Reference ID: MET-A3CE | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The equilibrium relationship in leaching, or solid-liquid extraction, is a fundamental concept in process engineering used to determine the distribution of a solute between a solid matrix and a liquid solvent. This calculation is critical for designing separation stages in industries such as food processing (e.g., sugar extraction from beet cossettes), hydrometallurgy, and pharmaceutical manufacturing.
In a single-stage ideal leaching process, the goal is to predict the final concentration of the extract and the composition of the liquid retained within the solid residue; by assuming perfect contact and immediate equilibrium, engineers can model the efficiency of the extraction process and determine the mass of solvent required to achieve specific purity targets, a calculation that is detailed in the material balance for single‑stage extraction methodology.
Methodology & Formulas
The calculation relies on mass balances for the solute and solvent, combined with the physical constraints of the solid matrix. The following variables are defined: B (mass of inert solid), A (mass of retained solvent), C (mass of retained solute), Stotal (total solute mass), and Wtotal (total solvent mass).
The system is governed by the following algebraic relationships:
Total Liquid Retained: The amount of liquid held by the solid matrix is determined by the holding capacity N:
\[ L_{ret} = \frac{B}{N} \]
Mass Balance Constraints: The total liquid retained is the sum of the retained solvent and solute:
\[ L_{ret} = A + C \]
Equilibrium Relationship: Under ideal conditions, the concentration of the extract y* equals the concentration of the imbibed solution x:
\[ y^* = x = \frac{C}{A} = \frac{S_{total} - C}{W_{total} - A} \]
Derived Solute Distribution: Based on the equilibrium ratio, the retained solute is proportional to the retained solvent:
\[ C = A \cdot \left( \frac{S_{total}}{W_{total}} \right) \]
Final Retained Solvent Calculation: Substituting the distribution into the liquid retention equation:
\[ A = \frac{L_{ret}}{1 + \left( \frac{S_{total}}{W_{total}} \right)} \]
Parameter
Condition/Regime
Threshold/Range
Holding Capacity (N)
Empirical Validity
0.1 ≤ N ≤ 2.0
Total Solvent (Wtotal)
Physical Existence
Wtotal > 0
Equilibrium Assumption
Ideal Stage
y* = x
To determine the equilibrium stage, you must establish the relationship between the solute concentration in the solid phase and the liquid phase. Follow these steps:
Plot the experimental equilibrium data on a ternary diagram or an X-Y coordinate system.
Construct the operating line based on the mass balance of the solute across the system.
Step off the theoretical stages between the operating line and the equilibrium curve starting from the feed concentration.
Account for the underflow retention of the solvent to adjust the stage efficiency.
The equilibrium relationship is primarily governed by the physical and chemical properties of the solute-solvent-solid matrix. Key factors include:
Temperature, which significantly alters the solubility of the solute in the solvent.
The nature of the solid matrix, specifically its porosity and adsorption capacity.
The presence of other dissolved species that may cause common-ion effects or competitive adsorption.
The solvent-to-solid ratio, which dictates the maximum achievable concentration in the extract.
The assumption of constant underflow implies that the amount of liquid retained by the solid remains unchanged regardless of the solute concentration. In practice, this is rarely true because:
Changes in solute concentration often alter the density and viscosity of the liquid phase.
The solid matrix may swell or shrink depending on the solvent composition.
High solute concentrations can lead to precipitation or changes in the interfacial tension, affecting the drainage characteristics of the solid cake.
Worked Example: Single-Stage Leaching of Sugar Beet Cossettes
A batch of sugar beet cossettes is leached in a single ideal equilibrium stage with hot water to extract sugar. The goal is to determine the equilibrium sugar concentration in the liquid extract stream.
Knowns (Input Parameters):
Mass of solute (sugar) in solid feed, \( S_{in} = 14.0 \, \text{kg} \).
Mass of inherent solvent (water) in solid feed, \( W_{inherent} = 16.0 \, \text{kg} \).
Mass of added solute-free solvent (water), \( W_{added} = 200.0 \, \text{kg} \).
Mass of inert solid (fiber), \( B = 70.0 \, \text{kg} \).
Holding capacity of the solid matrix, \( N = 0.5 \, \text{kg inert solid per kg retained liquid} \).
Step-by-Step Calculation:
Calculate total solute and total solvent available:
Determine the total liquid retained with the solids using the holding capacity:
\[ L_{retained} = \frac{B}{N} = \frac{70.0 \, \text{kg}}{0.5} = 140.0 \, \text{kg} \]
This represents the mass of liquid (solvent plus solute) held in the solid residue after leaching.
Apply the ideal equilibrium relationship and overall mass balances. The system equations are:
Solute balance: \( S_{total} = C + S_{extract} \)
Solvent balance: \( W_{total} = A + W_{extract} \)
Equilibrium: \( y^* = x \), which gives \( \frac{C}{A} = \frac{S_{extract}}{W_{extract}} \)
Liquid retained: \( L_{retained} = A + C \)
Solving these equations simultaneously yields:
\[ A = 131.478 \, \text{kg}, \quad C = 8.522 \, \text{kg} \]
where \( A \) is the mass of solvent retained and \( C \) is the mass of solute retained in the solid residue.
Compute the mass of solute and solvent in the extract stream using the mass balances:
\[ S_{extract} = S_{total} - C = 14.0 \, \text{kg} - 8.522 \, \text{kg} = 5.478 \, \text{kg} \]
\[ W_{extract} = W_{total} - A = 216.0 \, \text{kg} - 131.478 \, \text{kg} = 84.522 \, \text{kg} \]
Calculate the equilibrium concentration \( y^* \), which equals the concentration \( x \) in the retained liquid:
\[ y^* = x = \frac{C}{A} = \frac{8.522 \, \text{kg}}{131.478 \, \text{kg}} = 0.065 \, \text{kg solute per kg solvent} \]
This is a dimensionless mass ratio.
Final Answer:
The equilibrium sugar concentration in the extract is \( y^* = 0.065 \, \text{kg sugar per kg water} \).
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
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