Reactive Crystallization Design

Reference ID: MET-DA57 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Reactive crystallization is a process where chemical reaction and crystallization occur simultaneously, enabling precise control over particle size distribution (PSD), morphology, and polymorphism. This calculation sheet focuses on the design of a reactive crystallization system for calcium carbonate (CaCO₃), a model system for studying precipitation kinetics, mixing effects, and particle engineering.

In Process Engineering, reactive crystallization is critical for:

Producing high-purity, uniform particles (e.g., pharmaceuticals, pigments, catalysts).
Controlling supersaturation to avoid fouling or agglomeration.
Optimizing energy input (mixing) to balance nucleation and growth.

Typical applications include:

Carbon capture (CO₂ mineralization via CaCO₃).
Pharmaceutical active ingredient (API) synthesis.
Specialty chemical manufacturing (e.g., nanoscale precipitates).

Methodology & Formulas

1. Temperature Conversion

Convert the process temperature from Celsius (\(T_C\)) to Kelvin (\(T_K\)): \[ T_K = T_C + 273.15 \]

2. Reaction Kinetics (Arrhenius Law)

The temperature-dependent reaction rate constant (\(k\)) is calculated using the Arrhenius equation: \[ k = k_{ref} \cdot \exp \left( -\frac{E_a}{R_g} \left( \frac{1}{T_K} - \frac{1}{T_{ref}} \right) \right) \] where:

\(k_{ref}\): Rate constant at reference temperature \(T_{ref}\).
\(E_a\): Activation energy (J/mol).
\(R_g\): Universal gas constant (8.314 J/mol·K).

The reaction rate (\(r\)) for a bimolecular reaction (e.g., Ca(OH)₂ + CO₂ → CaCO₃ + H₂O) is given by: \[ r = k \cdot C_A^{\alpha} \cdot C_B^{\beta} \] where:

\(C_A, C_B\): Concentrations of reactants A and B (mol/m³).
\(\alpha, \beta\): Reaction orders for A and B.

3. Mixing Characterization

The Damköhler number (Da) quantifies the competition between reaction and mixing timescales: \[ Da = r \cdot \tau_{mix} \] where \(\tau_{mix}\) is the mixing time (s), estimated from turbulent energy dissipation (\(\epsilon\)) and velocity fluctuations (\(u'\)): \[ \tau_{mix} = \frac{\epsilon^{1/3}}{u'^2} \]

The Reynolds number (Re) validates turbulent flow conditions: \[ Re = \frac{\rho \cdot u' \cdot L}{\mu} \] where:

\(\rho\): Fluid density (kg/m³).
\(\mu\): Dynamic viscosity (Pa·s).
\(L\): Characteristic length (m).

4. Supersaturation and Particle Size Control

Supersaturation (\(S\)) drives nucleation and growth. An empirical correlation links \(S\) to the target particle size (\(d_p\)): \[ d_p \propto S^{-2} \] Rearranged to estimate the actual supersaturation (\(S_{actual}\)) for a given \(d_p\): \[ S_{actual} = S_{target} \cdot \left( \frac{d_p}{d_{p,target}} \right)^{-1/2} \]

5. Regime Criteria and Validity Checks

Parameter	Symbol	Recommended Range	Regime Implications
Damköhler Number	\(Da\)	\(10^{-6} < Da < 1\)	\(Da \gg 1\): Reaction-limited (mixing is fast). \(Da \ll 1\): Mixing-limited (reaction is fast).
Supersaturation Ratio	\(S\)	\(1.1 < S < 2.0\)	\(S < 1.1\): Low nucleation; risk of slow growth. \(S > 2.0\): High nucleation; risk of fines/agglomeration.
Turbulent Energy Dissipation	\(\epsilon\)	\(0.1 < \epsilon < 10\) W/kg	\(\epsilon < 0.1\): Poor mixing; inhomogeneous supersaturation. \(\epsilon > 10\): Excessive energy; potential particle breakage.
Reynolds Number	\(Re\)	\(Re > 10,000\)	\(Re < 10,000\): Laminar flow; turbulent correlations invalid.
Particle Size	\(d_p\)	\(1 < d_p < 100\) μm	\(d_p < 1\) μm: Colloidal range; filtration challenges. \(d_p > 100\) μm: Settling risk; suspension instability.

Reactive crystallization combines a chemical reaction with simultaneous crystallization of the product. It is advantageous when:

The reaction generates a solid product that can be isolated directly from the reactor, reducing downstream processing.
High purity is required, because crystallization can act as an in-situ purification step.
The reaction is exothermic and the heat of crystallization can help control temperature spikes.
Solvent recovery is a priority, as the same solvent can serve both reaction medium and crystallization medium.

Selecting solvents follows a systematic approach:

Solubility profile: Choose a solvent where the reactants are soluble at reaction temperature but the product’s solubility drops sharply upon cooling or antisolvent addition.
Reaction compatibility: Ensure the solvent does not deactivate catalysts or interfere with reaction kinetics.
Safety and environmental impact: Prefer solvents with low toxicity, flammability, and easy recycle options.
Antisolvent selection: Pick a miscible liquid that dramatically reduces product solubility without reacting with any species in the system.

The most influential parameters include:

Supersaturation level: Drives nucleation and growth rates; must be controlled to balance crystal size distribution.
Temperature profile: Determines solubility and reaction rate; cooling ramps should be optimized for uniform crystal formation.
Residence time: Must be long enough for reaction completion but short enough to avoid over-growth or agglomeration.
Mixing intensity: Affects mass transfer of reactants and antisolvent, influencing local supersaturation zones.
Seeding strategy: Introducing seed crystals can control nucleation and produce consistent crystal habit.

Successful scale-up relies on preserving the critical dimensionless groups:

Maintain the same supersaturation ratio by matching temperature and concentration profiles.
Keep the mixing time relative to reaction time constant; adjust impeller speed or reactor geometry accordingly.
Scale the heat removal capacity proportionally to reaction exotherm and crystallization heat of formation.
Validate the seeding protocol at larger volumes to ensure consistent crystal size distribution.
Perform a step-wise increase (e.g., 2×, 5×, 10×) while monitoring key quality attributes (purity, particle size, yield) at each stage.

Worked Example – Sizing a Continuous Reactive Crystallizer for Precipitated Calcium Carbonate

A specialty-chemical plant must produce 150 kg h^-1 of precipitated CaCO₃ (mean crystal size 15 µm) by carbonating 0.1 mol L^-1 Ca(OH)₂ with 0.1 mol L^-1 CO₂ at 25 °C. The reaction is fast and mixing-controlled; we need to check whether the selected draft-tube crystallizer (L = 1 m, Re ≈ 10⁵) can deliver the target supersaturation ratio S = 1.5 while keeping the Damköhler number below the micro-mixing limit.

Knowns

Temperature: T = 25 °C (298.15 K)
Rate constant at 25 °C: k₂₅ = 0.005 s^-1
Activation energy: E_a = 30,000 J mol^-1
Reactant concentrations: C_A0 = C_B0 = 0.1 mol L^-1
Reaction orders: n_A = n_B = 1
Target supersaturation: S_target = 1.5
Target mean crystal size: d_p = 15 µm
Energy dissipation: ε = 1.0 m² s^-3
Kinematic viscosity: ν = 1 × 10^-6 m² s^-1
Diffusivity: D = 1 × 10^-9 m² s^-1
Reactor length scale: L = 1.0 m
Reynolds number: Re = 100,000

Step-by-step calculation

Compute the intrinsic rate constant at 25 °C (already given):
k = 0.005 s^-1
Estimate the micro-mixing time (Bourne model) to compare with reaction time:
\[ \tau_{\text{mix}} = \left( \frac{L^2}{\epsilon} \right)^{1/3} = \left( \frac{1.0^2}{1.0} \right)^{1/3} = 1.0\ \text{s} \]
Calculate the Damköhler number for the bulk reaction:
\[ Da = \frac{k\ C_{\text{A0}}^{n_A-1}\ C_{\text{B0}}^{n_B-1}}{\tau_{\text{mix}}^{-1}} = \frac{0.005 \times 1}{1.0} = 0.005 \] Since Da ≪ 1, mixing is faster than reaction and the system is kinetics-controlled.
Determine the actual supersaturation generated at the chosen energy dissipation:
\[ S_{\text{actual}} = 1 + \frac{Da}{1 + Da} = 1 + \frac{0.005}{1 + 0.005} = 1.005 \approx 1.01 \] To reach the target S = 1.5, raise the energy dissipation or reduce the mixing time. Re-arranging: \[ \tau_{\text{mix,calc}} = \frac{k}{S_{\text{target}} - 1} = \frac{0.005}{0.5} = 0.01\ \text{s} \] The required specific power is therefore: \[ \epsilon_{\text{req}} = \frac{L^2}{\tau_{\text{mix,calc}}^3} = \frac{1.0^2}{(0.01)^3} = 1 \times 10^6\ \text{m}^2\ \text{s}^{-3} \]
Check crystal size via the diffusion-limited growth expression:
\[ G = \frac{2\ D\ (S - 1)}{d_p} = \frac{2 \times 1 \times 10^{-9} \times 0.5}{15 \times 10^{-6}} = 6.7 \times 10^{-8}\ \text{m s}^{-1} \] For a mean residence time τ = 1800 s (30 min) the size increment is: \[ \Delta L = G\ τ = 6.7 \times 10^{-8} \times 1800 = 0.00012\ \text{m} = 120\ \text{µm} \] Hence the target 15 µm is easily met within the chosen residence time.

Final Answer

Under the specified conditions, the crystallizer must operate at an energy dissipation of 1 × 10⁶ m² s^-3 (≈ 1 kW kg^-1) to achieve the target supersaturation ratio of 1.5, yielding a 15 µm mean CaCO₃ crystal size with a 30-min residence time.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle