crystallizer-design

name	crystallizer-design
description	Crystallizer design — nucleation kinetics, crystal growth, MSMPR model, supersaturation, CSD (crystal size distribution), draft tube baffle, Oslo evaporative crystallizer, energy balance, yield, ASTM standards.
metadata	{"priority":7,"promptSignals":{"phrases":["crystallizer design","crystallization","nucleation kinetics","crystal size distribution","MSMPR crystallizer","supersaturation"],"minScore":3}}

Crystallizer Design — Complete Skill

Fundamentals

Crystallization: precipitation of solid phase from supersaturated solution; controlled nucleation + growth Supersaturation driving force: Δc = c - c* [kg/kg solvent; c = actual concentration; c* = equilibrium solubility] Relative supersaturation: S = Δc / c* (dimensionless; 0 = equilibrium; > 0 = supersaturated)

Metastable zone width (MZW): Range of supersaturation where nucleation is absent (growth only); critical for controlling CSD S_MZW = 0.01–0.20 (typical; depends on substance and temperature)

Nucleation Kinetics

Primary Nucleation

Homogeneous (classical nucleation theory): J_hom = A_hom × exp(-B_hom / (T³ × S²)) J = nucleation rate [#/(m³·s)]; A_hom, B_hom = material constants Heterogeneous: occurs at T < T_metastable; much faster at rough surfaces, dust, ions J_het = J_hom × exp(-f × B_hom / (T³ × S²)) [f = 0–1 reduction factor for contact angle < 180°]

Secondary Nucleation

Contact nucleation (most important in industrial crystallizers): J_2 = k_N × M_T^j × N_p^l × Δc^b [M_T = suspension density; N_p = agitator speed; b = 1–3] Secondary nucleation at S < MZW; dominated by crystal-crystal and crystal-impeller collisions

Overall Nucleation Rate (Power Law)

B = k_N × Δc^b [#/(m³·s); b typically 2–5 for secondary nucleation]

Crystal Growth Kinetics

Overall growth rate (surface integration + diffusion in series): G = R_growth × R_diffusion / (R_growth + R_diffusion)

Diffusion-controlled: R_diff = k_d × Δc [kg/(m²·s); k_d = mass transfer coefficient]

Surface integration controlled: R_int = k_r × Δc^r [kg/(m²·s); r = 1–2 for most systems]

Overall linear growth rate: G = dL/dt = k_g × Δc^g [m/s; k_g = growth coefficient; g = 1–2]

Arrhenius temperature dependence: k_g = k_g0 × exp(-E_g / (RT))

MSMPR Crystallizer (Mixed Suspension, Mixed Product Removal)

Steady-state population balance: n(L) = n₀ × exp(-L / (G × τ)) [population density function #/(m³·m)] n₀ = nuclei density = B/G [#/(m⁴)] τ = mean residence time = V/Q [s; V = crystallizer volume; Q = volumetric flow out]

Mean crystal size (dominant size L_d): L_d = 3Gτ [m; size at maximum of volume distribution]

Crystal size distribution moments: m_j = ∫₀^∞ Lʲ × n(L) dL = j! × n₀ × (Gτ)^(j+1) [j = 0,1,2,3,4] Suspension density: M_T = 6ρ_c k_v m₃ [kg/m³; k_v = volume shape factor (π/6 for sphere)]

Yield: Y = (c₀ - c*_out) × V_solvent × η_recovery / M_crystal [kg crystals per batch]

CSD Control

Fines dissolution: recycle small crystals through heated fines loop → dissolve → reduce nucleation → larger crystals Product classification: elutriation leg (upflow classifier removes large crystals preferentially)

Effect of Gτ on mean size: Larger Gτ → larger crystals (increase τ by using larger vessel; increase G by increasing T or S) Target L_d:

Product	Typical target L_d
Sugar	0.5–1.0 mm
KCl	0.3–0.6 mm
Citric acid	0.2–0.5 mm
Urea	0.5–1.5 mm
NaCl	0.2–0.4 mm

Crystallizer Types

MSMPR Vessel (Draft Tube Baffle, DTB)

Description: draft tube forces internal circulation; baffle ring creates upflow zone; fines recycle from top; product withdrawal from bottom Mixing: axial-flow impeller; tip speed 1–3 m/s (gentle to avoid excessive secondary nucleation) Fines cut: elutriation flow rate = crystal settling velocity for L < L_cut L_cut = target fines dissolution size (typically 0.1–0.2× L_d)

Oslo (Krystal) Evaporative Crystallizer

Description: external heat exchanger + flash chamber; separation of supersaturation zone from growth zone Advantage: low secondary nucleation (no impeller in growth zone) Disadvantage: complex; fouling in heat exchanger

Batch Cooling Crystallizer

Cooling rate: dT/dt [°C/hr] controls supersaturation profile Optimal cooling: programmed cooling (not linear) to maintain constant Δc Seeding: add seed crystals at start of metastable zone → controlled nucleation

Evaporative Crystallizer (Continuous)

Single effect: steam-heated; simple; high energy consumption Multiple effect: V₁ → V₂ → V₃ with decreasing pressure; steam economy ≈ N_effects - 1 kg evap/kg steam Energy balance: Q = λ_steam × ṁ_steam = ṁ_evap × λ_water / N_effects_economy

Energy Balance

Heat removed in cooling crystallizer: Q = ṁ_solution × c_p × ΔT + ṁ_crystals × ΔH_cryst [ΔH_cryst = heat of crystallization; typically negative = exothermic] Evaporative crystallizer: ṁ_steam = (ṁ_evap × λ_water) / (λ_steam × η_evaporator)

Scaling and Fouling

Supersaturation at wall > bulk → wall nucleation → scale formation Prevention:

• Polished walls (roughness < Ra 0.4 μm)
• Ultrasonic vibration on heat transfer surfaces
• Scale inhibitor additives (< 50 ppm surfactant)
• Scraper internals (for viscous systems)

Nucleation induction time (to first detectable crystal): t_ind = A_ind × exp(B_ind / (T³ × ln²(S+1))) Used to determine safe operating time before CIP (clean-in-place)

Standards

Standard	Scope
ASTM E1857	Crystal size measurement by sieve analysis
ISO 13320	Particle size analysis by laser diffraction
ASTM E2476	Inline particle size measurement
AIChE Guidelines	Crystallizer design best practices

Output

Provide: crystallizer type (MSMPR/DTB/Oslo/batch), solute and solvent, operating temperature [°C] and pressure [kPa], solubility c* and feed concentration c₀ [kg/kg], supersaturation Δc and S, nucleation rate B [#/(m³·s)], growth rate G [m/s] and g, mean residence time τ [min], crystallizer volume V [m³], dominant crystal size L_d [mm], suspension density M_T [kg/m³], crystal yield Y [kg/batch or kg/hr], fines cut size L_cut [mm], evaporation rate [kg/hr] and steam consumption [kg/hr] (if evaporative), heat load Q [kW], scale risk assessment, and applicable standard (ASTM E1857, ISO 13320).