Fluid Mechanics and Machines

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This section defines what a fluid is, the modelling ideas used throughout fluid mechanics (continuum, no-slip, Lagrangian vs Eulerian, control volume), the key fluid properties, and fluid statics: pressure, its measurement and hydrostatic force on plane surfaces.

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Basic Concepts • A fluid deforms continuously under a shear stress, however small; a solid resists shear by a finite static deformation.

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At rest, a fluid can sustain NO shear stress. • Continuum hypothesis: fluid treated as continuous matter so that properties (ρ, V, p) are point functions.

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Valid when molecular mean free path λ ≪ characteristic length L — Knudsen number Kn = λ/L < 0.01.

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Fails for rarefied gases (upper atmosphere, vacuum). • No-slip condition: fluid in contact with a solid wall has ZERO velocity relative to the wall.

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This produces velocity gradients near walls and is the origin of the boundary layer and viscous drag. • Lagrangian approach: follows individual fluid particles (like tracking one boat).

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Eulerian approach: observes flow properties at FIXED points in space as a field (like a speed camera) — used in most fluid mechanics. • Material (total) derivative:

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D/Dt = ∂/∂t (local) + V·∇ (convective) — links the two descriptions. • Control volume: fixed region in space through which fluid flows (open system) — used for pumps, turbines, pipes.

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System (control mass): fixed quantity of matter.

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The Reynolds transport theorem converts system laws into control-volume form.

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Fluid Properties Property Definition / typical value Density ρ Mass per unit volume.

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Water 1000 kg/m³; air ≈ 1.225 kg/m³ at sea level Specific weight γ Weight per unit volume = ρg.

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Water ≈ 9810 N/m³ Specific gravity S ρ/ρwater.

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Mercury 13.6 Dynamic viscosity μ τ = μ du/dy; unit Pa·s = N·s/m²;

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Water ≈ 1 × 10−3 Pa·s (1 cP) at 20 °C Kinematic viscosity ν ν = μ/ρ; unit m²/s;

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1 stoke = 1 cm²/s = 10−4 m²/s.

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Water ≈ 1 × 10−6 m²/s Surface tension σ Force per unit length (N/m).

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Water-air ≈ 0.073 N/m at 20 °C; decreases with temperature Bulk modulus K K = −dp/(dV/V).

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Compressibility = 1/K Vapour pressure pv Pressure at which liquid boils at given temperature; if local p < pv → cavitation • Effect of temperature on viscosity: liquids — viscosity DECREASES with temperature (cohesive forces weaken); gases — viscosity INCREASES with temperature (more molecular momentum exchange). • Ideal fluid: incompressible and zero viscosity (no shear).

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Real fluid: has viscosity.

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Newtonian and Non-Newtonian Fluids Type Behaviour (τ = K(du/dy)n) Examples Newtonian τ ∝ du/dy (n = 1, straight line through origin) Water, air, petrol, thin oils Pseudoplastic (shear-thinning) n < 1: apparent viscosity falls as shear rate rises Blood, paints, polymer solutions, ketchup Dilatant (shear-thickening) n > 1: apparent viscosity rises with shear rate Starch/corn-flour in water, wet sand, quicksand Bingham plastic Needs a yield stress τy before flowing, then linear Toothpaste, drilling mud, sewage sludge Thixotropic Viscosity decreases with TIME of shearing Printer ink, some paints, gels Rheopectic Viscosity increases with time of shearing Gypsum paste, some lubricants Surface Tension and Capillarity • Caused by cohesion between liquid molecules at the free surface. • Pressure excess inside: liquid droplet Δp = 4σ/d; soap bubble Δp = 8σ/d (two surfaces); liquid jet Δp = 2σ/d. • Capillary rise/fall h = 4σ cos θ/(ρ g d).

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Water wets glass (θ ≈ 0°, adhesion > cohesion) → RISES; mercury (θ ≈ 130–140°, cohesion > adhesion) → FALLS (capillary depression). h ∝ 1/d.

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Pressure and Its Measurement • Pascal's law: pressure at a point in a fluid at rest is the SAME in all directions. • Hydrostatic law: dp/dz = −ρg → p = ρgh (pressure increases linearly with depth; independent of container shape — hydrostatic paradox). • Absolute pressure = gauge pressure + atmospheric pressure; vacuum (negative gauge) pressure = atmospheric − absolute. • 1 atm = 101.325 kPa = 760 mm Hg = 10.33 m of water.

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Pressure head h = p/(ρg).

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Instrument Measures / feature Barometer Atmospheric pressure (mercury column ≈ 760 mm) Piezometer Gauge pressure of LIQUIDS only; simple tube; cannot measure gas or vacuum pressure Simple U-tube manometer Gauge and vacuum pressure of liquids/gases using heavier manometric liquid (Hg) Differential U-tube manometer Pressure DIFFERENCE between two points Inverted U-tube manometer Small pressure differences, uses lighter fluid (air/oil) on top Inclined manometer / micromanometer Very small pressures — better sensitivity (reading = L sin θ) Bourdon tube gauge Mechanical; elliptical curved tube straightens under pressure — high pressures Diaphragm, strain-gauge, piezoelectric transducers Electrical output, dynamic pressure measurement • Manometer equation: start at one point, add ρgh going DOWN, subtract going UP, equate at the other end. • Differential manometer with mercury under water: pA − pB = x(ρHg − ρw)g = 12.6 × ρwg·x (same level).

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Hydrostatic Force on a Plane Surface • Total pressure force F = ρ g A h̄ (h̄ = vertical depth of the CENTROID below free surface) — acts normal to the surface. • Centre of pressure (point of action): vertical surface h* = h̄ + IG/(A h̄); inclined surface h* = h̄ + IG sin²θ/(A h̄).

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The centre of pressure always lies BELOW the centroid (approaches it as depth increases). • Vertical rectangle (width b, depth d) with top edge at free surface:

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F = ρg b d²/2, h* = 2d/3 from the surface.

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IG of rectangle = bd³/12; circle = πD⁴/64. • Horizontal surface: pressure uniform, centre of pressure = centroid. • Related: buoyancy (Archimedes: upthrust = weight of fluid displaced, acts at centre of buoyancy).

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Floating body is stable if metacentre M lies above centre of gravity G (metacentric height GM > 0).

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Industrial Applications — Fluid Properties, Pressure and Instrumentation • Viscosity decides pumping power and pipe sizing for process fluids (oils, syrups, slurries, paints); non-Newtonian behaviour is common in industry — pseudoplastic (paint, ketchup, polymer melts), dilatant (starch slurry), Bingham plastic (toothpaste, sewage sludge, cement paste) — so equipment such as positive-displacement pumps and anchor-type agitators is used instead of centrifugal machines. • Pressure measurement in plants:

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Bourdon gauges, manometers, diaphragm and piezo-resistive pressure transmitters (4–20 mA signal to PLC/SCADA); vacuum gauges on filters and evaporators; differential pressure across filters and bag houses indicates clogging (a maintenance trigger, Chapter 8). • Hydrostatic principles in industry: level measurement in tanks by hydrostatic head, forces on tank walls and gates, hydraulic presses and jacks (Pascal's law, F2 = F1A2/A1) in forming, moulding and baling machines, buoyancy in flotation and level floats. • Surface tension and capillarity matter in coating, printing, dyeing, wetting agents and in spray droplet formation.

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Kinematics describes fluid motion without considering the forces causing it: types of flow, the Reynolds number, flow lines, stream function, velocity potential, rotation, vorticity and circulation.

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Types of Fluid Flow Flow type Condition Example Steady Properties at a point do NOT change with time: ∂V/∂t = 0 Constant discharge through a pipe Unsteady ∂V/∂t ≠ 0 Flow while a valve is being closed; emptying tank Uniform Velocity does NOT change with position (at an instant): ∂V/∂s = 0 Flow in a long straight pipe of constant diameter Non-uniform ∂V/∂s ≠ 0 Flow through a tapering pipe or around a bend Laminar Particles move in smooth layers; viscous forces dominate Oil in thin tubes, blood in capillaries Turbulent Random, eddying motion with strong mixing Rivers, most pipe flows, atmosphere Compressible Density varies (ρ ≠ const); significant when Mach > 0.3 High-speed gas flow, nozzles, jets Incompressible ρ = constant Liquids; low-speed air (Ma < 0.3) Rotational Particles rotate about their own axis (vorticity ≠ 0) Forced vortex, flow near walls (boundary layer) Irrotational No rotation of particles (vorticity = 0) Free vortex, flow outside boundary layer 1-D / 2-D / 3-D Number of space coordinates needed to describe velocity Pipe (1-D), flow past long cylinder (2-D) • Combinations: steady-uniform (constant Q in constant-diameter pipe); steady-non-uniform (constant Q in tapering pipe); unsteady-uniform (accelerating flow in constant-diameter pipe); unsteady-non-uniform (varying Q in tapering pipe, waves). • Mach number Ma = V/c, speed of sound c = √(γRT) (≈ 343 m/s in air at 20 °C).

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Subsonic Ma < 1; transonic ≈ 0.8–1.2; supersonic 1–5; hypersonic > 5.

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Mach angle sin α = 1/Ma.

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Laminar and Turbulent Flow;

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Reynolds Number • Reynolds number Re = ρVD/μ = VD/ν = inertia force ÷ viscous force.

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Demonstrated by Osborne Reynolds' dye experiment (1883). • Pipe flow: laminar Re < 2000 (≈ 2300); transitional 2000–4000; turbulent Re > 4000. • Flat plate boundary layer: transition at Rex ≈ 5 × 105.

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Open channel: laminar Re < 500 (based on hydraulic radius). • Turbulent flow has higher friction losses, flatter velocity profile and better mixing/heat transfer.

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Streamline, Pathline and Streakline • Streamline: imaginary line tangent to the velocity vector at every point at an instant.

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No flow crosses a streamline; two streamlines cannot intersect.

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A bundle of streamlines forms a stream tube. • Pathline: actual path traced by a single particle over time. • Streakline: locus of all particles that have passed through a fixed point (e.g., dye or smoke injected at a point). • In steady flow, streamlines, pathlines and streaklines coincide. • Acceleration = local (temporal) (∂V/∂t, zero in steady flow) + convective (V ∂V/∂s, zero in uniform flow).

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Continuity, Stream Function and Velocity Potential • Continuity (3-D incompressible): ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. • Stream function ψ (2-D): u = ∂ψ/∂y, v = −∂ψ/∂x.

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Defined for 2-D incompressible flow and automatically satisfies continuity. ψ = constant along a streamline; ψ2 − ψ1 = flow rate per unit width between the two streamlines.

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Satisfies Laplace's equation only if the flow is irrotational. • Velocity potential φ: u = −∂φ/∂x, v = −∂φ/∂y (sign convention varies; some texts use +).

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Exists ONLY for irrotational flow.

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For incompressible irrotational flow it satisfies Laplace's equation ∇²φ = 0. • Lines of constant φ (equipotential lines) are perpendicular to streamlines (ψ = const).

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Together they form a flow net (used for seepage under dams).

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Flow nets are valid only for irrotational (potential) flow.

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Rotation, Vorticity and Circulation • Rotation about z-axis: ωz = ½(∂v/∂x − ∂u/∂y). • Vorticity Ω = 2ω = ∇ × V.

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Flow is irrotational if vorticity is zero everywhere. • Circulation Γ = ∮ V·ds (line integral of tangential velocity around a closed curve) = ∫∫ Ω dA (Stokes' theorem).

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So vorticity = circulation per unit area. • Free vortex: irrotational, V·r = constant (whirlpool, flow into a sink/drain, flow in casing of pump).

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Forced vortex: rotational, V = ωr like a solid body (liquid in rotating cylinder, flow inside impeller) — free surface is a paraboloid with rise z = ω²r²/2g. • Lift on an aerofoil/rotating cylinder is linked to circulation:

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Kutta-Joukowski theorem L = ρVΓ per unit span;

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Industrial Applications — Flow Regimes and Process Flows • Reynolds number decides pipe friction and mixing: laminar (Re < 2000) in viscous oil and polymer lines, turbulent (Re > 4000) in most water, air and steam lines.

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Turbulence is wanted for mixing and heat transfer but costs pressure drop. • Steady vs unsteady flow: process plants are designed for steady operation, but start-up, shut-down, valve closure (water hammer) and batch operations are unsteady — surge tanks, air vessels and slow-closing valves protect the piping. • Streamlines and flow visualisation (smoke, dye, tracers, CFD) are used in duct, hood and mixer design; residence-time distribution studies in reactors and mixing tanks come from the same ideas. • Compressible flow appears in compressed-air and steam lines and in pneumatic conveying of powders; air can be treated as incompressible when velocity is low (Mach < 0.3).

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The governing equations of fluid flow are conservation of mass (continuity), energy (Euler and Bernoulli) and momentum.

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This section also covers flow-measuring devices based on Bernoulli, the momentum equation, and dimensional analysis and similitude.

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Continuity, Euler and Bernoulli Equations • Continuity: ρ1A1V1 = ρ2A2V2; incompressible Q = A1V1 = A2V2. • Euler's equation of motion (along a streamline): dp/ρ + V dV + g dz = 0.

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Assumptions: ideal (inviscid) fluid, steady flow, flow along a streamline; only pressure and gravity forces. • Bernoulli's equation (integrating Euler for incompressible fluid): p/ρg + V²/2g + z = H = constant — pressure head + velocity (kinetic) head + datum (potential) head.

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It is an energy equation (energy per unit weight). • Assumptions of Bernoulli: steady, incompressible, inviscid (no friction), along a single streamline, irrotational (for across streamlines), no shaft work or heat transfer. • Real fluids: p1/ρg + V1²/2g + z1 = p2/ρg + V2²/2g + z2 + hL (+ hpump/ − hturbine). • Energy grade line (EGL) = total head line; hydraulic grade line (HGL) = p/ρg + z.

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EGL lies above HGL by V²/2g;

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EGL always falls in the flow direction (without a pump). • Kinetic-energy correction factor α: laminar pipe flow 2.0; turbulent ≈ 1.03–1.1.

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Applications of Bernoulli's Equation Device Principle / formula Remarks Venturimeter Q = Cd A1A2√(2gh)/√(A1² − A2²); h = x(Sm/S − 1) with manometer Cd ≈ 0.95–0.99; convergent cone ~21°, divergent 5–7° (long) to reduce loss Orifice meter Same formula with orifice area Cd ≈ 0.6–0.65; cheap, compact, higher head loss Flow nozzle Intermediate between venturi and orifice Cd ≈ 0.95–0.98 Pitot tube Measures stagnation pressure:

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V = Cv√(2gh) Point velocity measurement; pitot-static tube (aircraft airspeed) Orifice / tank (Torricelli) Jet velocity V = √(2gH) Cc ≈ 0.64, Cv ≈ 0.97–0.98, Cd = Cc × Cv ≈ 0.62; jet smallest at vena contracta Rectangular notch/weir Q = (2/3) Cd L √(2g) H3/2 Large flows in channels Triangular (V) notch Q = (8/15) Cd tan(θ/2) √(2g) H5/2 More accurate for LOW flows;

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90° notch common Rotameter Variable-area meter with float in tapered tube Direct reading, vertical installation Momentum Equation • Impulse-momentum principle (Newton's 2nd law for a control volume): ΣF = ṁ(V2 − V1) = ρQ(V2 − V1) — applied separately in x and y directions. • Uses: force on pipe bends and reducers, jet propulsion, force of jets on vanes (turbines), rocket thrust.

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Momentum correction factor β: laminar 4/3. • Moment of momentum → torque T = ρQ(Vw1r1 − Vw2r2) — basis of Euler's turbomachine equation (lawn sprinkler, turbines, pumps).

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Jet striking Force in jet direction Stationary flat plate normal to jet F = ρaV² Stationary flat plate inclined at θ to jet Normal force Fn = ρaV² sin θ Stationary curved vane, jet deflected 180° (hemispherical) F = 2ρaV² (maximum) Jet striking Force in jet direction Single moving flat plate (velocity u) F = ρa(V − u)² Series of flat plates on a wheel F = ρaV(V − u); max efficiency 50% at u = V/2 Series of hemispherical vanes (Pelton ideal) Max efficiency 100% at u = V/2 Dimensional Analysis • Fundamental dimensions:

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M, L, T (and θ for temperature).

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Force = MLT−2; pressure = ML−1T−2; μ = ML−1T−1; ν = L²T−1; power = ML²T−3; σ = MT−2. • Dimensional homogeneity (Fourier): every term in a physically meaningful equation has the same dimensions. • Rayleigh's method: expresses a variable as a power product of others — suitable for ≤ 3–4 variables. • Buckingham π-theorem: n variables with m fundamental dimensions → (n − m) dimensionless π-terms.

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Choose m repeating variables (one geometric, one flow/kinematic, one fluid property — e.g., D, V, ρ); they must not form a dimensionless group themselves, and the dependent variable must not be a repeating variable.

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Number Definition Force ratio Important in Reynolds Re ρVL/μ Inertia / Viscous Pipe flow, aircraft (subsonic), submarines, fully submerged bodies Froude Fr V/√(gL) Inertia / Gravity (square root) Open channels, ships, spillways, free-surface flows Euler Eu V/√(p/ρ) Inertia / Pressure Pressure-dominated flows, cavitation studies Weber We V/√(σ/ρL) Inertia / Surface tension Droplets, capillary flow, thin films, sprays Mach Ma V/c Inertia / Elastic (square root) Compressible, high-speed flows (Cauchy number = Ma²) Similitude and Model Laws • Geometric similarity: all length ratios equal (Lr = Lm/Lp).

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Kinematic similarity: velocity/acceleration ratios equal.

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Dynamic similarity: ratios of all forces equal (requires geometric and kinematic similarity). • Reynolds model law (viscous forces dominant — pipes, submarines):

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Rem = Rep. • Froude model law (gravity dominant — ships, dams, spillways, rivers):

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Frm = Frp → Vr = √Lr; time Tr = √Lr;

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Qr = Lr 2.5; force Fr = Lr³; power Pr = Lr 3.5. • Mach model law for compressible flows;

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Weber law for surface-tension flows. • Distorted models: different horizontal and vertical scales (rivers, harbours) so that depth is measurable.

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Industrial Applications — Flow Measurement and Process Instrumentation • Bernoulli-based (differential-pressure) meters: orifice plate (cheap, high permanent pressure loss ≈ 50–70% of Δp, easily damaged by wear), venturi meter (low loss ≈ 10%, costly, used for large water lines), flow nozzle, Pitot tube (duct air velocity traverse).

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Q = CdA2√(2gΔh/(1 − (A2/A1)²)). • Other industrial flow meters: rotameter (variable area), electromagnetic (conductive liquids, no obstruction), ultrasonic/transit-time and Doppler (clamp-on, no cutting of pipe), Coriolis (direct mass flow, high accuracy for costly fluids), turbine, positive-displacement (fuel and oil custody), vortex-shedding, thermal-mass meters for gases; weirs and flumes for open channels (water supply and effluent discharge). • Metering is the basis of material balances, utility accounting and process control — energy and water audits depend on it (Chapter 9); calibration of flow meters is part of the measurement-system analysis in quality control (Chapter 7). • Momentum equation is used for forces on pipe bends and anchor blocks, jet-cleaning and hydro-cutting nozzles, and for sizing thrust supports; dimensional analysis and similitude underlie model testing of turbines, pumps, fans, mixers and ship/aero models, and the affinity laws used in plant pump/fan selection.

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Viscous effects cause energy losses in pipes and drag on bodies.

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This section covers exact laminar solutions for pipes and parallel plates, major and minor losses, the boundary layer and its thicknesses, and flow separation.

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Laminar Flow in a Circular Pipe (Hagen-Poiseuille) • Velocity profile is parabolic: u = (−dp/dx)(R² − r²)/(4μ). umax = 2 Vavg (at centre). • Shear stress varies linearly with radius: zero at the centre, maximum at the wall: τ = (−dp/dx)·r/2. • Hagen-Poiseuille equation: Δp = 32 μ V L / D²; head loss hf = 32μVL/(ρgD²); discharge Q = πΔp D⁴/(128 μ L). • Darcy friction factor for laminar flow: f = 64/Re (independent of roughness). • Kinetic-energy correction factor α = 2; momentum correction factor β = 4/3.

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Laminar Flow between Parallel Plates • Both plates fixed (plane Poiseuille flow): parabolic profile, umax = 1.5 Vavg; pressure drop Δp = 12 μ V L / t² (t = gap). • Couette flow: one plate moving, no pressure gradient → linear velocity profile, τ = μU/t — models journal bearings (Petroff's equation). • Laminar flow through narrow gaps is used for viscometers, dashpots and bearings.

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Turbulent Flow and Major Losses • Darcy-Weisbach equation (major loss due to friction): hf = f L V²/(2 g D).

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Darcy f = 4 × Fanning friction factor f′ (hf = 4f′LV²/2gD). • Moody chart: f vs Re and relative roughness ε/D.

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Laminar zone f = 64/Re; smooth turbulent (Blasius): f = 0.316/Re0.25 (Re < 105); fully rough zone: f depends only on ε/D (Colebrook-White covers the transition). • Turbulent velocity profile is flatter (1/7th power law); umax ≈ 1.2 Vavg.

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A thin laminar (viscous) sub-layer exists near the wall — pipe is hydraulically smooth if roughness is buried in it. • hf ∝ V (laminar) but ∝ V1.75–2 (turbulent).

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Chezy formula for channels:

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V = C√(m i) (m = hydraulic mean depth = A/P). • Pipes in series: same Q, head losses add.

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Pipes in parallel: same head loss in each branch, discharges add.

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Equivalent pipe (Dupuit):

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L/D5 = ΣLi/Di 5. • Power transmission through a pipe is maximum when hf = H/3 → maximum efficiency 66.7%. • Water hammer: pressure surge from sudden valve closure; rise p = ρVc (rigid pipe, rapid closure).

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Controlled by slow closure, surge tanks, relief valves.

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Minor Losses Loss Head loss Sudden expansion h = (V1 − V2)²/2g (Borda-Carnot) Sudden contraction h = (1/Cc − 1)² V2²/2g ≈ 0.5 V2²/2g Entrance to pipe (sharp) 0.5 V²/2g Exit from pipe into tank V²/2g (K = 1) Bends, elbows, valves, fittings K V²/2g (K from tables); gate valve open ≈ 0.2, globe valve ≈ 10 Obstruction in pipe (A/(Cc(A − a)) − 1)² V²/2g • Minor losses are small in long pipes (L/D > 1000 — often neglected) but important in short pipes with many fittings.

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Equivalent length Le = K D/f.

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Boundary Layer • Boundary layer (Prandtl, 1904): thin region near a surface where velocity rises from zero (no-slip) to ~99% of free-stream velocity U; viscous effects are confined to it.

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Outside, flow is treated as inviscid. • Boundary layer thickness δ: distance where u = 0.99U. • Displacement thickness δ* = ∫(1 − u/U)dy — distance the wall is effectively displaced due to reduced flow. • Momentum thickness θ = ∫(u/U)(1 − u/U)dy — loss of momentum; used for drag. • Energy thickness δe = ∫(u/U)(1 − u²/U²)dy.

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Shape factor H = δ*/θ (laminar ≈ 2.59, turbulent ≈ 1.3–1.4).

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Flat plate Laminar (Rex < 5 × 105) Turbulent Thickness δ/x = 5/√Rex (Blasius, 4.91) — δ ∝ x0.5 δ/x = 0.37/Rex 0.2 — δ ∝ x0.8 Local skin friction Cf 0.664/√Rex 0.0592/Rex 0.2 Average drag coefficient CD 1.328/√ReL 0.074/ReL 0.2 Velocity profile Parabolic-like, gradual Fuller, steep near wall (laminar sub-layer) Flow Separation • Separation occurs under an adverse pressure gradient (dp/dx > 0) — pressure increasing in the flow direction (e.g., rear of a cylinder, diffuser).

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Fluid near the wall, already slowed by friction, cannot overcome the rising pressure and reverses. • At the separation point (∂u/∂y)wall = 0 (wall shear stress = 0).

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Downstream: reverse flow, eddies and a wake → large pressure (form) drag. • Favourable pressure gradient (dp/dx < 0) keeps the boundary layer attached.

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Turbulent boundary layers resist separation better than laminar ones (more momentum near wall) — reason for dimples on golf balls. • Control methods: streamlining the body, suction of boundary layer, blowing/injection of high-energy fluid, vortex generators, slotted wings and flaps, rotating cylinder at the surface. • Total drag = skin-friction drag + pressure (form) drag.

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FD = CD·½ρV²A; lift FL = CL·½ρV²A.

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Streamlined bodies: mostly skin friction; bluff bodies: mostly pressure drag.

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Industrial Applications — Piping Systems, Pumping Cost and Duct Design • Pumping energy cost is a major plant expense: power P = ρgQH/η (W).

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Head loss in pipes (Darcy-Weisbach hf = fLV²/2gD, or Hazen-Williams for water) rises with V², so oversizing the pipe one size usually pays for itself through lower friction; economic pipe diameter balances pipe capital cost against pumping cost (typical design velocities: water ≈ 1–3 m/s, compressed air ≈ 6–10 m/s, steam ≈ 15–40 m/s). • Minor losses (bends, valves, fittings, sudden expansions) can dominate in short plant piping — h = KV²/2g, or equivalent length; smooth long-radius bends and correctly chosen valves (gate/butterfly for isolation, globe for throttling) reduce losses.

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Throttling a valve to control flow wastes energy — a VFD is the efficient alternative. • Laminar viscous flow theory covers lubrication of bearings, oil films, viscometers and the flow of viscous products in pipes and dies; the Hagen-Poiseuille equation gives pressure drop in small tubes. • Boundary layer and separation explain drag on bodies, losses in diffusers and hoods, and the design of local exhaust ventilation (capture velocity, hood entry loss), cyclones and dust-collection ducts. • Fluid-system maintenance: scaling and fouling raise roughness and friction; filters and strainers must be cleaned; leak surveys (water and compressed air) are standard energy-audit items.

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Hydraulic turbines convert the energy of water (head and flow) into mechanical shaft power — the core of hydropower, Nepal's key energy resource.

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This section covers classification, working principles, components, governing, cavitation, performance curves and draft tubes.

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Classification of Turbines Basis Types Energy at inlet Impulse: all available head converted to KE in nozzle; runner at atmospheric pressure (Pelton, Turgo, cross-flow/Banki).

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Reaction: water enters with pressure + KE; pressure drops through runner; runner fully submerged and enclosed (Francis, Kaplan, propeller, Deriaz, bulb) Direction of flow Tangential (Pelton); radial inward (old Francis); mixed (modern Francis); axial (Kaplan, propeller) Head High head > 250 m (Pelton); medium 30–250 m (Francis); low < 30 m (Kaplan/propeller) Specific speed Low (Pelton), medium (Francis), high (Kaplan) Shaft Horizontal or vertical Turbine Type / flow Head Specific speed Ns (SI, kW) Pelton (single jet) Impulse, tangential High (> 250 m) ≈ 8.5–30 Pelton (multi-jet) / Turgo Impulse High–medium 30–60 Francis Reaction, mixed/radial Medium (30–250 m) ≈ 50–300 Kaplan / Propeller Reaction, axial Low (< 30 m), large flow ≈ 300–1000 • Specific speed of a turbine Ns = N√P / H5/4 — speed of a geometrically similar turbine producing unit power under unit head.

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Used to select turbine type. • Nepal examples:

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Kulekhani-I and Upper Tamakoshi use Pelton turbines (high head);

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Kaligandaki-A uses Francis turbines (medium head). • Hydropower P = η ρ g Q H (W).

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Working Principle and Components • Pelton wheel: water from penstock → nozzle with spear (needle) regulates flow and forms a jet → jet strikes double-cup buckets with a central splitter; jet deflected by ≈ 160–165°.

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Casing has no hydraulic function (prevents splashing, guides water to tailrace).

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Breaking jet stops the runner; jet deflector diverts the jet during sudden load rejection (avoids water hammer from fast spear closure). • Pelton design: jet ratio m = D/d (≈ 12; minimum ≈ 10); number of buckets Z ≈ 15 + D/(2d); speed ratio u/V ≈ 0.45–0.47 (theoretical 0.5);

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Max hydraulic efficiency = (1 + k cos φ)/2. • Francis turbine: spiral (scroll) casing with decreasing area gives uniform flow → stay vanes (fixed, structural) → guide vanes / wicket gates (adjustable — regulate flow and direct water at correct angle) → runner with fixed curved blades → draft tube → tailrace. • Kaplan turbine: axial-flow reaction turbine with 3–8 adjustable runner blades + adjustable guide vanes — double regulation → high efficiency over a wide load range.

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Propeller turbine: same but fixed blades. • Degree of reaction R = pressure energy change in runner / total energy change (R = 0 for impulse). • Euler's turbine equation: work per unit weight = (Vw1u1 ± Vw2u2)/g.

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Maximum when exit whirl Vw2 = 0 (radial/axial discharge). • Efficiencies: hydraulic ηh = runner power / water power (ρgQH); mechanical ηm = shaft power / runner power; volumetric ηv; overall ηo = ηh × ηm (× ηv) = shaft power / ρgQH. • Unit quantities (for predicting behaviour under different heads): unit speed Nu = N/√H, unit discharge Qu = Q/√H, unit power Pu = P/H3/2.

14

Turbine Governors • A governor keeps turbine speed (and hence generator frequency, 50 Hz) constant when load changes, by regulating water flow. • Oil-pressure governor components: speed-sensing element (centrifugal fly-balls or electronic speed sensor), distribution/relay valve, servomotor (oil-pressure piston), oil pump and sump, gate-operating mechanism. • Pelton: governor moves the spear slowly and the jet deflector quickly (to avoid water hammer).

15

Francis/Kaplan: governor moves the guide vanes (wicket gates); a relief valve or surge tank limits pressure rise.

16

Kaplan also adjusts runner blades. • Modern plants use electro-hydraulic / digital (PID) governors.

17

Cavitation • Cavitation: when local absolute pressure falls below the vapour pressure, vapour bubbles form; carried to high-pressure regions they collapse violently producing very high local pressures. • Effects: pitting and erosion of blades, noise, vibration, drop in efficiency and output. • In reaction turbines it occurs at the runner blade exit and draft tube inlet (lowest pressure); in pumps at the impeller eye (inlet).

18

Pelton turbines are relatively free from it (runner at atmospheric pressure). • Thoma's cavitation factor σ = (Hatm − Hv − Hs)/H (Hs = suction height of runner above tailrace).

19

Cavitation avoided if σ > σcritical. • Prevention: set the turbine low relative to tailrace (limit suction head), use cavitation-resistant materials (stainless steel, weld overlays), proper blade design, avoid part-load/overload operation, admit air.

20

Performance Curves • Main (constant head) characteristics:

21

Q, P and η plotted against unit speed Nu for different gate openings. • Operating characteristics (constant speed): efficiency vs load.

22

Kaplan and Pelton have flat efficiency curves (good at part load);

23

Francis and especially propeller turbines have peaky curves (efficiency drops at part load). • Constant-efficiency (iso-efficiency / 'hill' / muschel) curves: lines of equal efficiency on the Nu–Qu plane — show best operating range.

24

Draft Tube • A gradually diverging pipe connecting the outlet of a reaction turbine runner to the tailrace.

25

Not used for Pelton (impulse) turbines. • Functions:

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(1) allows the turbine to be placed ABOVE tailrace without losing that head (creates suction/negative pressure at runner exit);

27

(2) recovers kinetic energy of the exit water by converting it into pressure energy (acts as a diffuser);

28

(3) allows access for inspection. • Types: conical (straight divergent) — cone angle ≤ 8° to avoid separation, η up to ~90%; simple elbow (~60%); elbow with varying section (large Kaplan units, low excavation);

29

Moody spreading / hydracone (bell-mouthed). • Draft tube efficiency = (V2² − V3² − hf)/V2².

30

Height of draft tube is limited by cavitation (pressure at inlet must stay above vapour pressure).

31

Industrial Applications — Turbines, Hydropower and Plant Water Systems • Nepal's industry depends on hydro-electricity:

32

Pelton wheels for high-head schemes, Francis for medium head and Kaplan/propeller for low head; cross-flow (Banki-Mitchell) turbines are widely used in Nepali micro-hydro because they are simple and locally manufacturable. • Plant-level applications: micro-hydro and mini-hydro for captive power in hill industries, hydraulic turbines/PATs (pumps as turbines) and energy-recovery turbines in pressure-reducing stations, and water-wheel driven agro-processing (traditional ghatta). • Cavitation damages runners, valves and pump impellers — it is limited by keeping pressure above vapour pressure (setting level, Thoma's cavitation factor σ); draft tube recovers kinetic energy and allows the runner to be set above tailwater. • Performance curves, part-load efficiency and governing (speed control) decide how well a machine follows a varying industrial load; specific speed guides selection. • Compressed-air, steam and gas turbines in plants follow the same turbomachinery principles; turbine and generator maintenance (alignment, vibration monitoring) belongs to condition-based maintenance (Chapter 8).

1

Pumps add energy to liquids to raise them or move them through pipes.

2

This section covers pump classification, the centrifugal and reciprocating pump, components, priming, NPSH and performance curves.

3

Classification of Pumps Class Types Features Rotodynamic (kinetic) Centrifugal (radial flow), mixed flow, axial flow (propeller) Continuous smooth flow; high discharge, moderate head; most common Positive displacement — reciprocating Piston, plunger, diaphragm; single/double acting Low discharge, very high head; pulsating flow; self-priming Positive displacement — rotary Gear, screw, vane, lobe Viscous liquids (oil), metering, hydraulics Special Jet pump (ejector), air-lift, hydraulic ram (uses water-hammer energy, no external power), submersible, deep-well turbine pump Specific applications Centrifugal Pump — Working Principle and Components • Works on the principle of forced vortex: the rotating impeller imparts kinetic energy and pressure to the liquid; the casing converts KE into pressure.

4

A centrifugal pump is the reverse of a radial inward-flow reaction turbine. • Impeller: closed/shrouded (clean liquids, best efficiency), semi-open, open (liquids with solids, slurries).

5

Blades usually backward-curved (most stable, non-overloading power characteristic); radial and forward-curved also exist. • Casing: volute casing (spiral of increasing area — converts KE to pressure), vortex chamber casing, diffuser (turbine-pump) casing with guide vanes (highest efficiency). • Suction pipe with foot valve (non-return) and strainer; delivery pipe with delivery valve; shaft, bearings, stuffing box/mechanical seal, wear rings. • Euler head (radial entry, Vw1 = 0):

6

H = Vw2u2/g. • Manometric head Hm = hs + hd + hfs + hfd + Vd²/2g (static lift + losses + delivery velocity head).

7

Manometric efficiency = gHm/(Vw2u2).

8

Overall efficiency = ρgQHm / shaft power. • Minimum starting speed: flow begins when centrifugal head (u2² − u1²)/2g ≥ Hm. • Multistage pumps: impellers in series for high head (boiler feed pumps); in parallel for large discharge.

9

Priming • Priming = completely filling the suction pipe, casing and impeller with the liquid before starting, to expel air. • Reason: head developed by a centrifugal pump depends on the liquid's DENSITY.

10

With air inside (≈ 1/800 the density of water), the pressure produced is negligible and cannot lift water from the sump. • Methods: manual filling with foot valve holding the water; vacuum/ejector priming; self-priming pumps.

11

Reciprocating pumps are self-priming. • Start a centrifugal pump with the delivery valve closed (minimum power at zero discharge for radial/backward-curved impellers); open gradually.

12

Net Positive Suction Head (NPSH) • NPSH = absolute pressure head at pump suction minus vapour pressure head:

13

NPSHa = patm/ρg − pv/ρg − hs − hfs (hs = suction lift above sump, hfs = suction pipe losses). • NPSH available (NPSHa) depends on the installation;

14

NPSH required (NPSHr) is given by the manufacturer.

15

To avoid cavitation:

16

NPSHa > NPSHr (with a safety margin). • Theoretical maximum suction lift of water ≈ 10.3 m; practically limited to about 6–7 m. • To increase NPSHa: lower the pump / raise the sump level (or use flooded suction), larger and shorter suction pipe, fewer fittings, cooler liquid (lower pv), pressurise the tank. • Thoma's cavitation factor for pumps σ = NPSH/Hm.

17

Performance Curves • Main characteristics:

18

Hm, P and η vs Q at several constant speeds.

19

Operating characteristics (at rated speed):

20

H–Q curve falls, power rises with Q, efficiency peaks at the best efficiency point (BEP).

21

Head at zero flow = shut-off head.

22

Iso-efficiency curves join points of equal efficiency. • System curve:

23

The operating point is the intersection of pump H–Q curve and system curve. • Series operation: heads add at the same Q.

24

Parallel operation: discharges add at the same H. • Affinity laws (similar pumps):

25

Q ∝ ND³, H ∝ N²D², P ∝ N³D⁵.

26

Same pump, changed speed:

27

Q ∝ N, H ∝ N², P ∝ N³. • Specific speed of a pump Ns = N√Q / H3/4: low for radial (centrifugal) high-head pumps, medium for mixed flow, high for axial-flow (propeller) pumps.

28

Reciprocating Pump • Piston moves in a cylinder with suction and delivery valves.

29

Theoretical discharge: single-acting Qth = ALN/60; double-acting ≈ 2ALN/60. • Slip = (Qth − Qact)/Qth.

30

Negative slip (Qact > Qth, Cd > 1) occurs with long suction pipe, short delivery pipe and high speed (delivery valve opens before end of suction stroke). • Air vessels (on suction and delivery sides): give nearly uniform flow, reduce acceleration head and friction work, prevent separation, allow higher speed. • Indicator diagram (pressure head vs stroke) area ∝ work done.

31

Feature Centrifugal pump Reciprocating pump Discharge Large, continuous, smooth Small, pulsating Head Moderate (high with multistage) Very high Priming Required Self-priming Liquids Clean, dirty, slurries (open impeller) Clean liquids (valves may clog) Size, cost, maintenance Compact, cheaper, low maintenance Bulky, costly, more wear parts Starting Delivery valve closed Delivery valve OPEN (else pressure builds dangerously) Industrial Applications — Pumps, Fans and Fluid Power in Plants • Pump selection: centrifugal pumps for clean low-viscosity liquids at high flow; positive-displacement pumps (gear, screw, lobe, vane, diaphragm, peristaltic, reciprocating) for viscous fluids, slurries, metering and high pressure.

32

Special types: submersible and borewell (deep-tube-well irrigation and water supply), self-priming, vacuum pumps, slurry pumps, dosing pumps. • Affinity (similarity) laws:

33

Q ∝ N, H ∝ N², P ∝ N³ — the basis of energy saving by speed control (VFD) on pumps and fans; trimming impellers achieves a similar effect permanently. • System curve and duty point: the pump should operate near its best efficiency point (BEP); running far off BEP causes recirculation, vibration and seal/bearing failures — a common cause of plant breakdowns. • NPSH:

34

NPSH available (from suction conditions) must exceed NPSH required (from the pump curve) with a margin, or the pump cavitates; keep suction lines short, straight and flooded, and prime centrifugal pumps before starting. • Fans and blowers follow the same laws and serve ventilation, dust extraction, pneumatic conveying, boiler draught (FD/ID fans) and drying; correct duct design and damper-free control save large amounts of energy. • Hydraulic and pneumatic power in factories: hydraulic presses, injection moulding and clamping (high force, precise control) and pneumatic cylinders, actuators and tools (fast, clean, low force) — an industrial engineer compares their energy cost, safety and maintenance.