# Agitated Liquid-Liquid Extraction Columns: Sizing Karr, Scheibel, and Kühni Columns for Continuous API Workup

# An Industrial Design Guide to Counter-Current Multi-Stage Hydrodynamics, Sauter Mean Diameter ($d_{32}$), Minard-Johnson Flooding Models, and Kremser Transfer Unit Sizing

# Executive Summary & Industrial Context

In pharmaceutical API development, liquid-liquid extraction (LLE) is performed to partition target products away from inorganic salts, polar catalysts, and reaction byproducts.

In traditional batch facilities, this unit operation is executed in a jacketed stirred-tank reactor:

1. Charge aqueous wash solvent,
2. Agitate for 15–30 minutes,
3. Stop agitation and allow phases to settle under gravity for 1–4 hours,
4. Manually observe the sight glass to execute a phase cut, and
5. Repeat for 2 to 4 consecutive washes.

This batch approach incurs severe operational bottlenecks: phase cuts rely on operator eyesight, persistent emulsions or rag layers can halt plant operations for an entire shift, and multiple sequential washes occupy an expensive multi-thousand-liter reactor that should be running chemical reactions.

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│                             BATCH VS. MIXER-SETTLERS VS. AGITATED COLUMNS                        │
├────────────────────────────────┬────────────────────────────────┬────────────────────────────────┤
│ Engineering Characteristic     │ Batch Stirred Reactor Wash     │ Agitated Column (Karr/Kühni)   │
├────────────────────────────────┼────────────────────────────────┼────────────────────────────────┤
│ Operating Mode                 │ Discrete Batch (Multi-step)    │ Uninterrupted Continuous Train │
│ Theoretical Extraction Stages  │ 1.0 Stage per batch wash       │ 5 – 15 Stages in Single Shell  │
│ Phase Decantation Time         │ 1 – 4 Hours                    │ 30 – 120 Seconds (Fast!)       │
│ Physical Plant Footprint       │ 15 – 25 m²                     │ 0.8 – 1.5 m² (Vertical tower)  │
│ Emulsion / Rag Layer Risk      │ High (Severe downtime)         │ Low (Controlled shear input)   │
│ Solvent Holdup Inventory       │ 2,000 – 6,000 L                │ 25 – 150 L (-97%)              │
│ Cleanroom Height Requirement   │ Standard single floor (4 m)    │ Vertical space (5 – 9 m)       │
└────────────────────────────────┴────────────────────────────────┴────────────────────────────────┘

Mechanically Agitated Extraction Columns replace bulky mixer-settler trains and slow batch washes. By introducing mechanical energy (reciprocation or rotation) into a vertical counter-current tower, they continuously break dispersed droplets to maximize interfacial area while preventing axial backmixing.

This guide provides the complete mathematical framework for selecting, sizing, and operating Karr reciprocating plate, Scheibel rotary, and Kühni agitated extraction columns for commercial pharmaceutical manufacturing.

# 1. Column Types & Mechanical Agitation Mechanisms

                  THREE MAJOR AGITATED COLUMN ARCHITECTURES
 ┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
 │                                                                                                  │
 │  TYPE 1: KARR RECIPROCATING       TYPE 2: SCHEIBEL ROTARY          TYPE 3: KÜHNI TURBINE         │
 │          PLATE COLUMN                     AGITATED COLUMN                  AGITATED COLUMN       │
 │                                                                                                  │
 │    Heavy Phase In (Aqueous)         Heavy Phase In (Aqueous)         Heavy Phase In (Aqueous)    │
 │              │                                │                                │                 │
 │              ▼                                ▼                                ▼                 │
 │    ┌───────────────────┐            ┌───────────────────┐            ┌───────────────────┐       │
 │    │ Top Settling Zone │            │ Top Settling Zone │            │ Top Settling Zone │       │
 │    ├───────────────────┤            ├───────────────────┤            ├───────────────────┤       │
 │    │ ══ Perforated ══  │            │  [Turbine Impeller]│           │  [Radial Turbine] │       │
 │    │   Plate Stack     │            │ ░░ Wire Mesh ░░░░░│            │ ── Stator Plate ──│       │
 │    │   (Oscillates     │            │  (Calming Packing)│            │  Compartment Box  │       │
 │    │    Up & Down)     │            │  [Turbine Impeller]│           │  [Radial Turbine] │       │
 │    │ ══ Amplitude A ══ │            │ ░░ Wire Mesh ░░░░░│            │ ── Stator Plate ──│       │
 │    │    Frequency f    │            │  [Turbine Impeller]│           │  [Radial Turbine] │       │
 │    ├───────────────────┤            ├───────────────────┤            ├───────────────────┤       │
 │    │Bottom SettlingZone│            │Bottom SettlingZone│            │Bottom SettlingZone│       │
 │    └─────────┬─────────┘            └─────────┬─────────┘            └─────────┬─────────┘       │
 │              ▲                                ▲                                ▲                 │
 │              │                                │                                │                 │
 │     Light Phase In (Org.)            Light Phase In (Org.)            Light Phase In (Org.)      │
 │                                                                                                  │
 └──────────────────────────────────────────────────────────────────────────────────────────────────┘

# 1.1 Karr Reciprocating Plate Column (RPC)

• Mechanics: A central shaft suspends a stack of perforated plates with large free cross-sectional areas ($\approx 55 - 65\%$). The entire plate stack oscillates vertically with stroke amplitude $A = 6 - 15\,\text{mm}$ and frequency $f = 1 - 5\,\text{Hz}$.
• Agitation Intensity Parameter: Defined as the stroke-frequency product ($A \cdot f$, $\text{m/s}$ or $\text{cm/min}$).
• Strengths: Excellent handling of slurries, suspended catalyst particles, and fouling systems. No internal seals or bearings inside the active process fluid.

# 1.2 Scheibel Column (Rotary Agitated with Calming Mesh)

• Mechanics: Alternates rotating turbine mixing compartments with quiescent calming zones filled with knitted wire mesh packing (Yorkmesh).
• Strengths: Ideal for low interfacial tension systems ($\gamma < 5\,\text{mN/m}$) that form fine hazes. The calming wire mesh coalesces micro-droplets between mixing stages, minimizing axial backmixing.

# 1.3 Kühni Column (Compartmented Radial Turbine)

• Mechanics: Precision-engineered mixing compartments separated by perforated stator plates. Each compartment contains a shrouded radial turbine impeller.
• Strengths: High extraction efficiency (very low Height Equivalent to a Theoretical Stage, $HETS = 15 - 35\,\text{cm}$). The stator plates strictly isolate compartment eddies, preventing inter-stage backmixing even at high throughputs.

# 2. Two-Phase Hydrodynamics: Droplet Size & Flooding Velocity

Counter-current extraction is driven by the density differential ($\Delta\rho = |\rho_{\text{heavy}} - \rho_{\text{light}}|$) between continuous and dispersed phases. For stable operation, $\Delta\rho$ should be at least $50\,\text{kg/m}^3$.

# 2.1 Sauter Mean Droplet Diameter ($d_{32}$)

Interfacial area for mass transfer ($a = 6\phi / d_{32}$, $\text{m}^2/\text{m}^3$) is governed by the Sauter mean droplet diameter ($d_{32}$), which balances mechanical shear breakup against surface tension coalescence:

Where:

• $\gamma$ = Interfacial tension between phases ($\text{N/m}$)
• $\rho_c$ = Continuous phase density ($\text{kg/m}^3$)
• $P/M$ = Specific power dissipation rate per unit mass of fluid ($\text{W/kg}$)
• $C_1$ = Empirical column constant ($\approx 0.35 - 0.50$)

For a Karr Column, power input is proportional to $(A \cdot f)^3$:

       AGITATION INTENSITY (A·f) VS. MASS TRANSFER & FLOODING
    HETS / Efficiency
     ▲
     │  Operating "Sweet Spot" (Maximum Efficiency)
     │        │
     │        ▼
     │       ╭───╮                                      ╭─── Total Flooding Limit!
     │      ╭╯   ╰╮                                    ╭╯    (Emulsion / Entrainment)
     │     ╭╯     ╰╮                                  ╭╯
     │    ╭╯       ╰──────────────────────────────────╯
     │   ╭╯ Under-agitated
     │  ╭╯  (Large droplets, low mass transfer area)
     └──┴───────────────────────────────────────────────────────► Agitation Intensity (A·f)

# 2.2 Flooding Velocity: The Minard-Johnson Equation

If agitation intensity ($A \cdot f$ or $N_{\text{rpm}}$) or volumetric flow rates exceed column hydraulic capacity, droplets can no longer counter-currently separate, and the column floods.

The total superficial flooding velocity ($v_{\text{flood}} = v_c + v_d$, $\text{m/s}$) is modeled using the Minard-Johnson model:

Where $\bar{u}_0$ is the single-droplet terminal characteristic velocity, and $\phi_{\text{flood}}$ is the dispersed phase holdup at the flooding point:

# 3. Mass Transfer Kinetics & Sizing Equations (Kremser & NTU/HTU)

# 3.1 The Extraction Factor ($E$)

For a solute partitioning between feed solvent ($F$) and extract solvent ($S$), the Distribution Coefficient ($K_D$) is:

The dimensionless Extraction Factor ($E$) governs separation feasibility:

• If $E < 1.0$: Complete extraction is thermodynamically impossible regardless of column height.
• If $E = 1.3 - 2.5$: The optimal economic and operational design regime.
• If $E > 5.0$: Extraction requires minimal stages, but consumes excessive solvent.

# 3.2 The Kremser-Brown-Souder Equation for Number of Theoretical Stages ($N_{\text{stages}}$)

For linear equilibrium ($Y^* = K_D \cdot X$), the required number of theoretical stages ($N_{\text{stages}}$) to achieve a target solute recovery fraction is:

# 3.3 Height Equivalent to a Theoretical Stage ($HETS$)

The active column height ($H_{\text{active}}$) is calculated from $N_{\text{stages}}$ and $HETS$:

For continuous agitated columns:

┌─────────────────────────────────┬─────────────────────────────────┬─────────────────────────────────┐
│ Column Architecture             │ Typical Laboratory HETS         │ Industrial Scale HETS (D=150mm) │
├─────────────────────────────────┼─────────────────────────────────┼─────────────────────────────────┤
│ Karr Reciprocating Column       │ 12 – 25 cm                      │ 35 – 65 cm                      │
│ Scheibel Rotary Column          │ 8 – 18 cm                       │ 25 – 45 cm                      │
│ Kühni Compartmented Turbine     │ 6 – 14 cm                       │ 18 – 35 cm                      │
│ Unagitated Packed Column        │ 50 – 120 cm                     │ 150 – 300 cm (Impractical!)     │
└─────────────────────────────────┴─────────────────────────────────┴─────────────────────────────────┘

Total overall column height includes top and bottom phase-disengagement settling sumps:

# 4. Complete Industrial Sizing Case Study: Continuous API Washing

• Process Task: Remove an organic acid catalyst byproduct from a toluene-rich stream ($45.0\,\text{L/h}$) into an aqueous sodium bicarbonate wash stream ($18.0\,\text{L/h}$).
• Operating Conditions: $25^\circ\text{C}$, atmospheric pressure.
• Solute Distribution Coefficient: $K_D = 4.2$ (prefers aqueous bicarbonate).
• Solute Reduction Target: Lower byproduct concentration from $12,500\,\text{ppm}$ down to $< 10\,\text{ppm}$ ($99.92\%$ extraction).

# 4.1 Step-by-Step Sizing Calculations

1. Extraction Factor ($E$):

(Condition $E > 1.3$ is verified).

2. Theoretical Stage Count ($N_{\text{stages}}$):

3. Column Diameter ($D_{\text{column}}$):
   • Total volumetric throughput: $Q_{\text{total}} = 45.0 + 18.0 = 63.0\,\text{L/h} = 1.75 \times 10^{-5}\,\text{m}^3/\text{s}$.
   • Allowable design superficial velocity (at $65\%$ of flooding): $v_{\text{design}} = 0.0035\,\text{m/s}$.
   • Required cross-sectional area:

• Calculated internal diameter:

4. Active Column Height ($H_{\text{active}}$) using a Kühni Column ($HETS = 25\,\text{cm}$):

• With $0.6\,\text{m}$ settling sumps at top and bottom:

# 5. Operational Troubleshooting & Control Strategy

┌─────────────────────────┬─────────────────────────┬─────────────────────────┬─────────────────────────┐
│ Operational Anomaly     │ Root Cause Physics      │ Impact on Extraction    │ Engineered Mitigation   │
├─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┤
│ Column Flooding at Low  │ Trace surfactant or     │ Phase inversion, severe │ Reduce agitator RPM by  │
│ Throughput              │ fine solids causing     │ carryover of aqueous in │ 20%; pulse clean solvent│
│                         │ premature coalescence   │ organic product stream  │ wash through column     │
├─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┤
│ Dispersed Phase Hazing  │ Agitation intensity too │ Sub-micron droplets     │ Increase wire mesh      │
│ (Milky Discharge)       │ high ($A \cdot f > (A \cdot f)_{\text{crit}}$)│ bypass settling sumps   │ packing depth; install  │
│                         │ creating micro-emulsion │                         │ coalescing filter pad   │
├─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┤
│ Interfacial Level Drift │ Interfacial level       │ Sudden loss of light or │ Deploy dual-flange      │
│ in Bottom Sump          │ transmitter drift (dirty│ heavy phase purity      │ differential pressure or│
│                         │ capacitance sensor)     │                         │ guided-wave radar probe │
├─────────────────────────┼─────────────────────────┼─────────────────────────┼─────────────────────────┤
│ Entrained Precipitate   │ API crystallization     │ Plugs plate perforations│ Increase operating      │
│ in Column Internals     │ triggered by localized  │ causing liquid backup   │ temperature by 5°C; add │
│                         │ pH or solubility shift  │ and flooding            │ 5% co-solvent to feed   │
└─────────────────────────┴─────────────────────────┴─────────────────────────┴─────────────────────────┘

# 6. Key Chemical Engineering Rules of Thumb

1. Selection Criteria: Choose the Karr Column if your process carries solid particulates, precipitates, or catalyst traces. Choose the Kühni Column when cleanroom vertical ceiling height is limited and high stage efficiency is paramount.
2. Phase Choice for Dispersion: As a rule of thumb, disperse the phase with the higher volumetric flow rate, unless interfacial tension or viscosity dictates otherwise. Dispersing the higher-flow phase maximizes total interfacial surface area.
3. Pilot Plant Scale-Up: Sizing agitated extraction columns requires a small-scale laboratory column ($D = 25 - 50\,\text{mm}$) test. Scale up to commercial diameter ($D > 100\,\text{mm}$) by maintaining equal specific power dissipation ($P/M$) and applying the scale-up factor:

Published by the PharmaChemEng Technical Editorial Board for chemical process development engineers, separation specialists, and industrial scale-up teams.