Mass Transfer

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This section covers steady state molecular diffusion in gases and liquids, Fick's laws of diffusion, correlations for diffusivity in gases and liquids for binary and multi-component systems, the measurement and prediction of diffusivity, and diffusion in solids with its various types.

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Fick's Laws • Fick's first law describes steady diffusion:

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JA = −DAB(dCA/dz), where JA is the molar flux relative to the molar average velocity and DAB is the diffusivity or diffusion coefficient in m²/s.

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The negative sign shows that diffusion proceeds down the concentration gradient, exactly as heat flows down a temperature gradient. • Fick's second law describes unsteady diffusion: ∂CA/∂t = DAB(∂²CA/∂z²), which is mathematically identical to the unsteady heat conduction equation with D in place of the thermal diffusivity α. • The essential distinction between flux definitions, which is a regular examination question:

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JA is the diffusive flux relative to the molar average velocity, while NA is the total flux relative to fixed coordinates, and the two are related by NA = JA + yA(NA + NB) — the second term being the bulk flow or convective contribution. • Symmetry: for a binary system DAB = DBA, so a single diffusivity describes the pair.

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The Two Standard Cases • Case 1 — equimolar counter-diffusion, as in the distillation of a binary mixture with equal molar latent heats:

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NA = −NB, so the bulk-flow term vanishes and • NA = DAB(pA1 − pA2)/(RTz).

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The partial pressure profile is linear. • Case 2 — diffusion of A through stagnant, non-diffusing B, as in absorption, evaporation and humidification:

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NB = 0, so the bulk flow term survives and • NA = DABP(pA1 − pA2)/(RTz pB,lm), where pB,lm is the logarithmic mean partial pressure of B. • The comparison that is asked about: the ratio P/pB,lm is the drift or bulk-flow factor and is always greater than one, so the flux in stagnant-B diffusion always exceeds that in equimolar counter-diffusion under the same conditions.

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The partial pressure profile of A is then logarithmic rather than linear. • Stefan tube (Arnold diffusion cell) is the classical apparatus for measuring gas diffusivity, and works on exactly this second case: a liquid evaporates at the bottom of a vertical tube and diffuses through stagnant air, the diffusivity being obtained from the rate of fall of the liquid level.

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Diffusivity Values and Correlations Medium Typical D (m²/s) Dependence on temperature Dependence on pressure Gases 10−5 (about 10−5 to 10−4) D ∝ T1.5 to T1.75 D ∝ 1/P Liquids 10−9 (about 10−10 to 10−9) D ∝ T/μ, so strongly increasing Essentially none Solids 10−12 and below Arrhenius:

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D = D₀exp(−E/RT) Negligible • The four orders of magnitude between gas and liquid diffusivity are the reason the liquid-phase resistance so often dominates in gas-liquid contacting. • Prediction in gases: the Chapman-Enskog equation, from kinetic theory, and the simpler empirical Fuller-Schettler-Giddings correlation, which uses atomic diffusion volumes and gives D ∝ T1.75/P. • Prediction in liquids: the Stokes-Einstein equation for large spherical solutes, and the Wilke-Chang correlation for general use, which gives D ∝ T/(μ VA 0.6) with an association factor for the solvent — 2.6 for water, 1.9 for methanol, 1.0 for unassociated solvents. • Multi-component systems are handled for a dilute species by the Wilke equation, which combines the binary diffusivities of the species in each of the others into an effective diffusivity.

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Rigorous treatment requires the Maxwell-Stefan equations, in which diffusion can occur against a concentration gradient, and a species with zero gradient can still diffuse — behaviour that Fick's law alone cannot describe.

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Diffusion in Solids Type Mechanism Characteristics Fickian (true solid) diffusion Solute dissolves in the solid and moves through the lattice or polymer matrix Follows Fick's law;

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D independent of structure; leaching of metals, gas through rubber or metal Knudsen diffusion Pore diameter is smaller than the mean free path, so molecules collide with pore walls rather than each other D depends on pore radius and √(T/M), and is independent of pressure Bulk (ordinary) pore diffusion Pores are much larger than the mean free path; ordinary molecular diffusion occurs within them Corrected for porosity and tortuosity:

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Deff = Dε/τ Surface diffusion Adsorbed molecules migrate along the pore surface Contributes in parallel with pore diffusion, important for strongly adsorbed species Hydrodynamic (Poiseuille) flow Bulk flow through pores under a total pressure gradient Occurs only when a pressure difference is imposed • Effective diffusivity in a porous solid Deff = D ε/τ, where ε is the porosity (which reduces the available cross-section) and τ the tortuosity (which lengthens the path), typically 2 to 6.

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Deff is therefore always smaller than the free-fluid diffusivity. • Knudsen diffusion is distinguished by being independent of pressure and proportional to the pore radius, since collisions are with the wall rather than with other molecules — a standard identification question.

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It dominates in fine-pored catalysts and molecular sieves at low pressure.

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This section covers the concepts of molecular diffusion and the mass transfer coefficient, coefficients in laminar and turbulent flow, the analogies between mass, heat and momentum transfer including the Reynolds and Chilton-Colburn analogies, the dimensionless groups, simultaneous heat and mass transfer, the equilibrium curve, diffusion between phases, overall mass transfer coefficients, two-film theory, steady state co-current and counter-current processes, and stages and multistage cascades.

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The Mass Transfer Coefficient • By analogy with Newton's law of cooling, the convective mass flux is written NA = kc(CA1 − CA2) or, for a gas, NA = kG(pA1 − pA2).

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The coefficient depends on which driving force is used, so its units and value change accordingly — a point that must be checked in every problem. • Primed and unprimed coefficients: k′ denotes a coefficient for equimolar counter-diffusion and k an uncorrected coefficient for diffusion through stagnant B, the two being related by the drift factor.

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Group Definition Heat transfer analogue Meaning Sherwood, Sh kcL/DAB Nusselt Dimensionless mass transfer coefficient Schmidt, Sc μ/(ρDAB) = ν/DAB Prandtl Momentum to mass diffusivity; a fluid property Lewis, Le α/DAB = Sc/Pr — Thermal to mass diffusivity; links the two transfers Stanton (mass), StM kc/v = Sh/(Re·Sc) Stanton (heat) Mass transferred to bulk flow Peclet (mass) Re × Sc Peclet (heat) Bulk to diffusive mass transport • Typical Schmidt numbers: about 1 for gases (so momentum and mass diffuse at similar rates and the two boundary layers are of similar thickness) and of the order 10³ for liquids (so the concentration boundary layer is far thinner than the velocity layer). • Correlations take the form Sh = f(Re, Sc) for forced convection, directly parallel to Nu = f(Re, Pr).

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For turbulent flow in a pipe, Sh = 0.023 Re0.83Sc1/3 is the mass-transfer counterpart of the Dittus-Boelter equation. • Theories of the coefficient: film theory gives k ∝ D; penetration theory (Higbie) and surface renewal theory (Danckwerts) both give k ∝ D0.5.

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Experiment usually shows an exponent between 0.5 and 1, so the truth lies between the models, but the power-of-D dependence is the standard way of distinguishing them in an examination.

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The Analogies • Reynolds analogy is the simplest, assuming Pr = Sc = 1, and gives StH = StM = f/2.

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It is reasonably good for gases, where both groups are near unity, but poor for liquids. • Chilton-Colburn analogy is the practical form, introducing the j-factors: jH = StHPr2/3 and jD = StMSc2/3, with the result jH = jD = f/2.

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It is valid over 0.6 < Pr < 100 and 0.6 < Sc < 2500, and its great value is that it lets a mass transfer coefficient be predicted from a measured heat transfer coefficient, or either from a friction factor. • The essential caution: the analogy holds only for skin friction, not for form drag, so it fails for flow around bluff bodies and through packed beds where pressure drag dominates. • Prandtl and von Kármán analogies refine Reynolds by treating the laminar sublayer and buffer layer separately.

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Simultaneous Heat and Mass Transfer • Where a liquid evaporates into a gas, the latent heat is drawn from the liquid, cooling it, so heat and mass transfer are coupled.

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The wet-bulb temperature is the steady temperature reached when the sensible heat arriving from the gas exactly balances the latent heat of evaporation. • The Lewis relation: when Le = 1, which is very nearly true for the air-water system, h/(kYcs) = 1.

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The important consequence is that for air-water the wet-bulb temperature and the adiabatic saturation temperature are practically the same, which is why a simple sling psychrometer can be used to read humidity.

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For other systems the two differ and must be distinguished. • Psychrometry terms: absolute humidity (kg water per kg dry air), relative humidity (partial pressure as a percentage of the saturation value), dew point (the temperature at which condensation begins on cooling at constant humidity) and humid heat.

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Interphase Mass Transfer and Two-Film Theory • Equilibrium between the phases, most often described by Henry's law pA = H xA or by y* = m x, sets the limit of any separation.

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Mass transfer ceases when the two phases reach equilibrium, not when their concentrations are equal — a point that must be firmly held. • Two-film (Whitman) theory assumes that the whole resistance lies in two stagnant films, one on each side of the interface, that equilibrium prevails at the interface itself, and that the bulk of each phase is well mixed. • Individual and overall coefficients:

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NA = kG(pA − pAi) = kL(CAi − CA) = KG(pA − pA*) = KL(CA* − CA), with the resistances adding in series: • 1/KG = 1/kG + m/kL and 1/KL = 1/(m kG) + 1/kL. • Which film controls — the central result: for a highly soluble gas, m is very small, so the liquid-film term is negligible and the process is gas-film controlled (ammonia or hydrogen chloride in water).

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For a sparingly soluble gas, m is very large, so the gas-film term is negligible and the process is liquid-film controlled (oxygen or carbon dioxide in water).

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Effort spent improving the non-controlling film is wasted, so identifying which one controls is the first step in design. • Other models: penetration theory pictures fluid elements arriving at the interface and remaining for a fixed exposure time; surface renewal theory replaces the fixed time with a distribution of ages.

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Both predict k ∝ D0.5 against film theory's k ∝ D.

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Stages and Cascades • An ideal or theoretical stage (equilibrium stage) is one in which the two streams leaving are in equilibrium with each other.

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Stage efficiency — the Murphree efficiency being the commonest measure — compares the actual change in composition with the change that would occur in an ideal stage, so actual stages = theoretical stages / efficiency. • Counter-current contact gives a larger average driving force and permits a much closer approach to equilibrium than co-current contact.

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In the limit, a counter-current cascade can bring the exit of one phase into equilibrium with the entering other phase, which co-current operation can never do — exactly parallel to the temperature cross in heat exchangers. • Multistage cascades repeat the contact, each stage supplied with the streams leaving the neighbouring ones.

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The operating line comes from a material balance and the equilibrium curve from thermodynamics; the number of stages is found by stepping between them, which is the graphical method common to distillation, absorption, extraction and leaching.

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This section covers the types of distillation including flash and batch distillation, binary distillation with the McCabe-Thiele method, an introduction to multi-component distillation, the principles of liquid-liquid extraction with phase equilibrium diagrams, and the principles of leaching.

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Vapour-Liquid Equilibrium and Relative Volatility • Raoult's law for an ideal solution gives pA = xAPA°, and with Dalton's law yA = pA/P. • Relative volatility αAB = (yA/xA)/(yB/xB), which for a binary system rearranges to the most useful single equation in distillation: y = αx/[1 + (α − 1)x]. • For an ideal mixture α = PA°/PB°, the ratio of the pure-component vapour pressures. • The decisive facts: separation by ordinary distillation is impossible when α = 1, since vapour and liquid then have the same composition; the greater the departure of α from unity, the easier the separation; and α generally decreases as the pressure rises, which is one reason vacuum distillation is used for close-boiling and heat-sensitive mixtures. • Azeotropes form when α passes through 1 at some composition, so that vapour and liquid have identical composition and no further separation is possible by simple distillation.

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A minimum-boiling azeotrope arises from positive deviation from Raoult's law (ethanol-water, at about 95.6 per cent ethanol by mass); a maximum-boiling azeotrope from negative deviation (nitric acid-water).

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They are broken by changing the pressure, or by azeotropic, extractive or pressure-swing distillation, or by membranes such as pervaporation.

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Types of Distillation Type Description Typical use Differential (simple batch) Liquid boiled, vapour removed and condensed continuously; described by the Rayleigh equation Small batches, large volatility difference, laboratory work Flash (equilibrium) Feed is heated and expanded through a valve into a drum; a single equilibrium stage Crude pre-separation, rough splits, feed preparation Batch with reflux Batch still with a rectifying column; operated at constant reflux or constant distillate composition Fine chemicals, multiple products from one campaign Continuous fractional Multistage column with rectifying and stripping sections The industrial standard for large throughput Steam distillation Steam lowers the partial pressure so the mixture boils well below its normal boiling point Heat-sensitive, high-boiling, water-immiscible materials such as essential oils Azeotropic An entrainer forms a new azeotrope that is removed overhead Breaking azeotropes, for example ethanol dehydration with benzene or cyclohexane Extractive A high-boiling solvent alters relative volatility and leaves with the bottoms Close-boiling and azeotropic mixtures; solvent is recovered separately Vacuum Reduced pressure lowers the boiling temperature and usually raises α Heat-sensitive and high-boiling materials, e.g. vacuum gas oil • Rayleigh equation for differential distillation: ln(F/W) = ∫xW xF dx/(y − x), relating the quantity distilled to the change in still composition. • Flash distillation is a single equilibrium stage: with f the fraction vaporised, the operating line is y = −[(1−f)/f]x + xF/f, and its intersection with the equilibrium curve gives the product compositions.

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It can never achieve a sharp separation, which is why it is used only for a rough split.

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The McCabe-Thiele Method • The method rests on the assumption of constant molal overflow — that the molar liquid and vapour flows are constant in each section of the column, which is acceptable when the components have similar molar latent heats and the column is nearly adiabatic.

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This makes the operating lines straight. • Rectifying section operating line: y = [R/(R+1)]x + xD/(R+1), where R = L/D is the reflux ratio.

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Its slope is R/(R+1) and its intercept on the y-axis is xD/(R+1), and it passes through the point (xD, xD) on the diagonal. • Stripping section operating line passes through (xW, xW) on the diagonal and through the intersection of the rectifying line with the q-line. • The q-line describes the feed condition, with q = (heat required to vaporise one mole of feed)/(molar latent heat), and has the equation y = [q/(q−1)]x − xF/(q−1), passing through (xF, xF).

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Feed condition Value of q Slope of q-line Direction of the q-line Cold (sub-cooled) liquid q > 1 Positive, greater than 1 Steep, leaning to the right Saturated liquid at its bubble point q = 1 Infinite Vertical Partially vaporised (two-phase) 0 < q < 1 Negative Sloping backwards to the left Saturated vapour at its dew point q = 0 Zero Horizontal Superheated vapour q < 0 Positive, less than 1 Shallow, leaning right • Stepping off stages: starting at (xD, xD), draw horizontal and vertical steps between the equilibrium curve and the operating lines, switching operating lines at the q-line intersection, until xW is passed.

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Each triangle is one theoretical stage, and the reboiler counts as one, so the number of plates in the column is one less than the number of stages counted. • The two limits, which are asked about constantly: at total reflux (R = ∞) the operating lines coincide with the 45° diagonal, the driving force is maximal, and the number of stages is the minimum possible — given by the Fenske equation Nmin = ln[(xD/(1−xD))((1−xW)/xW)]/ln α.

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At minimum reflux the operating line touches the equilibrium curve, creating a pinch point of zero driving force, and an infinite number of stages is required. • The economic optimum lies between: a higher reflux ratio means fewer stages (lower capital cost) but more vapour, hence a larger column diameter, condenser, reboiler and steam bill (higher operating cost).

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Practical designs use R = 1.2 to 1.5 Rmin. • The Underwood equations give minimum reflux and the Gilliland correlation relates the actual number of stages to the minimum, making up the Fenske-Underwood-Gilliland shortcut method used for multi-component systems. • Multi-component distillation introduces the idea of the light key and heavy key — the two components between which the split is specified — with components lighter than the light key going almost entirely overhead and those heavier than the heavy key to the bottoms.

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A single column can produce only two products, so N − 1 columns are needed for N products, and the number of possible sequences grows rapidly with N.

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Liquid-Liquid Extraction • Extraction separates components by contacting the feed with a partially miscible solvent, and is preferred to distillation when the components are close boiling or form an azeotrope, when the solute is heat sensitive, when it is present in dilute solution, or when the mixture is non-volatile. • Terminology: the extract is the solvent-rich product, the raffinate the solvent-lean residue. • Distribution coefficient K = y/x, the ratio of solute concentration in extract to that in raffinate, and selectivity (separation factor) β = (yA/yB)/(xA/xB).

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Extraction is feasible only if β is greater than 1, exactly as distillation requires α ≠ 1. • Solvent selection weighs high selectivity and capacity, a large density difference for ease of settling, low mutual solubility with the feed, an adequate interfacial tension (too low and emulsions form, too high and dispersion is difficult), ease of recovery — usually by distillation — plus low cost, low toxicity, chemical stability and non-corrosiveness. • Ternary phase diagrams, drawn on a right-triangular or equilateral-triangular plot, show the binodal (solubility) curve separating the one-phase from the two-phase region, tie lines joining equilibrium extract and raffinate compositions, and the plait point where the tie line shrinks to zero and the two phases become identical.

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The lever rule applies along a tie line. • Equipment: mixer-settlers (high stage efficiency, large footprint), spray, packed and sieve-plate columns, agitated columns (rotating disc, Scheibel, Kühni, pulsed) and centrifugal extractors for very short contact time or a small density difference. • Leaching (solid-liquid extraction) dissolves a soluble constituent out of a solid with a solvent, as in sugar from beet, oil from seeds, and metals from ore.

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The rate is governed by particle size, solvent choice, temperature and agitation; the standard analysis assumes the underflow retains a fixed quantity of solution, either constant or variable with concentration.

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This section covers the introduction and principles of absorption and desorption, the equilibrium solubility of gases in liquids, isothermal and adiabatic gas-liquid contact, packings and solvent selection, material balances in an absorber, counter-current multistage operation, and the principles of adsorption with selection criteria for adsorbents.

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Principles and Equilibrium • Absorption transfers a solute from a gas into a liquid; desorption or stripping is the reverse.

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Physical absorption relies on solubility alone; chemical absorption adds a reaction in the liquid, which greatly increases both capacity and rate — as in carbon dioxide absorption in amine solutions. • Henry's law pA = H xA describes equilibrium for dilute, sparingly soluble gases.

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A large Henry constant means low solubility. • The effect of conditions, which is asked directly: solubility increases with pressure and decreases with temperature.

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Hence absorption is favoured by high pressure and low temperature, and stripping by low pressure and high temperature — the basis of every absorber-stripper loop. • Isothermal versus adiabatic operation: absorption releases the heat of solution (and any heat of reaction), so the liquid warms as it descends.

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In dilute systems the rise is small and isothermal design is adequate; in concentrated systems the temperature rise reduces solubility and can seriously limit performance, so interstage cooling or a cooled absorber is used.

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Solvent Selection and Packings • Solvent selection criteria: high solubility for the solute (reducing the liquid rate required), high selectivity, low volatility (to limit solvent loss), low viscosity (for good mass transfer and low pumping cost), non-corrosiveness, non-toxicity, non-flammability, chemical stability, ready recoverability and low cost.

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Water is the first choice wherever it will serve. • Packings are classed as random (dumped) — Raschig rings, Pall rings, Berl and Intalox saddles — and structured, made of corrugated sheets or gauze.

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The progression from Raschig rings through Pall rings to modern saddles and structured packing gives steadily higher capacity, lower pressure drop and better efficiency at higher cost. • Requirements of a good packing: large surface area per unit volume, high void fraction for low pressure drop, good wetting characteristics, low weight, corrosion resistance, mechanical strength and low cost. • Flooding is the condition at which the upward gas flow prevents the liquid from flowing down, so liquid accumulates and the pressure drop rises sharply.

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It sets the upper hydraulic limit, and columns are designed to operate at 50-70 per cent of the flooding velocity.

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Loading is the lower point at which the gas begins to impede liquid flow and hold-up starts to rise. • Channelling — liquid migrating to the wall and leaving the core dry — is countered by liquid redistributors at intervals of a few column diameters, and by keeping the packing size below about one eighth of the column diameter.

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Material Balance and Stage Calculation • Using mole ratios Y = y/(1−y) and X = x/(1−x) makes the operating line straight even for concentrated systems, because the carrier gas and solvent flows are then constant: • Gs(Y₁ − Y₂) = Ls(X₁ − X₂), so the operating line has slope Ls/Gs. • For absorption the operating line lies above the equilibrium curve (the gas is richer than equilibrium, so solute moves into the liquid); for stripping it lies below.

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Recognising which is which is a standard question. • Minimum liquid rate occurs when the operating line just touches the equilibrium curve, giving a pinch of zero driving force and requiring infinite stages or infinite packing height; practical designs use 1.2 to 1.5 times (Ls/Gs)min.

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Reducing the liquid rate saves solvent and pumping but demands a taller column — the same capital-versus-operating trade-off as the reflux ratio in distillation. • Kremser equation gives the number of theoretical stages analytically when both the operating and equilibrium lines are straight, in terms of the absorption factor A = L/(mG).

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A greater than 1 favours absorption and A less than 1 favours stripping; the stripping factor is S = 1/A = mG/L.

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Packed Column Height • For a packed column the design is expressed as Z = HTU × NTU: • Height of a transfer unit, HOG = G/(KGa·P), which measures the efficiency of the packing and has units of length. • Number of transfer units, NOG = ∫dy/(y − y*), which measures the difficulty of the separation and is dimensionless. • The distinction between HTU and NTU is examined repeatedly: the NTU depends only on the required separation and the equilibrium relationship, while the HTU depends only on the packing, the flow rates and the physical properties. • The interfacial area a is lumped with the coefficient as Ka, because the area per unit volume cannot be measured independently in a packed bed — another point commonly asked. • For a tray column the corresponding measure is the HETP, the height equivalent to a theoretical plate, and the packed height = HETP × number of theoretical stages.

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Adsorption Feature Physical adsorption Chemisorption Forces van der Waals Chemical bonds Heat of adsorption Low, typically under 40 kJ/mol; comparable with latent heat High, 80-400 kJ/mol; comparable with heat of reaction Layers Multilayer possible Monolayer only Specificity Non-specific Highly specific Reversibility Readily reversible Often irreversible Temperature Favoured by low temperature Requires activation; occurs at higher temperature • Isotherms: the Langmuir isotherm q = qmKC/(1 + KC) assumes monolayer coverage on a uniform surface with no interaction between adsorbed molecules, and saturates at high concentration.

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The Freundlich isotherm q = KC1/n is empirical and suits heterogeneous surfaces; it does not saturate.

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The BET isotherm extends Langmuir to multilayers and is used to measure surface area. • A favourable isotherm is convex upward (concave to the concentration axis), giving a sharp, self-sharpening mass transfer zone; an unfavourable isotherm gives a spreading front.

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This determines how efficiently the bed can be used. • Breakthrough: in a fixed bed the mass transfer zone travels through the bed and breakthrough occurs when it reaches the outlet; the bed is then regenerated.

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A narrower mass transfer zone means a sharper breakthrough curve and better utilisation of the bed capacity. • Regeneration is by temperature swing (TSA), pressure swing (PSA), purge or displacement with another fluid, or chemical means.

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PSA is the standard route to industrial hydrogen purification and to oxygen and nitrogen from air.

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Adsorbent Character Principal use Activated carbon Non-polar, hydrophobic; area 500-1500 m²/g Organic vapour recovery, decolourising, water treatment, air purification Silica gel Polar, hydrophilic Drying gases and liquids at moderate temperature Activated alumina Polar; withstands higher temperature than silica gel Gas drying, fluoride and arsenic removal from water Adsorbent Character Principal use Molecular sieve (zeolite) Polar, crystalline, uniform pore size Deep drying, separations by molecular size and shape, PSA for O₂ and H₂ Polymeric resins Tailored surface chemistry Specialty separations, pharmaceutical recovery • Adsorbent selection criteria: high capacity and selectivity for the target species, a suitable pore size distribution, favourable adsorption and desorption kinetics, ease and cost of regeneration, mechanical strength and attrition resistance, thermal and chemical stability, and low cost.

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The unique feature of a molecular sieve is its uniform, crystallographically fixed pore size, which allows separation strictly by molecular dimension — the property that gives it its name.

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This section covers nucleation and crystal growth, batch crystallization and crystallization equipment, drying equilibria, the drying rate curve and the calculation of drying time.

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Supersaturation • Supersaturation is the driving force for crystallization, and is expressed as a concentration difference C − C*, a ratio C/C*, or a degree of undercooling.

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Without supersaturation no crystallization can occur, however long the solution is held. • Methods of generating supersaturation: cooling (for solutes whose solubility rises steeply with temperature, such as potassium nitrate), evaporation (for solutes of nearly flat solubility, such as common salt), vacuum flash cooling (which combines the two), salting out or drowning out with an anti-solvent, and chemical reaction (reactive crystallization or precipitation). • The Miers diagram divides the concentration-temperature plane into three: the stable (undersaturated) region, in which crystals dissolve; the metastable region, in which existing crystals grow but spontaneous nucleation does not occur; and the labile region, in which spontaneous nucleation is rapid. • The key operating principle, which is the most examined idea in the section: to grow large crystals, operate in the metastable zone with seeding and low supersaturation; to produce many small crystals, operate in the labile zone with high supersaturation.

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Rapid cooling or rapid addition of anti-solvent produces a shower of fine crystals; slow, controlled cooling produces few large ones.

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Nucleation and Growth Type Description Primary homogeneous nucleation Spontaneous formation of nuclei in a clear solution; requires very high supersaturation Primary heterogeneous nucleation Nucleation on dust, container walls or other foreign surfaces; needs less supersaturation Secondary nucleation New nuclei generated by existing crystals, mainly by contact with the impeller, walls and other crystals; dominates in industrial crystallizers at low supersaturation • Crystal growth proceeds in two steps in series: diffusion of solute from the bulk to the crystal surface, followed by surface integration into the lattice.

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Either may control — diffusion control is favoured by poor agitation and surface-integration control by vigorous agitation. • The ΔL law (McCabe) states that geometrically similar crystals of the same material growing in the same solution grow at the same rate, independent of their size — so all crystals gain the same increment of linear dimension in a given time.

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This is the standard basis for crystal size distribution calculations. • Crystal habit — the external shape — is governed by the relative growth rates of the different faces, and can be deliberately altered by impurities, additives and the choice of solvent; polymorphism, in which the same substance crystallises in different lattice forms, is critically important in pharmaceuticals, because polymorphs differ in solubility and bioavailability. • Yield is calculated from a material balance using the solubility at the final temperature, and must allow for the water of crystallization in the crystals and for any water evaporated.

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Forgetting the water of crystallization is the commonest error in yield calculations.

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Crystallization Equipment Type Principle Notes Tank crystallizer Batch cooling, with or without agitation Simplest and cheapest; large irregular crystals; poor control Swenson-Walker Open trough with a cooling jacket and a slow helical scraper Continuous, simple, low capacity Forced circulation evaporator-crystallizer Supersaturation by evaporation with a circulating pump High capacity; smaller crystals because of pump-induced secondary nucleation Draft tube baffle (DTB) Internal circulation with a settling zone and fines destruction Produces large, uniform crystals; the standard for controlled size Oslo (Krystal) crystallizer Supersaturated liquor fed beneath a suspended fluidized bed of crystals Classifying action gives large, uniform crystals Scraped-surface (Votator) Scrapers keep the chilled wall clear Viscous materials, waxes, fats, ice cream Drying:

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Equilibria and Terminology • Equilibrium moisture content is the moisture a solid retains when in equilibrium with air of a given humidity and temperature; it can never be removed by drying with that air. • Free moisture is the moisture above the equilibrium value — and only free moisture can be removed.

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This distinction underlies every drying calculation and is asked directly. • Bound moisture is held by capillary action, solution in cell walls or chemical combination, and exerts a vapour pressure below that of pure water; unbound moisture exerts the full vapour pressure of pure water. • A hygroscopic material has a significant equilibrium moisture content and is far harder to dry to low moisture than a non-hygroscopic one.

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The Drying Rate Curve • Plotting drying rate against free moisture content gives the characteristic curve with the following periods: • Initial adjustment period: the solid comes to the steady surface temperature; short and usually ignored. • Constant-rate period: the surface is covered by a continuous film of free water, so it behaves as a pool of liquid.

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The rate is controlled entirely by the external conditions — air temperature, humidity, velocity and the heat transfer to the surface — and not at all by the nature of the solid.

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The surface stays at the wet-bulb temperature, and the rate is given by Nc = h(T − Tw)/λw. • Critical moisture content marks the end of the constant-rate period, when the surface can no longer be kept fully wetted.

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It is not a fixed property: it depends on the thickness of the bed and on the drying rate itself, rising as drying is made faster. • First falling-rate period: the wetted surface area decreases progressively as dry patches appear; the rate falls roughly linearly. • Second falling-rate period: evaporation occurs inside the solid and the vapour must diffuse out, so internal moisture movement controls and the external conditions become almost irrelevant.

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This period usually takes most of the total drying time even though it removes little water. • Drying time: for the constant-rate period, t = ms(X₁ − Xc)/(A Nc); for the falling-rate period with a rate proportional to moisture content, t = msXc ln(Xc/X₂)/(A Nc).

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Because the second expression is logarithmic, the last traces of moisture take a disproportionately long time to remove — the practical reason very low final moisture specifications are expensive. • Case hardening occurs when drying is too rapid at the start: the surface dries and shrinks into an impermeable skin that traps moisture inside.

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It is avoided by using a high humidity and a gentle rate in the early stages.

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This section covers gas-dispersed equipment including bubble columns, spray towers, tray towers and packed towers, continuous contact equipment, the design of packed and trayed absorption towers, drying equipment and its selection, and cooling towers.

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Gas-Liquid Contacting Equipment Equipment Dispersed phase Characteristics Typical use Bubble column Gas in liquid Simple, no moving parts, high liquid hold-up and residence time, low gas-side pressure drop; considerable back-mixing Slow reactions, fermentation, oxidation, hydrogenation Spray tower Liquid in gas Very low pressure drop, handles solids, cheap; poor efficiency, entrainment, extensive back-mixing Gas cooling, dust and acid-mist scrubbing, quenching Tray (plate) tower Gas in liquid on each tray Stagewise contact; handles large liquid rates, solids and heat exchange; cleanable Distillation and absorption at large diameter Packed tower Liquid film on packing Continuous differential contact; low pressure drop, good for corrosive and foaming systems Absorption, small columns, vacuum distillation Venturi scrubber Liquid in high-velocity gas Very high efficiency on fine particles; very high pressure drop Particulate removal with simultaneous gas absorption Wetted wall column Film on a tube wall Known interfacial area; laboratory instrument Measurement of mass transfer coefficients • Packed versus tray towers — the comparison most often asked: • Choose a packed tower for corrosive service (ceramic and plastic packings are cheap), for small diameters below about 0.6 m, where a low pressure drop is essential as in vacuum distillation, for foaming systems (less agitation), and where a low liquid hold-up is wanted with heat-sensitive material. • Choose a tray tower for large diameters, for high liquid rates, where solids are present or fouling is likely (trays are easier to clean), where side draws or interstage heating or cooling are needed, where a large liquid hold-up is required for a slow reaction, and where the turndown must be wide. • The fundamental difference: a tray tower gives stagewise contact and is analysed in theoretical stages, while a packed tower gives continuous differential contact and is analysed by transfer units — although the HETP allows a packed column to be quoted in equivalent stages.

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Tray Types and Hydraulics • Sieve tray: simple perforated plate.

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Cheapest, lowest pressure drop, but poor turndown, since liquid weeps through the holes at low vapour rates. • Valve tray: liftable caps over the holes.

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Excellent turndown over a wide range of loads at moderate cost — the modern general-purpose choice. • Bubble cap tray: a riser and slotted cap.

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Cannot weep at all, so it gives the best turndown of the three and can handle very low liquid rates; but it is the most expensive, has the highest pressure drop and fouls most easily, and is now used only in special service. • Operating limits: flooding at high vapour rate (liquid cannot descend, often because the downcomer backs up); entrainment or jet flooding, where droplets are carried to the tray above, reducing efficiency; weeping at low vapour rate, where liquid falls through the openings; and dumping in the extreme case.

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The region between these limits is the operating window or turndown range. • Efficiency: overall column efficiency = theoretical trays / actual trays, typically 50-80 per cent for distillation and rather lower for absorption.

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The Murphree efficiency is defined for a single tray, and point efficiency for a single location on it.

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O'Connell's correlation relates overall efficiency to the product of relative volatility and viscosity.

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Column Design Essentials • Diameter is set by the flooding velocity, estimated from the Souders-Brown equation umax = C√[(ρL − ρV)/ρV] for trays or from the Eckert generalised pressure drop correlation for packing.

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Design is at 70-85 per cent of flooding for trays and 50-70 per cent for packing. • Height is set by the number of trays × tray spacing (typically 0.45-0.6 m), or by HTU × NTU for packing. • A packed tower's efficiency falls off badly in large diameters because of liquid maldistribution, which is why trays are preferred above roughly 1 m diameter unless there is a specific reason to use packing. • Minimum wetting rate must be maintained in a packed tower or the packing dries out and the effective area collapses.

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Drying Equipment Dryer Mode Best suited to Tray (shelf) dryer Batch, convective Small quantities, valuable or fragile products, wide variety of materials Rotary dryer Continuous, convective, direct or indirect Free-flowing granular solids in large tonnage: fertiliser, sand, ores Spray dryer Continuous, convective Solutions, slurries and suspensions to a free-flowing powder in seconds: milk, detergent, ceramics, pharmaceuticals Fluidized bed dryer Continuous or batch, convective Free-flowing granular solids; uniform temperature and very high rates Drum dryer Continuous, conductive Pastes and slurries dried as a thin film on a heated roll: starch, baby food Freeze dryer (lyophiliser) Batch, sublimation under vacuum Very heat-sensitive and high-value products: vaccines, biologicals, instant coffee Vacuum / tumble dryer Batch, conductive Heat-sensitive, solvent-wet and oxygen-sensitive materials Flash (pneumatic) dryer Continuous, convective Surface-moist powders needing only a few seconds of contact • Selection criteria: the physical form of the feed (solution, slurry, paste, granule, sheet), the heat sensitivity of the product, the required throughput and final moisture, whether the operation is batch or continuous, whether solvent or dust must be contained, and cost. • Direct (convective) dryers transfer heat by contact with hot gas, so the product is exposed to the gas and dust or solvent must be handled; indirect (conductive) dryers transfer heat through a wall, which suits vacuum operation, solvent recovery and oxygen-sensitive or dusty products, and is more energy efficient because no gas carries heat away. • Spray drying is distinguished by very short residence time — a few seconds — so a heat-sensitive product survives even in a hot gas, since evaporative cooling keeps the droplet near the wet-bulb temperature until it is nearly dry.

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This is the reason milk powder can be made in air at 200 °C. • Freeze drying removes water by sublimation of ice under vacuum, so the material never passes through a liquid phase; it gives the best possible product quality and a porous, readily rehydrated structure, at the highest cost.

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Cooling Towers • A cooling tower cools water by evaporating a small part of it into the air stream.

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Most of the cooling — typically 70-80 per cent — comes from the latent heat of evaporation rather than from sensible heat transfer, which is why a cooling tower works even when the air is warmer than the water. • The limit of cooling is the wet-bulb temperature of the entering air, not the dry-bulb temperature — the single most examined fact about cooling towers. • Approach = outlet water temperature − wet-bulb temperature of the air.

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A smaller approach requires a disproportionately larger tower, and an approach below about 3 °C is not economic. • Range = inlet water temperature − outlet water temperature, which is fixed by the process heat load and the circulation rate, not by the tower. • Water losses: evaporation (about 1 per cent of the circulation for every 5.5-6 °C of range), drift or windage (entrained droplets, minimised by drift eliminators) and blowdown, deliberately withdrawn to limit the build-up of dissolved solids.

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Cycles of concentration = concentration in the circulating water / concentration in the make-up, and make-up = evaporation + drift + blowdown. • Types: natural draught (hyperbolic, very large, no fans), mechanical draught — forced (fan at the air inlet) or induced (fan at the top, the commoner arrangement) — and cross-flow or counter-flow, with counter-flow giving the better thermal performance. • Fill or packing increases the air-water contact area, and drift eliminators recover entrained droplets.

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Legionella control through water treatment and regular cleaning is an important operational responsibility.