# Continuous Crystallization: MSMPR & Tubular Oscillatory Baffled Crystallizers (OBC)

# Population Balance Modeling, Residence Time Distributions (RTD), Nucleation vs Growth Control, and Polymorph Locking

Batch cooling crystallization in stirred tanks is notorious for batch-to-batch polymorphic variance, unpredictable oiling-out, and wide particle size distributions (PSD).

To satisfy FDA Quality by Design (QbD) mandates, pharmaceutical manufacturing is transitioning toward Continuous Crystallization. By decoupling fluid mixing from mean residence time, continuous systems establish stable, steady-state supersaturation profiles, producing identical crystals and consistent dissolution performance continuously.

# 1. Crystallization Kinetics & The Metastable Zone Width (MSZW)

All crystallization processes are driven by relative supersaturation ($\sigma$):

Where:

• $C$: Actual liquid solute concentration ($\text{kg/m}^3$).
• $C^*$: Saturation equilibrium solubility at process temperature ($\text{kg/m}^3$).

                      METASTABLE ZONE & CRYSTALLIZATION REGIMES
   Solute Conc. (C)
      ▲
      │                              Labile Zone (Uncontrolled Spontaneous Nucleation)
      │                     ------------------------------------------------------- (Spinodal / MSZW Limit)
      │                              Metastable Growth Zone (Target Operating Space)
      │                     ─────────────────────────────────────────────────────── (Solubility Curve C*)
      │                              Undersaturated Stable Solution (Dissolution Zone)
      └─────────────────────────────────────────────────────────────────────────────►
                                                                   Temperature (T)

# 1.1. Kinetic Equations

• Primary Spontaneous Nucleation Rate ($B_0$):

• Secondary Contact Nucleation (Seed-Induced):

• Linear Crystal Growth Rate ($G$):

In continuous crystallizers, maintaining steady-state seed magma keeps the operating point strictly within the metastable growth zone, channeling all available solute into controlled crystal growth without sparking unwanted fines.

# 2. Mixed-Suspension Mixed-Product-Removal (MSMPR) Modeling

An ideal MSMPR operates at steady state where feed solution continuously enters and an unclassified magma slurry is continuously discharged.

# 2.1. One-Dimensional Population Balance Equation (PBE)

Assuming size-independent growth ($G \neq f(L)$), the population density function integrates to:

Where:

• $n(L)$: Crystal population density ($\# / (\text{m}^3 \cdot \mu\text{m})$).
• $n_0 = B_0 / G$: Population density at zero crystal size.
• $\tau = V / Q$: Mean hydrodynamic residence time.

# 2.2. Method of Moments & Crystal Properties

The $j$-th moment of the population density distribution is:

From the moments:

• Total crystal count: $N_{tot} = \mu_0 = n_0 \cdot G \cdot \tau$.
• Total crystal surface area: $A_{tot} = k_a \cdot \mu_2 = 2 \cdot k_a \cdot n_0 \cdot (G \cdot \tau)^3$.
• Total crystal mass: $M_T = k_v \cdot \rho_c \cdot \mu_3 = 6 \cdot k_v \cdot \rho_c \cdot n_0 \cdot (G \cdot \tau)^4$.
• Mass-Weighted Mean Size ($L_{43}$):

# 2.3. The Cascaded MSMPR Advantage

A single MSMPR has an exponential residence time distribution with a broad coefficient of variation ($CV = \sigma_{PSD} / \bar{L} = 50\%$). Operating a cascade of 2 or 3 MSMPRs in series narrows the crystal size distribution dramatically ($CV \approx 30\%$).

flowchart LR
    A["Hot Saturated API Solution"] --> B["Stage 1 MSMPR (Nucleation / Seed Zone)"]
    B --> C["Stage 2 MSMPR (Intermediate Growth)"]
    C --> D["Stage 3 MSMPR (Final Deep Cooling / Depletion)"]
    D --> E["Continuous Centrifuge or Filter"]

    style A fill:#fee2e2,stroke:#dc2626
    style B fill:#fef3c7,stroke:#d97706
    style C fill:#e0f2fe,stroke:#0284c7
    style D fill:#dcfce7,stroke:#16a34a
    style E fill:#f3e8ff,stroke:#9333ea

# 3. Tubular Oscillatory Baffled Crystallizers (OBC)

For processes requiring true plug-flow residence time distributions with zero axial backmixing, the Oscillatory Baffled Crystallizer (OBC) is the premier modern continuous platform.

# 3.1. Hydrodynamic Mechanism

An OBC consists of a jacketed tube fitted with internal orifice baffles. A mechanical diaphragm or bellows oscillates the slurry back and forth:

The interaction of fluid oscillation with the sharp orifice edges generates intense toroidal vortex rings, achieving complete crystal suspension even at extremely low net forward velocities.

                      OBC VORTEX MIXING FLUID DYNAMICS
 ──► Net Flow Direction
 ──────────────────────────────────────────────────────────────────────────
           ▲                  ▲                  ▲                  ▲
        [Baffle]           [Baffle]           [Baffle]           [Baffle]
           │    ╭──────╮      │    ╭──────╮      │    ╭──────╮      │
           │   │ Vortex │     │   │ Vortex │     │   │ Vortex │     │
           │    ╰──────╯      │    ╰──────╯      │    ╰──────╯      │
           ▼                  ▼                  ▼                  ▼
 ──────────────────────────────────────────────────────────────────────────

# 4. Comprehensive Worked Case Study: Sizing a 50 kg/h Continuous Crystallizer

# Problem Statement:

A pharmaceutical API is to be crystallized continuously at $50\text{ kg/h}$ dry product output:

• Feed concentration at $60^\circ\text{C}$: $C_{in} = 120\text{ kg/m}^3$.
• Final solubility at $10^\circ\text{C}$: $C_{out} = 20\text{ kg/m}^3$.
• Theoretical yield: $\Delta C = 100\text{ kg/m}^3$ ($0.10\text{ kg/L}$).
• Required slurry flow rate:

• Desired mass-weighted crystal size: $L_{43} = 200\,\mu\text{m}$.
• Experimentally measured linear growth rate: $G = 2.5 \times 10^{-8}\text{ m/s}$ ($0.090\text{ mm/h} = 90\,\mu\text{m/h}$).

# Step 1: Calculate Required Residence Time ($\tau$)

Using the MSMPR moment formula:

# Step 2: Sizing a 2-Stage MSMPR Cascade

• Total working liquid volume:

• Dividing equally across 2 stages:

• Stage 1: Cooled from $60^\circ\text{C} \to 35^\circ\text{C}$ (handles $60\%$ of crystallization).
• Stage 2: Cooled from $35^\circ\text{C} \to 10^\circ\text{C}$ (polishing and yield harvest).

# Step 3: Sizing an Equivalent Tubular OBC

• For the OBC, near-ideal plug flow ($Bo > 80$) achieves the same target crystal size in approximately $50\%$ of the volume due to eliminated backmixing bypass:

• Selecting a standard DN50 ($D = 50\text{ mm} = 0.050\text{ m}$) jacketed tube:
  • Cross-sectional area: $A_c = \frac{\pi}{4} \cdot (0.050)^2 = 0.001963\text{ m}^2$.
  • Required tube length:

• Configured as a compact serpent bundle: 12 passes of $6.0\text{ m}$ length with $180^\circ$ return bends.

# 5. Process Analytical Technology (PAT) & Real-Time Feedback

flowchart TD
    A["Continuous Magma Stream"] --> B["Inline FBRM Probe (Chord Length PSD)"]
    A --> C["Inline ATR-FTIR (Liquid Concentration C)"]
    A --> D["Inline PVM (High-Speed Imaging)"]

    B --> E["DCS / Advanced Process Control (APC)"]
    C --> E
    D --> E

    E --> F["Modulate Jacket Cooling Valve"]
    E --> G["Adjust Continuous Seed Addition Rate"]
    E --> H["Alter Oscillation Frequency (f)"]

    style A fill:#e0f2fe,stroke:#0284c7
    style E fill:#fef3c7,stroke:#d97706
    style F fill:#dcfce7,stroke:#16a34a
    style G fill:#f3e8ff,stroke:#9333ea

# 6. Industrial Encrustation & Fouling Mitigation

# Applicable Engineering Standards & Codes Used

• FDA Guidance for Industry: Continuous Manufacturing for Drug Substances and Products (2023).
• ICH Q8(R2): Pharmaceutical Development (Design Space & Process Analytical Technology).
• ASME B31.3: Process Piping Code for High-Pressure Continuous Reactor Tubes.
• ISPE Baseline Guide Volume 1: Active Pharmaceutical Ingredients.