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Contents Page
Contents Page .................................................................................................................................................. 1
Index Page ........................................................................................................................................................ 6
Symbols ............................................................................................................................................................ 7
Approximations ................................................................................................................................................ 8
Fundamentals ................................................................................................................................................... 9
Common Equations & Values ......................................................................................................................... 10
Nature of Fluids .............................................................................................................................................. 11 Continuum Concept ................................................................................................................................................................................................ 11 Fluid Properties ........................................................................................................................................................................................................ 11
Density ................................................................................................................................................................................................................................ 11 Gas Law .............................................................................................................................................................................................................................. 12 Viscosity .............................................................................................................................................................................................................................. 12 Surface Tension .............................................................................................................................................................................................................. 12
Types of Flow ............................................................................................................................................................................................................ 12 Laminar flow .................................................................................................................................................................................................................... 12 Turbulent Flow ............................................................................................................................................................................................................... 13 Temporal Variation ...................................................................................................................................................................................................... 13 Spatial Variation ............................................................................................................................................................................................................ 13
Governing Principles .............................................................................................................................................................................................. 13 Continuity .......................................................................................................................................................................................................................... 13 Momentum ........................................................................................................................................................................................................................ 13 Energy ................................................................................................................................................................................................................................. 13 Applications of Governing Principles .................................................................................................................................................................... 14
Example 2 – Types of Flow .................................................................................................................................................................................. 14
Hydrostatic Pressure ...................................................................................................................................... 15 Pressure ....................................................................................................................................................................................................................... 15
Absolute Pressure .......................................................................................................................................................................................................... 15 Gauge Pressure ............................................................................................................................................................................................................... 15 Atmospheric Pressure .................................................................................................................................................................................................. 15
Pressure Transmission .......................................................................................................................................................................................... 15 Pascal’s Law ..................................................................................................................................................................................................................... 15
Equation of Fluid Statics ....................................................................................................................................................................................... 15 Pressure and Temperature in the Atmosphere ........................................................................................................................................... 16 Manometers ............................................................................................................................................................................................................... 16
U-tube Manometers ...................................................................................................................................................................................................... 16 Differential Manometer .............................................................................................................................................................................................. 17
Horizontal Acceleration Effects ......................................................................................................................................................................... 17 Vertical Acceleration Effects ............................................................................................................................................................................... 17 Combined Acceleration ......................................................................................................................................................................................... 17 Example 3 - Manometer ........................................................................................................................................................................................ 18 Summary ..................................................................................................................................................................................................................... 18
Hydrostatic Forces on Surfaces ....................................................................................................................... 19 Forces on Horizontal and Surfaces ................................................................................................................................................................... 19 Line of Action – Centre of Pressure & Centre of Gravity .......................................................................................................................... 19
Complex Shapes .............................................................................................................................................................................................................. 19 Parallel Axis Theorem .................................................................................................................................................................................................. 20
Example 4 - Hydrostatics ...................................................................................................................................................................................... 20 Summary ..................................................................................................................................................................................................................... 21
Hydrostatics - Buoyancy ................................................................................................................................. 22 Forces on Floating or Submerged Bodies ...................................................................................................................................................... 22
Archimedes Principle ................................................................................................................................................................................................... 22
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Stability of Submerged Bodies ........................................................................................................................................................................... 22 Stability of Floating Bodies .................................................................................................................................................................................. 22
Stability .............................................................................................................................................................................................................................. 23 Determination of Metacentric Height .............................................................................................................................................................. 23 Example 5 – Buoyancy ........................................................................................................................................................................................... 24 Summary ..................................................................................................................................................................................................................... 24
Dynamics – Fluid Motion ................................................................................................................................ 25 Tools for Describing Fluid Motion .................................................................................................................................................................... 25
Control Surface ............................................................................................................................................................................................................... 25 Control Volume ............................................................................................................................................................................................................... 25 Continuity .......................................................................................................................................................................................................................... 25 Flow Visualization ......................................................................................................................................................................................................... 25 Streamlines ....................................................................................................................................................................................................................... 25
1, 2 & 3 Dimensional Flows ................................................................................................................................................................................. 25 Continuity – Discharge & Mean Velocity ........................................................................................................................................................ 26 Reference Frames .................................................................................................................................................................................................... 26
Eulerian .............................................................................................................................................................................................................................. 26 Lagrangian ....................................................................................................................................................................................................................... 26
Flow Acceleration .................................................................................................................................................................................................... 26 Equation of Motion .................................................................................................................................................................................................. 27
Acceleration ..................................................................................................................................................................................................................... 27 Differential form of the Continuity Equation .................................................................................................................................................... 27
Example 6 – Wave Runup ..................................................................................................................................................................................... 27 Summary ..................................................................................................................................................................................................................... 29
Dynamics – Energy Equation ........................................................................................................................... 30 Potential Energy (PE) .................................................................................................................................................................................................. 30 Kinetic Energy (KE) ...................................................................................................................................................................................................... 30 Pressure Energy .............................................................................................................................................................................................................. 30 General Energy Equation ........................................................................................................................................................................................... 30
Energy Equation {E} or Bernoulli Equation {B} .......................................................................................................................................... 30 Steady Flow Energy Equation .................................................................................................................................................................................. 30
Piezometric and Energy/Total Head Lines ................................................................................................................................................... 30 E-line V P-Line ................................................................................................................................................................................................................. 31 Piezometric Pressure Equation ............................................................................................................................................................................... 31
Pressure & Velocity ................................................................................................................................................................................................. 31 Flow Meters ................................................................................................................................................................................................................ 32
Pitot or Total Head Tubes .......................................................................................................................................................................................... 32 Venturi Meter ................................................................................................................................................................................................................... 32 Orifice Meter .................................................................................................................................................................................................................... 32
Discharge Equation for Venturi Meter or Orifice Meter .......................................................................................................................... 33 Example 7 – Flow Meters – Venturi & Orifice .............................................................................................................................................. 33 Kinetic Energy Correction .................................................................................................................................................................................... 34 Fluid Power & Work ............................................................................................................................................................................................... 35
Power between two Points ........................................................................................................................................................................................ 35 Radial Flow & the Energy Equation ................................................................................................................................................................. 35 Jet Discharge .............................................................................................................................................................................................................. 35 Example 8 – Energy & Piezometric Lines ...................................................................................................................................................... 35 Summary ..................................................................................................................................................................................................................... 36
Dynamics – Momentum Equation .................................................................................................................. 37 Momentum Equation {M} ..................................................................................................................................................................................... 37
Convective Acceleration .............................................................................................................................................................................................. 37 Temporal Acceleration ................................................................................................................................................................................................ 37
Applications ............................................................................................................................................................................................................... 37 Force on a Pipe Bend .................................................................................................................................................................................................... 37
Example 9 – Force on a Contraction ................................................................................................................................................................ 38 Jet Impact .................................................................................................................................................................................................................... 39 Jet Reaction or Thrust ............................................................................................................................................................................................ 39 Euler’s Equation ....................................................................................................................................................................................................... 39
Euler’s Equation for Steady Fluid Motion .......................................................................................................................................................... 40
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Euler & Bernoulli ............................................................................................................................................................................................................ 40 Euler’s Equation for Unsteady flow ....................................................................................................................................................................... 40
Example 10 – Head Loss at Sudden Expansion ........................................................................................................................................... 41 Summary ..................................................................................................................................................................................................................... 41
General ................................................................................................................................................................................................................................ 42
Real & Ideal Fluids .......................................................................................................................................... 43 Viscous Flow .............................................................................................................................................................................................................. 44 Laminar Flow ............................................................................................................................................................................................................. 44 Stress/Strain Relationship: Newton’s Law of Viscosity ........................................................................................................................... 44 Viscous & Inertial Forces ...................................................................................................................................................................................... 44
Turbulent Flow ............................................................................................................................................................................................................... 45 Reynolds Number ........................................................................................................................................................................................................... 45
Laminar & Turbulent Flow Regimes ................................................................................................................................................................ 46 Example 11 – Reynolds Number ....................................................................................................................................................................... 46 Parallel Flow .............................................................................................................................................................................................................. 47
General Governing Equation for Steady Parallel Laminar Flow ............................................................................................................. 48 Example 12 – Parallel Flow ................................................................................................................................................................................. 48 Laminar Flow in Pipes – Velocity Profile ....................................................................................................................................................... 49
Parabolic Profile............................................................................................................................................................................................................. 50 Laminar Flow in Pipes ........................................................................................................................................................................................... 50 Steady Flow in Pipes ............................................................................................................................................................................................... 50
Momentum Equation ................................................................................................................................................................................................... 50 Head Loss – Hagen Poiseuille Equation .............................................................................................................................................................. 51
Summary – Viscous fluids & Head Loss .......................................................................................................................................................... 51
Turbulent Flow ............................................................................................................................................... 53 Darcy-Weisbach Equation for Flow in Pipes ................................................................................................................................................ 53 Example 13 – Laminar & Turbulent Flow Head Loss in Pipes .............................................................................................................. 53 Velocity Profile in a Pipe ....................................................................................................................................................................................... 54
Viscous or Laminar Sub Layer ................................................................................................................................................................................. 55 Eddy Viscosity .................................................................................................................................................................................................................. 55
Development of Velocity Profiles in Pipes – Laminar Flow ................................................................................................................... 56 Development of Velocity Profiles in Pipes – Turbulent Flow ................................................................................................................ 56 Boundary Roughness ............................................................................................................................................................................................. 56 Friction Factors: Ks V L ....................................................................................................................................................................................... 56
Smooth Turbulent Flow .............................................................................................................................................................................................. 57 Transitional Flow .......................................................................................................................................................................................................... 57 Rough Turbulent Flow ................................................................................................................................................................................................. 57
Example 14 – Pipe Friction Factors .................................................................................................................................................................. 57 Summary ..................................................................................................................................................................................................................... 58
Pipeline & Pipe Network Design ..................................................................................................................... 59 Local Losses in Pipe Flow – Flow Separation Losses ................................................................................................................................ 59 Local Losses at an Expansion .............................................................................................................................................................................. 59
Head Loss at Sudden Expansion.............................................................................................................................................................................. 59 Local Losses – Empirical Relationships .......................................................................................................................................................... 59
Intakes................................................................................................................................................................................................................................. 60 Exits/Outfalls ................................................................................................................................................................................................................... 60 Expansion/Contraction ............................................................................................................................................................................................... 60 Bends, Valves .................................................................................................................................................................................................................... 60
Pipeline Design ......................................................................................................................................................................................................... 60 Two Cases .......................................................................................................................................................................................................................... 60
Example 15 – Pipelines.......................................................................................................................................................................................... 61 Example 16 – Pipelines.......................................................................................................................................................................................... 62 Pipe Networks ........................................................................................................................................................................................................... 62
Pipes in Series .................................................................................................................................................................................................................. 62 Pipes in Parallel .............................................................................................................................................................................................................. 62
Example 17 – Cofferdam ....................................................................................................................................................................................... 63 Hardy-Cross Technique ......................................................................................................................................................................................... 64
The Three Relationships to Satisfy ........................................................................................................................................................................ 64 Example 18 – Hardy-Cross Method .................................................................................................................................................................. 65
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Summary ..................................................................................................................................................................................................................... 66
Dynamic Fluid Loading .................................................................................................................................... 67 Design Requirements ............................................................................................................................................................................................. 67
Example Wave Conditions ......................................................................................................................................................................................... 67 Currents .............................................................................................................................................................................................................................. 67
Design Requirements - Loading Regimes ...................................................................................................................................................... 67 Ideal Flow Approach .................................................................................................................................................................................................... 68 Real Fluids & Viscous Fluids ...................................................................................................................................................................................... 68
Fluid Loading ............................................................................................................................................................................................................. 68 Pressure Drag .................................................................................................................................................................................................................. 68 Total Drag ......................................................................................................................................................................................................................... 69 Drag & Lift Force............................................................................................................................................................................................................ 69
Boundary Layer Separation ................................................................................................................................................................................. 70 Terminal Velocity ..................................................................................................................................................................................................... 70 Example 19 – Terminal Velocity ........................................................................................................................................................................ 70 Skin Friction ............................................................................................................................................................................................................... 71 Boundary Layer Drag – Surface Roughness .................................................................................................................................................. 71 Loading Regimes – Real & Viscous Flows ...................................................................................................................................................... 72 Spin & Lift – Magnus Effect .................................................................................................................................................................................. 72 Unsteady Fluid Loading – Total in-line force ............................................................................................................................................... 72
Morison equation ........................................................................................................................................................................................................... 72 Hydrodynamic mass (added mass) and an Accelerating Body ............................................................................................................. 72 Example 20 – Unsteady Loading ....................................................................................................................................................................... 73 Summary ..................................................................................................................................................................................................................... 74
Dimensional Analysis & Similarity .................................................................................................................. 75 Importance of Dimensions ................................................................................................................................................................................... 75
Fundamental Dimensions .......................................................................................................................................................................................... 75 Buckingham Method ........................................................................................................................................................................................... 75 Step-by-step Method .............................................................................................................................................................................................. 75 Common Groups ....................................................................................................................................................................................................... 76 Example 21 – Dimensional Analysis ................................................................................................................................................................ 77 Similarity ..................................................................................................................................................................................................................... 78
Geometric Similarity .................................................................................................................................................................................................... 78 Dynamic Similarity ....................................................................................................................................................................................................... 78 Perfect Similarity ........................................................................................................................................................................................................... 79
Froude Scaling ........................................................................................................................................................................................................... 79 Froude Similitude – Undistorted Models ....................................................................................................................................................... 79 Reynolds Scaling – Acceleration & Viscous Forces .................................................................................................................................... 80 Reynolds Similitude – Undistorted Models ................................................................................................................................................... 80 Scale Modelling ......................................................................................................................................................................................................... 81
Problems ............................................................................................................................................................................................................................ 81 Other Scale Relationships .......................................................................................................................................................................................... 81
Example 22 – Similitude ....................................................................................................................................................................................... 81 Summary ..................................................................................................................................................................................................................... 82
Unsteady Flow in Pipe Systems ...................................................................................................................... 84 Unsteady Flows in Closed Conduits ................................................................................................................................................................. 84 Analysis technique .................................................................................................................................................................................................. 84 Slow Variations in Discharge .............................................................................................................................................................................. 84 Time Required for Head Change ....................................................................................................................................................................... 85 More Rapid Changes in Discharge .................................................................................................................................................................... 85 Rigid Column Theory.............................................................................................................................................................................................. 85 Example 23 – Rigid Column Theory ................................................................................................................................................................. 86 Unsteady Energy Equation .................................................................................................................................................................................. 89 Effect of Acceleration ............................................................................................................................................................................................. 89 Applications of Unsteady Energy Equation .................................................................................................................................................. 89 Example 24 – Unsteady Flow with Acceleration: Flow between two Reservoirs ......................................................................... 90 Summary ..................................................................................................................................................................................................................... 91
Surge Shafts ................................................................................................................................................... 92 Analysis ........................................................................................................................................................................................................................ 92
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Frictionless Solution ............................................................................................................................................................................................... 93 Solution Including Friction .................................................................................................................................................................................. 93 Types ............................................................................................................................................................................................................................. 94 Surge Protection in Pumped Mains .................................................................................................................................................................. 94 Example 25 – Surge Shaft ..................................................................................................................................................................................... 95 Summary ..................................................................................................................................................................................................................... 96
Waterhammer – Unsteady Compressible Flow ............................................................................................... 97 Shock Waves .............................................................................................................................................................................................................. 97 Shock Wave Propagation ...................................................................................................................................................................................... 97
Steps ..................................................................................................................................................................................................................................... 98 Shock Wave Velocity ............................................................................................................................................................................................... 99 Waterhammer Pressure ........................................................................................................................................................................................ 99 Pressure Variation with Time .......................................................................................................................................................................... 100 Friction Effects ....................................................................................................................................................................................................... 100 Rate of Valve Closure ........................................................................................................................................................................................... 100 Waterhammer Theory Summary ................................................................................................................................................................... 101 Example 26 - Waterhammer ............................................................................................................................................................................ 101 Summary .................................................................................................................................................................................................................. 102
Appendix ...................................................................................................................................................... 103 Lecture Schedule ................................................................................................................................................................................................... 103 Experiment 1 – Flow Meters ............................................................................................................................................................................ 104 Experiment 2 – Sluice Gate ............................................................................................................................................................................... 105 Experiment 3 – Pipe Flow ................................................................................................................................................................................. 106 Experiment 4 – Drag on Cylinder ................................................................................................................................................................... 107 Worked Lecture Examples ...................................................................................................................... Error! Bookmark not defined. Moody Diagram ........................................................................................................................................... Error! Bookmark not defined. Drag Coefficient Charts ............................................................................................................................. Error! Bookmark not defined. AS2200-2006: Resistance Coefficients (k) of Valves and Fittings .......................................... Error! Bookmark not defined.
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Index Page accelerate total mass, 71 Acceleration, 28 added mass coefficient, 71 Adiabatic expansion, 13 Archimedes Principle, 23 atmospheric pressure variation, 17 Bernoulli Equation, 15 Boundary Layer Drag, 70 Boundary Layer Separation, 69 Buckingham Method, 74 bulk modulus of the fluid, 12 Buoyancy, 23 CD, 69 Celerity, 98 centre of buoyancy, 23 centre of gravity, 20, 23 centre of pressure, 20 Chezy equation, 52 Compressibility, 12 Continuity, 14, 26 Continuum Concept, 12 Control Surface, 26 Control Volume, 26 Convective Acceleration, 27 Currents, 66 Darcy-Weisbach, 52 Density, 12 dimensionless groups, 75 Discharge Equation for Venturi
Meter, 33 Drag, 68 Dynamic Similarity, 77 Dynamic viscosity, 13 Eddy Viscosity, 54 E-line, 32 Energy Equation, 15, 31 energy loss, 50 Euler & Bernoulli, 40 Euler’s Equation for Steady Fluid
Motion, 40 Euler’s Equation for Unsteady flow,
40 Euler’s formula, 18 Eulerian, 27 Flow Regimes, 46 Flow separation, 58 force on a vertical surface, 20 friction drag, 70 Friction Factors, 55 Froude Scaling, 78 Froude Similitude, 78 Gas Law, 17 Geometric Similarity, 77 Hagen Poiseuille Equation, 50 Hardy-Cross Technique, 63
Head Loss, 50 Head Loss at Sudden Expansion, 58 hydraulic diameter, 52 Hydraulically rough, 55 Hydraulically smooth, 55 Hydrodynamic mass, 71 I00, 24 inertia coefficient, 71 instantaneous complete closure, 99 Isothermal expansion, 13 jet discharge, 35 Jet Reaction, 39 Kinematic viscosity, 13, 44 Kinetic Energy, 14 Kinetic Energy Correction, 35 Ks V L, 55 ks values, 55 Lagrangian, 27 Laminar Flow in Pipes, 49 Laminar Sub Layer, 54 Lapse rate, 17 Lift, 68 Lift force, 68 Loading Regimes, 66 Local Losses, 58 Logarithmic Velocity profile, 54 Mach number, 75 Magnus Effect, 71 Manometers, 17 Mass flow rate, 14 metacentric height, 23 Momentum, 14 Momentum Equation, 37 Momentum flux, 14 More Rapid Changes in Discharge,
84 Morison equation, 71 Moving control volumes, 42 Orifice Meter, 33 Parabolic Profile, 49 Parallel Axis Theorem, 21 Parallel Flow, 47 Pascal’s Law, 16 penstock, 91 Perfect Similarity, 78 physical parameters, 74 Piezometric Pressure Equation, 32 Pipes in Parallel, 61 Pipes in Series, 61 Pitot or Total Head Tubes, 32 P-line, 32 Potential Energy, 14 Power, 35 Pressure, 16 Pressure Drag, 67 Pressure Energy, 14
Pressure Transmission, 16 Rate of change of momentum, 14 rate of strain of fluid, 48 Rate of Valve Closure, 99 Reference Frames, 27 Reynolds Number, 45 Reynolds Scaling, 79 Reynolds Similitude, 79 Rigid Column Theory, 84 Scale Modelling, 80 shear stress velocity, 54 shock wave, 96 Shock Wave Velocity, 98 skin coefficient, 70 Skin Friction, 70 Spatial Variation, 14 Specific Gravity, 12 Stability of submerged bodies, 23 Stagnation pressure, 68 Steady Flow Energy Equation, 31 Steady Flow in Pipes, 50 Steady Parallel Laminar Flow, 48 Step-by-step Method, 74 streakline, 26 Streamlines, 26 Streamtubes, 26 Stress/Strain Relationship, 44 Strouhal number, 71 sudden expansion, 58 Surface Tension, 13 Surge Pressure, 85 Surge shafts, 91 Temperature in the Atmosphere, 17 Temporal (local) Acceleration, 27 Temporal Variation, 14 Terminal Velocity, 69 Time Required for Head Change, 84 Total Drag, 68 Unsteady Energy Equation, 88 Unsteady Flows in Closed Conduits,
83 Unsteady Fluid Loading, 71 valve closure, 99 Variations in Discharge, 83 velocity head, 32 Velocity Profile in a Pipe, 53 Venturi Meter, 33 Viscosity, 13, 44 Viscous force, 79 Von Karman constant, 53 vortex shedding frequency, 71 Wake, 68 waterhammer theory, 96 Wave Conditions, 66 Weber number, 75 wetted area, 70 Work, 35
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Symbols
~ centre of pressure
a ~ acceleration
A ~ cross sectional area (m2)
B ~ centre of buoyancy
BG ~ height of G (centre of gravity) above B (centre
of buoyancy)
BM ~ height of M (metacentre) above B (centre of
buoyancy)
c ~ celerity
Cc ~ coefficient of contraction
Cd ~ actual discharge correction factor
CD ~ drag coefficient
Cf ~ average skin coefficient
cf ~ skin coefficient
Cm ~ inertia coefficient
CP ~ centre of pressure
Cp ~ elastic celerity
CV ~ control volume
D ~ diameter
DH ~ hydraulic diameter (used when non circular
pipes)
Ev ~ bulk (or volume) modulus of the fluid
FD ~ drag force
FL ~ lift force
G ~ centre of gravity
GM ~ metacentric height
H ~ height
hf ~ head loss due to friction
hL ~ head loss (for turbulent flow, includes friction
factor from hf)
K ~ bulk modulus of the fluid (N/m2, Pa) –
measures substance resistance to uniform
compression
k ~ Von Karman constant 0.4
kM ~ added mass coefficient
ks ~ roughness of the boundary
M ~ metacentre
p ~ pressure (N/m2, Pa)
Pw ~ wetted perimeter
Q ~ flow rate
Q ~ volumetric flow rate or discharge
R ~ gas constant for a particular gas (J Kg-1 K-1)
R ~ upthrust
S ~ specific gravity
T ~ temperature (C)
U ~ local velocity, mean flow velocity
u* ~ shear stress velocity
V ~ velocity
z ~ height
~ lapse rate
~ unit weight of fluid
H ~ surge pressure
~ turbulent eddy viscosity
~ dynamic viscosity, absolute coefficient of
viscosity (Ns/m2) or (kg/ms)
~ kinematic viscosity (nu) (m2/s)
~ density (kg/m3)
Q ~ mass flow rate (kg/s)
~ surface tension (N/m)
~ shear stress (N/m2)
~ angular velocity
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Approximations
Velocity in a river
o Non-uniform flow cross section
Width constantly changing flow velocity constantly changing
o Boundary layer
Indicated by velocity profile close to bed
Not constant; mud, sand, vegetation etc
Difficult to calculate
o Turbulent flow
o Varying bed roughness
Turbulent Pipe flow
o Steady flow to start
o Turbulent eddies created at bends
Solution
o Ignore details by time averaging
o Ignore details inside Control Volume (CV)
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Fundamentals
Main principles
o Continuity, {C}
Q UA
discharge or volume flow rate, (m3/s)
o Momentum, {M}
F ma QU = gV
Force = mass x acceleration (N)
o Energy, {E}
H p1
g
U1
2
2g z1
Sum of potential, kinetic and pressure energy = total head (m)
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Common Equations & Values
dh
dtU
1 ton = 8896N
10hPa = 1kPa
1g/cm3 = 1000kg/ m3
1L = 0.001m3 = 10-6cm3
Acceleration
a v 2
r w 2r
Atmospheric pressure = bar, atm
Centroid of a Parabola
3
8X
cos2 = 1 – sin2
dA = 2rdr
Density
o water = 1000kg/m3
o seawater = 1025kg/m3
o air = 1.2kg/m3
Differentiation of cos -sin
Differentiation of sin cos
Efficiency: Power required = power/%
Extrapolating
Y YA X XA
XB XA
YB YA
Gas Law PV = mRT
Head =
h p1 p2
g
Power = gQh = force x speed = work/unit time
Pressure
o Patmos = 101.3kPa
o Pwater = 50kPa
o Pwater vapour = 2.3kPa
o Pair = 60Pa
Rotation rate = rpm – revolutions per minute
Steady flow = no acceleration
dU
dt 0
Temperature Absolute = T+ 273
Velocity
v wr
Viscosity
o water=1x10-3kg/ms
o seawater=1.13x10-3kg/ms
o vwater=1x10-6 m2/s
o vair = 1.5x10-5 m2/s
Volume sphere =
D3
6
4
3r3
Wave Period =
2
t
Wave velocity = U cos (wt)
Weight = F = gV
Youngs Modulus
o Ewater = 2x109
o Esteel = 2.1x1011
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Nature of Fluids
Fluids = liquids & gases
Fluids can
o Flow
o Change shape
o Take up the shape of the boundaries Strong intermolecular forces at boundary
o Deform continuously and permanently under application of a shearing stress,
Fluid at rest
o no shearing forces acting
o Therefore, all forces in the fluid are perpendicular to the surfaces on which they act
Fluids in motion
o Molecules adhere to the boundaries – no slip condition
o Velocity varies away from the boundary
o Fluid element deforms
o
Continuum Concept
If there are enough molecules, the average conditions (pressure, density etc) are considered constant or change
smoothly
o Valid – liquids and most gases
Tightly packed molecules
o Not valid – rarefied gases, small number of molecules
Large spaces between molecules
Fluid Properties
Density
- Mass per unit volume, kg/m3
Typical values
o air = 1.2 kg/m3
o water = 1000 kg/m3 (15C)
Specific Gravity
o S =
fluid
water
Compressibility
o Density varies with pressure and temperature
Change in volume with a change in pressure depends on the bulk modulus of the fluid, K (N/m2)
p
K where Kwater = 2.05x109 N/m2
Force per unit area exerted on the fluid by the boundary and vice-versa is the shear stress,
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o Generally, water can be treated as incompressible
Except for very large changes in pressure (i.e. a waterhammer)
Gas Law
Air is more compressible but can be treated as incompressible at velocities much lower than the speed of sound
Ideal Gas Law
o
p RT
R = gas constant for a particular gas
Rair = 287 J Kg-1 K-1
T = temperature
Isothermal expansion – no change in temperature
Adiabatic expansion – no heat exchange out of the system
Viscosity
Measure of how easily a fluid flows
o Dynamic viscosity, , Ns/m2
Also called absolute coefficient of viscosity (kg/ms)
water = 1x10-3 kg/ms (20C)
o Kinematic viscosity, (nu) alternative, frequently used
(m2/s)
water = 1x10-6 m2/s (20C)
Newtons Law of Viscosity
o
F
A
du
dy
Newtonian fluids obey this law – most common fluids
Surface Tension
, N/m
o arises from elasticity of the surface
o reduces surface area to a minimum
o Causes capillary rise between surfaces
Manometer tubes
Errors in readings
o
Types of Flow
Laminar flow
Smooth, uniform, regular
Weight of column of fluid = surface tension force acting on wetted length
gD2
4H D cos
H 4 cos
gD so small, so cos = 1
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Turbulent Flow
Chaotic, random, dispersive
Temporal Variation
Flow variation with time
o Steady flows
Velocity and depth constant with time
o Unsteady flows
Velocity and depth vary with time
Spatial Variation
Flow variation with space
o Uniform
Flow properties constant in direction
o Non-uniform
Flow properties vary in flow direction
Governing Principles
Continuity
Continuity, {C}
Conservation of mass
o Mass flow rate = Q (kg/s)
No storage
Q1 = Q2
Incompressible fluid, or no change in density
Q1 = U1A1 = U2A2 = Q2 Continuity equation {C}
o A = cross sectional area
o U = mean flow velocity
Momentum
Momentum, {M}
o Conservation of momentum Newton’s Second Law
F = ma
o Rate of change of momentum = sum of forces
o Momentum flux = mass flow rate x velocity
Or rate at which momentum passes through a cross section
QU UA U
o Rate of change between cross sections = QU
Requires a resultant force F in the direction of motion
F QU Q U2 U1 Momentum equation {M}
Energy
Energy, total head, head, H
14
o The sum of three forms of energy
Kinetic Energy, ½U2
Potential Energy, gz
Pressure Energy, p
o Conserved when no energy lost
p1 1
2U1
2 gz1 const
when divided through by g gives H in dimensions of length (m)
H p1
g
U1
2
2g z1 const
Energy Equation {E}
o also known as Bernoulli Equation {B}
Applications of Governing Principles
{C}
o eliminate unknowns
o write velocity in terms of area
{E}
o determine how velocity and pressure vary in the flow
o to find energy losses
{M}
o find forces
o determine how pressure and velocity vary if there are energy losses
Example 2 – Types of Flow
Steady flow
o Flow from a reservoir with constant head and boundary conditions (does not vary with time)
Steady & uniform flow
o Flow from a reservoir with constant head and boundary conditions into a long straight pipe (cross section not
changing, no change in flow)
Steady, non-uniform flow
o Flow from a reservoir with constant head and boundary conditions into a converging pipe or through an orifice
(conditions change along the pipe)
Unsteady, uniform flow
o Flow in an oil pipeline controlled by a variable speed pump (time varying)
Unsteady, non-uniform flow
o Wave motion
o Arterial flow
o Surge towers
o Domestic plumbing
Compressible unsteady flow or shockwave
o Power trip in hydro-electric plant (worst possible problem)
Rotational or vortex flow
o Flow into a vertical orifice
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Hydrostatic Pressure
Pressure
Fluid at rest
o Static equilibrium
o No shearing forces perpendicular or normal forces only
Shear only occurs when moving
Scalar quantity
Intensity & magnitude equal in all directions
p F
A (N/m2, Pa)
o Small forces over small areas give large pressures
Hydraulic presses, stiletto heels
o Small forces over large areas give large forces
Wind flow over roof
Absolute Pressure
Pressure in a vacuum = absolute zero
Gauge Pressure
Measured relative to local atmospheric pressure and can be positive or negative
Most common
Example
o Pressure reading is 50kPa where atmospheric pressure is 100kPa
Gauge pressure = 50kPa
Absolute pressure = 150kPa
Atmospheric Pressure
105N/m2 1 atm or 1 bar or 10m H2O
Pressure Transmission
Occurs in closed systems and can be used to amplify forces – hydraulic
controls
p1 = p2
F2 A2
A1
F1 ; where A2/A1 is the amplifier
Varying pressures (dynamic pressures) may be damped by viscous effects, but mean or static pressures are not
Pascal’s Law
o Pressure change at one point in a system is transmitted through the entire system
Equation of Fluid Statics
A fluid at rest is in equilibrium
Hydrostatic relationship for an incompressible fluid
o Vertical equilibrium
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dp
dz g
p gz const
o Horizontal equilibrium
dp
dx 0,
dp
dy 0
Implies equality of pressure at the same level in the same static fluid or in combinations of
static fluids
o PRESSURE IS ADDITIVE in the vertical for immiscible fluids (fluids that don’t mix)
Pressure and Temperature in the Atmosphere
Pressure, density & temperature vary with elevation in the atmosphere
Troposphere
o Sea level 13000m
o Temperature drops linearly with increasing elevation
Lapse rate, = drop in T with z
avg conditions = 5.87K/km
With the Gas Law, pressure and density can be calculated at any elevation
o
p RT
T T0 (z zo)
Using the hydrostatic equation gives
o
p p0
T0 z z0 T0
g
R
atmospheric pressure variation in the atmosphere
Manometers
Head & pressure measurement
o Fluid most often water, but can be any fluid
water = 1000kg/m3
= g
= unit weight of fluid
o Manometer Equation h = (S-1)h or h = (1-S)h
U-tube Manometers
A type of manometer with immiscible fluids
(will not mix), used for measuring pressures in
gases and higher pressures in liquids
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Differential Manometer
Created by connecting both ends of the manometer to the pipe
Similar in principle to the U-tube manometer
p = (2 - 1)gh = (2 - 1)h
Horizontal Acceleration Effects
A particle on the surface of a fluid under constant horizontal acceleration, a
Surface is a plane at angle to the horizontal
Vertical acceleration = 0
o No additional vertical forces, hence hydrostatic
pressure
Planes of equal pressure lie parallel to the free surface
Vertical Acceleration Effects
Pressure increases with upward acceleration, similar to going up in an elevator
Oscillatory flows (waves, surge towers) have non-hydrostatic pressure below the accelerating free surface not pure
hydrostatic
Combined Acceleration
Total acceleration = pressure gradient
o Euler’s formula
p
x ax
p
z g az
o if az is small, pressure is hydrostatic - the free surface slope gives the total horizontal acceleration
For rotating flows, the horizontal acceleration varies with radial distance from the centre of rotation – free surface
becomes parabolic – a forced vortex
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Example 3 - Manometer
In general
o
p1 p2
g z1 z2
1 2
h
S1 S2 h
Summary
Fluid at rest = no shearing forces
Hydrostatic pressure increases linearly with depth
Pressure is equal along lines of constant elevation
Pressures additive in the vertical
Manometers measure gage or relative pressures
Acceleration gives rise to additional pressure gradients
Surface slopes indicate pressure gradients