Reference ID: MET-7CC1 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Electrodialysis (ED) is a membrane-based separation process driven by an electric potential gradient. In the context of Process Engineering, it is primarily utilized for the selective removal of ionic species from aqueous solutions. A critical application is the desalting of whey permeate, where the removal of minerals (salts) is necessary to produce high‑value food ingredients. For detailed guidance on whey demineralization using electrodialysis, see whey demineralization by electrodialysis. The process relies on the migration of ions through alternating cation‑exchange membranes (CEM) and anion‑exchange membranes (AEM), effectively concentrating ions in a brine stream while depleting them in the product stream.
Methodology & Formulas
The design of an ED stack is governed by the relationship between mass transport, electrical current, and membrane area. The following formulas define the transport physics and the sizing requirements for a desalting system.
1. Mass Removal Rate
The molar removal rate of salt required to achieve a specific concentration change is determined by the volumetric flow rate and the concentration gradient:
\[ \dot{n}_{\text{salt}} = Q \cdot \Delta c \]
2. Total Current Requirement
Based on Faraday's Law, the total current required to achieve the desired mass removal is a function of the ion valence, the Faraday constant, and the current efficiency of the stack:
\[ I_{\text{total}} = \frac{z \cdot F \cdot \dot{n}_{\text{salt}}}{\eta} \]
3. Number of Cell Pairs
The number of cell pairs (\(N\)) required to meet the process demand is calculated by dividing the total current by the product of the design current density and the active area per cell pair:
\[ N = \frac{z \cdot F \cdot \dot{n}_{\text{salt}}}{\eta \cdot i \cdot A_{\text{cell}}} \]
4. Actual Current Density
Once the number of cell pairs is determined (typically rounded up to the nearest integer), the actual operating current density is verified as:
The ion transport principle relies on the application of an external direct current electric field across a stack of alternating cation-exchange membranes and anion-exchange membranes. The process involves:
Ions migrate toward the electrode of opposite charge.
Cations move toward the cathode, passing through cation-exchange membranes but being blocked by anion-exchange membranes.
Anions move toward the anode, passing through anion-exchange membranes but being blocked by cation-exchange membranes.
This selective migration results in the formation of alternating dilute and concentrated streams.
The limiting current density represents the threshold where the rate of ion transport through the membrane equals the rate of ion supply from the bulk solution. Exceeding this value leads to:
Water splitting at the membrane interface.
A significant increase in electrical resistance.
Potential pH fluctuations and scaling issues within the stack.
Permselectivity is a critical parameter for process engineers as it dictates the ability of the membrane to exclude co-ions while allowing counter-ions to pass. High permselectivity ensures:
Reduced energy consumption by minimizing current leakage.
Higher purity levels in the dilute product stream.
Improved current efficiency during high-salinity operations.
Concentration polarization occurs due to the difference in transport numbers between the membrane and the solution. Key contributing factors include:
Low fluid velocity leading to a thick boundary layer.
High current density relative to the ion concentration.
Inadequate turbulence promotion within the flow spacers.
Worked Example: Sizing an Electrodialysis Stack for Whey Permeate Desalting
Scenario: A dairy processing plant needs to desalt 1.0 m³/h of whey permeate, reducing NaCl concentration from 10 kg/m³ to 2 kg/m³. An electrodialysis (ED) stack is to be designed with ideal cation- and anion-exchange membranes operating at constant current density and maximum current efficiency. The design uses the migration flux principle together with Faraday’s law.
Required total molar salt removal rate (from mass balance):
\[ \dot{n}_{\text{salt}} = Q_{\text{m}^3/\text{s}} \cdot \Delta c = 2.778 \times 10^{-4}\ \text{m}^3/\text{s} \times 136.90\ \text{mol/m}^3 \approx 0.03803\ \text{mol/s} \]
Calculate number of cell pairs (\(N_{\text{cells}}\)) using Faraday’s law and the applied current density:
\[ N = \frac{z \cdot F \cdot \dot{n}_{\text{salt}}}{\eta \cdot i_{\text{applied}} \cdot A_{\text{cell}}} = \frac{1 \times 96485 \times 0.03803}{0.90 \times 80.0 \times 0.5} = \frac{3669.0}{36.0} \approx 101.92 \rightarrow \mathbf{N = 102}\ \text{cell pairs} \] (rounded up to the nearest integer)
Calculate total current required for this removal rate:
\[ I_{\text{total}} = \frac{z \cdot F \cdot \dot{n}_{\text{salt}}}{\eta} = \frac{1 \times 96485 \times 0.03803}{0.90} \approx 4076.7\ \text{A} \]
Verify actual current density with the integer number of cell pairs:
\[ i_{\text{actual}} = \frac{I_{\text{total}}}{N_{\text{cells}} \cdot A_{\text{cell}}} = \frac{4076.7}{102 \times 0.5} = \frac{4076.7}{51} \approx 79.93\ \text{A/m}^2 \]
Final Answer:
Number of cell pairs required: 102 (designed with 0.5 m² per cell pair). Total current: 4076.7 A. Actual current density: 79.93 A/m², which is within 0.1 % of the applied assumption (80.0 A/m²) — design consistency is verified. The stack desalts 1.0 m³/h of whey permeate from 10 kg/m³ to 2 kg/m³ NaCl under ideal conditions.
Note: Empirical ranges (current density 50–150 A/m², cell voltage 1–2 V, current efficiency 0.85–0.95) are all satisfied.
"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
