Fluid Mechanics and Mechanical operations

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This section covers the types of fluids — ideal and actual, compressible and incompressible, Newtonian and non-Newtonian including time-dependent and time-independent behaviour — together with Newton's law of viscosity, surface tension and its effects, and fluid pressure and its measurement.

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What a Fluid Is • A fluid is a substance that deforms continuously under the action of a shear stress, however small that stress may be.

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This is the defining distinction from a solid, which deforms by a finite amount and then stops.

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Both liquids and gases are fluids; the difference between them is that a liquid has a definite volume and a free surface, while a gas expands to fill its container. • Continuum hypothesis: fluid mechanics treats the fluid as a continuous medium whose properties are defined at every point, ignoring its molecular structure.

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This is valid when the Knudsen number (mean free path divided by characteristic length) is much less than one, which holds for essentially all process equipment but fails in high-vacuum systems and in flow through very fine pores.

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Classification of Fluids Basis Type Description Viscosity Ideal fluid Zero viscosity, incompressible, no shear stress; a hypothetical fluid used to simplify analysis Real (actual) fluid Possesses viscosity, so shear stresses exist and friction losses occur; all practical fluids Density Incompressible Density essentially constant with pressure; all liquids and gases below about Mach 0.3 Compressible Density varies with pressure; gases at high velocity, steam, compressed air lines Stress-strain Newtonian Shear stress directly proportional to velocity gradient; water, air, most gases, light oils Non-Newtonian Relation not linear; slurries, polymer melts, paints, blood, pulp suspensions Newton's Law of Viscosity • τ = μ (du/dy), where τ is the shear stress, du/dy the velocity gradient (also called the rate of shear strain), and μ the dynamic or absolute viscosity.

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A fluid obeying this relation, with μ constant at a given temperature and pressure, is Newtonian. • Units: dynamic viscosity in SI is Pa·s = N·s/m² = kg/(m·s); the CGS unit is the poise, and 1 Pa·s = 10 poise, so 1 centipoise = 10−3 Pa·s.

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Water at 20 °C has a viscosity of very nearly 1 cP, which is the standard reference value. • Kinematic viscosity ν = μ/ρ, in m²/s; the CGS unit is the stoke, with 1 stoke = 10−4 m²/s. • Effect of temperature — an examination favourite because the two cases are opposite: the viscosity of a liquid decreases with rising temperature (cohesive forces between molecules weaken), while the viscosity of a gas increases with rising temperature (viscosity arises from molecular momentum transfer, which intensifies as molecules move faster).

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Pressure has little effect on liquid viscosity and almost none on gas viscosity at moderate pressures.

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Non-Newtonian Fluids Class Behaviour Model / index Examples Pseudoplastic Shear thinning: apparent viscosity falls as shear rate rises Power law τ = K(du/dy)ⁿ with n < 1 Polymer solutions, paints, blood, paper pulp Dilatant Shear thickening: apparent viscosity rises with shear rate Power law with n > 1 Starch in water, wet sand, concentrated slurries Bingham plastic Behaves as solid until a yield stress is exceeded, then flows like a Newtonian fluid τ = τ₀ + μ(du/dy) Toothpaste, drilling mud, sewage sludge, chocolate Herschel-Bulkl ey Yield stress plus power-law behaviour τ = τ₀ + K(du/dy)ⁿ Food pastes, cement slurries Thixotropic Time dependent: viscosity falls with time at constant shear, recovers on rest Hysteresis loop Paints, ketchup, some gels Rheopectic Time dependent: viscosity rises with time at constant shear Reverse hysteresis Gypsum pastes, some lubricants (rare) • The distinction between time-independent fluids (pseudoplastic, dilatant, Bingham — apparent viscosity depends only on the current shear rate) and time-dependent fluids (thixotropic, rheopectic — apparent viscosity also depends on how long shear has been applied) is regularly examined as a definition question. • Viscoelastic fluids, such as polymer melts and dough, show both viscous and elastic behaviour, recovering part of their deformation when the stress is removed; the Weissenberg effect — climbing of the fluid up a rotating rod — is the classic demonstration.

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Surface Tension and Capillarity • Surface tension σ arises because molecules at a liquid surface have unbalanced cohesive forces, making the surface behave like a stretched membrane.

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Units are N/m (or J/m², since it is also the free energy per unit area).

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For water at 20 °C, σ ≈ 0.073 N/m; it decreases as temperature rises and becomes zero at the critical point, and it is strongly reduced by surfactants. • Pressure inside a curved surface exceeds that outside: for a droplet or a liquid jet with one surface, Δp = 2σ/R for a droplet and σ/R for a cylindrical jet, while for a soap bubble with two surfaces, Δp = 4σ/R.

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The general result is the Young-Laplace equation Δp = σ(1/R₁ + 1/R₂). • Capillary rise h = 2σ cos θ /(ρgR), where θ is the contact angle.

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Water wets glass (θ < 90°) and rises; mercury does not wet glass (θ ≈ 130°) and is depressed.

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The rise is inversely proportional to the tube radius, which is why manometer tubes are made wide enough to keep capillary error small. • Cohesion is attraction between like molecules and adhesion between unlike ones; the balance of the two determines wetting behaviour.

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Fluid Pressure and Its Measurement • Pascal's law: at a point in a fluid at rest, the pressure is the same in all directions.

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This is the basis of the hydraulic press and the hydraulic jack. • Hydrostatic pressure variation: dp/dz = −ρg, integrating for a constant-density fluid to p = p₀ + ρgh.

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Pressure depends only on depth, not on the shape or the volume of the vessel — the hydrostatic paradox, in which vessels of very different shape but equal base area and liquid depth exert the same force on the base. • Absolute, gauge and vacuum pressure: absolute = atmospheric + gauge, and a vacuum (negative gauge) pressure is measured below atmospheric.

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Standard atmospheric pressure is 101.325 kPa = 1.01325 bar = 760 mm Hg = 10.33 m of water.

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Device Principle Typical use Piezometer Open vertical tube; liquid rises to a height proportional to gauge pressure Small positive pressures of a liquid only Device Principle Typical use U-tube manometer Balance of the unknown pressure against a column of manometric fluid Moderate pressures, gases or liquids Differential manometer Measures the difference between two points Pressure drop across orifices, filters, pipe runs Inclined-tube manometer Lengthens the reading for a given height difference by the factor 1/sin θ Very small pressure differences, draught in ducts Inverted U-tube Lighter fluid above; used where the manometric fluid must be lighter Small differences in liquid lines Bourdon gauge Curved flattened tube tends to straighten under internal pressure Industrial gauge pressure, robust and direct reading Diaphragm / bellows Elastic deflection of a membrane, often with electrical transduction Transmitters, low pressures, corrosive service Barometer Mercury column balanced against the atmosphere Absolute atmospheric pressure • For a simple U-tube manometer, the working equation is obtained by equating pressures at the lowest common level in the two limbs.

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For a manometer reading R with manometric fluid of density ρm and process fluid of density ρ, the pressure difference is Δp = R g (ρm − ρ) — note the difference of densities, which is a common slip in examinations. • Mercury is the traditional manometric fluid (density 13 600 kg/m³) but is progressively being replaced on toxicity grounds; carbon tetrachloride, dibutyl phthalate and coloured water or oil are used for smaller differences.

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This section covers fluid statics, the kinematics of fluid flow, viscous flow, an introduction to compressible flow, the basic equations of fluid flow, the velocity field and stream function, irrotational flow, the integral and differential analysis of fluid motion through the Reynolds transport theorem, the Euler and Bernoulli equations, and dimensional analysis and similitude.

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Fluid Statics • Force on a plane submerged surface: the total hydrostatic force F = ρg h̄ A, where h̄ is the depth of the centroid, and it acts at the centre of pressure, which lies at hcp = h̄ + IG sin²θ/(A h̄) — that is, always below the centroid, approaching it as the surface is submerged more deeply.

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For a vertical rectangular gate of depth H with its top at the surface, the centre of pressure is at 2H/3 from the surface. • Buoyancy — Archimedes' principle: a body immersed in a fluid experiences an upward force equal to the weight of the fluid displaced, acting through the centre of buoyancy, which is the centroid of the displaced volume. • Stability of floating bodies depends on the metacentre M, the point where the line of action of the buoyant force crosses the centreline when the body is tilted.

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Stable if M lies above the centre of gravity G (positive metacentric height GM); unstable if M is below G; neutral if they coincide.

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The metacentric height is GM = I/V − BG, where I is the second moment of the waterline area and V the displaced volume.

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For a fully submerged body, stability requires simply that the centre of buoyancy lie above the centre of gravity.

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Describing the Flow Term Meaning Steady flow Properties at a point do not change with time (∂/∂t = 0) Uniform flow Properties do not change with position at a given instant (∂/∂s = 0) Laminar flow Fluid moves in orderly layers;

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Re below about 2100 in a pipe Turbulent flow Random eddying motion with strong cross-mixing;

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Re above about 4000 Streamline Line everywhere tangent to the velocity vector at a given instant Pathline Actual trajectory traced by an individual fluid particle over time Streakline Locus of particles that have passed through a given point, as in a dye trace Streamtube Bundle of streamlines forming a tube through which no fluid crosses • In steady flow, streamlines, pathlines and streaklines coincide — a favourite one-line question.

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They differ only in unsteady flow. • Lagrangian description follows an individual fluid particle;

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Eulerian description, which is the one almost always used in engineering, fixes attention on a point in space and records what passes through it. • Substantial (material) derivative:

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D/Dt = ∂/∂t + (v·∇), connecting the two descriptions.

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The first term is the local acceleration (present only in unsteady flow) and the second the convective acceleration, which exists even in steady flow whenever the fluid passes through a contraction or an expansion.

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Continuity, Stream Function and Irrotational Flow • Continuity equation: in integral form for steady flow, ρ₁A₁v₁ = ρ₂A₂v₂, reducing for an incompressible fluid to A₁v₁ = A₂v₂ = Q.

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In differential form, ∂ρ/∂t + ∇·(ρv) = 0, which for an incompressible fluid becomes simply ∇·v = 0. • Stream function ψ is defined for two-dimensional incompressible flow by u = ∂ψ/∂y, v = −∂ψ/∂x.

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It satisfies continuity automatically, and lines of constant ψ are streamlines; the difference in ψ between two streamlines equals the volumetric flow rate between them. • Velocity potential φ is defined by u = −∂φ/∂x, v = −∂φ/∂y and exists only for irrotational flow.

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Where both exist, lines of constant φ and constant ψ are mutually orthogonal, forming the flow net, and both satisfy the Laplace equation ∇²φ = ∇²ψ = 0. • Rotational versus irrotational: flow is irrotational when the vorticity ∇ × v is zero, meaning fluid elements translate and deform but do not spin.

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Real viscous flow near a solid boundary is always rotational, because the no-slip condition creates a velocity gradient; flow far from boundaries is often nearly irrotational. • Circulation Γ is the line integral of velocity around a closed curve, Γ = ∮ v·dl, and equals the flux of vorticity through the enclosed area by Stokes' theorem.

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It is zero for irrotational flow in a simply connected region.

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Viscous Flow • No-slip condition: a real fluid in contact with a solid boundary has zero velocity relative to that boundary.

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This is the origin of the boundary layer and of all wall friction. • Laminar flow in a circular pipe (Hagen-Poiseuille): the velocity profile is parabolic, the maximum velocity is twice the average, and Q = πΔP R⁴/(8μL) = πΔP D⁴/(128μL).

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The pressure drop is therefore directly proportional to velocity and viscosity and inversely proportional to the fourth power of diameter. • Laminar flow between parallel plates gives a parabolic profile with maximum velocity 1.5 times the average. • Turbulent flow in a pipe has a much flatter, roughly one-seventh-power profile, with maximum velocity about 1.2 times the average; the pressure drop varies approximately as v1.75 to v². • Boundary layer: the thin region adjacent to a surface in which the velocity rises from zero at the wall to the free-stream value.

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On a flat plate it is laminar up to Rex ≈ 5 × 10⁵ and turbulent beyond.

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Boundary layer separation occurs when an adverse pressure gradient reverses the flow near the wall, producing a wake and greatly increased form drag; this is why diffusers are given small included angles and why streamlined shapes have low drag.

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Euler and Bernoulli Equations • Euler's equation along a streamline for an inviscid fluid: dp/ρ + v dv + g dz = 0. • Bernoulli's equation, its integral for steady, incompressible, inviscid, irrotational flow along a streamline: p/ρg + v²/2g + z = constant, the three terms being the pressure head, velocity head and elevation head, with the sum called the total head. • Assumptions worth memorising, because they are asked directly: steady flow, incompressible fluid, no friction (inviscid), no shaft work or heat transfer, and along a single streamline. • Modified (engineering) Bernoulli equation adds the real terms: p₁/ρg + α₁v₁²/2g + z₁ + hpump = p₂/ρg + α₂v₂²/2g + z₂ + hturbine + hf, where α is the kinetic-energy correction factor — 2 for laminar flow and very nearly 1 for turbulent flow. • Reynolds transport theorem is the bridge between a system (fixed mass) and a control volume (fixed region): the rate of change of any extensive property of the system equals the rate of change within the control volume plus the net flux out through the control surface.

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Applying it to mass gives continuity, to momentum gives the momentum equation, and to energy gives the steady-flow energy equation.

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Introduction to Compressible Flow • Compressibility matters when the Mach number Ma = v/c is significant, where c = √(γRT) is the speed of sound.

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The conventional threshold is Ma = 0.3, below which density variation is under about 5 per cent and the flow may be treated as incompressible. • Regimes: subsonic (Ma < 1), sonic (Ma = 1), supersonic (Ma > 1) and hypersonic above about 5. • Area-velocity relation: for subsonic flow a converging duct accelerates the fluid, exactly as for an incompressible liquid, but for supersonic flow the behaviour reverses and a diverging duct accelerates it.

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Hence the converging-diverging (de Laval) nozzle, in which Ma = 1 occurs at the throat. • Choked flow: once the throat reaches sonic velocity, further lowering of the downstream pressure cannot increase the mass flow rate.

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The critical pressure ratio p*/p₀ = [2/(γ+1)]γ/(γ−1), which is about 0.528 for air with γ = 1.4.

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This value is worth memorising. • A normal shock wave can occur only in supersonic flow; across it the flow becomes subsonic, the pressure, temperature and density rise, the stagnation pressure falls, and the entropy increases — the process is irreversible.

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Dimensional Analysis and Similitude • Buckingham π theorem: a physical relation among n variables involving m fundamental dimensions can be reduced to a relation among n − m independent dimensionless groups.

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For most fluid problems m = 3 (mass, length, time). • Similitude requires geometric similarity (same shape, constant scale ratio), kinematic similarity (similar velocity fields) and dynamic similarity (constant ratio of corresponding forces).

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For a model test to be valid, the governing dimensionless group must be equal in model and prototype.

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Group Definition Ratio of forces Governs Reynolds, Re ρvD/μ Inertial to viscous Pipe flow, most closed-conduit and submerged flow Froude, Fr v/√(gL) Inertial to gravity Open channels, spillways, ship resistance, wave motion Mach, Ma v/c Inertial to elastic Compressible and high-speed gas flow Euler, Eu Δp/(ρv²) Pressure to inertial Cavitation, pressure-drop correlations Weber, We ρv²L/σ Inertial to surface tension Droplet and bubble formation, atomisation, thin films Power number, NP P/(ρN³D⁵) Drag to inertial Power drawn by an agitator

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This section covers the friction factor, energy losses in fittings and valves, friction in pipes and channels, flow measuring devices, the pumping of fluids, and the agitation and mixing of liquids including impeller types and power consumption in agitated vessels.

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Friction in Pipes • Darcy-Weisbach equation: hf = fD (L/D)(v²/2g), the universal expression for friction head loss in a pipe.

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Written with the Fanning friction factor it becomes hf = 4fF(L/D)(v²/2g). • The four-times trap: fD = 4 fF.

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In laminar flow, fF = 16/Re while fD = 64/Re.

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Chemical engineering texts (McCabe, Coulson) generally use the Fanning factor and mechanical/civil texts the Darcy; always check which is intended. • In laminar flow the friction factor is independent of pipe roughness and the head loss is directly proportional to velocity.

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In fully turbulent (rough-pipe) flow the friction factor depends only on relative roughness ε/D and becomes independent of Reynolds number, and the head loss varies approximately as v². • Moody chart plots friction factor against Reynolds number for a family of relative-roughness curves and is the standard design tool.

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The Colebrook-White equation is its implicit algebraic form; the Blasius equation fF = 0.079 Re−0.25 is a convenient explicit approximation for smooth pipes with Re between 4 000 and 10⁵. • Equivalent (hydraulic) diameter De = 4 × flow area / wetted perimeter, used for non-circular ducts, annuli and open channels.

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For a circular pipe running full it returns the actual diameter; for a square duct of side a it gives a; for an annulus it gives Do − Di.

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Minor Losses • Losses in fittings, bends and valves are expressed either as h = K v²/2g, with K a loss coefficient, or as an equivalent length Le/D of straight pipe.

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They are called minor losses but in a short, fitting-rich line they can far exceed the straight-pipe friction.

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Fitting Typical K Equivalent L/D Sudden enlargement (into a large tank) 1.0 — Sudden contraction (from a large tank) 0.5 — Standard 90° elbow 0.75 30-40 Long-radius 90° elbow 0.45 20 Tee, flow through run 0.4 20 Gate valve, fully open 0.17 7-10 Globe valve, fully open 6.0-10 300-340 Check (non-return) valve 2.0-2.5 100-135 • Sudden enlargement loss is given exactly by the Borda-Carnot expression h = (v₁ − v₂)²/2g; when a pipe discharges into a large tank, v₂ ≈ 0 and the entire velocity head is lost, K = 1. • A gate valve has a very low loss when fully open and is used for isolation; a globe valve has a high loss because of its tortuous path but gives good throttling control.

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This comparison is regularly examined.

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Flow in Open Channels • Chezy's formula: v = C√(m i), and Manning's formula: v = (1/n) m2/3 i1/2, where m is the hydraulic mean depth (area over wetted perimeter), i the bed slope and n Manning's roughness coefficient. • Froude number Fr = v/√(gy) classifies open-channel flow: subcritical (tranquil) below 1, critical at 1, supercritical (rapid) above 1.

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A hydraulic jump, which dissipates energy, occurs on the transition from supercritical to subcritical flow. • The most economical section is the one with the smallest wetted perimeter for a given area: for a rectangular channel the depth equals half the width, and for a trapezoidal channel the sloping side equals half the top width.

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Flow Measuring Devices Device Principle Permanent loss Notes Orifice meter Differential pressure across a thin plate High (40-90 %) Cheap, compact, easily replaced;

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Cd ≈ 0.61 Venturi meter Gradual convergence and divergence Low (about 10 %) Expensive and long but accurate;

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Cd ≈ 0.98 Flow nozzle Contoured inlet, abrupt exit Intermediate Between orifice and venturi in cost and loss Pitot tube Difference of stagnation and static pressure Negligible Gives point velocity, not flow rate; used for traverses Rotameter Variable area, float in tapered tube Low and constant Constant pressure drop, variable area; direct reading Weir / notch Head over a crest in an open channel — Rectangular Q ∝ H¹.5, triangular Q ∝ H².5 Magnetic flowmeter Faraday induction by a moving conductor None Conductive liquids only; no obstruction, handles slurries Ultrasonic / Coriolis Transit time or vibrating tube None Coriolis measures true mass flow directly • The head meters all follow the same form:

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Q = Cd A₂ √[2gΔh/(1 − β⁴)], where β = d₂/d₁ is the diameter ratio.

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Flow rate is proportional to the square root of the pressure drop, so a four-fold increase in Δh doubles the flow — a very common numerical question. • The essential contrast: the venturi meter's gradual divergence recovers most of the pressure, so its permanent loss is small; the orifice's abrupt expansion dissipates most of it.

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Against that, the orifice is far cheaper, shorter and easier to maintain, which is why it remains the commonest industrial element. • Rotameter is the odd one out in the family: it operates at essentially constant pressure drop with a variable flow area, the reverse of the head meters, which have a fixed area and a variable pressure drop. • Pitot tube gives v = √(2gΔh) at a point; to obtain the flow rate a velocity traverse must be integrated across the section.

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Pumping of Fluids Class Examples Characteristics Centrifugal Radial, mixed and axial flow; single and multistage Smooth flow, no pulsation, handles suspensions, needs priming, head falls as flow rises Reciprocating Piston, plunger, diaphragm Positive displacement, high head, self-priming, pulsating flow, needs relief valve Rotary positive displacement Gear, lobe, screw, vane, peristaltic Steady flow, good for viscous liquids, self-priming Special Jet ejector, air lift, vacuum pumps No moving parts in the liquid; low efficiency • Centrifugal pump relations: the head developed is H = (u₂² − u₁²)/2g ideally, and the affinity laws state that at constant impeller diameter Q ∝ N, H ∝ N² and P ∝ N³.

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Doubling the speed therefore doubles the flow, quadruples the head and increases the power eight-fold. • A centrifugal pump must never be started against a closed suction valve but is normally started against a closed discharge valve, because its power demand is lowest at zero flow. • NPSH: net positive suction head available (NPSHA) = (patm − pvapour)/ρg − zsuction lift − hf,suction.

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Cavitation is avoided by keeping NPSHA greater than NPSHR, the value required by the pump.

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Cavitation — formation and violent collapse of vapour bubbles — causes noise, vibration, loss of head and pitting of the impeller, and is combated by lowering the pump, shortening and enlarging the suction line, or cooling the liquid. • Priming is necessary for a centrifugal pump because the head it generates is proportional to the fluid density, so a gas-filled casing produces almost no head.

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Positive-displacement pumps are self-priming. • Pump power: hydraulic power = ρgQH, and shaft (brake) power = ρgQH/η.

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Agitation and Mixing of Liquids Impeller Flow pattern Best suited to Propeller (marine type) Axial Low-viscosity liquids, high speed, blending large volumes Flat-blade turbine (Rushton) Radial Gas dispersion, liquid-liquid dispersion, moderate viscosity Pitched-blade turbine Mixed axial and radial Solid suspension, general blending, heat transfer Anchor / gate Tangential, close to wall High viscosity, scraping heat-transfer surfaces Helical ribbon Axial in a viscous bulk Very high viscosity, laminar regime, pastes and polymers • Standard tank geometry: impeller diameter Da = T/3, liquid depth equal to tank diameter T, impeller clearance T/3 from the bottom, and four baffles each T/10 wide. • Baffles are fitted to prevent the swirling vortex motion that would otherwise form, in which the liquid rotates as a body with little mixing and air is drawn into the impeller.

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Vortexing can alternatively be avoided by mounting the agitator off-centre or at an angle. • Impeller Reynolds number Re = ρNDa²/μ, with N the rotational speed in revolutions per second.

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Flow is laminar below about 10 and fully turbulent above about 10⁴. • Power number NP = P/(ρN³Da⁵).

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In the laminar region NP ∝ 1/Re, so P ∝ μN²Da³; in the fully turbulent region with baffles NP is constant, so P ∝ ρN³Da⁵.

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For a standard six-blade Rushton turbine the turbulent power number is about 5 to 6. • The Froude number appears only in unbaffled tanks, where vortexing makes gravity relevant; in a fully baffled tank the power number depends on Reynolds number alone.

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This section covers the characteristics of solid particles, the standard screen series and sieve analysis, size reduction and enlargement, crushers, grinders and disintegrators for coarse and intermediate duty, wet and dry grinding, energy and power requirements with the laws of crushing and the work index, and mechanical and pneumatic conveyors and elevators.

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Characteristics of Solid Particles • Sphericity ψs = surface area of a sphere of the same volume / actual surface area of the particle.

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It is 1 for a sphere and less than one for every other shape — about 0.81 for a cube, 0.7 to 0.8 for crushed rock and sand, and as low as 0.3 for mica flakes.

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For a non-spherical particle, specific surface a = 6/(ψs Dp). • Equivalent diameters: the volume equivalent (the diameter of a sphere of the same volume), the surface equivalent, the sieve diameter (the aperture the particle just passes) and the Stokes diameter (from settling velocity).

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A quoted particle size is meaningless unless the basis is stated. • Mean diameters: the volume-surface or Sauter mean diameter D̄s = 1/Σ(xi/Dpi) is the one used for surface-dependent calculations such as pressure drop and mass transfer; the arithmetic mean and the mass mean are also defined but are less useful in transport calculations. • Bulk density (mass of loosely packed solid per unit total volume) is always less than true density; the ratio is fixed by the voidage or porosity ε, with ρbulk = ρtrue(1 − ε).

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Random packing of uniform spheres gives ε of about 0.38 to 0.40. • Angle of repose, the steepest angle of a freely formed heap, measures flowability: free-flowing solids give 25-30° and cohesive ones over 45°.

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Standard Screens and Sieve Analysis • Tyler standard screen series is built on a 200-mesh screen of 0.074 mm aperture, with successive apertures in the ratio √2 = 1.41; where a closer classification is needed, intermediate screens in the ratio ⁴√2 = 1.189 are inserted.

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US standard sieves use the same principle with slightly different designations. • Mesh number is the number of openings per linear inch, so a higher mesh number means a smaller aperture — and the aperture is not simply the reciprocal of the mesh, because the wire diameter must be subtracted. • Sieve analysis stacks screens with the coarsest at the top, shakes for a standard time, and weighs the retained fractions.

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Results are reported as differential analysis (mass fraction in each size interval) or cumulative analysis (mass fraction finer or coarser than each aperture). • Notation: a fraction written as −14 + 20 mesh means the material passed the 14-mesh screen and was retained on the 20-mesh screen.

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Reading this notation correctly is a standard examination requirement.

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Principles and Laws • Objectives of size reduction are to increase surface area for reaction, dissolution or drying, to liberate a valuable mineral from its gangue, to meet a product specification, and to make solids easier to handle and mix. • Mechanisms are compression (for coarse reduction), impact (general-purpose), attrition or rubbing (for fine products), and cutting (for a definite size with few fines). • Size reduction is extremely inefficient: usually less than 1 per cent of the energy supplied creates new surface, the rest appearing as heat, noise and elastic deformation. • Reduction ratio = feed size / product size; it is typically 3-6 for coarse crushers, up to 100 for fine grinders.

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Law Statement Energy relation Applies to Kick's law Energy is proportional to the size-reduction ratio E = KK ln(Df/Dp) Coarse crushing, large feed Bond's law Energy is proportional to the new crack length produced E = KB(1/√Dp − 1/√Df) Intermediate grinding — most industrially useful Rittinger's law Energy is proportional to the new surface created E = KR(1/Dp − 1/Df) Fine grinding, small product size • All three are special cases of the general differential expression dE/dD = −C/Dn, with n = 1 giving Kick, n = 1.5 giving Bond and n = 2 giving Rittinger.

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The mnemonic is the order Kick-Bond-Rittinger for coarse-intermediate-fine. • Bond work index Wi is the energy in kWh per tonne required to reduce a very large feed to a product of which 80 per cent passes 100 micrometres.

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Bond's law is then written W = 10 Wi(1/√Dp − 1/√Df) with sizes in micrometres, where Df and Dp are the 80 per cent passing sizes of feed and product.

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Crushing and Grinding Equipment Equipment Duty Mechanism Notes Jaw crusher Coarse (primary) Compression Fixed and swinging jaw; reduction ratio about 6; blake and dodge types Gyratory crusher Coarse (primary) Compression Continuous discharge, higher capacity than a jaw crusher for the same opening Smooth roll crusher Intermediate Compression Limited reduction ratio of about 4; angle of nip governs feed size Toothed roll crusher Coarse to intermediate Compression and shear Handles friable and sticky materials, higher reduction ratio Hammer mill Intermediate to fine Impact Swinging hammers, versatile, produces fines and heat Ball mill / tube mill Fine Impact and attrition Rotating cylinder with balls; wet or dry; the commonest fine grinder Rod mill Intermediate Impact and attrition Rods give a more uniform product with fewer fines than a ball mill Fluid energy (jet) mill Ultrafine Attrition by particle collision No moving parts and little heat; used for pharmaceuticals and pigments Attrition (disc) mill Intermediate to fine Shear and rubbing Grooved discs, one or both rotating; for tough and fibrous solids Cutter / knife mill Definite size Cutting For tough, fibrous, rubbery solids; gives few fines • Angle of nip is the angle formed by the tangents to the two rolls at the point of contact with the particle.

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The particle is gripped only if the angle of nip does not exceed twice the angle of friction; typical values are about 31 degrees, and for smooth rolls the largest feed particle is roughly 1/20 of the roll diameter. • Ball mill critical speed is the speed at which the outermost ball is held against the shell by centrifugal force and ceases to fall:

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Nc = (1/2π)√(g/(R − r)) in revolutions per second.

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Operation is normally at 65-80 per cent of the critical speed, since grinding stops entirely at and above it — a very frequently examined fact. • Wet grinding gives lower power consumption per tonne, no dust, a finer product and easier handling, but causes more wear and requires subsequent drying.

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Dry grinding avoids the drying step but creates dust hazards and generally consumes more energy for the same fineness. • Open-circuit grinding passes the material once; closed-circuit grinding returns oversize from a classifier to the mill and is more efficient, avoiding over-grinding of material already fine enough. • Size enlargement — the reverse operation — includes granulation, pelletizing, briquetting, extrusion, sintering and spray drying, and is used to improve flow, reduce dust, control dissolution rate and prevent segregation.

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Conveying of Solids Conveyor Principle Suited to Belt conveyor Endless belt over idlers Long horizontal or gently inclined runs; large tonnages; cheapest per tonne-kilometre Screw conveyor Rotating helix in a trough Short distances, horizontal or inclined; dusty or hot solids; some mixing occurs Bucket elevator Buckets on a belt or chain Vertical lift of free-flowing bulk solids Chain / apron conveyor Slats or chain-drawn flights Hot, heavy, abrasive lumps Vibrating conveyor Oscillating trough Gentle handling; conveys and screens or cools at the same time Pneumatic conveyor Solids carried in a moving air stream Flexible routing, dust-free and enclosed, multiple sources and destinations • Pneumatic conveying is divided into dilute phase (low solids loading, high velocity, solids fully suspended) and dense phase (high loading, low velocity, solids moving as plugs or a moving bed).

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Dense phase is gentler on friable materials and less abrasive but needs higher pressure. • Pneumatic systems may be negative pressure (vacuum), useful for drawing from several points to one destination; positive pressure, for delivering from one point to several; or combined.

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The main advantages are complete enclosure, flexible routing and freedom from dust; the main drawbacks are high power consumption, wear at bends and attrition of the solids.

1

This section covers screen analysis, the estimation of particle size, surface area and particle population from screen data, ideal and actual screens with screen effectiveness, and the principles of elutriation, flotation, jigging, electrostatic and magnetic separation.

2

Screening • Screening separates particles solely on the basis of size, by presenting them to an aperture.

3

The overflow or oversize is the material retained; the underflow or undersize passes through.

4

The term cut diameter Dpc denotes the aperture size that defines the separation. • Industrial screens include the grizzly (parallel bars, for coarse run-of-mine material), the trommel (rotating cylindrical screen), the vibrating screen (the workhorse of the industry, giving high capacity and good efficiency) and the gyratory or reciprocating screen for finer duty.

5

Capacity and effectiveness always work against each other: feeding a screen faster raises throughput but lowers the fraction of the undersize actually recovered.

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Ideal and Actual Screens • An ideal screen would make a sharp separation at the cut diameter, sending every particle smaller than the aperture to the underflow and every larger one to the overflow.

7

An actual screen does not, because near-mesh particles take time to find an aperture, some fines adhere to coarse lumps, apertures blind, and the screening time is finite. • Screen effectiveness is the standard measure.

8

With F, D and B the mass rates of feed, overflow and underflow and xF, xD, xB the mass fractions of material finer than the cut in each stream: • Effectiveness based on the undersize, EA = B xB /(F xF) — the fraction of the undersize in the feed that is actually recovered in the underflow. • Effectiveness based on the oversize, EB = D(1 − xD)/[F(1 − xF)]. • Overall effectiveness E = EA × EB, the product of the two, which is the value normally quoted. • A material balance supplies the stream rates:

9

F = D + B and F xF = D xD + B xB, so D/F = (xF − xB)/(xD − xB) and B/F = (xD − xF)/(xD − xB).

10

Numerical questions on effectiveness are common and follow this route exactly.

11

Estimating Surface Area and Particle Population • From a differential screen analysis with mass fractions xi in size intervals of mean diameter D̄pi: • Specific surface area Aw = Σ 6xi/(ψs ρp D̄pi) per unit mass of solid. • Number of particles per unit mass Nw = Σ xi/(a ρp D̄pi³), where a is a shape factor equal to π/6 for a sphere. • Volume-surface (Sauter) mean diameter D̄s = 1/Σ(xi/D̄pi), which is the diameter to use whenever surface area controls the process. • Note that surface area and particle number are dominated by the finest fractions, because they enter as 1/D and 1/D³ respectively — a small mass of fines contributes a disproportionate share of both.

12

Other Separation Methods Method Property exploited Principle Typical application Elutriation Terminal settling velocity Upward fluid stream carries particles with settling velocity below the fluid velocity; heavier ones sink Size classification of fine powders; removing fines from a product Jigging Density (and size) Pulsating vertical water current stratifies a bed, dense particles reaching the bottom Coal washing, ore concentration of coarse particles Froth flotation Surface wettability Air bubbles in a conditioned pulp attach to hydrophobic particles and carry them to a froth Sulphide ore concentration, de-inking, coal cleaning Magnetic separation Magnetic susceptibility Ferromagnetic or paramagnetic particles are deflected or retained by a field Tramp iron removal, magnetite and ilmenite recovery Electrostatic separation Electrical conductivity Charged particles behave differently on a grounded rotor according to conductivity Rutile and zircon beach sands, plastic recycling Cyclone separation Inertia and density Centrifugal field in a vortex throws particles to the wall Dust collection, classification, catalyst recovery Sink-and-float (dense medium) Density Particles are immersed in a medium of intermediate density and either float or sink Coal preparation, mineral pre-concentration Froth Flotation in More Detail • Flotation is the most widely used mineral concentration process and its reagent vocabulary is examined directly: • Collectors (xanthates, fatty acids) adsorb on the mineral surface and make it hydrophobic so it attaches to bubbles. • Frothers (pine oil, methyl isobutyl carbinol) stabilise the bubbles and the froth layer. • Activators (copper sulphate for sphalerite) make a mineral respond to a collector it would otherwise ignore. • Depressants (lime, cyanide, sodium silicate) prevent an unwanted mineral from floating. • pH regulators (lime, soda ash, sulphuric acid) set the pulp chemistry, since flotation selectivity is highly pH dependent. • The valuable mineral is normally floated into the concentrate and the gangue left in the tailings, although reverse flotation — floating the gangue — is used for iron ore and potash.

13

Elutriation and Classification • In an elutriator, fluid flows upward at a controlled velocity u.

14

Particles whose terminal velocity is less than u are carried out in the overflow; those with a greater terminal velocity settle and are collected below.

15

Because terminal velocity depends on both size and density, elutriation separates cleanly by size only when the material has uniform density. • Free settling occurs in dilute suspensions where particles do not interfere with one another; hindered settling occurs in concentrated suspensions, where the effective density and viscosity of the medium rise and settling velocities fall, but the separation becomes more sensitive to density differences — which is why hindered-settling classifiers are used in gravity concentration.

1

This section covers sedimentation and settling velocity, flocculation, fluidization including dense-phase and boiling beds, minimum fluidization velocity, minimum porosity and bed height, batch and continuous fluidization, and filtration with filter media, filter aids, batch and continuous operation, vacuum filters and rotary drum filters.

2

Settling and Terminal Velocity • A particle falling through a fluid accelerates until gravity, buoyancy and drag are in balance; the constant velocity then reached is the terminal or free-settling velocity. • General expression: vt = √[4g Dp(ρs − ρ)/(3 CD ρ)], where CD is the drag coefficient, itself a function of the particle Reynolds number.

3

Regime Particle Re Drag coefficient Terminal velocity Dependence on Dp Stokes (laminar) Below 1 CD = 24/Re vt = gDp²(ρs − ρ)/18μ Proportional to Dp² Intermediate (Allen) 1 to 1000 CD ≈ 18.5/Re0.6 Empirical Roughly Dp 1.14 Newton (turbulent) 1000 to 200 000 CD ≈ 0.44 vt = 1.75√[gDp(ρs − ρ)/ρ] Proportional to √Dp • Stokes' law is the one that must be memorised: vt = g Dp²(ρs − ρ)/18μ.

4

Note that the terminal velocity is proportional to the square of the diameter and inversely proportional to viscosity, so doubling the particle diameter quadruples the settling velocity. • Flocculation is the deliberate aggregation of fine particles into larger flocs so that they settle at a practical rate.

5

Coagulants such as alum and ferric chloride neutralise the surface charge (compressing the electrical double layer), while polyelectrolyte flocculants bridge the destabilised particles into large flocs.

6

Rapid mixing is used for coagulation and gentle slow mixing for flocculation, since vigorous agitation would tear the flocs apart. • Batch sedimentation of a concentrated suspension in a cylinder shows four zones — clear liquid A, uniform initial concentration B, a transition zone C and the sediment or compression zone D — with the interface between A and B falling at a constant rate until the critical settling point, after which compression takes over. • Continuous thickeners concentrate a slurry, producing a clear overflow and a thickened underflow.

7

The required area is fixed by the settling rate of the most difficult (usually the most dilute) layer, computed by the Coe and Clevenger or Kynch method, while the depth is fixed by the residence time needed for compression.

8

Fluidization • When a fluid flows upward through a bed of particles, the pressure drop rises with velocity while the bed remains fixed.

9

At the minimum fluidization velocity umf the drag on the particles equals the net weight of the bed, the particles become suspended, and the bed behaves like a liquid — it has a level surface, exerts a hydrostatic pressure, and objects float or sink in it according to density. • The characteristic signature of fluidization is that beyond umf the pressure drop stays essentially constant at the value ΔP = (L)(1 − ε)(ρs − ρ)g — that is, the buoyed weight of the bed per unit area — while the bed expands.

10

This constant-pressure-drop plateau is the most examined fact about fluidization. • Minimum fluidization velocity is obtained by equating the Ergun pressure-drop equation to the buoyed bed weight.

11

For small particles in the laminar regime this reduces to umf = Dp²(ρs − ρ)g εmf³ ψs²/[150 μ (1 − εmf)]. • Ergun equation for pressure drop through a packed bed combines a viscous (Kozeny-Carman) term proportional to velocity and an inertial (Burke-Plummer) term proportional to velocity squared; the viscous term dominates at low Reynolds number and the inertial term at high. • Bed expansion: since the total mass and the pressure drop stay fixed, L(1 − ε) is constant, giving L₂/L₁ = (1 − ε₁)/(1 − ε₂) — the standard route for bed-height questions. • Terminal velocity sets the upper limit: above vt the particles are carried out of the vessel (entrainment or pneumatic transport).

12

The operating range of a fluidized bed therefore lies between umf and vt.

13

Type Description Particulate (dense-phase, homogeneous) fluidization Smooth, uniform expansion with no bubbles; typical of liquid-solid systems and fine light particles Aggregative (bubbling, heterogeneous) fluidization Gas passes as bubbles through a dense emulsion phase; typical of gas-solid systems; also called a boiling bed Slugging In a tall narrow vessel bubbles coalesce to fill the cross-section and lift the bed in slugs Channelling Gas bypasses through preferred paths, leaving much of the bed unfluidized; caused by sticky or wide-size-range solids Spouting A single central jet penetrates the bed; used for coarse particles that fluidize poorly • Advantages of fluidized beds: excellent solid mixing giving near-uniform temperature, very high heat transfer coefficients to immersed surfaces, the ability to handle solids like a fluid, and large gas-solid contact area.

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They are the basis of fluid catalytic cracking, fluidized-bed combustion, roasting and drying. • Disadvantages: attrition of the particles and erosion of internals, entrainment of fines requiring cyclones, bypassing of gas in bubbles, difficulty of scale-up, and near-complete back-mixing of the solids, which is a drawback where a narrow residence-time distribution is wanted.

15

Filtration • Filtration separates solids from a fluid by passing the suspension through a permeable medium that retains the solids.

16

The retained solid is the cake and the clarified liquid the filtrate. • Cake filtration builds a layer of solids that itself becomes the filtering medium; clarifying (depth) filtration traps small amounts of solid within the medium; crossflow filtration sweeps the surface with the feed to limit cake build-up, as in membrane processes. • The basic rate equation is dV/dt = A ΔP/[μ(α c V/A + Rm)], where α is the specific cake resistance, c the mass of solids per unit volume of filtrate and Rm the medium resistance. • Constant-pressure filtration integrates to t/V = (μαc/2A²ΔP)V + μRm/(AΔP).

17

A plot of t/V against V is a straight line whose slope gives the specific cake resistance and whose intercept gives the medium resistance — a result asked about very frequently, both as theory and as a calculation. • Incompressible cakes have a specific resistance independent of pressure, so doubling the pressure doubles the rate; compressible cakes (most flocculated and biological solids) follow α = α₀ ΔPs with the compressibility index s between 0 and 1, so the gain from increasing pressure is much less than proportional and may even be negative. • Filter aids such as diatomaceous earth (kieselguhr) and perlite are used either as a precoat, to protect the medium, or as a body feed mixed into the slurry, to make a rigid, porous, incompressible cake.

18

They are used only where the filtrate is the valuable product, since they contaminate the cake.

19

Filter Operation Driving force Features Plate-and-frame press Batch Pressure Cheap, flexible, high pressure and dry cake, but labour-intensive to dismantle Leaf (Kelly, Sweetland) filter Batch Pressure Enclosed, easier cake discharge than a press Rotary drum vacuum filter Continuous Vacuum Drum partly submerged; zones for cake formation, washing, drying and discharge by scraper, string or roll Filter Operation Driving force Features Rotary disc filter Continuous Vacuum High area per floor space; poor washing Horizontal belt filter Continuous Vacuum Excellent washing and free-draining cakes Centrifugal filter Batch or continuous Centrifugal Produces the driest cake; basket and pusher types Nutsche filter Batch Vacuum or pressure Simple tank with a false bottom; common in fine chemicals and pharmaceuticals • The rotary drum vacuum filter is the standard continuous machine and its zones are regularly examined: as the drum rotates the submerged surface performs cake formation (pick-up), then emerging passes through washing, then drying (dewatering), and finally cake discharge, before returning to the trough.

20

Submergence is typically 30-40 per cent, and since vacuum can provide at most about 1 bar of driving force it is unsuitable for hot liquids near their boiling point or for very fine, high-resistance cakes. • Washing removes mother liquor retained in the cake; it is far more effective when displacement washing is used on a cake that has not cracked, which is why cracked cakes are compressed before washing.