Whey Demineralization by Electrodialysis

Reference ID: MET-863C | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Whey demineralization by electrodialysis (ED) is a critical unit operation in the dairy industry, specifically for the production of infant formula and high‑value protein concentrates. The process utilizes an electrochemical potential gradient to selectively migrate ions across ion‑exchange membranes, effectively reducing the ash content of sweet whey. The demineralization efficiency calculation is essential for process engineers to determine the required number of stages, energy requirements, and operational feasibility while avoiding membrane fouling and water splitting.

Methodology & Formulas

The demineralization process is governed by the relationship between the electrical current applied to the stack and the mass transfer of ionic species, a concept detailed in the ion transport principle in electrodialysis. The following formulas define the core physics of the system:

a) Current Efficiency for Ion Species:

\[ \eta_{i} = \frac{z_{i} \cdot F \cdot \Delta N_{i}}{I} \]

Where \(\eta_{i}\) is the current efficiency for species \(i\), \(z_{i}\) is the ion valence, \(F\) is the Faraday constant, \(\Delta N_{i}\) is the molar removal rate (mol/s), and \(I\) is the total stack current. For a comprehensive overview of the whey demineralization process design, see whey demineralization process design.

b) Batch Demineralization Kinetics (Constant Current):

\[ C(t) = C_{0} \cdot \exp\left( -\frac{\eta \cdot I}{\bar{z} \cdot F \cdot V} \cdot t \right) \]

Where \(C(t)\) is the diluate concentration at time \(t\), \(C_{0}\) is the initial concentration, \(\eta\) is the overall current efficiency, \(I\) is the total stack current, \(\bar{z}\) is the average ion valence, \(F\) is the Faraday constant, and \(V\) is the diluate volume. This first-order model is widely used as an engineering approximation, implicitly accounting for the decline in current efficiency as the diluate becomes depleted.

c) Number of Stages Required:

\[ n = \frac{\ln(C_{\mathrm{target}} / C_{\mathrm{initial}})}{\ln(1 - R_{\mathrm{stage}})} \]

Where \(n\) is the number of stages and \(R_{\mathrm{stage}}\) is the fractional ash removal efficiency per stage.

Operational Constraints and Regimes

Parameter	Operational Range	Consequence of Violation
Current Density	20 – 50 mA/cm²	Water splitting, pH swings, and membrane scaling
Temperature	10 – 30 °C	Viscosity issues (low) or protein denaturation (high)
Feed Ash Content	5 – 12 % (dry basis)	Process becomes economically unviable
Stage Removal Efficiency	0.35 – 0.50	Current inefficiency and excessive membrane area
Voltage per Cell Pair	0.5 – 1.5 V	Negligible current (low) or water electrolysis (high)

To ensure optimal performance and membrane longevity, process engineers should monitor the following variables:

Current density to prevent polarization and water splitting.
Feed flow rate and pressure drop across the stack to detect fouling.
Conductivity of the diluate stream to track demineralization efficiency.
Temperature of the whey feed to maintain viscosity and ionic mobility.
pH levels to prevent protein precipitation on the membrane surface.

Protein fouling is a common challenge in electrodialysis. You can mitigate this by:

Implementing periodic polarity reversal to dislodge deposits.
Maintaining the feed pH away from the isoelectric point of the whey proteins.
Utilizing high-turbulence spacers to minimize the boundary layer thickness.
Performing regular Clean-in-Place (CIP) cycles using enzymatic or alkaline detergents.

Operating above the limiting current density leads to water splitting, which generates H⁺ and OH⁻ ions. This results in:

Significant pH shifts in the diluate and concentrate streams.
Increased energy consumption due to ohmic resistance.
Potential damage to the ion-exchange membranes.
Reduced current efficiency for the transport of target minerals.

Pre-filtration is essential to protect the stack from physical damage and premature fouling. Key reasons include:

Removal of suspended solids that can block the narrow flow channels.
Reduction of lipid content, which can coat the membrane surface and block ionic sites.
Prevention of particulate accumulation that causes uneven flow distribution across the membrane area.

Worked Example: Stage Calculation for Whey Demineralization by Electrodialysis

Scenario: A whey demineralization plant must reduce the ash content from 8.0% to 4.0% (dry basis) for infant formula production. The plant operates a batch electrodialysis (ED) system with constant current control. The following known parameters are provided.

Knowns (Input Parameters)

\(C_{\mathrm{initial}}\) (initial ash) = 8.0 % (dry basis)
\(C_{\mathrm{target}}\) (target ash) = 4.0 % (dry basis)
\(\eta\) (current efficiency) = 0.85
\(\bar{z}\) (average valence) = 1.5
\(R_{\mathrm{stage}}\) (stage removal efficiency) = 0.4
\(i\) (current density) = 35.0 mA/cm²
\(T\) (temperature) = 20.0 °C
\(V_{\mathrm{pair}}\) (voltage per cell pair) = 1.2 V
\(t_{\mathrm{life}}\) (membrane lifetime) = 3.0 years

Step-by-Step Calculation

Verify operational bounds:
- Current density \(i = 35.0\ \mathrm{mA/cm^2}\) is within 20–50 mA/cm² ✓
- Temperature \(T = 20.0\ \mathrm{°C}\) is within 10–30 °C ✓
- Stage removal efficiency \(R_{\mathrm{stage}} = 0.4\) is within 0.35–0.50 ✓
- Voltage per cell pair \(V_{\mathrm{pair}} = 1.2\ \mathrm{V}\) is within 0.5–1.5 V ✓
All parameters are within acceptable operating ranges.
Compute the natural logarithm of the target-to-initial ash ratio:
\(\ln\left(\frac{C_{\mathrm{target}}}{C_{\mathrm{initial}}}\right) = \ln\left(\frac{4.0}{8.0}\right) = \ln(0.5) = -0.693\).
Compute the natural logarithm of the retention factor \((1 - R_{\mathrm{stage}})\):
\(\ln(1 - R_{\mathrm{stage}}) = \ln(0.6) = -0.511\).
Calculate the theoretical number of stages as a floating-point value:
\(n_{\mathrm{float}} = \frac{\ln(C_{\mathrm{target}}/C_{\mathrm{initial}})}{\ln(1 - R_{\mathrm{stage}})} = \frac{-0.693}{-0.511} = 1.357\).
Round up to the next integer, since a fraction of a stage is not possible in a batch system:
\(n_{\mathrm{required}} = \lceil 1.357 \rceil = 2\).
Verify the stage sequence:
- After Stage 1: \(C_{\mathrm{stage1}} = C_{\mathrm{initial}} \cdot (1 - R_{\mathrm{stage}}) = 8.0 \cdot 0.6 = 4.8\) %.
- After Stage 2: \(C_{\mathrm{stage2}} = C_{\mathrm{stage1}} \cdot (1 - R_{\mathrm{stage}}) = 4.8 \cdot 0.6 = 2.88\) %.

Final Answer
The required number of stages is 2. The final ash content after two stages is 2.88%, which is below the target of 4.0%. Therefore, the process is feasible (\(C_{\mathrm{stage2}} = 2.88 \le 4.0\)). The additional parameters (\(\eta\), \(\bar{z}\), \(i\), \(T\), \(V_{\mathrm{pair}}\), \(t_{\mathrm{life}}\)) are provided for reference and would be used for current requirement, batch time, and energy calculations using formulas (a) and (b) from the methodology.

"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