Contents Page... 1 Index Page... 6 Symbols... 7 Approximations... 8 Fundamentals... 9 Common Equations & Values Nature of Fluids...

Transcription

1 Cntents Page Cntents Page... 1 Index Page... 6 Symbls... 7 Apprximatins... 8 Fundamentals... 9 Cmmn Equatins & Values Nature f Fluids Cntinuum Cncept Fluid Prperties Density Gas Law Viscsity Surface Tensin Types f Flw Laminar flw Turbulent Flw Tempral Variatin Spatial Variatin Gverning Principles Cntinuity Mmentum Energy Applicatins f Gverning Principles Example 2 Types f Flw Hydrstatic Pressure Pressure Abslute Pressure Gauge Pressure Atmspheric Pressure Pressure Transmissin Pascal s Law Equatin f Fluid Statics Pressure and Temperature in the Atmsphere Manmeters U-tube Manmeters Differential Manmeter Hrizntal Acceleratin Effects Vertical Acceleratin Effects Cmbined Acceleratin Example 3 - Manmeter Summary Hydrstatic Frces n Surfaces Frces n Hrizntal and Surfaces Line f Actin Centre f Pressure & Centre f Gravity Cmplex Shapes Parallel Axis Therem Example 4 - Hydrstatics Summary Hydrstatics - Buyancy Frces n Flating r Submerged Bdies Archimedes Principle

2 Stability f Submerged Bdies Stability f Flating Bdies Stability Determinatin f Metacentric Height Example 5 Buyancy Summary Dynamics Fluid Mtin Tls fr Describing Fluid Mtin Cntrl Surface Cntrl Vlume Cntinuity Flw Visualizatin Streamlines , 2 & 3 Dimensinal Flws Cntinuity Discharge & Mean Velcity Reference Frames Eulerian Lagrangian Flw Acceleratin Equatin f Mtin Acceleratin Differential frm f the Cntinuity Equatin Example 6 Wave Runup Summary Dynamics Energy Equatin Ptential Energy (PE) Kinetic Energy (KE) Pressure Energy General Energy Equatin Energy Equatin {E} r Bernulli Equatin {B} Steady Flw Energy Equatin Piezmetric and Energy/Ttal Head Lines E-line V P-Line Piezmetric Pressure Equatin Pressure & Velcity Flw Meters Pitt r Ttal Head Tubes Venturi Meter Orifice Meter Discharge Equatin fr Venturi Meter r Orifice Meter Example 7 Flw Meters Venturi & Orifice Kinetic Energy Crrectin Fluid Pwer & Wrk Pwer between tw Pints Radial Flw & the Energy Equatin Jet Discharge Example 8 Energy & Piezmetric Lines Summary Dynamics Mmentum Equatin Mmentum Equatin {M} Cnvective Acceleratin Tempral Acceleratin Applicatins Frce n a Pipe Bend Example 9 Frce n a Cntractin Jet Impact Jet Reactin r Thrust Euler s Equatin Euler s Equatin fr Steady Fluid Mtin

3 Euler & Bernulli Euler s Equatin fr Unsteady flw Example 10 Head Lss at Sudden Expansin Summary General Real & Ideal Fluids Viscus Flw Laminar Flw Stress/Strain Relatinship: Newtn s Law f Viscsity Viscus & Inertial Frces Turbulent Flw Reynlds Number Laminar & Turbulent Flw Regimes Example 11 Reynlds Number Parallel Flw General Gverning Equatin fr Steady Parallel Laminar Flw Example 12 Parallel Flw Laminar Flw in Pipes Velcity Prfile Parablic Prfile Laminar Flw in Pipes Steady Flw in Pipes Mmentum Equatin Head Lss Hagen Piseuille Equatin Summary Viscus fluids & Head Lss Turbulent Flw Darcy-Weisbach Equatin fr Flw in Pipes Example 13 Laminar & Turbulent Flw Head Lss in Pipes Velcity Prfile in a Pipe Viscus r Laminar Sub Layer Eddy Viscsity Develpment f Velcity Prfiles in Pipes Laminar Flw Develpment f Velcity Prfiles in Pipes Turbulent Flw Bundary Rughness Frictin Factrs: Ks V L Smth Turbulent Flw Transitinal Flw Rugh Turbulent Flw Example 14 Pipe Frictin Factrs Summary Pipeline & Pipe Netwrk Design Lcal Lsses in Pipe Flw Flw Separatin Lsses Lcal Lsses at an Expansin Head Lss at Sudden Expansin Lcal Lsses Empirical Relatinships Intakes Exits/Outfalls Expansin/Cntractin Bends, Valves Pipeline Design Tw Cases Example 15 Pipelines Example 16 Pipelines Pipe Netwrks Pipes in Series Pipes in Parallel Example 17 Cfferdam Hardy-Crss Technique The Three Relatinships t Satisfy Example 18 Hardy-Crss Methd

4 Summary Dynamic Fluid Lading Design Requirements Example Wave Cnditins Currents Design Requirements - Lading Regimes Ideal Flw Apprach Real Fluids & Viscus Fluids Fluid Lading Pressure Drag Ttal Drag Drag & Lift Frce Bundary Layer Separatin Terminal Velcity Example 19 Terminal Velcity Skin Frictin Bundary Layer Drag Surface Rughness Lading Regimes Real & Viscus Flws Spin & Lift Magnus Effect Unsteady Fluid Lading Ttal in-line frce Mrisn equatin Hydrdynamic mass (added mass) and an Accelerating Bdy Example 20 Unsteady Lading Summary Dimensinal Analysis & Similarity Imprtance f Dimensins Fundamental Dimensins Buckingham Methd Step-by-step Methd Cmmn Grups Example 21 Dimensinal Analysis Similarity Gemetric Similarity Dynamic Similarity Perfect Similarity Frude Scaling Frude Similitude Undistrted Mdels Reynlds Scaling Acceleratin & Viscus Frces Reynlds Similitude Undistrted Mdels Scale Mdelling Prblems Other Scale Relatinships Example 22 Similitude Summary Unsteady Flw in Pipe Systems Unsteady Flws in Clsed Cnduits Analysis technique Slw Variatins in Discharge Time Required fr Head Change Mre Rapid Changes in Discharge Rigid Clumn Thery Example 23 Rigid Clumn Thery Unsteady Energy Equatin Effect f Acceleratin Applicatins f Unsteady Energy Equatin Example 24 Unsteady Flw with Acceleratin: Flw between tw Reservirs Summary Surge Shafts Analysis

5 Frictinless Slutin Slutin Including Frictin Types Surge Prtectin in Pumped Mains Example 25 Surge Shaft Summary Waterhammer Unsteady Cmpressible Flw Shck Waves Shck Wave Prpagatin Steps Shck Wave Velcity Waterhammer Pressure Pressure Variatin with Time Frictin Effects Rate f Valve Clsure Waterhammer Thery Summary Example 26 - Waterhammer Summary Appendix Lecture Schedule Experiment 1 Flw Meters Experiment 2 Sluice Gate Experiment 3 Pipe Flw Experiment 4 Drag n Cylinder Wrked Lecture Examples... Errr! Bkmark nt defined. Mdy Diagram... Errr! Bkmark nt defined. Drag Cefficient Charts... Errr! Bkmark nt defined. AS : Resistance Cefficients (k) f Valves and Fittings... Errr! Bkmark nt defined. 5

6 Index Page accelerate ttal mass, 71 Acceleratin, 28 added mass cefficient, 71 Adiabatic expansin, 13 Archimedes Principle, 23 atmspheric pressure variatin, 17 Bernulli Equatin, 15 Bundary Layer Drag, 70 Bundary Layer Separatin, 69 Buckingham Methd, 74 bulk mdulus f the fluid, 12 Buyancy, 23 CD, 69 Celerity, 98 centre f buyancy, 23 centre f gravity, 20, 23 centre f pressure, 20 Chezy equatin, 52 Cmpressibility, 12 Cntinuity, 14, 26 Cntinuum Cncept, 12 Cntrl Surface, 26 Cntrl Vlume, 26 Cnvective Acceleratin, 27 Currents, 66 Darcy-Weisbach, 52 Density, 12 dimensinless grups, 75 Discharge Equatin fr Venturi Meter, 33 Drag, 68 Dynamic Similarity, 77 Dynamic viscsity, 13 Eddy Viscsity, 54 E-line, 32 Energy Equatin, 15, 31 energy lss, 50 Euler & Bernulli, 40 Euler s Equatin fr Steady Fluid Mtin, 40 Euler s Equatin fr Unsteady flw, 40 Euler s frmula, 18 Eulerian, 27 Flw Regimes, 46 Flw separatin, 58 frce n a vertical surface, 20 frictin drag, 70 Frictin Factrs, 55 Frude Scaling, 78 Frude Similitude, 78 Gas Law, 17 Gemetric Similarity, 77 Hagen Piseuille Equatin, 50 Hardy-Crss Technique, 63 Head Lss, 50 Head Lss at Sudden Expansin, 58 hydraulic diameter, 52 Hydraulically rugh, 55 Hydraulically smth, 55 Hydrdynamic mass, 71 I00, 24 inertia cefficient, 71 instantaneus cmplete clsure, 99 Isthermal expansin, 13 jet discharge, 35 Jet Reactin, 39 Kinematic viscsity, 13, 44 Kinetic Energy, 14 Kinetic Energy Crrectin, 35 Ks V L, 55 ks values, 55 Lagrangian, 27 Laminar Flw in Pipes, 49 Laminar Sub Layer, 54 Lapse rate, 17 Lift, 68 Lift frce, 68 Lading Regimes, 66 Lcal Lsses, 58 Lgarithmic Velcity prfile, 54 Mach number, 75 Magnus Effect, 71 Manmeters, 17 Mass flw rate, 14 metacentric height, 23 Mmentum, 14 Mmentum Equatin, 37 Mmentum flux, 14 Mre Rapid Changes in Discharge, 84 Mrisn equatin, 71 Mving cntrl vlumes, 42 Orifice Meter, 33 Parablic Prfile, 49 Parallel Axis Therem, 21 Parallel Flw, 47 Pascal s Law, 16 penstck, 91 Perfect Similarity, 78 physical parameters, 74 Piezmetric Pressure Equatin, 32 Pipes in Parallel, 61 Pipes in Series, 61 Pitt r Ttal Head Tubes, 32 P-line, 32 Ptential Energy, 14 Pwer, 35 Pressure, 16 Pressure Drag, 67 Pressure Energy, 14 Pressure Transmissin, 16 Rate f change f mmentum, 14 rate f strain f fluid, 48 Rate f Valve Clsure, 99 Reference Frames, 27 Reynlds Number, 45 Reynlds Scaling, 79 Reynlds Similitude, 79 Rigid Clumn Thery, 84 Scale Mdelling, 80 shear stress velcity, 54 shck wave, 96 Shck Wave Velcity, 98 skin cefficient, 70 Skin Frictin, 70 Spatial Variatin, 14 Specific Gravity, 12 Stability f submerged bdies, 23 Stagnatin pressure, 68 Steady Flw Energy Equatin, 31 Steady Flw in Pipes, 50 Steady Parallel Laminar Flw, 48 Step-by-step Methd, 74 streakline, 26 Streamlines, 26 Streamtubes, 26 Stress/Strain Relatinship, 44 Struhal number, 71 sudden expansin, 58 Surface Tensin, 13 Surge Pressure, 85 Surge shafts, 91 Temperature in the Atmsphere, 17 Tempral (lcal) Acceleratin, 27 Tempral Variatin, 14 Terminal Velcity, 69 Time Required fr Head Change, 84 Ttal Drag, 68 Unsteady Energy Equatin, 88 Unsteady Flws in Clsed Cnduits, 83 Unsteady Fluid Lading, 71 valve clsure, 99 Variatins in Discharge, 83 velcity head, 32 Velcity Prfile in a Pipe, 53 Venturi Meter, 33 Viscsity, 13, 44 Viscus frce, 79 Vn Karman cnstant, 53 vrtex shedding frequency, 71 Wake, 68 waterhammer thery, 96 Wave Cnditins, 66 Weber number, 75 wetted area, 70 Wrk, 35 6

7 Symbls ~ centre f pressure a ~ acceleratin A ~ crss sectinal area (m 2 ) B ~ centre f buyancy BG ~ height f G (centre f gravity) abve B (centre f buyancy) BM ~ height f M (metacentre) abve B (centre f buyancy) c ~ celerity Cc ~ cefficient f cntractin Cd ~ actual discharge crrectin factr CD ~ drag cefficient Cf ~ average skin cefficient cf ~ skin cefficient Cm ~ inertia cefficient CP ~ centre f pressure Cp ~ elastic celerity CV ~ cntrl vlume D ~ diameter DH ~ hydraulic diameter (used when nn circular pipes) Ev ~ bulk (r vlume) mdulus f the fluid FD ~ drag frce FL ~ lift frce G ~ centre f gravity GM ~ metacentric height H ~ height hf ~ head lss due t frictin hl ~ head lss (fr turbulent flw, includes frictin factr frm hf) K ~ bulk mdulus f the fluid (N/m 2, Pa) measures substance resistance t unifrm cmpressin k ~ Vn Karman cnstant 0.4 km ~ added mass cefficient ks ~ rughness f the bundary M ~ metacentre p ~ pressure (N/m 2, Pa) Pw ~ wetted perimeter Q ~ flw rate Q ~ vlumetric flw rate r discharge R ~ gas cnstant fr a particular gas (J Kg -1 K -1 ) R ~ upthrust S ~ specific gravity T ~ temperature (C) U ~ lcal velcity, mean flw velcity u* ~ shear stress velcity V ~ velcity z ~ height ~ lapse rate ~ unit weight f fluid H ~ surge pressure ~ turbulent eddy viscsity ~ dynamic viscsity, abslute cefficient f viscsity (Ns/m 2 ) r (kg/ms) ~ kinematic viscsity (nu) (m 2 /s) ~ density (kg/m 3 ) Q ~ mass flw rate (kg/s) ~ surface tensin (N/m) ~ shear stress (N/m 2 ) ~ angular velcity 7

8 Apprximatins Velcity in a river Nn-unifrm flw crss sectin Width cnstantly changing flw velcity cnstantly changing Bundary layer Indicated by velcity prfile clse t bed Nt cnstant; mud, sand, vegetatin etc Difficult t calculate Turbulent flw Varying bed rughness Turbulent Pipe flw Steady flw t start Turbulent eddies created at bends Slutin Ignre details by time averaging Ignre details inside Cntrl Vlume (CV) 8

9 Fundamentals Main principles Cntinuity, {C} Q UA Mmentum, {M} F ma QU = gv discharge r vlume flw rate, (m 3 /s) Energy, {E} H p 1 g U 2 1 2g z 1 Frce = mass x acceleratin (N) Sum f ptential, kinetic and pressure energy = ttal head (m) 9

10 Cmmn Equatins & Values dh dt U 1 tn = 8896N 10hPa = 1kPa 1g/cm 3 = 1000kg/ m 3 1L = 0.001m 3 = 10-6 cm 3 a v 2 Acceleratin r w 2 r Atmspheric pressure = bar, atm Centrid f a Parabla cs 2 = 1 sin 2 da = 2rdr Density water = 1000kg/m X seawater = 1025kg/m 3 air = 1.2kg/m 3 Differentiatin f cs -sin Differentiatin f sin cs Efficiency: Pwer required = pwer/% Extraplating Gas Law PV = mrt Head = h p 1 p 2 g Y Y A X X A X B X A Y B Y A Pwer = gqh = frce x speed = wrk/unit time Pressure Patms = 101.3kPa Pwater = 50kPa Pwater vapur = 2.3kPa Pair = 60Pa Rtatin rate = rpm revlutins per minute Steady flw = n acceleratin Temperature Abslute = T+ 273 Velcity v wr Viscsity water=1x10-3 kg/ms seawater=1.13x10-3 kg/ms vwater=1x10-6 m 2 /s vair = 1.5x10-5 m 2 /s Vlume sphere = D r 3 Wave Perid = 2 t Wave velcity = U cs (wt) Weight = F = gv Yungs Mdulus Ewater = 2x10 9 Esteel = 2.1x10 11 du dt 0 10

11 Nature f Fluids Fluids = liquids & gases Fluids can Flw Change shape Take up the shape f the bundaries Strng intermlecular frces at bundary Defrm cntinuusly and permanently under applicatin f a shearing stress, Fluid at rest n shearing frces acting Therefre, all frces in the fluid are perpendicular t the surfaces n which they act Fluids in mtin Mlecules adhere t the bundaries n slip cnditin Velcity varies away frm the bundary Fluid element defrms Frce per unit area exerted n the fluid by the bundary and vice-versa is the shear stress, Cntinuum Cncept If there are enugh mlecules, the average cnditins (pressure, density etc) are cnsidered cnstant r change smthly Valid liquids and mst gases Tightly packed mlecules Nt valid rarefied gases, small number f mlecules Large spaces between mlecules Fluid Prperties Density - Mass per unit vlume, kg/m 3 Typical values air = 1.2 kg/m 3 water = 1000 kg/m 3 (15C) Specific Gravity S = fluid water Cmpressibility Density varies with pressure and temperature Change in vlume with a change in pressure depends n the bulk mdulus f the fluid, K (N/m 2 ) p K where Kwater = 2.05x10 9 N/m 2 11

12 Generally, water can be treated as incmpressible Except fr very large changes in pressure (i.e. a waterhammer) Gas Law Air is mre cmpressible but can be treated as incmpressible at velcities much lwer than the speed f sund Ideal Gas Law p RT R = gas cnstant fr a particular gas Rair = 287 J Kg -1 K -1 T = temperature Isthermal expansin n change in temperature Adiabatic expansin n heat exchange ut f the system Viscsity Measure f hw easily a fluid flws Dynamic viscsity,, Ns/m 2 Als called abslute cefficient f viscsity (kg/ms) water = 1x10-3 kg/ms (20C) Kinematic viscsity, (nu) alternative, frequently used (m2 /s) water = 1x10-6 m 2 /s (20C) Newtns Law f Viscsity F A du dy Newtnian fluids bey this law mst cmmn fluids Surface Tensin, N/m arises frm elasticity f the surface reduces surface area t a minimum Causes capillary rise between surfaces Manmeter tubes Errrs in readings Weight f clumn f fluid = surface tensin frce acting n wetted length g D2 4 H D cs H 4 cs gd s small, s cs = 1 Types f Flw Laminar flw Smth, unifrm, regular 12

13 Turbulent Flw Chatic, randm, dispersive Tempral Variatin Flw variatin with time Steady flws Velcity and depth cnstant with time Unsteady flws Velcity and depth vary with time Spatial Variatin Flw variatin with space Unifrm Flw prperties cnstant in directin Nn-unifrm Flw prperties vary in flw directin Gverning Principles Cntinuity Cntinuity, {C} Cnservatin f mass Mass flw rate = Q (kg/s) N strage Q1 = Q2 Incmpressible fluid, r n change in density Q1 = U1A1 = U2A2 = Q2 Cntinuity equatin {C} A = crss sectinal area U = mean flw velcity Mmentum Mmentum, {M} Cnservatin f mmentum Newtn s Secnd Law F = ma Rate f change f mmentum = sum f frces Mmentum flux = mass flw rate x velcity Or rate at which mmentum passes thrugh a crss sectin QU UA U Rate f change between crss sectins = QU Requires a resultant frce F in the directin f mtin F Mmentum equatin {M} QU Q U 2 U 1 Energy Energy, ttal head, head, H 13

14 The sum f three frms f energy Kinetic Energy, ½U 2 Ptential Energy, gz Pressure Energy, p Cnserved when n energy lst p U 2 1 gz 1 cnst when divided thrugh by g gives H in dimensins f length (m) H p 1 g U 2 1 2g z 1 cnst Energy Equatin {E} als knwn as Bernulli Equatin {B} Applicatins f Gverning Principles {C} {E} {M} eliminate unknwns write velcity in terms f area determine hw velcity and pressure vary in the flw t find energy lsses find frces determine hw pressure and velcity vary if there are energy lsses Example 2 Types f Flw Steady flw Flw frm a reservir with cnstant head and bundary cnditins (des nt vary with time) Steady & unifrm flw Flw frm a reservir with cnstant head and bundary cnditins int a lng straight pipe (crss sectin nt changing, n change in flw) Steady, nn-unifrm flw Flw frm a reservir with cnstant head and bundary cnditins int a cnverging pipe r thrugh an rifice (cnditins change alng the pipe) Unsteady, unifrm flw Flw in an il pipeline cntrlled by a variable speed pump (time varying) Unsteady, nn-unifrm flw Wave mtin Arterial flw Surge twers Dmestic plumbing Cmpressible unsteady flw r shckwave Pwer trip in hydr-electric plant (wrst pssible prblem) Rtatinal r vrtex flw Flw int a vertical rifice 14

15 Hydrstatic Pressure Pressure Fluid at rest Static equilibrium N shearing frces perpendicular r nrmal frces nly Shear nly ccurs when mving Scalar quantity Intensity & magnitude equal in all directins p F A (N/m2, Pa) Small frces ver small areas give large pressures Hydraulic presses, stilett heels Small frces ver large areas give large frces Wind flw ver rf Abslute Pressure Pressure in a vacuum = abslute zer Gauge Pressure Measured relative t lcal atmspheric pressure and can be psitive r negative Mst cmmn Example Pressure reading is 50kPa where atmspheric pressure is 100kPa Gauge pressure = 50kPa Abslute pressure = 150kPa Atmspheric Pressure 10 5 N/m 2 1 atm r 1 bar r 10m H2O Pressure Transmissin Occurs in clsed systems and can be used t amplify frces hydraulic cntrls p1 = p2 F 2 A 2 A 1 F 1 ; where A2/A1 is the amplifier Varying pressures (dynamic pressures) may be damped by viscus effects, but mean r static pressures are nt Pascal s Law Pressure change at ne pint in a system is transmitted thrugh the entire system Equatin f Fluid Statics A fluid at rest is in equilibrium Hydrstatic relatinship fr an incmpressible fluid Vertical equilibrium 15

16 dp dz g Hrizntal equilibrium dp dx 0, dp dy 0 p gz cnst Implies equality f pressure at the same level in the same static fluid r in cmbinatins f static fluids PRESSURE IS ADDITIVE in the vertical fr immiscible fluids (fluids that dn t mix) Pressure and Temperature in the Atmsphere Pressure, density & temperature vary with elevatin in the atmsphere Trpsphere Sea level 13000m Temperature drps linearly with increasing elevatin Lapse rate, = drp in T with z avg cnditins = 5.87K/km With the Gas Law, pressure and density can be calculated at any elevatin p RT T T 0 (z z ) Using the hydrstatic equatin gives Manmeters T p p 0 z z 0 0 T 0 g R atmspheric pressure variatin in the atmsphere Head & pressure measurement Fluid mst ften water, but can be any fluid water = 1000kg/m 3 = g = unit weight f fluid Manmeter Equatin h = (S-1)h r h = (1-S)h U-tube Manmeters A type f manmeter with immiscible fluids (will nt mix), used fr measuring pressures in gases and higher pressures in liquids 16

17 Differential Manmeter Created by cnnecting bth ends f the manmeter t the pipe Similar in principle t the U-tube manmeter p = (2-1)gh = (2-1)h Hrizntal Acceleratin Effects A particle n the surface f a fluid under cnstant hrizntal acceleratin, a Surface is a plane at angle t the hrizntal Vertical acceleratin = 0 N additinal vertical frces, hence hydrstatic pressure Planes f equal pressure lie parallel t the free surface Vertical Acceleratin Effects Pressure increases with upward acceleratin, similar t ging up in an elevatr Oscillatry flws (waves, surge twers) have nn-hydrstatic pressure belw the accelerating free surface nt pure hydrstatic Cmbined Acceleratin Ttal acceleratin = pressure gradient Euler s frmula p z g a z p x a x if az is small, pressure is hydrstatic - the free surface slpe gives the ttal hrizntal acceleratin Fr rtating flws, the hrizntal acceleratin varies with radial distance frm the centre f rtatin free surface becmes parablic a frced vrtex 17

18 Example 3 - Manmeter In general p 1 p 2 g 1 2 h S 1 S 2 h z 1 z 2 Summary Fluid at rest = n shearing frces Hydrstatic pressure increases linearly with depth Pressure is equal alng lines f cnstant elevatin Pressures additive in the vertical Manmeters measure gage r relative pressures Acceleratin gives rise t additinal pressure gradients Surface slpes indicate pressure gradients 18