23. Head Loss: Minor Losses potential thermal Vehicle drag Hydraulic jump Vena contracta Minor losses! ä Head loss due to outlet, inlet, bends, elbows, valves, pipe size changes ä Flow expansions have high losses ä Kinetic energy decreases across expansion ä Kinetic energy  ________ and _________ energy ä Examples – ________________________________ __________________________________________ ä Losses can be minimized by gradual transitions ä Head loss due to outlet, inlet, bends, elbows, valves, pipe size changes ä Flow expansions have high losses ä Kinetic energy decreases across expansion ä Kinetic energy  ________ and _________ energy ä Examples – ________________________________ __________________________________________ ä Losses can be minimized by gradual transitions

24. Minor Losses ä Most minor losses can not be obtained analytically, so they must be measured ä Minor losses are often expressed as a loss coefficient, K, times the velocity head. ä Most minor losses can not be obtained analytically, so they must be measured ä Minor losses are often expressed as a loss coefficient, K, times the velocity head. High Re

25. Head Loss due to Sudden Expansion: Conservation of Energy 1 2 z 1 = z 2 What is p 1 - p 2 ?

26. Apply in direction of flow Neglect surface shear Divide by (A 2  ) Head Loss due to Sudden Expansion: Conservation of Momentum Pressure is applied over all of section 1. Momentum is transferred over area corresponding to upstream pipe diameter. V 1 is velocity upstream. Pressure is applied over all of section 1. Momentum is transferred over area corresponding to upstream pipe diameter. V 1 is velocity upstream. 1 2 A1A1A1A1 A2A2A2A2 x

27. Energy Head Loss due to Sudden Expansion Momentum Mass

28. Contraction V1V1 V2V2 EGL HGL vena contracta ä losses are reduced with a gradual contraction Expansion!!!

29. Entrance Losses ä Losses can be reduced by accelerating the flow gradually and eliminating the vena contracta

30. Head Loss in Valves ä Function of valve type and valve position ä The complex flow path through valves often results in high head loss ä What is the maximum value that K v can have? _____ ä Function of valve type and valve position ä The complex flow path through valves often results in high head loss ä What is the maximum value that K v can have? _____ 

31. Questions ä What is the head loss when a pipe enters a reservoir? ä Draw the EGL and HGL ä What is the head loss when a pipe enters a reservoir? ä Draw the EGL and HGL V EGL HGL

32. Questions ä Can the Darcy-Weisbach equation and Moody Diagram be used for fluids other than water? _____ Yes No Yes ä What about the Hazen-Williams equation? ___ ä Does a perfectly smooth pipe have head loss? _____ ä Is it possible to decrease the head loss in a pipe by installing a smooth liner? ______

33. Example D=40 cm L=1000 m D=40 cm L=1000 m D=20 cm L=500 m D=20 cm L=500 m valve 100 m Find the discharge, Q. What additional information do you need? Apply energy equation How could you get a quick estimate? _________________ Or spreadsheet solution: find head loss as function of Q. Find the discharge, Q. What additional information do you need? Apply energy equation How could you get a quick estimate? _________________ Or spreadsheet solution: find head loss as function of Q. Use S-J on small pipe cs 1 cs 2

34. Non-Circular Conduits: Hydraulic Radius Concept ä A is cross sectional area ä P is wetted perimeter ä R h is the “Hydraulic Radius” (Area/Perimeter) ä Don’t confuse with radius! ä A is cross sectional area ä P is wetted perimeter ä R h is the “Hydraulic Radius” (Area/Perimeter) ä Don’t confuse with radius! For a pipe We can use Moody diagram or Swamee-Jain with D = 4 R h !

35. Quiz ä In the rough pipe law region if the flow rate is doubled (be as specific as possible) ä What happens to the major head loss? ä What happens to the minor head loss? ä Why do contractions have energy loss? ä If you wanted to compare the importance of minor vs. major losses for a specific pipeline, what dimensionless terms could you compare? ä In the rough pipe law region if the flow rate is doubled (be as specific as possible) ä What happens to the major head loss? ä What happens to the minor head loss? ä Why do contractions have energy loss? ä If you wanted to compare the importance of minor vs. major losses for a specific pipeline, what dimensionless terms could you compare?