| 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).
