Liquid-Liquid Extraction Equilibrium & Stage Calculations - Chemical Engineering Knowledge Platform

Typical Scale

Industrial plants handle 10–500 m³/h aqueous feed (e.g., INCO’s nickel SX plant: 320 m³/h)

Key Standards

ASTM D1193 (reagent water), ISO 10303-238 (process data exchange for SX/EW)

Industry Applications

Copper SX-EW, rare earth element separation, penicillin G recovery, spent nuclear fuel reprocessing (PUREX)

⚠️ Why It Matters

1

Inaccurate K_D estimation

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2

Wrong solvent-to-feed ratio selection

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3

Insufficient theoretical stages

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4

Low solute recovery (<90%)

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5

Increased downstream separation load

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6

Higher operating cost & environmental footprint

📘 Definition

Liquid-liquid extraction (LLE) is a mass transfer unit operation in which a solute is selectively partitioned between two immiscible or partially miscible liquid phases based on relative solubility and chemical affinity. Equilibrium is governed by the distribution coefficient (K_D), and stage-wise design relies on material balances coupled with equilibrium relationships to achieve target recovery and purity. It is widely applied in hydrometallurgy, pharmaceutical purification, nuclear fuel reprocessing, and fine chemical synthesis.

🎨 Concept Diagram

AI-generated illustration for visual understanding

💡 Engineering Insight

Never assume equilibrium is instantaneous — especially with viscous solvents or slow-dissociating metal complexes (e.g., Cu(II)-LIX984). Always measure effective K_D under dynamic mixing conditions (not just shaken-flask), because mass transfer resistance can depress apparent K_D by up to 40% versus static equilibrium values.

📖 Detailed Explanation

At its core, liquid-liquid extraction exploits differences in molecular polarity and coordination chemistry: a polar solute (e.g., acetic acid) partitions into a polar phase (water), while a nonpolar solute (e.g., caffeine) prefers a nonpolar phase (dichloromethane). Equilibrium is reached when the chemical potential of the solute is equal in both phases — a condition mathematically expressed by the distribution law.

Beyond simple partitioning, real systems involve complex equilibria: association (e.g., dimerization of carboxylic acids in organic phase), ion-pair formation (e.g., FeCl₄⁻ with quaternary amines), and pH-dependent speciation (e.g., UO₂²⁺ extraction only above pH 1.8). These require extended models like the mass action approach or NRTL-SAC for activity coefficient prediction — not just K_D.

Advanced design accounts for nonequilibrium effects: axial dispersion in columns, holdup variation with flow rate, and transient behavior during feed composition shifts. Modern practice couples population balance models (for droplet size distribution) with computational fluid dynamics (CFD) to predict mixing intensity, interfacial area, and local mass flux — enabling digital twin deployment for predictive maintenance and solvent inventory optimization.

🔄 Engineering Workflow

Step 1

Step 1: Define separation objective (recovery %, purity spec, throughput)

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Step 2

Step 2: Screen solvents via ternary phase diagram & measure K_D/β at process T & pH

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Step 3

Step 3: Construct equilibrium curve and operating line; determine minimum solvent-to-feed ratio (S/F)_min

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Step 4

Step 4: Calculate theoretical stages using McCabe–Thiele or Kremser equation (for dilute systems)

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Step 5

Step 5: Size equipment (mixer-settler, column, or centrifugal contactor) using mass transfer coefficients (k_La) and residence time requirements

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Step 6

Step 6: Validate with pilot-scale testing (≥48 h continuous run) and analyze raffinate/extract composition drift

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Step 7

Step 7: Implement online monitoring (conductivity, UV-Vis, NIR) and feedback control on S/F ratio and pH

📋 Decision Guide

Rock/Field Condition	Recommended Design Action
K_D < 0.3 and β > 15	Use countercurrent multistage cascade with refluxed solvent; avoid single-stage batch extraction.
K_D ≈ 1.0–3.0 and μ < 3 cP	Optimize for 3–5 theoretical stages in pulsed sieve-plate column; minimal energy input required.
γ < 2 mN/m and β < 4	Add demulsifier or switch to ionic liquid solvent; install coalescer section or electrostatic separator.

📊 Key Properties & Parameters

Distribution Coefficient (K_D)

0.1–100 (dimensionless)

Ratio of solute concentration in the extract phase to that in the raffinate phase at equilibrium (K_D = C_ext / C_raff).

⚡ Engineering Impact:

Directly determines minimum solvent flow rate and governs feasibility of single-stage extraction.

Selectivity (β)

2–50 (dimensionless)

Ratio of distribution coefficients for two solutes (e.g., β = K_D,A / K_D,B), quantifying relative preference of solvent for target vs. impurity.

⚡ Engineering Impact:

Dictates number of stages required for separation; β < 2 often necessitates >10 stages or alternative separation methods.

Interfacial Tension (γ)

0.5–30 mN/m

Energy per unit area at the interface between two immiscible liquids, influencing droplet formation, coalescence, and mass transfer rate.

⚡ Engineering Impact:

Low γ (<5 mN/m) promotes stable emulsions and poor phase disengagement; high γ (>20 mN/m) limits interfacial area and slows extraction kinetics.

Solvent Viscosity (μ)

0.5–15 cP at 25°C

Resistance of the extracting solvent to flow under shear, affecting mixing efficiency and phase separation time.

⚡ Engineering Impact:

Viscosity >8 cP significantly increases mixer power demand and settling time, limiting throughput in centrifugal contactors.

📐 Key Formulas

Distribution Coefficient

K_D = C_{ext} / C_{raff}

Quantifies equilibrium partitioning of solute between extract and raffinate phases.

Variables:

Symbol	Name	Unit	Description
K_D	Distribution Coefficient		Ratio of solute concentration in extract phase to solute concentration in raffinate phase at equilibrium
C_{ext}	Concentration in Extract Phase	mol/L or g/L	Equilibrium concentration of solute in the extract phase
C_{raff}	Concentration in Raffinate Phase	mol/L or g/L	Equilibrium concentration of solute in the raffinate phase

Typical Ranges:

Copper in LIX84/kerosene–H₂SO₄ leachate

8–25

Uranium in D2EHPA/kerosene–H₂SO₄

5–20

⚠️ K_D < 0.2 indicates impractical solvent; K_D > 50 may cause third-phase formation

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Minimum Solvent-to-Feed Ratio (S/F)_min

(S/F)_min = (x_F - x_R) / (x_E - x_F)

Lowest solvent flow needed for infinite stages; derived from tie-line intersection on McCabe–Thiele plot.

Variables:

Symbol	Name	Description
S/F_min	Minimum Solvent-to-Feed Ratio	Lowest solvent flow rate to feed flow rate required for extraction with infinite theoretical stages
x_F	Feed Solute Mass Fraction	Mass fraction of solute in the feed stream
x_R	Raffinate Solute Mass Fraction	Mass fraction of solute in the raffinate (extracted) stream
x_E	Extract Solute Mass Fraction	Mass fraction of solute in the extract stream

Typical Ranges:

Dilute pharmaceutical API extraction

0.8–2.5 (kg solvent/kg feed)

Hydrometallurgical copper SX

1.2–3.0 (L/L)

⚠️ Operate at 1.2–1.5× (S/F)_min to ensure robustness against feed variability

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Kremser Equation (extraction factor E = K_D × S/F)

Fraction extracted = (E^N - 1) / (E^{N+1} - 1)

Predicts overall solute recovery for N ideal stages under constant K_D and flow rates.

Variables:

Symbol	Name	Unit	Description
E	Extraction factor		Dimensionless ratio defined as K_D × S/F, where K_D is the distribution coefficient, S is solvent flow rate, and F is feed flow rate
N	Number of ideal stages		Number of theoretical equilibrium stages in the extraction process
K_D	Distribution coefficient		Ratio of solute concentration in solvent phase to that in feed phase at equilibrium
S	Solvent flow rate	kg/s or mol/s	Mass or molar flow rate of solvent
F	Feed flow rate	kg/s or mol/s	Mass or molar flow rate of feed solution