CIVE2400: Pipeflow - Lecture 409/04/2009 School of Civil Engineering FACULTY OF ENGINEERING Local Head Losses • Local head losses are the “loss” of energy at point where the pipe changes dimension (and/or direction). Fluid Flow in Pipes: Lecture 4 Dr Andrew Sleigh Dr Ian Goodwill CIVE2400: Fluid Mechanics www.efm.leeds.ac.uk/CIVE/FluidsLevel2 Pipe Expansion Pipe Contraction Entry to a pipe from a reservoir Exit from a pipe to a reservoir Valve (may change with time) Orifice plate Tight bends • They are “velocity head losses” and are represented by u2 hL k L 2g Fluid Mechanics: Pipe Flow – Lecture 4 2 Value of kL • For junctions and bends we need experimental measurements • kL may be calculated analytically for Losses at an Expansion • As the velocity reduces (continuity) • Then the pressure must increase (Bernoulli) • So turbulence is induced and head losses occur Turbulence and losses Expansion Contraction • By considering continuity and momentum exchange and Bernoulli Fluid Mechanics: Pipe Flow – Lecture 4 3 Fluid Mechanics: Pipe Flow – Lecture 4 4 Value of kL for Expansion • Apply the momentum equation from 1 to 2 p1 A1 p2 A2 Q u2 u1 Value of kL for Expansion • Combine p2 g p1 u2 u1 u2 g and hL 2 2 u12 u2 2g p2 g p1 • Using the continuity equation we can eliminate Q p p u u u 2 1 2 hL u1 u2 2g g g 1 2 • Using the continuity equation again u1A1 = u2A2 A1 A2 2 u2=u1A1/A2 kL 1 A1 A2 2 hL 1 u12 2g • From Bernoulli 1 2 hL Fluid Mechanics: Pipe Flow – Lecture 4 2 u12 u2 2g p2 g p1 5 • A1 >> A2, kL = 1 Fluid Mechanics: Pipe Flow – Lecture 4 exit loss 6 1 CIVE2400: Pipeflow - Lecture 4 09/04/2009 Losses at an Contraction • Flow converges as the pipe contracts • Convergence is narrower than the pipe Losses at an Contraction • Apply the general local head loss equation between 1’ and 2 hL Due to vena contractor A1’ = 0.6A2 1 • Experiments show for common pipes A1' A2 2 u12' 2g • Can ignore losses bewteen 1 and 1’ 1 Using A1’ = 0.6A2 And Continuity u1' A2u2 A1' 0.44 2 u2 2g A2u2 0.6 A2 u2 0.6 1 1’ 2 1’ 2 As Convergent flow is very stable 7 hL Fluid Mechanics: Pipe Flow – Lecture 4 Fluid Mechanics: Pipe Flow – Lecture 4 8 Other Losses • Whenever there is expansion • Pressure increases down stream • Danger of boundary layer separation as the fluid near the walls had little momentum Losses: Junctions Reduced velocity Reduced velocity Increased pressure Increased pressure Fluid Mechanics: Pipe Flow – Lecture 4 9 Fluid Mechanics: Pipe Flow – Lecture 4 Losses: Sharp bends kL values Bell mouth Entry kL valu e Bellmouth entry Sharp entry Sharp exit 0.10 0.5 0.5 0.4 T-branch kL = 1.5 kL = 0.1 Sharp Entry/Exit kL = 0.5 Reduced velocity 90 bend 90 tees In-line flow Branch to line Gate value (open) 0.4 1.5 0.25 T-inline Increased pressure kL = 0.4 Fluid Mechanics: Pipe Flow – Lecture 4 Fluid Mechanics: Pipe Flow – Lecture 4 2 CIVE2400: Pipeflow - Lecture 4 09/04/2009 Pipeline Analysis • Bernoulli Equation Bernoulli Graphically • Reservoir • Pipe of Constant diameter Pressure head • No Flow p/ g p/ g z A= H p/ g H z z Datum line Elevation pA g 2 uA zA 2g H equal to a constant: Total Head, H pA g 2 uA zA 2g Total Head Line H • Applied from one point to another (A to B) With head losses pA g 2 uA zA 2g H 2 pB u B z B hL h f g 2g z Fluid Mechanics: Pipe Flow – Lecture 4 Fluid Mechanics: Pipe Flow – Lecture 4 Bernoulli Graphically • Constant Flow • Constant Velocity • No Friction pA g Total Head Line Velocity head Hydraulic Grade Line 2 uA zA 2g Bernoulli Graphically H • Constant Flow • Constant Velocity • No Friction Change of Pipe Diameter pA g 2 uA zA 2g H Total Head Line Velocity head u22/ g Hydraulic Grade Line u2/ g p/ g H Pressure head u2/ g p/ g z A= H z A= H Wider Pipe H Pressure head z Datum line Fluid Mechanics: Pipe Flow – Lecture 4 Elevation Datum line z Elevation Fluid Mechanics: Pipe Flow – Lecture 4 Bernoulli Graphically • Constant Flow • Constant Velocity • With Friction Total Head Line u2/ g p/ g H-hf Reservoir Feeding Pipe Example pA g 2 uA zA 2g H 2 pB u B zB g 2g hL h f Hydraulic Grade Line u2/ g u2/ g • • • • • • d = 0.1m Find Length A-C = L = 15m a) Velocity in pipe Length A-B = L = 1.5m b) Pressure at B f = 0.08 kL entry = 0.5 Sharp kL exit = 0 Opens to atmosphere B A zB-zA = 1.5m zA-zC = 4m z A= H zA z Datum line Fluid Mechanics: Pipe Flow – Lecture 4 Fluid Mechanics: Pipe Flow – Lecture 4 zB pc = Atmospheric C zC 3 CIVE2400: Pipeflow - Lecture 4 09/04/2009 Reservoir Feeding Pipe Example • Apply Bernoulli with head losses pA g 2 uA zA 2g 2 pC uC zC g 2g Reservoir Feeding Pipe Example • Find pressure at B: Apply Bernoulli A-B pA g 2 uA zA 2g 2 pB u B z B hL h f g 2g hL h f pA= pc = Atmospheric uA= negligible pA= Atmospheric = treat as 0 uA= negligible z A zC 2 u2 uC 0.5 2g 2g 4 fLu 2 2 gd 4 fL 1.0 0.5 2g d u 1.26m / s u2 2 u 2 4 fLABu 2 pB u B 0.5 g 2g 2g 2 gd zB zA u uB 1.26m / s 1.5 4 A u2 4 0.08 15 1.5 2 9.81 0.1 B pB 1000 9.81 A 1.26 4 0.08 5.0 1.5 2 9.81 0.1 B 2 pB 28.58 103 N / m2 Negative i.e. less than Atmospheric pressure zA zB pc = Atmospheric C zC zA zB pc = Atmospheric C zC Fluid Mechanics: Pipe Flow – Lecture 4 Fluid Mechanics: Pipe Flow – Lecture 4 Today’s lecture: • Local head losses Expansion loss Contraction loss Junction + other minor losses Total Head Line Hydraulic Grade Line hL kL u2 2g hL 1 A1 A2 2 u12 2g • Graphical representation of Bernoulli • Analysis of pipeline, including losses 21 4