# Liquid-Liquid Extraction (LLE) & Mixer-Settler Sizing in API Downstream Isolation

# Advanced Partition Thermodynamics, Hydraulic Dispersion & Counter-Current Stage Sizing for Pharmaceutical Workups

In active pharmaceutical ingredient (API) synthesis, natural product extraction, and fermentation broth recovery, Liquid-Liquid Extraction (LLE) is the quintessential unit operation for separating heat-labile molecules from aqueous reaction matrices, unreacted reagents, and polar inorganic salts.

Unlike thermal distillation, which risks thermal degradation, racemization, or product charring, liquid extraction operates at mild or ambient temperatures by exploiting differences in chemical potential and solubility between two immiscible liquid phases.

# 1. Thermodynamic Fundamentals: Distribution Coefficient & Selectivity

The core thermodynamic driving force of liquid-liquid extraction is the Distribution Coefficient (Partition Ratio), $K_D$, defined as the equilibrium ratio of solute mass fraction in the extract phase ($y_i^*$) to the raffinate phase ($x_i$):

Where:

• $\gamma_{i,R}^\infty$: Activity coefficient of solute in the raffinate phase at infinite dilution.
• $\gamma_{i,E}^\infty$: Activity coefficient of solute in the extract phase at infinite dilution.
• $M_R, M_E$: Average molecular weight of raffinate and extract phases.

# 1.1. Separation Factor (Selectivity $\beta$)

When separating a target API ($A$) from an undesirable process impurity ($B$), the solvent selectivity $\beta_{A/B}$ dictates thermodynamic feasibility:

# 1.2. The Extraction Factor ($E$)

To evaluate equipment sizing feasibility, process engineers evaluate the dimensionless Extraction Factor ($E$):

Where:

• $S$: Solvent mass flow rate (kg/h).
• $F$: Feed mass flow rate (kg/h).

                         THE EXTRACTION FACTOR REGIME SPECTRUM
   E < 1.0                     1.3 <= E <= 2.5                     E > 3.5
  ◄───────────────────────────┼─────────────────────────────────┼──────────────────────────►
   Thermodynamically Limited    Optimal Industrial Sizing Zone    Solvent Inefficiency Zone
   Incomplete recovery even     Balanced stage count & solvent    High recovery per stage,
   with infinite stages.        recovery utility costs.           but extreme downstream evap.

# 2. Multi-Stage Counter-Current Cascade Sizing: The Kremser Model

In multi-stage counter-current extraction batteries, fresh solvent enters at the opposite end of the feed stream, maintaining a nearly uniform chemical potential gradient across every stage.

flowchart LR
    F["Aqueous Feed (F, x_f)"] --> M1["Stage 1 (Mixer-Settler)"]
    M1 --> M2["Stage 2 (Mixer-Settler)"]
    M2 --> Mn["Stage N (Mixer-Settler)"]
    Mn --> R["Exhausted Raffinate (R, x_n)"]

    S["Fresh Solvent (S, y_s)"] --> Mn
    Mn --> M2
    M2 --> M1
    M1 --> E["Rich Extract (E, y_1)"]

    style F fill:#e0f2fe,stroke:#0284c7
    style S fill:#fef3c7,stroke:#d97706
    style E fill:#dcfce7,stroke:#16a34a
    style R fill:#fee2e2,stroke:#dc2626

# 2.1. Analytical Kremser Equation

For linear equilibrium operating lines ($y = K_D \cdot x$), the required number of theoretical stages ($N_{theor}$) is derived via the Kremser-Brown-Souders equation:

# 2.2. Murphree Stage Efficiency ($E_M$)

Because liquid-liquid diffusion is significantly slower than vapor-liquid diffusion, practical stage efficiency in mixer-settlers is between $75\% \text{ to } 90\%$:

# 3. Mixer Hydrodynamics, Power Input & Droplet Breakup

The mixer tank must disperse the solvent into droplets fine enough to maximize specific interfacial area ($a$), but coarse enough to prevent terminal emulsification.

                  MIXER HYDRODYNAMICS & AGITATOR SCHEMATIC
               ┌──────────────────────────────────────────────┐
               │              Drive Motor & VFD               │
               └──────────────────────┬───────────────────────┘
                                      │  Shaft
                                  ┌───┴───┐
                                  │       │
             Solvent In ──► ──────┤       ├────── ◄── Aqueous In
                                  │   ▲   │
                                  │  / \  │
                                  │ /   \ │
                                  │       │
                                  └───┬───┘
                                 ┌────┴────┐
                                 │ Impeller│ (Curved Pumping Turbine)
                                 └────┬────┘
                                      ▼
                           Discharge to Settler Chute

# 3.1. Specific Interfacial Mass Transfer Area ($a$)

The interfacial area per unit mixer volume is governed by the dispersed phase hold-up ($\phi$) and Sauter mean diameter ($d_{32}$):

# 3.2. Sauter Mean Diameter ($d_{32}$) via Hinze-Kolmogorov Theory

In fully turbulent liquid dispersion, droplet diameter is determined by the balance of turbulent kinetic energy dissipation ($\varepsilon$) and interfacial tension ($\sigma$):

Where the Impeller Weber number is:

# 3.3. Pumping Turbine Power & Tip Speed Criteria

To pump and disperse without external inter-stage transfer pumps, plants employ curved-blade bottom-suction turbines:

• Power Density Target: $0.8 \text{ to } 1.5\text{ kW/m}^3$ of mixer volume.
• Impeller Tip Speed ($v_{tip} = \pi D_{imp} N$): Kept strictly within $2.5 \text{ to } 3.8\text{ m/s}$. Speeds $>4.2\text{ m/s}$ shatter droplets below $10\,\mu\text{m}$, creating stubborn emulsions.

# 4. Gravity Settler Sizing, Coalescence & Hydraulic Balance

The settler must provide laminar horizontal flow allowing droplets to rise or settle to the interface.

                      SETTLER COALESCENCE WEDGE PROFILE
 ┌────────────────────────────────────────────────────────────────────────┐
 │                      Light Phase Layer (Extract)                       │
 │~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~│
 │ ◄── Coalescence Wedge (Dispersed Band) ──►                             │
 │........................................................................│
 │                      Heavy Phase Layer (Raffinate)                     │
 └────────────────────────────────────────────────────────────────────────┘

# 4.1. Stokes' Law Terminal Velocity

For rigid dispersed spherical droplets in the continuum:

# 4.2. Settler Area Sizing via Specific Settling Flux

In industrial batteries, settlers are sized using the Total Specific Volumetric Flux ($\Phi_{settler}$):

# 4.3. Hydraulic Underflow / Overflow Weir Equation

To maintain a stable liquid-liquid interface inside the settler without active electronics, an external adjustable inverted loop (heavy phase underflow leg) is balanced by hydrostatic head:

# 5. Comprehensive Worked Industrial Case Study: 1,500 kg/h API Extraction

# Problem Statement:

An antibiotic intermediate ($MW = 385\text{ g/mol}$) is synthesized in an aqueous broth at $C_f = 4.0\text{ wt}\%$ ($40\text{ g/kg}$).

• Feed rate: $F = 1,500\text{ kg/h}$ ($\rho = 1,020\text{ kg/m}^3$).
• Extractant: Pure Ethyl Acetate ($S/F = 0.8$, $\rho = 900\text{ kg/m}^3$).
• Partition coefficient: $K_D = 6.2$.
• Target extraction recovery: $\ge 99.0\%$ ($x_n \le 0.04\text{ wt}\%$).

# Step 1: Calculate Extraction Factor ($E$)

# Step 2: Determine Theoretical Stages via Kremser Formula

With an 85% Murphree stage efficiency ($E_M = 0.85$):

# Step 3: Mixer Dimensions & Agitation Power

• Total volumetric flow:

• Target mixer residence time $\tau = 90\text{ seconds}$:

• Assuming square geometry ($H = D_{tank}$):

• Impeller diameter ($D_{imp} = D_{tank}/3 = 150\text{ mm}$):
  • Operating speed $N = 350\text{ rpm} = 5.83\text{ rps}$.
  • Tip speed $v_{tip} = \pi \cdot 0.150 \cdot 5.83 = 2.75\text{ m/s}$ (well within the safe non-emulsifying $2.5 - 3.8\text{ m/s}$ window).
• Motor Power ($N_p = 2.2$, $\rho_{mix} \approx 960\text{ kg/m}^3$):

With motor losses and viscous margins, install a $0.25\text{ kW}$ motor with VFD.

# Step 4: Settler Area & Dimensions

• Using design flux $\Phi_{settler} = 20\text{ m}^3/(\text{m}^2\cdot\text{h})$:

• Applying $3.5:1$ length-to-width aspect ratio:

Depth $H = 300\text{ mm}$ (liquid depth $250\text{ mm}$).

# 6. Industrial Troubleshooting & Operational Mitigations

# Applicable Engineering Standards & Codes Used

• ISO 5167: Measurement of fluid flow in closed conduits.
• ASME BPE: Bioprocessing Equipment cGMP design for pharmaceutical extraction skids.
• API RP 520 / 521: Sizing pressure relief valves for solvent extraction systems.
• ASTM D971: Standard Test Method for Interfacial Tension of Liquid against Water.