π Lesson 8
D4
Kremser Equation Derivation and Multi-Stage Cascade Simulation
The Kremser equation is a math tool that tells us how well a multi-stage liquid-liquid extraction system separates one chemical from another using repeated contact between two liquids.
π― Learning Objectives
- β Calculate fractional solute recovery using the Kremser equation for given stage count, distribution coefficient, and solvent-to-feed ratio
- β Analyze trade-offs between number of stages and solvent consumption by comparing Kremser predictions with graphical McCabeβThiele analysis
- β Explain the physical meaning of the absorption factor (A) and stripping factor (S), and diagnose when the Kremser equation becomes invalid due to nonlinearity or variable flows
- β Apply the Kremser equation to design a minimum solvent flow rate for a target recovery in a pharmaceutical LLE purification step
π Why This Matters
In hydrometallurgy, nuclear fuel reprocessing, and pharmaceutical manufacturing, extracting valuable metals or active ingredients from aqueous solutions often relies on liquid-liquid extraction (LLE) β think separating uranium from acid leachates or purifying antibiotics using organic solvents. Doing this efficiently requires balancing capital cost (number of mixer-settlers or columns) against operating cost (solvent use and recycling). The Kremser equation gives engineers a rapid, algebraic way to size extraction cascades *before* building expensive pilot plants β turning weeks of trial-and-error into minutes of calculation.
π Core Principles
The Kremser method assumes idealized, steady-state, countercurrent multistage extraction with constant flow rates and linear equilibrium: y = KΒ·x, where y and x are solute concentrations in raffinate and extract phases. Starting from component mass balances over each stage and applying geometric series summation, we derive closed-form expressions for solute recovery. Two forms exist: one for absorption-type (solute transferred *into* solvent) and one for stripping-type (solute removed *from* solvent). The key dimensionless group is the absorption factor A = KΒ·S/F (for extraction), where S is solvent flow rate and F is feed flow rate β when A > 1, high recovery is achievable with few stages; when A < 1, many stages or excess solvent are needed. Deviations occur with nonlinear isotherms, emulsion formation, or significant phase volume change β signaling need for simulation or graphical methods.
π Key Calculation
The Kremser equation for fractional recovery (R) of solute in a countercurrent N-stage extractor is: R = (1 β A^βN) / (1 β A^β(N+1)) for A β 1, where A = KΒ·S/F. For stripping (e.g., back-extraction), replace A with S = 1/A and use the stripping form. This equation applies only under constant-flow, linear-equilibrium conditions β always verify K from lab shake-flask tests and confirm flow stability before application.
π‘ Worked Example
Problem: A copper sulfate leach solution (F = 100 L/min, 5 g/L Cu) is extracted using kerosene-based solvent (S = 60 L/min). Lab tests give K = 3.2 (Cu concentration ratio: organic/aqueous). Design a 4-stage countercurrent mixer-settler. What % of Cu is recovered?
1.
Step 1: Compute absorption factor A = KΒ·S/F = 3.2 Γ (60/100) = 1.92
2.
Step 2: Apply Kremser equation: R = (1 β A^βN) / (1 β A^β(N+1)) = (1 β 1.92^β4) / (1 β 1.92^β5)
3.
Step 3: Calculate: 1.92^β4 β 0.074; 1.92^β5 β 0.039 β R = (1 β 0.074) / (1 β 0.039) = 0.926 / 0.961 β 0.964
4.
Step 4: Convert to %: 0.964 Γ 100 = 96.4% recovery
Answer:
The result is 96.4%, which falls within the typical high-recovery range of 95β98% for well-designed 4-stage copper SX circuits.
ποΈ Real-World Application
In the Rio Tinto Kennecott Utah Copper solvent extractionβelectrowinning (SX-EW) plant, the Kremser equation guides initial design of the copper extraction cascade (using LIX 984N in kerosene). With K β 2.8β3.5 (pH-dependent), F β 1200 mΒ³/h leach solution, and target >99% Cu recovery, engineers used Kremser to compare 6-stage vs. 8-stage configurations at varying S/F ratios β identifying that 6 stages at S/F = 0.45 achieved 99.2% recovery, minimizing capital cost while meeting purity specs for electrowinning cathodes. Field validation showed <2% deviation from Kremser-predicted recovery β validating its utility for preliminary sizing.
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