PROPERTIES
1hp = 746 watts
For gas → ρ ∝ P ∝ 1/T ∝ 1/vol
Fathometer → Ocean depth
Hygrometer → Water vapour
isentropic process → Always frictionless and adiabatic
Specific heat ratio at constant pressure and constant volume = Always constant for a gas
Wt/mass/density/vol
γ = ρ g = mg/V
Wt. = mg → 1 kg wt = 9.81 N = 1 kgf → 1 gm = 981dynes → 1N = 10⁵ Dyne
γw = 1g/cc = 1000 kg/m³ = 9.81 KN/m³ ≈ 10 KN/m³
G → Mercury = 13.6, Glycerine = 1.26
Specific vol. = 1/ρ or 1/
Specific volume of water when heated from 0°C → first decreases and then increases
Water has max density at 4°C = 9810N/m³ = 9.81kN/m³
Compressibility → =1/K=(-dV/V)/dp → Variation in volume of liquid with variation in pressure
Incompressibility → Water > kerosene oil > gasoline > helium
Note
Fluid statistics → Study of fluid at rest
Dynamics → Study of fluid in motion Considering force
Kinematics → Study of fluid in motion Without considering force
Flow of fluid →Due to its deformation under shear force → Fluid in equilibrium can't sustain shear stress
A fluid is a substance that cannot remain at rest when subjected to a sharing stress
Solid → Stress ∝ strain → Resists shear stress by elastic deformation
Fluid → Stress ∝ strain rate
Continuity eqn → Based on conservation of mass
Free vortex eqⁿ → Conservation of momentum
Continuity eqⁿ → Relates mass rate of flow along streamline
Rayleigh lines → Use of Momentum and continuity eqn
Free surface → No shear stress
For liquid → ρ = constant
Types of fluid
Real fluid
Practically all fluid are real
has μ, σ, real fluid are compressible
ideal fluid/Perfect gas
μ = ST(σ) = τ = 0
Friction less, Non - viscous, incompressible(k = ∞), Viscous force is negligible
Always satisfy continuity equation
Ideal Gas → Pv = nRT
Newtonian fluid
Obeys Newton's law of viscosity → τ ∝ Rate of change of shear strain (dθ/dt) and τ ∝ velocity gradient
Ex. Water, air, gas, diesel, kerosine
τ = μ du/dy = μ dθ/dt → Newtons law of viscosity
μ = Constant → Viscosity doesn't change with the rate of deformation for Newtonian fluid and shear stress
Rate of angular deformation (dθ/dt) = rate of change of gradient(du/dy)
Non Newtonian Fluids
Doesn't follow Newton's law of viscosity → Shear stress is not proportional to the rate of shear strain
Rheology → Study of non-Newtonian fluid
Eg. Thixotropic, Bingham plastic, Pseudo, Dilatant
Thixotropic fluid
Printers ink, ketchup
Bingham plastic or ideal plastic fluid
Shear stress(τ) ∝ Rate of shear strain and τ ∝ velocity gradient
Shear stress > Yield value
Ex. Toothpaste, Cream
Pseudo plastic
Apparent viscosity decreases with increasing deformation
Ex. Paint, Blood, Milk
Dilatant
Ex. Sugar in water, Butter, Starch solⁿ
Viscosity
Property which characterises the resistance which fluid offers to applied shear force
Viscosity of liquid doesn't affected by Pressure
μwater = 55 x μair , v air = 15.2 v water→ At 20°C
μ of water is due to cohesion
μ of air is due to molecular momentum transfer
μ Hg> μwater> alcohol>air
vhg < vwater< vair
μ water ∝ 1/T → Higher temp → Lesser cohesion
μ air ∝ T → Higher temp → More energy → greater momentum of colliding gases
Dynamic/Absolute/coeff of viscosity (μ)
τ = μ du/dy = μ dθ/dt → [ML⁻¹ T⁻¹]
SI → N-s/m² = Pa-sec = kg/m.s, CGS → Poise
1Pa-sec = 10 Poise = 1N-s/m² = 10 Dyne-sec/cm²
μ = 0 → Perfect gas
Kinematic viscosity or Momentum diffusivity
v = μ/ρ → v = m²/sec(si), μ = NS/m², ρ = kg/m³
SI → m²/sec, CGS → Stoke
1 Stoke = 1 cm²/sec → 1 m²/sec = 10⁴ stoke
Red wood →To determine Kinematic Viscosity
Engler viscometer → Viscosity of lubricant oils
Say bolt → Viscosity of petroleum products
Surface Tension(Cohesion)
ST is caused by the force of Cohesion → Resist tensile stress
Due to ST → Wetting property, Spherical shape of drop, Liquid surface causes contraction
ST = Surface Energy/Work done/area = Force/Length (N/m) → [MT-2]
ST water/air = 0.073N/m (At critical point it becomes zero)
Temp ↑es → σ ↓e liquid
Salt or Soluble matter added → σ ↑es
insoluble or impurities added → σ ↓es
At 20°C for water → μ = 0.01 poise, σ = 0.075N/m
Cohesion → Molecule attract by their own (Hg)
Adhesion → Molecule of diff type (H2O)
Excessive Pressure(ΔP) → Bubble = 8σ/D, Drop = 4σ/D, Jet = 2σ/D
Capillarity
Due to adhesion & ST(Cohesion) both
Capillary tension → Stress which is responsible for retaining water in capillary tube above the free water surface of the water body
Tube → hc = 4σcosθ/ρliquidgd = 4σcosθ/γliquidd = 4σcosθ/Gγwd = 2σ/rγ
Soil → hc = C/eD10 = 0.3/dcm→ C = 0.1 - 0.5 cm²
Angle of contact with glass → Water = 0°, Kerosene = 26°, Hg = 130°
| ϕ < 90° |C < A | Wetting of surface | Concave top | rise in capillary tube | Water |
| ϕ > 90° |C > A | No wetting of surface | Convex top | fall in capillary tube | Hg |
Hg → G = 13.6, Doesn't stick to glass due to Cohesion
The flow in a capillary tube is laminar because the diameter of capillary tube is very small
Critical velocity(Vc)
Max Velocity up to which fluid motion is streamlined
Vc=Re/r
Re = Reynolds no, μ = coeff of viscosity, ρ = density, r = radii of capillary tube
Thermodynamics
Isentropic process → Frictionless & adiabatic
Control volume → A fixed region in space
Biot no → related to Heat Conduction
Isentropic flow of air
Critical pressure ratio = 0.528
Critical Temperature ratio = 0.833
Critical Density ratio = 0.634
PRESSURE
Stagnation pressure = Static + Dynamic pressure = P + ρV²/2
Altitude increases → Patm decreases slowly then Steeply
Pressure variation of air above sea level → Exponentially decreasing with height
Pressure on liquid > Vapour pressure → Liquid is said to be not in a boiling or vaporised state
Units of pressure
Si unit = N/m² or Pascal → 1 Pascal = 1 N/m²
1 kgf/m² = 9.81N/m²
1 Torr = 1 mm of Hg = 133 N/m² = 133 Pa
At MSL → 1 Patm = = 76 cm Hg = 10.33 m water = 1.0132 bar = 101.32kPa = 1.01 x 10⁵ Pascal → 1 bar = 0.9869 Patm
Pascal Law
Px = Py = Pz → Pressure intensity is same in all direction at a point when fluid is at rest
Fluid at Rest → No relative motion exists between different fluid layer, Frictionless, Shear stress = 0
Application → Hydraulic pressure, hydraulic jack, hydraulic lift, hydraulic brakes
Note → If there are shear stresses the pressure at a point in a fluid is not equal in all direction
Note → For ideal flow Pressure at a point is same in all direction when fluid is in motion
WtRam/ARam=Fat plunger/Aplunger
Atmospheric pressure
P exerted by Weight of Atmosphere on a given point
Measured by Barometer
Barometer → High density fluid & very low vapour pressure ex. Hg.
Gauge pressure
Measured → w.r.t. Atmospheric pressure as datum
-ve, 0, +ve
Measured by → Manometer or Bourdon Gauge
if suffix or prefix given treat it as gauge pressure for numerical problems
Bourdon tube pressure Gauge → Mechanical gauge used for measuring the pressure of fluids in pipe flow
Absolute pressure
it is actual Pressure
Pabs = Patm + Pgauge = PAtm- Psuction/vacuum
Measured → w.r.t. Absolute zero or Complete Vacuum
measured by Aneroid Barometer
Aneroid Barometer also used to measure. local Patm
Vacuum/Suction/Negative pressure
Pressure below atmospheric pressure
Measurement of Pressure
Piezometer
Gauge pressure, Pressure head measurement, Static pressure in a pipe
Piezometric head = Gauge Pressure head + datum head
The line of constant piezometric head passes through 2 point which have the same velocity
Piezometer tapping → Static pressure in a pipe
Piezometer is not used for pressure measurement in pipe → when fluid in pipe is gas
Manometr
Measure pressure in water channels, pipes .etc
U-tube Manometer
Micro manometer is used to determine Low pressure difference
Property of manometric liquid
High → Chemical stability, Density
Low → Viscosity, capillary constant, volatility, vapour pressure, Coeff of thermal expansion
Provide suitable meniscus for the inclined tube
Should completely immiscible with liquid
ex. Mercury (high pressure), Alcohol & water ( Low Pressure)
Mercury is generally used in manometers for measuring all pressure accept the smallest ones
Differential manometer
Difference of pressure between two points in a pipe
H= (Pa - Pb)/G1 γw
Difference in P Head wrt liquid 1 → Hwrt to 1 = (G2/G1 -1)h
h = diff in fluid height, G2 = Specific gravity of manometric fluid
Multi U-tube Manometer
For High pressure → Different Fluids are used
HYDROSTATIC FORCES
Hydrostatic paradox → The Apparent contradiction in hydrostatic force on the base of liquid container and the weight of liquid in the container, with reference to containers of different shapes having the same base area and filled with the same liquid for equal depth
Hydrostatic law
Rate of increase of pressure in the vertical direction is equal to weight density of fluid at that point
dp/dh = ± ρg=h → at any point
Downward = +ve
Upward = -ve
(ρgh)liquid1=(ρgh)liquid2
Total Hydrostatic force(F)
F = pA = ρgh̅A = γh̅A → Horizontal or Vertical or inclined surface
Total water force at bottom of tank = Water pressure at bottom x Area of tank bottom = ρghA
The vertical component of the hydrostatic force on a submerged curved surface = the weight of liquid vertically above it
Total pressure (p)
p = F/A = γ h̅
Centre of pressure
The resultant hydrostatic force or pressure on a submerged body act through
hp = h̅ + Isin²θ / Ah̅ → From free surface
Vertical plane surface(θ=90) → hp = h̅ + I/Ah̅
Horizontal plane surface(θ=0) → hp = h̅ → Centre of pressure will coincide with the centre of gravity
Centre of pressure → Always lies below the centroid of area & moves towards the centroid as depth increases
LIQUID IN RELATIVE EQUILIBRIUM
i. At Rest
P = ρgh
ii. Fluid moves with constant acceleration
P = ρ(g ± a)h=ρgh(1 a/g) → (+) = Upward acceleration, (-) = Downward acceleration
P = 0 → P = Atmospheric pressure
iii. Constant horizontal a in x-directⁿ (Tank in truck)
The level of water rises on the back/rear side and fall on the front side
Tanθ = a/g = h / ½D = 2h/D
When half the water spills out → θ = 45, Pressure at centre of bottom = 0
P ∝ r²
Central Depth = H + h = r²ω²/2g + h
H = r²ω²/2g = rise + fall = Height of Paraboloid of revolution
h = Rise above original water level for no Spilling = Height of tank - Water depth = Fall below water level(H)
D = Tank dia, r = Tank radius, g = 9.81m/s²
ω is angular velocity in radian/sec
BOUNCY & FLOTATION
Wtair= Wtwater+wVbody → body=Wtair/Vbody
Buoyant/submerged unit weight = Saturated unit weight - Weight of water = γ sat - γw = ½ γ sat
Density of Sea > River → Ship enters Sea from river, will rise a little
Floating logs of wood tend to move to mid-river reach on the water surface → Due to Near-symmetry of the isovels across the section is conductive to principle of least work
When a block of ice floating on water in the container melts the level of water in the container remains the same
Archimedes Principle
For wholly or partially submerged Body in Liquid
Buoyant force = Net upward force = wt. of liquid displaced
Fb = γ body Vbody= γw Vol water displaced
Density of body > Density of fluid → Sinks → Wt of body > Wt of fluid displaced
Centre of Buoyancy
Through which force of buoyancy is supposed to act → Buoyant force always acts upward
COB of a submerged body → Coincides with the centroid of the volume of fluid displaced
Resultant pressure of the liquid in case of an immersed body acts through the centre of buoyancy
Stability of a completely submerged body → Determined by considering centre of buoyancy and centre of gravity
Direction of total liquid pressure on submerged body → Normal to the surface
Metacentre
Point about which a floating body starts oscillating when the body is → tilted by a small angle or Give a nice small angular displacement
Metacentric height (GM)
Distance b/w meta-centre of floating body & the centre of gravity of the body
GM increases → increases Stability, Decreases comfort
GM = BM - BG
Neutral equilibrium → BM = BG and GM = 0
BM = I/V
I = MOI of top view
V = Vol liquid displaced
T=2K2/g.GM
DIMENSIONAL-ANALYSIS
Strouhal number = fd/V → Dimensionless value
Dimensions
Angular velocity ()→ [T-1]
Methods
Rayleigh method
Use → Max 3-4 variables
Buckingham π-theorem
π-terms/Dimensionless group [M0L0T0] = n - m
n → No of variables
m → 3(M,L,T) → Fundamental dimensions/Primary variable/Repeating variables
Repeating variables → Should not contain dependent variables, No two RV have the same dimensions, Have basic dimensions
Dimensionless group [M0L0T0] → D2/, (dP/dx)D4/Q,
Force acting on Fluid mass
Inertia force = ρV²
2. Viscous
3. Pressure
4. Gravity
5. Surface tension
6. Elasticity =
Pressure coefficient → Cp = Pressure force/inertia force
inertia forces unimportant → Flow through a long capillary tube
★ Rose For Every Worry Man
I V
G P
S E
a) Reynolds no
Re = ρ V D / μ = inertia force/Viscous Force
Characteristic length(D) → Circular pipe = Dia
Use → Submerged Body, Aeroplane, Submarine, Pipe, incompressible flow, Valve
Ship → Re & Fr both used
Supersonic missile → Both Re & M is used
Stanton diagram → log-log plot of airflow Friction Coefficient vs The Reynold’s number → log friction factor vs log Re
Re < 1 → Viscous force are very strong as compared to inertia force
b) Frauds no
Fr = V/√(gD) = √(inertia force/Gravity Force)
D = Area/top width
Use → OCF, spillway, weir, jet, hydraulic jump
c) Euler's no
Pressure is predominant
E = √(inertia force/pressure Force)
Use → Cavitation, Water hammer, High pressure flow in pipe
d) Weber no
Surface tension is predominant
W = √(inertia force/surface tension force)
Use → Capillary tube flow, Formation of Liquid droplet
Velocity of sound/Pressure wave
C = K/ρ =KRT
ρ=1000 kg./m3, K → N/m2, C → m/sec
In Fluid medium → Velocity of Elastic pressure wave = Sonic velocity
e) Mach no
Compressibility forces are predominant (M ≥ 0.3)
M = inertia force/Elastic Force =ρV²/K
M=Vbody/C=V/K/ρ =Stream speed/Acoustic or sound speed
Use → Compressibility, Aerodynamic testing, Rocket, missile, Aircraft
M < 0.2 → incompressible
M < 0.8 → Subsonic
0.8 < M < 1.3 → Trans-sonic
1< M < 5 → Supersonic
M >> 5 → Hypersonic
Normal shock wave
Approaching flow is supersonic → Supersonic flow to subsonic flow
Sudden change in pressure, temperature and density → The pressure and temperature rises
Irreversible process
Shock waves will not occur in the divergent section of a nozzle
Supersonic flow
The Velocity increases as the area increases → Conservation of masses Momentum and energy require DA/A should increase with dM/M
Diverging passes → Decrease in pressure and density in Supersonic flow
Mach cone
Possible in Supersonic flow
Zone of silence → The region outside the mach cone
Diffuser conduit → Gradually decreasing area in a Supersonic flow
Similitude
Similarity b/w model & prototype in every aspect
To design & testing of prototype based on results of model
Geometric → Similarity of linear dimensions, Similar streamlines
Kinematics → Similarity of motⁿ → V, a
Dynamic → Similarity of Force → ST, wt., μ
Model Can be Smaller or Larger than prototype
Prototype → Actual structure
Two geometrically similar units are homologous → if they have similar streamlines
Reynolds law
(ρ V D / μ)prototype = (ρ V D / μ)model
Frauds law
(V/√gy)p = (V/√gy)m
Acceleration → Remains Same
Scale ratio
Scale = Model/Prototype → m/p
Scale effects in models can be → Both positive and negative
Horizontal scale ration → Lrh=Lp/Lm=Bp/Bm
Vertical scale ratio → Lrv=Hp/Hm=Dp/Dm
FLUID DYNAMICS
Study of motion of fluid with force causing the motion
Analysed by Newton's 2nd law
? subjected to uniform acceleration are analysed by Newton's second law of motion
Schlieren flow visualisation technique → Operates by employing density gradients(d/dy) in flow
Fx=Q(Vx) → The flow is steady and velocity of flow is constant over the end cross sections
Control volume → Volume fixed in space → while applying the momentum and energy equation in FM
Momentum equation
Completely independent of →
Von-Karman → o/=Uo2(d/dx) → Momentum integral equation for dp/dx = 0, flow is steady
Moment of momentum equation is based on Momentum conservation
The change in moment of momentum of fluid due to flow along a curved path results in torque
Impulse momentum eqn → F = d(mv)/dt = dM/dt → F = ma
Momentum flux → Mx=QV → Steady incompressible with uniform velocity distribution
Navier-Stoke equation
Fg + Fp + Fv = ma → Derived from the Newton's second law of motion
Laminar flow of real fluid Given by NS equation
Governor the motion of incompressible viscous fluid in laminar motion
Euler's eqn
Fg + Fp = ma → Based on momentum conservation → 2nd law of motⁿ
Force of compressibility and force of turbulence are neglected
Zero viscous force, ideal flow, incompressible, homogeneous
Differential form → dp/+g.dz+V.dv=0
Cannot be applied to a fluid at rest
3D equation of motion based on Momentum conservation
Note
Velocity/Dynamic/Kinetic head → h=V2/2g → as we know V = √(2gh) → Kinetic energy per unit Weight
Pressure/Static head → h = (P1 - P2)/ρg = H1 - H2 → as P = ρgH
Potential/Datum head → h = Z
Piezometric head = Pressure + Datum head
Stagnation head = Pressure + Velocity head → Stagnation pressure = Static pressure + Dynamic pressure
Total head = Piezometric head + Dynamic head
Hydraulic gradient(dP/dx) → Change in piezometric head per unit length of pipe
Normal acceleration = 0 (When particles move in a straight line) → Then the Piezometric head is a constant
Bernoulli Eqn
Only gravitational force is considered
Follows law of conservation of energy → The total energy of fluid at point is constant or Based on energy or head (H1 = H2)conservation.
Assumption → Along streamline, ideal flow (μ = f = 0, inviscous flow) , Steady(time variation = 0), incompressible(ρ = constant) & irrotational(ωx = ωy = ωz = 0).
P1/g+V12/2g+Z1=P2/g+V22/2g+Z2+Hf → (Direction of Flow is 1 → 2)
BE → Derived above the velocity head of the kinetic energy per unit weight of the fluid
Equation of motion for 1D steady flow
Original BE is Energy per unit mass, which is integration of Euler's eqⁿ, but it can be represented as below
Energy Per unit mass → P/+V2/2+gZ=Constant
Energy per unit weight (N-m/N) → P/g+V2/2g+Z=Constant → Conventional form of BE
Energy per unit Vol → P+V2/2+gZ=Constant
If the flow is a irrotational the Bernoulli constant for points lying on the same streamline and those which lie on other stream lines will have the same value
Steady rotational flow → BE Can be derived for the points lying on the same streamline
Coeff
Cd = Cc x Cv
Cv > Cc > Cd
Coeff of Discharge
Cd = Qa/Qt = Cc x Cv
More Cd → More Discharge
Venturi meter = 0.98
Orifice = 0.64 - 0.76
Internal Mouthpiece(Borda mouthpiece) → Running full = 0.707, running free = 0.50
external mouthpiece = 0.855, Convergent mouthpiece = 0.95
Weir & Notch ≈ 0.6 → For all type/shape
Cipolletti weir(Trapezoidal weir) = 0.61
Coeff of velocity
Cv = Va/Vt = 2ghvena contracta/2gh < 1
Avg value = 0.97
Orifice = 0.97 - 0.98
Venturi meter = 0.98
Pitot tube = 0.98
Pitot static tube = 0.99
Wier & notch ≈ 0.97
Borda's mouthpiece → Running full = 0.707, Running Free = 1 (No loss of Head)
Cv → Totally submerged Orifice < Orifice discharging Free
Coeff of Contraction
Cc = Ac/A = Area vena contracta/Area orifice
Venturi meters & External cylinder mouthpiece = 1
Sharp edge orifice = 0.611
External Cylindrical mouthpiece → Cd = Cv = 0.855
At Vena Contracta → Max Velocity, Least Dia/Area of stream, Pressure intensity = Atmospheric, contraction is maximum, Steam lines are parallel throughout the jet
Application of Energy Eqn
Airplane works on Bernoulli eqn
Flow ratio = Velocity of flow at exit /Theoretical velocity of Jet corresponding to manometric head
The shape of fire hose nozzle is generally kept convergent
Loss of energy → Orifice meter > Venturi Meter → Because of sudden obstruction
In the category of flow meters head loss is least for venturimeter
Discharge(Rate of flow) → Orifice Meter, venturi meter
Velocity → Pitot tube,
1. Orifice meter
For discharge or Rate of flow → Only for pipe
it is pipe having circular plate with a hole inside it
Qactual=CdAoA12gh / A21-A2o
Head loss → Hf=H(1-CV2 )
CV =x/2yH
Orifice Dia = Pipe dia/2
Level angle = 30 - 45
Large orifice → If Water head = 5 x (diameter of the Orifice)
2. Venturi meter
For Discharge(Rate of flow) through pipe → Large Q of large dia pipe
Floor takes place it atmospheric pressure
Can install H , V & inclined
Size is specified by both dia of pipe & Throat dia
Convergent = 22° & L = 2.5d
Divergent = 5° - 7° & L = 7.5d
Length divergent cone > Convergent → To increase the velocity, to avoid the tendency of breaking away the stream of liquid, to minimise energy losses
Pipe dia (D) = (2 - 3) throat dia (d)
Qactual=CdAoA12gh / A21-A2o
Cd = √[(h - hL)/ h]
Cc = 1 → Cd = Cv = 0.98
h = ∆Vi²/2g
Cd Increases up to certain value of Reynolds number and then become constant
VM in pipe → Pressure is Max at midpoint of convergent section and velocity is maximum at throat section
Reading of Differential manometer of a venturimeter → independent of angle of inclination of venturimeter
Venturi Flume/Throat flume
Max Flow → Depth at throat = 2/3rd
Flow takes place at Patm
Meas Q ∝ H^3/2
Venturi flumes → Q for very large flow rates
Standing wave flume
Modified version of venturi flume
it is a critical depth flume
Q ∝ H^3/2
Nozzle meter
it is a Venturi meter if pipe is not contracted (Cc = 1)
Cheaper but more losses → Energy loss in Nozzle > Venturi meter
Q is independent of orientation of venturimeter whether it is horizontal, vertical or inclined
A nozzle is generally made of convergent shape
3. Pitot tube
Velocity of fluid & Flow stagnation pressure
Va²/2g = h
V actual = Cv.√(2gh)
Cv = 0.98
Alignment → Opening faces upstream and the horizontal leg is perfectly aligned with the direction of flow
Nose Towards Flow → Liquide Rise by V²/2g
Nose Facing Downstream → Liquide Fall by V²/2g
4. Pitot-static tube (Prandtl tube)
Velocity of flow at the required point in a pipe → By measuring Stagnation pressure, Also Dynamic pressure, Static pressure
Vactual=Cv2gh
Cv = 0.99
At V = 0 → P ↑es due to conservation of KE
For Non-uniform rotational flow → Tip piezometric reading varies only across the flow, while the side opening/piezometric reading varies only along the (indirection of) flow
5. Elbow meter or Bend meter
measure Q → Through pipes
6. Rotameter
meas Q → Through pipes
7. Current meter
V of stream flow or velocity in open channel
Calibration → in Towing tank
Has rotation elements → the speed of rotation is the function of velocity of flow
8. Hot wire anemometer
instantaneous velocity & temp at a point in flow
Velocity of gases
9. Mouth piece
Meas Rate of flow → Q(discharge)
Tube fixed at Circular opening of tank
L = (2 - 3)d
Coefficient of discharge depends on length of mouthpiece
Vena contracta → 1/4 (The diameter of the Orifice) → In a short cylindrical external mouthpiece
FLUID KINEMATICS
Motⁿ of fluid without considering force causing it
Coriolis method → Flow rate of liquid
Fluid continuum → Fluid flow analysis is valid as long as the smallest length dimension of the problem is much larger than the distance between molecules
Two concepts are used
Lagrangian → Study of motion of single particle → Pathline
Eulerian → Particular section, in FM Eulerian method is generally used bcz it is difficult to keep track of a single particle
Types of Fluid
Surge wave → Unsteady and non-uniform flow, rapidly varied flow
Sink flow → A flow in which fluid moves rapidly inwards towards a point where it disappears it constant rate
Flow through a long pipe of constant dia and constant rate → Steady + Uniform flow
Flow through tapering pipe at constant rate → Steady + Non Uniform flow
Increasing/decreasing rate of fluid flow through a constant dia → Unsteady + Uniform
Increasing/decreasing rate of fluid flow through a diverging pipe → Unsteady + Non Uniform
2D laminar flow under steady and uniform → d/dy = dP/dx
Steady
At any given location fluid properties(V, P, T) doesn't change wrt time, otherwise unsteady
δv/δt=δp/δt=δT/δt=0
Eg. Flow through a tapering pipe
Flow in a river during flood → Gradually varied Unsteady flow
In unsteady flow the velocity changes in magnitude or direction or both wrt times
Uniform
At any given time fluid properties doesn't change wrt location, otherwise non Uniform
δv/δx=δp/δx=δT/δx=0
dp/dx = d1/d2
Compressible
Density changes with time otherwise incompressible
Rotational flow
When particle rotate about their mass centre during motion otherwise irrotational
Forced vortex flow → flow inside Boundary layer
Rotation of the fluid is always associated with shear stress
irrotational flow (Potential flow)
The net rotation of fluid particles about their mass centre = 0
Velocity potential (ϕ) exists → Existence of velocity potential implies that fluid is a irrotational
Free vortex → Flow outside boundary layer, wash basin
Non-viscous fluid can never be rotational
dp/dx = dz/dy
Laplace equation for irrotational flow → d²ϕ /dx² + d²ϕ /dy² = du/dy - dv/dx = 0
Vortex Flow
Flow revolves around an axis line, which may be straight or curved is known as vortex flow
Cylindrical vortex motion → Fluid mass rotate in Concentric circle
spiral vortex motion → fluid mass moves spirally outward or spirally inwards
Radial/Angular acceleration = V2/R
i. Free Vortex Motion (irrotational flow)
V ∝ 1/R , P ∝ 1/R
it is a irrotational flow, Fluid may rotate without external force
Ex. Wash basin, Whirlpool in a river, Flow of liquid in centrifugal pump casing and circular bend in a pipe,
Radial component of velocity = 0
ii. Force Vortex Motion (Rotational flow)
it is a rotational flow → Surface profile is parabolic
Ex. Rotational vortex, rotating cylinder, washing machine, Centrifugal pump
rotate by external force or power →
V = ω R → V ∝ R
Radial Direction → Horizontal plane → dp/dr=v2/r, Vertical plane → dp/dz=g
h = ω²R²/2g → P ∝ R²
Air mass motion in a tornado → Forced vortex at centre & free vortex at Edge/Outside
Rankine Vortex Motion
Combination of force & free vortex flow
No spelling case → rise above original water level = fall below original water level
Radial flow → fluid particle flow along the radius of rotation.
Flow Lines
Streamline Eqn → dx/u=dy/v=dz/w
Streamline → Direction of motion of a particle at that instant, Tangential to the velocity vector everywhere at a given instant
There can be no flow across the streamline, dr x V = 0
Streamline flow → Each liquid particle as a definite path and path of individual particle do not cross each other
Streak line → lines formed by particle rejected from nozzle, Line that is traced by a fluid particle passing through a fixed point, Curve formed by the dye in the flow field constantly injected at a single point
Path line → Trajectory of fluid Particle, Path traced by a single particle, Constitute feature of the lagrangian approach
Potential line → Equal potential on adjacent flow line
For steady flow → Streamline, Streakline & Path lines always coincide or are identical
Bluff body surface doesn't coincide with streamline
For 2D flow → Stream line is represented by a curve
Dividing streamline for a uniform flow superimposed over a 2D droplet → Circle
Flow net
Streamline & Equipotential are mutually perpendicular → Intersect Each other orthogonally curvilinear squares
Flow should be Steady, irrotational & Not be governed by the Force of Gravity
Used to design the Hydraulic structure
Observation of a flow net enable us to estimate the velocity variations, Determining energy loss in flow
Continuity Eqn
Based on conservation of mass
Continuity equation → Should follow Laplace Equation
ideal flow of fluid obeys Continuity eqn
Fow is possible → if continuity equation is satisfied
1D → 1A1V1=2A2V2Q1=Q2dV/V+dA/A+d/=0 → Steady Compressible flow
2D → du/dx+dv/dy=0
3D → dρ/dt + d(ρu)/dx + d(ρv)/dy + d(ρw)/dz = 0
Steady flow (dρ/dt = 0) → d(ρu)/dx + d(ρv)/dy + d(ρw)/dz = 0
Steady, incompressible flow → du/dx + dv/dy + dw/dz = 0
incompressible flow → ρ = Constant
Velocity
U = ui + vj + wk + t
Stagnation point → Where velocity = 0 → → U = 0 → u = v = w = 0
Shear strain rate = (1/2)(∂v/∂x + ∂u/∂y)
Acceleration
Total acceleration = Convective (wrt space) + Local or temporal(wrt time)
Steady flow of a fluid → The total acceleration of any fluid particle can be zero
Steady Flow → Temporal or Local acceleration = 0, Only convective acc exist
Uniform Flow → Convective acceleration = 0 = Spatial rate of the change of velocity
Convectional tangential acceleration = Vdv/ds
Vorticity
Curl of velocity vector is called verticity → indicates the rate of deformation
Vorticity = 2 x Angular velocity
irrotational flow → Vorticity = 0, Angular Velocity (ωx = ωy = ωz = 0)
Circulation = Vorticity x Area
Vortices → for both rotational and irrotational flow
Velocity Potential / Potential fⁿ
ϕ = f(x,y,z,t) → Velocity of flow is in direction of decreasing Potential fⁿ
Exist only for ideal, Flow must be irrotational and Should satisfies Laplace eqⁿ
ϕ satisfies Laplace eqⁿ (d²ϕ /dx² + d²ϕ /dy² = 0) → for Steady + incompressible + irrotational flow
For a source → =qloge(r)/2
Free vortex → =Гθ//2 → Function of angle
Equipotential line → Same potential fⁿ
An equipotential line has no velocity component tangent to it
ϕ → Only for irrotational flow
Stream fⁿ
Discharge per unit width Q = | ψ2 - ψ1 |
Ψ fn → Define when flow is continuous
ψ = Constant → if two points lie on the same straight line
ψ satisfy laplace eqⁿ (d²ψ/dx² + d²ψ/dy² = 0) → Then flow is irrotational + continuous and Continuity is satisfied
ϕ-line & ψ-lines are orthogonal wherever they meet
Ψ = xy → Equation to idealise in impingement of a jet on a flat plate
Ψ → for both rotational and irrotational flow
Doublet
Formed when the source and sink approach each other and approach distance become zero
ψ=sin/2r, =cos/2r
Source → R=m/2U, ψ=0
Cauchy-Riemann eqⁿ
for incompressible & irrotational flow
u = -dϕ/dx = -dψ/dy
v = -dϕ/dy = dψ/dx
dψ=vdx-udyψ=∫vdx-∫udy+c
Magnitude of V = u2 +v2
PIPE FLOW
Bourdon tube pressure gauge → Pressure of fluid in pipe
Practically all flow in pipe is turbulent
Absolute Roughness of pipe increases with time
Nominal size of the inlet pipe < Nominal size of the discharge pipe of pump
Max efficiency of power transmission through pipe = 66.67%
Pipe condition
Hydrodynamically smooth Pipe → Ref/(R/K) <17
Boundary transition Pipe→ 17<Ref/(R/K) <400
Hydrodynamical rough Pipe → Ref/(R/K) >400 → Turbulent flow
And cast iron pipes carrying fluid under pressure regarded as hydraulically smooth → when the roughness element are completely covered by the laminar sublayer
Momentum correction factor (β)
β = Momentum based on actual Velocity/ based on avg velocity
Used in account for non uniform distribution of velocity at inlet and outlet section
Kinetic energy factor (α)
α = KE based on actual Velocity/ based on avg velocity
α is included in Bernoulli equation if the flow is an unsteady
α ≥ β ≥ 1
When velocity distribution is uniform → V = Vavg
Hydraulic gradient & Total energy line
HGL = p/γ + z → Piezometric head in direction of flow
TEL = HGL + V²/2g = p/γ + V²/2g + z
TEL → Always drops in the direction of flow bcz of loss of head
TEL → Horizontal in case of idealised Bernoulli flow bcz losses are zero
TEL → Locus of elevation that water will rise
HGL → May rise or fall, HGL is velocity head below the EGL
Pressure intensity < Atmospheric → HGL is below pipeline
OCF → HGL coincides with free surface
Head loss
Turbulent > Laminar flow
U-Band causes maximum had loss
Formula used → Darcy-weisbach formula, Hazen-Williams formula, Lea formula. chezy's formula
i. Frictional or Major loss
hf = 80 - 90%
hf = H/3 → Max Transmission of Power
Darcy weisbach eqn
Only for pipe & laminar flow
hf = f'LV²/2gD = 4fLV²/2gD = f'LQ²/12.1D⁵
frictⁿ coeff → f = 2τo / ρV² = f’/4 → Dimensionless
f' ∝ f ∝ 1/Q² ∝ 1/V² ∝ 1/Re
hf ∝ 1/D⁵ → if Q = Constant
hf ∝ 1/D → if V = Constant
Friction factor (f’)
Friction factor → Depends on size of the pipe, rate of flow, age of the pipe
Moody equation → Used to find frictⁿ factor → Based on colebrook-white data on commercial pipes
f' = 4f
Laminar flow → f' = 64/Re → f = 16/Re
Turbulent flow → f' = 0.316/Re1/4 = Roughness ht(ε) / dia
Fully develop rough-turbulent resume in pipe flow → Friction factor is independent of the Reynold's number
f’ = 0.032 → Minimum value that can be occur in laminar flow through a circular pipe
Chezy's Formula
For both pipe & OCF
hf=LS S=h/L
V = C √RS
C = 8g/f' [L1/2T⁻¹]
Hazen-Williams formula
Hazens Williams : Velocity of water supply
hf=(10.67LQ1.852)/(C1.852D4.87)
V=0.85CR0.63S0.54 → Water supply mains or Pipe flow
ii. Minor losses
Caused by local disturbance due to pipe fittings
in pipe fitting = 10 - 20%
Momentum & Bernoulli eqⁿ are used in derivation of losses
Always expressed in terms of Velocity of smaller dia pipes.
In general → hL = k V² / 2g
a) Sudden Expansion
HGL ↑es & TEL ↓es
hL = (V1 - V2)²/2g = (V1²/2g)(1 - A1/A2)² = (V2²/2g)(A2/A1 - 1)²
Loss expension >> Loss contraction → When the flow contract it tends to become rotational
b) Sudden contraction
Loss is due to expansion of flow after sudden contraction
hL=(Vc - V2)²/2g=(V2²/2g)(1/Cc-1)2
hL=0.5V2²/2g= Entry loss → if Cc not given
c) Exit or impact loss
hL = V²/2g
d) Entry loss
hL = 0.5V²/2g
Obstruction
hL = V²/2g [(A / Cc(A-a)) -1]²
Pipe Bends
hL = KV²/2g
K → 90° = 1.2, 45° = 0.4
Force exerted by fluid → Rx=(P1A1)x-(P2A2)x-∫Q(V2x-V1x)
Pressure → Outer radii > inner radius
Parallel pipe connection
Q = Q1 + Q2 + Q3...
H = H1 = H2 = H3 = f’LV²/2gd
Deq=n2/5dd=D/n2/5
Series pipe connection
Q = Q1 = Q2 = Q3
H = H1 + H2 + H3 = Σ(f’LV²/2gd)
Equivalent pipe system
Same H & Q
Series → L/D⁵ = Σ Li/Di⁵
Parallel → Leq/D⁵ = L1/D1⁵ = L2/D2⁵ =....
Q ↑es by 26.53% if adding a pipe of same dia in mid way & keeping head constant
Compound pipe → If pipes of different lens and diameter are connected with one another to form a pipeline
Equivalent pipe → If compound pipeline is replaced by a single pipe of same diameter with the same rate of flow, same loss of head and the length
L equivalent = L compound
Power transmitted through pipe
P = Q γ (H - hf)
for max P → hf = H/3 ( H = total head)
Max efficiency = 66.67% → Max power lost = 33.33%
Pipe Network
Σ Piezometric head = 0 → Around each elementary circuit
inflow = outflow → At a junction or Node
Cost of pumping ∝ Hf
Hardy-cross → To analyse the flow in pipe networks
Nozzle
Convert the total energy to velocity
Placed at the end of water pipeline → Discharge water at high velocity
At critical pressure ratio → Velocity at the throat of nozzle = Sonic speed
Discharge pressure < Critical pressure → Convergent divergent nozzle is used
Angle of elevation = 45 → HGL for jet coincides with the centre line of the jet
Cavitation
Pressure of flow decreases to a value close to its vapour pressure, cavitation is caused by low pressure
P absolute < P vapour & σ = 0
Water Vapour pressure → At 100°C = P atm, 20°C = 17.54 torr = 2.34 kPa
Temp ↑es → P vapour ↑es
Collapse pressure of vapour bubbles < Vapour pressure
Cavitation and Pitting can be prevented by Reducing the velocity head
Effect → Lower efficiency, Damage to flow passages, Noise and vibrations
Cavitation parameter = 0 → Boiling of liquid start, Cavitation starts, Local pressure reduced to vapour pressure
Surface tension control the cavitation
Flow through Syphon
Use → Hill & Raised ground level
P summit < P atmospheric
Max vacuum = 7.4m of water → max ht of summit = 7.5m → To prevent Cavitation
For no vaporisation → P syphon > P vapour → Otherwise flow stops
Pipe is said to be a syphon → if it has sub atmospheric pressure in it
At the summit an air vessel is provided to avoid interruption in flow
Inverted syphon or depressed sewer
Sewer line crosses a river
An inverted syphon is designed generally for three pipes
Water hammer Pressure
Due to sudden closer of pipe
Surge tanks(Hydraulic shock) → used to minimise water hammer pressure,
Prevention → Using pipe of Greater wall thickness
Magnitude of water hammer(pressure) depends on → Velocity of flow, length of pipe, time taken to close valve, elastic properties of material of pipe
Elementary wave in still water → V =gy
Water wave Velocity → C = √(K/ρ)
Intensity of pressure wave → P =VK = ρVC
Pressure/Inertia Head at valve → h=CV/g=LV/gt= (Vwave x Vwater) / 9.81
Critical time (To)
To = 2L/C
T = 4L/C → For complete cycle of water hammer
L = 1m → if not given
T ≤ To → T < 2L/C → Sudden/Rapid closer
T > To → T > 2L/C → Gradual/Slow closure
To << T → Slow closure
To < T ≤ 1.5 To → Rapid closer
LAMINAR & TURBULENT FLOW
Boundary shear stress → =(-dp/dx)(R/2) → For Both Laminar & Turbulent flow
Value of Re for Transition flow
Re = ρ V D / μ
Pipe = 2000 - 4000 → Critical flow Re = 2000
Parallel plate = 1000 - 2000
Open Channel = 500 - 2000
Soil = 1 - 2
Critical Reynolds number → Below which flow is laminar
At lower critical velocity → Laminar flow stops
At Critical velocity → Laminar flow changes to Turbulent flow
LAMINAR/VISCOUS/STREAMLINE FLOW
Frictional resistance → Velocity and temperature of flow, it is proportional to surface area of contact for both laminar and turbulent flows, it is independent of the pressure for both laminar and turbulent flow
Fully developed laminar flow → the velocity profile does not change in the direction of flow
i. Circular pipe (Steady uniform flow)
V max = 2 V avg =(-dp/dx)(R2/4)
Hagen-Poiseuille formula → Q = (-dp/dx)(πD⁴/128μ) Q ∝ 1/
Pressure Gradient → (-dp/dx) ∝ 1/D4 → if Q is constant
hL = (P2 -P1)/γ = 32μVL / γD²
Hydraulic gradient → (i) = hL/L = 32μV / γD² → Pressure drop per unit length
hf = f'LV²/2gD = 4fLV²/2gD = fLQ²/12.1D⁵
f' = 2τo / ρV² & f = 64/Re
Velocity distribution → U=Umax(1-(r/R)2)
V distribution → Parabolic → Zero at edge & max at centre
τ & Power ditⁿ → Linear → max at Edge & Zero at Centre
f for laminar flow depends on Re & for Turbulent it depends on Roughness of pipe
at y = 0.29R → Vavg = Vlocal
ii. Two parallel fixed plate
Vmax = (3/2) x V avg
hL = 12μVL / γh²
τ variatⁿ → linear → Max at boundaries & 0 at centre
V variation → Parabolic → Max at centre & 0 at boundaries
U = 1/2μ (-dp/dx) (By-y²)
Hele shaw flow → Laminar flow between two parallel plates (both stationary)
Couette flow → Laminar flow between two parallel plates (One plate moving & other is at rest)
TURBULENT FLOW
Frictional Resistance ∝ density
Diffusion is more vigorous → Flute particles movie zigzag way
τ at boundary turbulent > laminar
Re ↑es → Velocity profile become more Flatter
Radial distance = 0.223R → Local velocity = mean velocity
Pressure gradient → Varies linearly with distance Are developed turbulent flow In horizontal pipe
Re > 4000
For TF → τ total = τ laminar + τ Turbulent = μ.du/dy
Eddy viscosity → For turbulent flow it is dependent on the flow
Turbulent flow → friction factor depends only on Reynold’s number
Turbulent pipe flow Is said to be in the transition regime → if the friction factor dependent on Reynolds number and relative roughness
Entrance/Establishment Length for turbulent flow → L ∝ Re0.25
Turbulence intensity = Root mean square value of velocity fluctuations
Velocity and pressure at a point → Exhibit irregular fluctuation of high frequency
Laser-Doppler anemometer → The turbulent velocity fluctuations in a flow, Also used to measure velocity without any obstruction to the flow in a pipe
Velocity distribution for Turbulent flow
Velocity ditⁿ → Logarithmic, But inside sublayer of laminar flow → linear
Shows increasing fullness with increase in Renauld's number
U/Us = 5.75log10(UsR/ν)+1.75=5.75log10(Usy/ν)
Increasing ageing of pipes → The proportion between maximum velocity and the main velocity in turbulent flow → Decreases and then increases
BOUNDARY LAYER
Developed by Prandtl →introduced Concept of boundary layer
At Boundary layer → The effect of viscosity is confined
y = 2R/3 → τ = τo/3 → Turbulent shear stress = Wall shear stress/3
The Prandtl mixing length → Pipe wall = 0
Laminar sub layer exists → in all turbulent boundary layers
a) Boundary layer thickness (δ)
depth(y) = δ → u = 0.99U(99% of free stream velocity) ≈ U, du/dy = 0
Boundary layer max thickness = R (Pipe radii)
b) Displacement thickness (δ*)
* = 0(1-V/Vo)dy
c) Momentum thickness (θ)
=0(V/Vo)(1-V/Vo)dy
d) Energy thickness ( δε )
=0(V/Vo)(1-V2/Vo2)dy
V → Velocity at any distance y from boundary
Vo → Free stream velocity
if not given assume V/Vo = y/δ
δε > δ* > θ → edm*
Separation of boundary layer
Flow separation takes place → Where Pressure Gradient changes Abruptly and Boundary layer comes to rest, dp/dx < 0, (∂u/∂y)y=0=0, Shear stress = 0
(+ve) or adverse Pressure gradient (dp/dx > 0) helps in BL separation → Velocity gradient becomes (-ve)(dv/dx < 0
Wake → Region b/w separation of streamline & boundary surface of solid body, Always occur after Separation Point, Occurs in bluff body
Boundary layer takes place for real fluid
Change in boundary layer from laminar to turbulent is directly affected by velocity of flow
Thickness of boundary layer at the leading/entrance edge = 0
The boundary layer flow over a sufficient large flat plate is laminar over a short initial length and therefore turns turbulent
Trip wire → To delay the point of separation is mounted near the leading is of a body
Consequences of boundary layer separation
internal flow like pipes → increases flow losses
External flow → increase in pressure drag
Methods to control Separation
Rotating boundary in flow direction, Stream lining the body, Suction of fluid from boundary layer, Supplying additional energy from blower, Providing a bypass in the slotted wing, Accelerating the fluid in boundary layer by injecting fluid, Providing guide blades on bends
Entrance Length
Where boundary layer increases & flow is fully developed
Laminar flow = 0.07ReD
Turbulent flow = 50D
Nikurde's experiment Boundary classification
Hydrodynamical smooth → k/δ ≤ 0.25
Boundary transition condition → 0.25 < k/δ < 6
Hydrodynamical rough → k/δ > 6
Blassius Slotⁿ for smooth plate
Local Reynolds number → Rex=Vx/
Critical Reynolds no. → Rex = 5 x 10⁵
Rex > 5 x 10⁵ → Turbulent boundary layer
1. Laminar flow
=5x/Rex → δ ∝ √x ∝ 1/√Re
Cfx ∝ 1/√x
2. Turbulent flow
δ ∝ x4/5
Cfx ∝ 1 / x^⅘
u/U = (y/δ)^1/7
Relative thickness
Laminar → /x=5/Rex
Turbulent → /x ∝ 1/Rex1/5 → Relative thickness on flat plate decreases with distance x
Streamlined Body
Body surface coincide with the streamline → Flow separation is suppressed
Airfoil → A small wake and consequently small pressure drag
The critical angle of attack of an airfoil → Where the lift begins to drop
Separation of flow takes place at the trailing edge or farthest downstream part of the body
Friction drag >> Pressure drag → Friction drag force is predominant
Bluff body
Body surface doesn't coincide with the streamline
Pressure drag >>> Friction drag → Pressure drag force is predominant
Rankine oval body
Stagnation point → two at θ = 0 and θ = π → One on front and other at back side
Force on Plate
Drag force → due to pressure, lift force → due to viscosity
Drag → Fd = CdρAVo²/2 → Parallel (∥), Component of resultant fluid dynamic force in the flow direction
A = Platform/projected area when the body is Flat like an airfoil
Lift → Fd = CvρAVo²/2 → Perpendicular (⊥) to velocity, Always opposite to the relative motion of body / lift
An immersed body in a flowing fluid the lift force is always in the opposite direction to gravity
Jet → F = ρAV² → on plate
Airfoil section at stall → Lift coeff/Drag coeff = 1.5
Drag Coefficient is reduced when the surface is smooth
If fluid is ideal and the body is Symmetrical(Sphere, Cylinder) → Both Drag & Lift will be zero
A Supersonic velocity drag Coefficient depends on mach number only
Designing of a body for drag and lift forces → Streamline body is one in which flow separation is suppressed
Magnus effect → Associate with rotation → Produced by a spinning cylinder
Robin effect → Related to creation of lift of a sphere
Both Lift + Drag → Motion of aeroplanes, Submarines, Torpedoes
Coeff of drag
Cd=24/Re → Re < 5 x 10⁵
Cd = 0.20 → If Re just greater than 5 x 10⁵
Turbulent → Cd = 0.074/Re1/5
Drag force
Drag total = Pressure drag(Form drag) + Friction drag(Skin/shear drag) = Vd + 2Vd
Friction drag/Pressure drag = 2
Deformation drag on sphere → 3Vd
Pressure/Form drag → Depends upon the separation of boundary layer and the size of wake
Surface/Skin/Shear/Friction drag → The tangential force exerted by the shearing stresses on a object submerged in a fluid, Major drag force experienced by the body at very small velocity, Primary due to shear stress generated due to viscous action
Drag force = Wt. of body → Net external force = 0, Acceleration ceases and Body will move at constant speed
Drag force on streamline shape is → Due to Primarily Shear stress
Plate parallel to flow → angle made by pressure with direction of motion = 90 → Pressure drag = 0
Plate perpendicular to flow → angle = 0 → friction drag = 0
Drag force on a cylinder → in Turbulent flow >>> Laminar flow
Bluff body → More pressure drag and less friction drag → Compare to a streamlined body
Largest total drag → Circular disc of plate held normal to flow → With the same C/S Area and immersed in same turbulent flow
Less Drag → Airfoil
Plate of negligible thickness is held perpendicular to the flow direction → Forces mainly due to form/Pressure drag
Total Drag is reduced if the boundary layer on the surface of a cylinder separates for the downstream of the leading point → As the separation point moves for the downstream form-drag is reduced and the skin drag is only marginally increased
WEIR & NOTCH
Crest/Sill → Top edge of weir/notch over which water flows
Nappe → Sheet of water flowing through a notch/weir
Weir is aligned at right angles → To ensures less length of weir, gives better discharging capacity, it is economical
Due to each and contraction the crest length is reduced by 0.1H
Aeration of Nappe is necessary for suppressed wear
Afflux → Rise in upstream water level due to an obstruction in the flow of water like weir, notch
Discharge Formulas
Q = kHⁿ → dQ/Q = n dH/H.
Q ∝ H → Proportional weir, Sutro
Q ∝ H^3/2 → For Rectangular, Cipolletti, Broad crest, Stepped, Ogee
Q = Cd A √(2gh) → For Orifice, Mouthpiece, Bordas
Q = ⅔Cd √(2g) LH^3/2 → For Rectangular, Cipolletti, Broad crest, Ogee
Sardha fall (vertical drop fall) → ht = up to 1.5m
Types of Weir
Concrete weir with sloping glacis → Excess energy of overflowing water dissipated by means of a hydraulic jump.
Type - D → A surplus weir of an earthen dam with stepped apron
A) Shape of opening
4 Types of weir
i) Rectangular sharp crested
Q = (2/3) Cd 2g L [(H + h)3/2 - h3/2 ]
dQ/Q = 3/2 dH/H + dL/L
Cd = 0.62
h = 0 → if velocity of approach is not considered
Due to each end contraction → Q decreases by 10 %
Contracted Rectangular → Crest length < width of channel
Suppressed Rectangular → Without end contraction
Suppressed weir → Crest length = width of channel
ii) Triangular(V)
Effectively measure low Q
Q = (8/15) Cd 2g tan(θ/2) [(H + h)5/2 - h5/2 ]
dQ/Q = 5/2 dH/H
for Max discharge → Angle of notch = 90
Right angle noth → Q = 1.416 H5/2 → if Cd = 0.6 & θ=90°
Advantage of V
Cd nearly constant, No effect of Viscosity & surface tension
Only one Reading/dimension is to measure → Hence More accurate
for small Q high H obtained
iii) Trapezoidal
Q = Q rect + Q triangle.
vi) Cipolletti
It is a trapezoidal weir/notch → Whose slopes are adjusted such that ↓es Q end contraction of rectangular weir = ↑ Q Triangular portion
Qcipolletti=Qrect = (2/3) Cd 2g L [(H + h)3/2 - h3/2 ]
Francis formula → 1.84LH3/2
Side slope → 1H : 4V → θ/2 = 14° → θ = 28°
Cd = 0.62
B) Shape of crest
i) Broad crested
B > H/2
Q ∝ H^3/2
Max Discharge → Depth of flow = 2H/3
ii) Narrow crested
Width < H
iii) Ogee-shaped
Spillway of Dam
Q = Q rectangular & Cd = 0.62
Q ∝ H^3/2
iv) Sharp edge crested
It is a standard Orifice
C) Meter of Discharge
i) Submerged/drowned weir
D/s WL is > Crest level
Suppressed weir
Crest length = Width of channel
Suppressed sharp crested weir → Cd = 0.602 + 0.083 x (Head/Height)
DOSE
Laser Doppler anemometer: Turbulent velocity
Too small dia pipe : power↑↑es
Liquid : No volume change
Angular velocity ω = 2πn → [T⁻¹]
Angular acceleration = Rad/T² → [T⁻²]
Angular momentum = moment of momentum = rotation momentum = mvr = I ω = mr²v/r
Compressibility: flight of supersonic aircraft
Gravity: OCF(hydraulic jump)
Viscosity: real fluid
Vapour pressure: cavitation
Hardy cross method : pipe network
Flow develop : Entrance Length
flow velocity = Sonic , at throat of a converging & diverging nozzle.
Subsonic: up to throat (converging)
Supersonic: after throat (diverging)
Prandtl's universal velocity distⁿ eqⁿ → used for both smooth & rough boundaries
Rayleigh lines → use of Momentum & continuity eqn.
OPEN CHANNEL FLOW
INTRO
In OCF we study rigid channels
Prismatic channel → c/s shape, size, and bed slope is constant.
All natural channels are non prismatic.
Rigid channel → Boundary is not deformable.
Degree of freedom → Rigid Channel = 1(depth) & Mobile channel = 4
Specific force = (Pressure force + Momentum flux) / γ = (P+M)/γ
P = γh ← Small slope
P = γhcosθ ← Large slope
Surge in OCF → uses Continuity eqn & Momentum eqn.
St Venant’s eqn for unsteady open channel flow → Continuity and Momentum eqn
Venturi Flume/meter → Most common device for measuring discharge through open channel
Floats → Used to measure velocity of stream
Eddies formation take place in turbulent flow
Stationery Shockwaves → Two stream saving same flow value but different density is meets
Steady non uniform flow in open channel → Occurs when for a constant discharge the liquid in the channel varies along its length
Bed load → Combination of contact load and siltation load
Float → Small object made of wood → Measure Approximate velocity of flow of water in rivers
Echo sounder → Measure depth of river
Q will be max only when slope will be maximum → When c/s of open channels constant
Section Factor(Z)
Critical flow → Z=AD=AA/T
Uniform Flow → Z=AR2/3=A(A/P)2/3
Hydraulic Depth → D= A/T = Area/Top width
Hydraulic mean depth/Hydraulic radius → R= A/P = Flow Area/Wetted Perimeter
Hydraulically equal → R = A/P is same
Circular channel running full → R = D/4
Circular channel running Half full→ R = D/4
Circular Partially full → P=2R, A=R2(-sin/2)
Rectangular channel → R = By/(B+2y), D=,Z = By3/2
Square Running full → R = y2/3y=y/3
Trapezoidal → R = y(B+my)/(B+2ym2+1 , D= y(B+my)/(B+2my)
Triangular → D = y/2
Triangular with corner rounded off → R = y/2
Note :- Non circular duct or Pipe → Hydraulic mean diameter = 4A/P
Froude number
Fr=V/gD=V/gA/T
Frtriangular =2Frrectangular
Frrectangular=V/gy,Frtriangular =V/gy/2
Subcritical/Streaming/Tranquil flow → y > yc, v < vc, Fr < 1
Critical → y = yc, v = vc, Fr =1
Supercritical/Torrential/Shooting/Rapid flow → y < yc, v > vc, Fr > 1
Rectangular channel for Subcritical flow → Width of channel decreases then depth of flow decreases
Velocity Ditⁿ in open channel
Velocity distribution is logarithmic
i) Avg velocity = V0.6y → Less better → 0.6y below the water surface or 0.4y above the stream bed
ii) V avg = (V0.2y + V0.8y)/2 → Much better
iii) V avg = K x surface Velocity ( K = 0.8 - 0.95)
Max velocity occur → at 0.05d - 0.15d → A little below the free surface
Velocity in a channel → By Current meter, Float, Pitote tube
Measurement of velocity
From froude number → V =FrgD
Rectangular channel → Q ∝ y5/3
i. Chezy's equation
V=CRS =Cmi
R = A/P
C = [L1/2T-1]
τo = γ R S = (K ρ V²)/2
ii. Manning's equation
Best → Design of lined canal or lined alluvial canal or impermeable soil
V = (1/n) R2/3 S
C=R1/6/n=8g/f'
f=8gn2/R1/3
n → Coeff of rugosity/manning's roughness coeff → [L-1/3T] → Depends on type of surface of channel
Design of earthen channel → Roughness coefficient(n) = 0.0225
Value of chezy’s constant
Mannings → C=R1/6/n
Kutters → C = [1/n + (23 + 0.00155/S)] / [1 + n(23 + 0.00155/S)/R ]
Bazin’s → C =157.6/(1.81+K/S
Chezy’s → C=V/RS
Elementary wave
Speed/Celerity of an elementary wave in still liquid = gy
Elementary wave travel upstream = gy -v
Elementary wave travel Downstream = gy +v
y = depth of flow, v = velocity of flow, g = 9.81
UNIFORM FLOW
At any given time fluid properties doesn't change with location
Normal depth(yn) → Depth of Flow that would occur if the flow was uniform and steady
Bed slope(S) = energy line slope = water surface slope = slope of HGL = slope of TEL → TEL, HGL, and Bottom of channel are all Parallel → dy/dx = 0
Acceleration = 0 ( V = Constant), ∆momentum = 0
Wide rectangular canals → Would the flow be practically uniform
Uniform flow is possible in the middle of a long Prismatic channel
Main characteristics curve → Curves at constant head
Economical & Efficient Channel
Best Hydraulic Channel → Minimum Wetted Perimeter → Maximum hydraulic radius
Efficient → Max Q for a given c/s area
Economical → min construction cost (dP/dy =0) for a given Q
Semicircle is the best hydraulic section
Most efficient section of a channel is → Semi-circular → But Due to practical limitation in maintaining the section Trapezoidal channels are usually employed
i. Rectangular section
y = B/2
R = y/2 = B/4
ii. Triangular sectⁿ
Half of a square
θ = 45° → m = 1 → Each sloping side makes with vertical
R = y/8=y/22
T = 2my = 2y
iii Trapezoidal
Most economical trapezoidal sectⁿ should be half of a regular Hexagon
Circle of radii (r = y) can be inscribed in trapezoidal sectⁿ
Case-i → Side slope is fixed
Top width = 2 x (Side slope length)
B + 2my = 2y m² + 1
If m is given → use above eqn otherwise below results
Case-ii → Side slope is variable
m = 1/√3 → θ = 60° → tan θ = 1/m
R = y/2
B = 2y/3=1.154
A=y23 , P=23 y
iv. Circular sectⁿ
For max V → 2θ = 257°27'56", y = 0.81D, A = R²/2(2θ - sin2θ)
For max Q → 2θ = 302°22', y = 0.938D
From Chezy's equation for Circular sectⁿ
for max V → 2θ = 257°27', y = 0.81D, P = 2.83D, R = 0.305D
for max Q → 2θ = 308° & y = 0.95D, R = 0.29D
ENERGY-DEPTH
Blench curves → Relationship between the loss of head and specific energy downstream → for a given discharge
Specific energy
Total energy at a sectⁿ wrt the channel bed/bottom as datum
SE = y + α V²/2g = Depth of Flow + Kinetic head= Potential head/energy + Kinetic E
Uniform flow(α = 1) → SE = y + V²/2g= y + Fr²y/2=y+Q2/2gA2
KE ∝ 1/y² → Parabolic curve
PE ∝ y → Linear or straight line curve
E < Ec → No flow
Alternate depth → Two possible depths for a Same SE for a given Q → Fr ∝ y3/2
Critical depth (yc) → Minimum Specific Energy → Can be produced by raising the bottom of channel or by decreasing the width of channel
The critical depth in a channel carrying a fixed discharge is → Function of cross sectional geometry only
For any channel → SE increases → increase in depth of subcritical flow & decrease in depth of supercritical flow
Solve channel problem with 2 section → By taking E1 = E2 → Also To find alternative depth
If two alternative depth are given use hydraulic jump formula to find discharge
Critical Flow Condition
Fr² = Q²T/gA³ = 1 → Fr = 1
Velocity head = 1/2 of Hydraulic depth → V²/2g = D/2
for a given Q → SE & SF is minimum
for a given SE or SF → Q will be maximum
Fr = V/√gD = 1 → D = V²/g
Assume Re = 2000 & find V & Fr
q = Q/B
Rectangular section
yc = (q²/g)1/3
Vc = (qg)1/3
Ec = 3yc/2
E/yc=yn/yc+1/2(yn/yc)2
Triangular section
yc = (2Q²/gm²)1/5
Ec = 5yc/4
Most economical → m = 1 → yc = (2Q²/g)1/5
Parabolic section
Ec = 4yc/3
Flow through hump/Local rise in bed
The depth of flow over the Hump < Upstream flow depth
Zm→ Minimum height of Hump for critical flow or maximum height of Hump for which a upstream flow is not affected
Subcritical flow → Depth of water < Upstream depth
Z <Zm→ The flow over Hump remains subcritical
Z >Zm → Flow is not possible (choking or critical flow condition) → The upstream flow condition should be changed for further increase in height of Hump → Drop in water level at of stream section of the Hump
Z=E1-Ec
E2=Ec+Z
Ehump=Ec
E2>E1 → Surge will travel upstream
Z ≥ (E1 + Emin) → For flow over a broadcasted weir to be critical
GVF & RVF
Length of backwater curve → Distance along the bad of channel between the section, where water starts rising to the section and where water as maximum depth
Gradually Varied Flow
it is Steady & Non-uniform flow
GVF Caused when the force causing the flow is equal to resistance force
Slope of energy grade line, Hydraulic grade line and Bottom of channel are all Different
If Slope of free water surface = 0 → Free water surface is parallel to bed of channel
Example → Back water curve due to any obstruction such as weir
GVF Eqn
dy/dx=(So-Sf)/(1-Fr2)=(So-Sf)/(1-Q²T/gA³)
Wide rectangular channel → dy/dx=So(1-(yn/y)3)/(1-(yc/y)3)
Total GVF Profiles = 12
In mild slope channel with uniform flow the HGL coincide with the free surface
Note → 1 → Subcritical, 3 → Supercritical, 2 → Subcritical (Except S2)
Back water curve profile → M1,S1,H3
Adverse (So < 0) → Bottom slope rises in the direction of flow, +ve slope in downstream direction
Fr < 1 if y > yc, Fr > 1 if y < yc, Fr = 1 if y = yc
y > yc & yn → Subcritical flow
yn > y > yc → Subcritical
yc > y > yn → Super critical
y < yc & yn → Super critical flow
Rapidly Varied Flow
Hydraulic jump, Surge in open channel
Hydraulic jump
HJ → Steady & Non uniform flow
Steep slope (Supercritical) to mild slope(Sub critical) → Below critical Depth to Above Critical depth → Rapidly flowing stream abruptly changes to a slowly flowing stream causing a distinct rise of liquid surface
HJ → Used to reduce the energy of flow in Hydraulic structure
Shooting flow can never occur directly after hydraulic jump
Jump formation → SF → Remains Constant & SE → ↓es
Sequent/Conjugate depth → Having same Specific Force
Specific force → F = Az̅ + Q²/gA³
In Concrete weirs with glacis excess energy of overflowing water dissipated by means of a hydraulic jump
Hydraulic jump is an analogous to normal shock wave
Depth < Critical depth → HJ in a control meter will be formed above the control
Hydraulic jump is also known as Standing wave
Strength of HJ → By the upstream froude number or Froude number at the beginning of the jump
Compare to a horizontal surface the hydraulic jump on a sloping glacis is always more definite and less efficient
HJ in horizontal frictionless rectangular channel
Relation → Continuity equation and Momentum equation are used
V1>Vc>V2 y1<y2
y2/2+q2/gy=Constant
Limiting flow velocity → VL= q/yc =Q/Byc→ For hydraulic jump to occur
y2/y1=(1/2)(-1+1+8F12 )
F=q2/gy3
2q2/g = y1y2(y1+y2)
yc3=q2/g=y1y2(y1+y2) /2
Energy loss → E=(y2-y1)3/4y1y2=(V1-V2)3/2g(V1-V2)
Power loss → P=QE
Efficiency → =E2/E1
Length of HJ → L=6.9(y2-y1)=(5-7)Ht of jump
Ht. of jump = (y2-y1) → Diff of Conjugate/Sequent depth
Ht of jump Without Causing Afflux = (E2-E1)
Surge in open channel
Sudden change of flow depth → Abrupt increase or decrease in depth
Unsteady and non-uniform, Rapidly varied flow
(+ve) surge → Abrupt increase → Moving hydraulic jump with a wave front moving a upstream or downstream
(-ve) surge → Abrupt decrease
Analysis of surge → By Continuity eqn and Momentum eqn
HYDRAULIC MACHINE
HYDROELECTRIC PLANT
Pressure-time method → Basic method to measure the flow rates in hydro power plants
Firm power → Net amount of power which is continuously available from a hydro power plant without any Break on firm
Underground power house are found to be more economical than an equivalent surface power station → Less amount of concrete is required compared to a surface power
The amount of electrical energy that can be generated by a hydroelectric power plant depends upon head of water
Hydroelectric power plant is conventional source of energy
Hydraulic torque converter → Transmitting increased or decrease torque to the driven shaft
Load factor = Avg load/Peak load
Capacity/Plant factor = Avg load output/installed capacity of plant
Utilisation factor = Water actually utilised for power/Water available in river = Peak load/installed capacity
Hydel/Hydro-electric Scheme Classification
Runoff river plants → A low head scheme, Suitable only on a perennial river, Defined minimum dry weather flow is available, utilise the flow as it occurs without any provision for storage
Pumped storage plants → Generates power only during the peak hours
Components
Reservoir → Sluice gate/gate opening) → Canal/Tunnel) → Penstock → Forebay/surge tank → intake str. → Powerhouse/turbine → Tailrace → River
Penstock → carry water from storage reservoir to the power house
Headrace tunnel → a channel of free-flow tunnel leading water to the fore bay or a pressure tunnel leading the water to the surge tank
Forebay
Issued at junction of powerchannel and penstock
It stores water temporarily when reject by plant (when electric load is reject)
Also to meet the instantaneous increased demand of water due to sudden increase in load
Draft tube
A pipe of gradually increasing area which is used for discharging water from exit of reaction turbine to tail race
Always immersed in water, ↑es head/pressure head, ↓es loss of K.E. at the outlet
Angle of taper on the draft tube < 8
For reaction turbine arranged to convert kinetic head into pressure head
Efficiency of draft tube = (V12-V22)/V12
Surge tank
Reservoir to ↓es water hammer pressure when suddenly closed, Restrict the water hammer effects to small length of penstock, To regulate flow of water to turbines by providing necessary regarding head of water, to control the pressure variation due to Rapid change in pipeline flow
When it is not possible to provide a forebay → We provide a surge tank
Hydraulic Rim
Work on the principle of water hammer
Device used to lift small quantity of water to a larger height when a large quantity is available at smaller height
It does not need any external power like electricity
TURBINE
Hydraulic energy → Mechanical energy → Electric Energy or Turbo machine that extracts energy from a liquid by virtue of rotating blade system
At Design speed → Turbine reaches its peak efficiency
Runway speed → at which turbine runs freely without load
Potential & pressure energy are the same
Design speed → Turbine reaches its peak efficiency
Governing of a turbine means the speed is kept constant under all condition of working
Specific speed of Turbine
Produce unit Power(1kW) for unit Head(1m)
Ns=NP / H5/4=Constant → [M1/2L-1/4T-5/2] or F1/2L-3/4T-3/2]
Multijet Pelton turbine
Ns= nNs of single
Power = n x Power single jet
Q = n x Discharge single
n → Number of jet
Higher specific speed turbines are more reliable to cavitation
Ns increases → Smaller size and necessitate proper design of draft tube to that losses due to whirl component at exit is less
Unit power of reaction turbine → 1st increases than decreases with the unit speed
In a reaction turbine the draft tube is used to increase the effective head of water
Number of blades = 15 + D/2d → Pelton wheel turbine
Mixed flow
Inward radial flow → The water enters the wheel at outer Periphery and then flows towards the centre of the wheel
Outward radial flow →
Old francis → ? .. Radial turbine
Francis turbine → Water flows out through a closed draft tube
Sardar sarovar dam → Percent of total power is generated using Francis turbine
Turbine head
Net head = Gross head - Frictional loss → Effective had used to calculate power production
Gross head → Difference in elevation between head race level of intake and the tail race level at discharge side
Performance characteristic curves
Pelton wheel → unit discharges is independent of unit speed
Main characteristic curve or Constant head curve → Helps in determining the overall efficiency of turbine
Constant efficiency curves → Muschel curves
Constant speed curve → Operating characteristic curves
Indicator diagram → (Pressure had in the cylinder) vs (the distance travel by the Piston from inner dead Centre for one complete revolution of Crank)
Efficiency
=Power/gQH
Shaft Power → P=gQH → H=(P1-P2)/g+(V12-V22)/2g
Overall(ηo ) = Output Water power/input Shaft power = Mechanical x Manometric/Hydraulic
ηo = ηmech ηmanηv or ηmech ηHηv
Mechanical efficiency = (Power available at the shaft)/(Power deliver by water to the runner)
Manometric/Hydraulic efficiency = Runner power/Hydraulic power = Vw1u1/gH
Max hydraulic efficiency → (1+cos)/2 → impulse turbine
Max efficiency for Impulse turbine → velocity of wheel = 1/2 that of the jet velocity
Max efficiency for Reaction turbine → Angle of absolute velocity vector at the outlet = 90, Velocity of Swirl at the outlet must be zero
At Part load → Efficiency → Kaplan > Pelton > Francis > Propeller
At full load →Efficiency → Francis > Kaplan = Propeller > Pelton
For max efficiency → Vane velocity = Jet velocity/2
Thoms cavitation factor < Critical factor → Overall efficiency will have sharp fall
Pelton wheel → ηo = (75-85)%
Unit Quantities of Turbine
Unit speed → Nu=N/H →N ∝ H
Unit Discharge → Qu=Q/H → Q ∝ H
Unit Power → Pu=P/H3/2 → P ∝ H3/2
Speed ratio
SR = Vtangential/Vjet=Vvane/Vjet
Vane Velocity → u=DN/60=2gH
Pelton wheel → SR = 0.45 - 0.50
Impact of jet on vane
Velocity triangle is based on Newton's law → F = ma
Rather than mass → Weight/g is used, Change of velocity normal to the direction of impacting inflow is also relevant to the computations
Jet ratio = (Pelton wheel diameter)/(The jet water)
Eulers eqn for turbine → Derived on the basis of rate of change of angular momentum
Work done by impeller → W=Vwu/g
Force jet strike → F=AV2 → Curved plate → F=AV2(1+cos)
Q1/Q2= (1+cos)/(1-cos) → (Q1 > Q2), After striking the plate the jet gets divided into two stream
Flow number → Q/ND3
Max number of Jet = 6 → in impulse turbine without Jet interference
Oil pressure governor in modern turbines
Component → Sarvomotor, Oilsump, Oil pump, Draft tube
PUMPS
Pump → Mechanical device to increase the pressure energy of liquid, To raising fluid from a lower to higher level
Pump is not a main component of hydroelectric plant
Counterbalance/Holding Valve → Used to control as vertical cylinder to prevent it from descending due to external load, Prevent load from falling unwantedly
Check valve → Provided immediately above the pump to reduce back surge and water hammer
Balanced system → Uniform rate of pumping is required
Suction lift of water supply pump = Height of pump above water level of storage reservoir
Pressure compensator control → Limit pump outlet pressure to predetermined level and adjust pump outlet flow to the level needed to maintain the set pressure
Pump operating point → Intersection of pumping head-discharge curve and the system head-discharge curve
Types
Rotodynamic pump → High Q, Low H or Low pressure → Centrifugal pump, propeller pump
Rotary/(+ve) displacement pump → High H or high pressure, Low Q → Reciprocating pump
Axial flow pump →Very High Q at low pressure → Flood control and irrigation applications, Water supply mains
Concrete pumping → Centrifugally operated with straight blades pump, Pneumatically operated pump
Same Pump → H/N^2D^2 = Constant
Centrifugal Pump
Principle of Working → Forced vortex motion
Convert mechanical energy into hydraulic energy by means of centrifugal energy
High discharge & Low Head, Can run at high speed, Liquid enters at the centre, Low initial cost and it is compacted, Priming required
Use → Lift highly viscous liquids e.g. Sewage water, Chemicals
Size → Given by horsepower
Energy is developed due to whirling motion
Max permissible suction lift for CP = 6m → At sea level and 30℃
Speed increases → Q increases, Head increases, Power increases
Installed → (-ve) Pressure doesn't reach as low the vapour pressure
Priming → Operation in which liquid is completely filled in the chamber of pump so that air or gas or vapour from the portion of pump is driven out & no air pocket is left
Type → Valued pump, Diffuser/Turbine pump, Vortex pump → Based on type of casing
P ∝ N3D5
Q ∝ ND3
H ∝ N2
When speed changes → Shape of velocity triangle and Various angle remain same, magnitude of velocity will change proportionately
Inlet angle of CP → Designed to have absolute velocity vector in the radial direction
Diffuser blade → Used to convert kinetic energy to pressure energy
impeller(backward curved blade) → Max vane exit angle = 20-25
Backward bent vanes are preferred over forward bent vanes
Regulating valve → Provided on the delivery pipe To control the flow from pump in to delivery pipe
Discharge increase with speed, Head increase with speed but Manometric head decreases with discharge
Multi-Stage CP
Give high discharge, Produce high heads, Pump viscous fluids
High lift Multi-Stage CP → Head > 40m
Reciprocating Pump
High Head, high efficiency & Low Discharge, Delivery of water need not to be continuous , Low maintenance cost, Pump need not run at high speed
Does not need priming
Ex. commonly used hand pump, Positive displacement pump like Rotary Pump
Power → P=QHs/75
Single Acting Reciprocating Pump → Q = ALN/60 → Water is delivered in delivery stroke
Double Acting Reciprocating Pump → Q = 2ALN/60
Q = m^3/sec, A = cylinder area (m^2), L = Cylinder/Stroke length (m), N = Crank speed (rpm)
Double Acting duplex RP → 2 Pistons
Hand Pump → operates through one way valve
Air vessel → For continuous supply of water at uniform rate, Smoothen the flow
Work saved by fitting an Air vessel → in SARP = 89.4%, DARP = 39.3%
Slip% = (Qth-Qactual)100/Qth=(1-Cd)100
Slip = (Qth-Qactual)
(-ve) Slip → Delivery pipe small and suction pipe long, Pump running at very high speed
Separation may take place at the beginning of suction stroke
Specific speed of Pump
Ns → Speed in revolution/minute → Deliver unit Discharge (1 cumec) of liquid against unit head (1m)
Ns=NQ / H3/4=Constant → [L3/4T-3/2]
For multi stage H = Total Head / No. of Stage
Centrifugal pump → Ns< 2000
Radial flow pump → Ns = 500 - 4000
Mixed flow pump → Ns = 4000 - 10000
Axial flow pump → Ns = 9000 - 15000
Specific speed for CP
Ns < 20 rpm → Slow speed and radial flow at outlet
Ns = 80-160 rpm → High speed with mixed flow at outlet
Ns = 160 - 500 rpm → High speed with axial flow at outlet
Pump Head
Static head(Hs) → Suction + Delivery head → Difference between water level in sump and top storage Reservoir Where water is told after pumping
Suction head(hs) → Vertical distance between free surface of liquid level in the sump and pump Central Line
Manometric head(Hm) → Difference of head as shown by manometers connected between the inlet and outlet flanges of pump → head against which the pump has to work
Hm= Hs+hfs+hfd=inlet-outlet head=Static head+ friction loss in section pipe+ friction loss in delivery pipe
Net positive suction head = Pressure head + Kinetic head - vapour pressure
Efficiency
Mechanical = Power at impeller / Power at shaft of Centrifugal pump
Manometric = Manometric Head / Head imparted by impeller to the water
Overall(η) = Mechanical x Manometric/Hydraulic → η = ηmech x ηmano
η ∝ 1/ input Power
Efficiency of a pumping set = 65% - 70%
Power required to drive a pump
P = ⍴Qg(H + hf)/η=QHm/η
Cavitation in Pump/Turbine
Formation of vapour bubbles of a flowing liquid in a region where the pressure of the liquid falls below its vapour pressure
Due to Higher Runner/Pump speed, High suction lift, High temp, Low pressure, Less available NPSH
Minimum NPSH → Prevents the Cavitation
To avoid Cavitation the suction pressure should be high
Effect → Loss in efficiency, production of the noise and vibration, material damage/loss due to erosion, Irregular collapse of vapour cavities
For No Cavitation → NPSH ≥ σcH, P ≥ Sat vapour pressure
Net positive Suction head → The absolute pressure head - minus the vapour pressure head + the velocity head
NPSHA=(Patm-Pvapour)/g-hs-hfs
Pvapour/g → at 20℃ = 2.5m, at 100℃ = 10.3m
Patm/g = 10.3m at MSL
σc → Thomas cavitation number parameter
Cavitation Parameter = (P-Pv) / 0.5V2
Multistage Pump
Qused=Qtotal/no.of pump used
Pump in Series
For High head
H = H1 + H2 + H3 …
Q1 = Q2 = Q3 …
Pump in Parallel
For high Discharge
Q = Q1 + Q2 + Q3 …
H1 = H2 = H3 …
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