Online Optimizing Control and Dynamic Operation and Design Optimization of a Batch Electrodialysis Process for Sulfuric Acid Recovery

Abstract

Electrodialysis is an efficient separation and recovery method for ionic species such as sulfate ions. The batch operation of the electrodialysis process unit involves several decisions that affect its separation and economic performance. An optimal control system is developed that monitors the separation efficiency under real-time conditions and identifies the most suitable operating profile for the applied current voltage and the recirculation flow rate. A dynamic model is employed for the electrodialysis process, which is subsequently utilized within a dynamic optimization framework that aims to meet the separation and recovery specifications in the most economical way while satisfying the operating constraints. A discretized model using orthogonal collocation on finite elements enables the calculation of the optimal profile for the current voltage using nonlinear programming techniques. The control system has been successfully applied in the compensation of process disturbances mainly attributed to the variation of the membrane activity and other factors. Under severe membrane-activity loss (50–65%), the adaptive control profile achieved an increase of 34.9% in the degree of separation while limiting the batch-time penalty to 15.5% at the expense of higher energy consumption. An optimization problem is further formulated that determines the optimal design and operational characteristics of an industrial-scale size unit. In addition to the control variable profiles, the membrane surface that minimizes a comprehensive objective function is calculated. The objective function incorporates several targets for the electrodialysis process, such as batch duration, energy requirements, achieved degree of separation, membrane size, and control action behavior. The obtained optimal solutions are analyzed by Pareto front methods to reveal the critical trade-offs among the various competing objective function terms. The proposed approach enables the efficient separation of ions by electrodialysis in a diversely operating environment.

1. Introduction

Electrodialysis (ED) is an efficient process for the effective separation and recovery of valuable ions from dilute solutions. There are numerous applications where ED could be highly beneficial, such as for the removal of heavy metals like lead from industrial water effluents, nitrate ions from polluted aquifers to produce potable water, and salts from brackish waters, to name but a few. It combines the use of membranes that are selective toward specific ionic species, along with an applied electric potential that is responsible for the migration of the ions toward electrodes of opposite charges.

The optimization of an ED process requires the minimization of the energy consumption while operating under conditions that will minimize membrane polarization and fouling as this is a key phenomenon that is the main cause for performance reduction. For example, Vermaas et al. estimated that the lack of antifouling strategies could result in 40% higher power density requirements even for short periods of operation. There are several operating and design parameters that play a significant role in the design of optimum ED systems, including the stack voltage, the membrane size, and the flow rate of the processed streams. ED operation is also subject to disturbances caused by membrane fouling, the variation in the concentration of ions in the treated stream, or the variation in the stack voltage in renewable energy-powered ED systems, which may deteriorate separation performance significantly. Unless such disturbances are addressed efficiently, they have detrimental effects on the requirement to meet stringent outlet composition and concentration specifications as well as the process economics due to increased power utilization.

Electrodialysis usually takes place in the batch operation mode. Therefore, dynamic optimization is essential in order to design both robust and flexible ED systems with sufficient process capacity to implement the appropriate operating policies for the desired resiliency to disturbances. In addition, the online calculation and adaptation of the optimal operating profiles based on the prevailing process conditions are crucial for the satisfaction of the separation system specifications despite the effects of the disturbances.

Dynamic optimization has been applied to the ED design and operation. The weighted-average resistance method has been used to optimize current and flow trajectories to reduce energy use and operating cost for salt removal. An OCFE-based model has been developed to obtain optimum operating conditions for hydrochloric acid recovery and to maximize ED performance. OCFE has also been used to optimize ED stacks operating in both forward and reverse modes, examining the influence of process parameters such as flow rate patterns on performance. The aforementioned studies indicated that reduced-order models, such as OCFE models, address the computational complexities of dynamic optimization problems effectively. The transformation of the ED dynamic model expressed in terms of differential and algebraic equations into a set of nonlinear equations enables the employment of nonlinear programming techniques for the calculation of optimal control profiles. In addition, OCFE models are ideal for the approximation of sharply changing concentration profiles and exploitation from the numerical point of view of mildly changing concentration profiles frequently occurring during operation. Comparative work has reported the superior performance of OCFE over control-vector parametrization and multiple shooting for ED dynamic optimization, identifying policies with lower energy consumption and higher acid recovery within acceptable computation times. OCFE models were able to identify operating policies that resulted in the lowest energy consumption with a maximum acid recovery within an acceptable computational time. Dynamic optimization can be directly linked to the operational costs of the ED process as disturbances and model uncertainty may cause performance degradation and potential failure to satisfy process constraints and product specifications.

To this end, advanced methods have been proposed in the published literature for applications other than ED that develop control schemes to account for uncertainties in system parameters. A semiautomated procedure for constrained dynamic optimization of uncertain large-scale processes uses iterative updates based on necessary conditions of optimality to maintain feasibility and performance. Online dynamic optimization frameworks have been shown more effective than tracking precomputed set points. Extensions include online optimizing control of polymerization reactors with the integrated state and parameter estimation and analogous strategies for semibatch polymerization.

Few control-based approaches have been reported that address uncertainties in the power source and the model parameters for the ED process. For batch ED desalination of brackish water, time-variant feed-forward voltage scheduling that tracks a safe fraction of the limiting current has been shown to keep stack current within roughly −15% to +20% of a target and to increase batch-ED production rate by up to 37% versus constant-voltage operation. , While their work indicated the benefits of a time-variant controller, the approach was not extended to the fluid flow rates assuming linear flow velocity models. Model-based cocontrol of voltage and flow (two degrees of freedom) has also been piloted to maximize desalination rate under variable power, reporting up to 45% higher production than an equivalently sized steady-operation baseline without disclosing details for the underlying methods. The goal was to improve the brackish water desalination rate on an ED system by controlling the stack voltage and flow rate. A model predictive control algorithm was implemented by Le Henaff. Later, Connors elaborated on Le Henaff’s work to adjust in real time the flow rate and voltage profiles and to maximize the desalination rate while using time-variant solar power. However, both studies did not consider parameter variations and process disturbances, which may strongly affect the operation of the ED unit. Under significant model parameter variations, this control scheme may lead to aggressive and less robust control changes, increasing the operational costs for the process.

Operating-regime analyses indicate that constant-current (CC) and constant-entropy-generation (CEG) operation can lower energy relative to constant-voltage (CV) at equal duty, with the advantage being most pronounced at higher salinity. Architectural choices can also reduce power demand; a two-stage ED configuration achieved up to 29% lower power at equal duty. For renewable direct-drive, a cascade-PID flow-commanded current control (FCCC) field pilot harnessed 94% of available PV energy while requiring >99% less battery than literature-median PV-desalination systems; related model-based time-variant EDR achieved 77% direct use of available solar energy (+91% vs conventional), a 92% reduction in battery reliance, and a 22% reduction in levelized cost of water. ,

The above studies considered ED systems of fixed capacity, where dynamic optimization and control approaches were implemented. It is important to be able to simultaneously address the calculation of operating policies and control actions with the design of system size characteristics. Such an approach adds extra flexibility in meeting the design and operating system constraints and may lead to considerably improved system performance. Connors approached this requirement for the design and control of ED systems through a two-level method. An MPC block was used to find the optimal control action based on the response of the system, whereas the system update block tried to estimate the optimal matching system parameters. While the use of an MPC has been proven to be a great tool to alleviate short-term disturbances, the computational complexities hinder its application. Furthermore, in the objective function of the design optimization block used by Connors, the relative weighting between reliability and cost terms was imposed by an external penalty term. The selection of this parameter was based on empirical and ad hoc rules and could be rigorously defined either by leveraging the designer’s experience or by considering a hypothetical cost attributed to potential deviations from the desired reliability levels. However, it is crucial to acknowledge that a more comprehensive approach should be implemented to thoroughly explore the various trade-offs between reliability and cost. By adoption of such an approach, a deeper understanding of the optimal balance between these two factors can be achieved.

The discussion described above indicates that the existing literature contains limited research on control strategies for ED systems. Online parameter estimation algorithms that specifically address the uncertainty arising from the unknown imposed disturbances and drifting of model parameters values have not been reported. Rare implementations of control approaches address the system design without considering dynamic optimization while also involving limitations in the objective function formulation. The objective of the present work is to highlight the usefulness of dynamic optimization in the control and design of ED systems. Initially, we developed a dynamic optimization approach for the online optimization of the batch ED stack. The aim is primarily to attain an efficient compensation of time-varying major disturbances, in combination with the estimation of the disturbance magnitude through an optimization-based estimator. The real-time estimated values expand the knowledge on the system and the effects of the disturbances, allowing the algorithm to modify its parameter values to match the real cases. Furthermore, the second part of this study implements dynamic optimization to simultaneously determine both the optimal design of the ED system sizes and its operating conditions. The proposed approach incorporates a multiobjective analysis to systematically determine the set of optimal designs for the ED. Finally, the designed system successfully compensates disturbances and achieves the desired targets.

2. Models and Mathematical Representations

2.1. ED Process Model

An ED unit that operates in the batch recirculation process mode is considered. A specific amount of contaminated solution is processed by multiple passes through the ion selective membranes under an applied electric field. The concentrated and the diluted solutions (called the concentrate and diluate) are gathered in two separate tanks and are continuously recirculated until the desired concentration is reached in the diluate as shown in Figure . The applied electric current to the system generates the required potential difference, resulting in the creation of the driving force for the motion of ions through the ion selective membranes. The cations seep through the negative charged cation exchange (CE) membrane (CEM) toward the cathode, whereas the anions through the positive charged anion exchange (AE) membrane (AEM) toward the anode (Figure ). The concentrate and diluate compartments are alternating in order at the two sides of the membranes. Additionally, CE and AE membranes are placed in an alternating configuration. Eventually, the concentration of the ions increases in the concentrate solution compartments and subsequently in the concentrate tank, while it decreases in the diluate solution compartments and tank.

1.

Electrodialysis (ED) flow diagram.

2.

Ion selective membrane configuration in an electrodialysis process.

The mathematical model used in the current study has been adopted by Rohman et al. and was validated against previous experimental results.

$$\frac{\mathrm{d}{C}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}}{\mathrm{d}t}=\frac{Q(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}-C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}})+\frac{N\varphi j_{\mathrm{s}\mathrm{t}}\mathrm{A}}{zF}-\frac{\mathrm{N}\mathrm{A}{\mathrm{D}}_{\mathrm{A}\mathrm{E}\mathrm{M}}(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{A}\mathrm{E}\mathrm{M}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{A}\mathrm{E}\mathrm{M}})}{l_{\mathrm{A}\mathrm{E}\mathrm{M}}}-\frac{\mathrm{N}\mathrm{A}{\mathrm{D}}_{\mathrm{C}\mathrm{E}\mathrm{M}}(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{C}\mathrm{E}\mathrm{M}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{C}\mathrm{E}\mathrm{M}})}{l_{\mathrm{C}\mathrm{E}\mathrm{M}}}}{\mathrm{N}{\mathrm{V}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}}$$ 1

$$\frac{\mathrm{d}{C}_{\mathrm{d}\mathrm{i}\mathrm{l}}}{\mathrm{d}t}=\frac{Q(C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}})-\frac{N\varphi j_{\mathrm{s}\mathrm{t}}\mathrm{A}}{zF}+\frac{\mathrm{N}\mathrm{A}{\mathrm{D}}_{\mathrm{A}\mathrm{E}\mathrm{M}}(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{A}\mathrm{E}\mathrm{M}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{A}\mathrm{E}\mathrm{M}})}{l_{\mathrm{A}\mathrm{E}\mathrm{M}}}+\frac{\mathrm{N}\mathrm{A}{\mathrm{D}}_{\mathrm{C}\mathrm{E}\mathrm{M}}(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{C}\mathrm{E}\mathrm{M}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{C}\mathrm{E}\mathrm{M}})}{l_{\mathrm{C}\mathrm{E}\mathrm{M}}}}{\mathrm{N}{\mathrm{V}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}}$$ 2

$$\frac{\mathrm{d}{C}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}}{\mathrm{d}t}=\frac{Q(C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}-C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}})}{V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}}$$ 3

$$\frac{\mathrm{d}{C}_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}}{\mathrm{d}t}=\frac{\mathrm{Q}(C_{\mathrm{d}\mathrm{i}\mathrm{l}}-C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}})}{V_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}}$$ 4

The mass balance equations express the ion transfer between the compartments (eq , ) and the tanks (eq , ). C _conc, C _dil, $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$ , and $C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$ correspond to the concentration of the anions (e.g., sulfuric anions) in the concentrate and dilute compartments and tanks, respectively, while V _comp, $V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$ , and $V_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}$ correspond to the volumes of the contained solutions. Q is the overall flow rate of the fluids that are fed into the ED system, whereas j _st and φ are the current density and efficiency, respectively. N is the number of cell pairs, A is the membrane area per cell pair, z is the ion charge, F is Faraday’s constant, D _AEM and D _CEM stand for the diffusion coefficients through AE and CE membranes, respectively, l _AEM and l _CEM represent the corresponding membrane thicknesses, and $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{A}\mathrm{E}\mathrm{M}}$ , $C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{A}\mathrm{E}\mathrm{M}}$ , $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{C}\mathrm{E}\mathrm{M}}$ , and $C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{C}\mathrm{E}\mathrm{M}}$ represent the concentration on the surface of the corresponding membranes. The mathematical representation of the operation of the ED plant is completed with the equation, which includes all the essential electrical quantities

$$E_{\mathrm{E}\mathrm{s}\mathrm{t}}=E_{\mathrm{e}\mathrm{l}}+N(E_{\mathrm{C}\mathrm{E}\mathrm{M}}+E_{\mathrm{A}\mathrm{E}\mathrm{M}})+N(R_{\mathrm{C}\mathrm{E}\mathrm{M}}+R_{\mathrm{A}\mathrm{E}\mathrm{M}}+R_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}+R_{\mathrm{d}\mathrm{i}\mathrm{l}})I_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}$$ 5

The above terms are derived by Ohm’s and Kirchhoff’s second laws by representing the ED stack as an electrical circuit. E _Est corresponds to the power source voltage, E _el corresponds to the difference among electrode potentials of the anode and cathode, E _CEM and E _AEM correspond to the membrane potentials, R _CEM, R _AEM, R _conc, and R _dil correspond to the membrane and tank resistances, whereas I _stack corresponds to the overall current flowing across the stack. More extensive details along with the analytic equations for the estimation of the model parameters can be found in a variety of sources. ,

The overall energy used by the system is calculated through the summation of the electrical energy consumed in the stack to achieve the ion transfer and the energy required for the pumps used for the recirculation of the solutions. However, the energy for liquid recirculation is considered negligible compared with the stack energy. The electrical energy consumption E _des is calculated as follows

$$E_{\mathrm{d}\mathrm{e}\mathrm{s}}=E_{\mathrm{E}\mathrm{s}\mathrm{t}}{\int }_{t_0}^{t_{\mathrm{f}}}\frac{j_{\mathrm{s}\mathrm{t}}*A_{st}}{V_{dil}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}}\mathbb{d}t$$ 6

where E _Est is the applied potential difference, j _st is the current density, and A _st is the membrane area.

2.2. Optimization-Based ED Operation Problem Formulation

The power source voltage E _Est and the fluid circulation flow rate Q are considered as the control variables due to the large impact they have on the separation efficiency. A systematic change of these variables can result in a shorter ED process duration to achieve the purity specifications for the diluted stream. The general definition of the dynamic optimization problem consists of the objective function, the ED process model expressed as a set of equality and inequality constraints, along with the bounds on the system variables that define the operating window of the ED process system. Therefore, the optimization problem can be structured as follows

$$\min /\max F(x(t),d,\epsilon (t),t)$$ 7

s.t.

$$\begin{array}{l}ẋ=f(x(t),u(t),\epsilon (t),d,t)\\ g(x(t),u(t),\epsilon (t),d,t)\le 0\\ \begin{array}{l}h(x(t),u(t),\epsilon (t),d,t)=0\\ x{(t)}^L\le x(t)\le x{(t)}^U\\ \begin{array}{l}u{(t)}^L\le u(t)\le u{(t)}^U\\ \mathrm{\Delta }u{(t)}^L\le \mathrm{\Delta }u(t)\le \mathrm{\Delta }u{(t)}^U\end{array}\end{array}\end{array}$$

where F(x(t),u(t),d,ε­(t),t) is the objective function expressing the optimization targets, x(t) denotes the vector of state variables, u(t) = [U _st,Q] denotes the vector of control variables, x(t) ^L and x(t) ^U denote the lower and upper bounds for the state variables, u(t) ^L and u(t) ^U denote the lower and upper bounds for the control variables, Δu(t) ^L and Δu(t) ^U denote the lower and upper bounds for the rate of change for the control variables, and g and h indicate the inequality and equality constraints, respectively. Vector d denotes the design variables that are constant for the entire batch duration. Design variables may also be part of the optimization problem as decision variables. Vector ε­(t) denotes the vector of time-varying disturbances affecting the plant.

2.3. Solution of the Optimization-Based ED Operation Problem

The nonlinear dynamic optimization problem defined by formulation () is solved through a suitable parametrization of the control vector and the time discretization of the process model. OCFE techniques enable the transformation of the differential equations in the process model to an approximate set of nonlinear algebraic equations. In this study, the OCFE technique divides the time domain into a number of finite elements. Within each element, a number of collocation points are selected (Figure ), with their location determined by the roots of Legendre orthogonal polynomials of order equal to the number of collocation points. The discretization of the time requires the introduction of the dimensionless variable τ

$$\tau =\frac{t-t_0}{t_{\mathrm{f}}-t_0}=\frac{t-t_0}{\delta t}$$ 8

so each finite element is normalized as τ ∈ [0,1].

3.

Schematic of the partition of the batch time horizon into finite elements and position of the collocation points within the finite elements.

Lagrange interpolation polynomials, φ _i (τ), of the (k + 1)^th degree, with k equal to the number of internal collocation points per finite element, are used for the approximation of the state variables within each finite element.

$${\phi }_i(\tau )=\prod _{\begin{array}{l}j=0\\ j\ne i\end{array}}^{NC}\frac{\tau -{\tau }_j}{{\tau }_i-{\tau }_j}$$ 9

The degree of Lagrange interpolation accounts for an additional interpolation point at the beginning of each finite element. The state variables within each element are approximated using the Lagrange interpolation

$$x(\tau )=\sum _{j=0}^{\mathrm{N}\mathrm{C}}{x}_j{\phi }_j(\tau ),\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s}x(\tau )=\mathrm{{\rm X}}{\phi }^T(\tau )$$ 10

where x _j is the value of the state variable at the jth collocation point X = [x ₀,x ₁,···,x _k ] and φ­(τ) = [ϕ_ο(τ),···,ϕ _k (τ)]. The derivative of the Lagrange polynomials is calculated as follows

$${\mathrm{\varphi ̇}}_j(\tau )=\sum _{\begin{array}{l}L=0\\ L\ne j\end{array}}^{\mathrm{N}\mathrm{C}}\frac{1}{{\tau }_j-{\tau }_L}\prod _{\begin{array}{l}i=0\\ i\ne j,L\end{array}}^k\frac{\tau -{\tau }_k}{{\tau }_j-{\tau }_{\mathrm{i}}}$$ 11

The derivative terms can then be approximated as

$$ẋ(\tau )=\sum _{j=0}^{\mathrm{N}\mathrm{C}}{x}_j{\mathrm{\varphi ̇}}_j(\tau )$$ 12

OCFE then assumes that the discretized differential equations are satisfied exactly only at the collocation points. The control variables are parametrized within each finite element to facilitate the solution of the optimization problem and bring in a form consistent with digital control system implementation. A piecewise constant control vector parametrization is considered, which assumes constant control variables during the time interval defined by each finite element.

3. Electrodialysis Process Dynamic Optimization

The work presented here investigates two different problems. The first problem includes the online operation optimization of an ED unit of given structural and capacity features (e.g., membrane area) under the influence of disturbances occurring during operation. The second problem includes the simultaneous design optimization of the ED process under the influence of a set of expected disturbance scenarios. The objective is to determine the capacity of the ED process (unit capacity and membrane area) that would enable the most efficient operation under a given set of disturbance scenarios. These two problems are presented in detail in Sections and 3.2, respectively.

3.1. Online Operation Optimization and Control

The schematic of the operation optimization and control problem is shown in Figure .

4.

Online optimization and control of an ED process unit.

The “Plant” block represents the real electrodialysis plant that provides measurements for the control system. The “Model” block represents the ED model as described by the system of equations (differential and algebraic) as well as its discretized form. In a simulated case study, the “Plant” block is represented by the ED model, with disturbances affecting its performance. Vector d ₁ is defined as a vector of disturbances applied to the system. The “Estimator” block considers the deviations between the plant measurements and the model predictions to update the process state variables and disturbance estimates so that the accuracy of the ED process model is increased. The state and disturbance estimation requires that the state variable vector, consisting of the electric density in the stack and the concentrate and dilute streams in the ED compartments and the summation tanks, is augmented by adding an additional state variable accounting for the disturbances in the ions transferring through the membranes in the form of the correction factor in the effective membrane area as follows

$$x^T=[j_{\mathrm{s}\mathrm{t}},C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}},C_{\mathrm{d}\mathrm{i}\mathrm{l}},C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}},C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}},A_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}]$$ 13

The objective function, F _est, in the “Estimator” blocks attempts to calculate the state estimates that minimize the difference between the process model and the plant measurements

$$\min F_{\mathrm{e}\mathrm{s}\mathrm{t}}(x(t))=\sum _{i=1}^{\mathrm{N}\mathrm{C}\mathrm{E}}{({C_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}}^{\mathrm{d}\mathrm{i}\mathrm{l},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(i)-C}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}^{\mathrm{d}\mathrm{i}\mathrm{l},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(i))}^2$$ 14

with NCE denoting the number of finite elements that coincide with the time that the most current measurements become available. For the last iteration, NCE = NE will apply, where NE is the number of finite elements.

The “Optimizer” block solves the optimization problem defined in formulation () based on the updated process model incorporating the most recent process measurements that calculate the optimal profile for the stack voltage and recirculation flow rate for the remaining batch time. Obviously, the efficiency of the optimization problem increases as the process model predictions are as close as possible to the ED plant response. However, a continuous variation of the process disturbances may drift the plant away from the target specifications for purity of the dilute solution. Slow varying disturbances in the effective membrane area are considered. The estimation of the current level of the disturbance enables the process optimization and control system to calculate an updated profile for the electric field and liquid circulation in the ED process. The control vector u(t) = [E _Est,Q] contains the voltage for the development of the electric field in the membrane stack and the liquid recirculation rate.

The objective function consists of two main parts; (a) the cost from possible target deviations at the final time of the batch and (b) a time-varying term that accounts for the total batch time, the energy consumption during the batch duration, the deviations of the state variables from a desired trajectory, and the control effort for the achievement of the control goals. Similar to the work of Rohman and Aziz, the objective function incorporates the minimization of the batch process time and energy consumption and the maximization of the achieved degree of separation, defined as the achieved ratio of the diluate’s final to initial ion concentrations. The objective function takes the following form

$$\min F(x,u)=-W_{\mathrm{Deg}}(1-\frac{C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(t_{\mathrm{f}})}{C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(t_0)})+W_t*{\int }_0^{t_{\mathrm{f}}}\mathbb{d}t+W_E*E_{\mathrm{E}\mathrm{s}\mathrm{t}}{\int }_0^{t_{\mathrm{f}}}\frac{j_{st}*A_{\mathrm{s}\mathrm{t}}}{V_{\mathrm{d}\mathrm{i}\mathrm{l}}}\mathbb{d}t+{\int }_0^{t_{\mathrm{f}}}\mathrm{\Delta }{u}^T{W}_R\mathrm{\Delta }u\mathrm{d}t$$ 15

Parameters ${W_{\mathrm{Deg}},W}_t,W_E$ , and W _R denote the weighting factors that determine the relative importance of each term in the objective function.

3.2. Simultaneous Process Design and Operation Optimization

In the second problem, the simultaneous ED process design and operation optimization under a set of expected disturbance scenarios is sought. The goal is to build an ED process unit that exhibits high flexibility and robust operation by alleviating process disturbances effectively with minimal operating cost. The reasoning is that ED process design decisions may have a favorable impact on the ability of the process system along with its associated optimization-based control system, as defined in Section , to compensate for the disturbances in the membrane effectiveness, the variability in the concentration of ions in the concentrated solution and so forth. In this way, the overall cost of the separation can be maintained at the lowest possible level while satisfying the separation targets. The design optimization variable that is considered is the total membrane surface area, A _st , available for the separation of the ions from the concentrated liquid. In the problem formulation, instead of the membrane area, a coefficient is used that defines the membrane surface area change from a nominal value

$$A_{\mathrm{s}\mathrm{t}}^{\prime }=m\text{\_}A·A_{st}$$ 16

where m_A is the membrane area increase/decrease factor and $A_{\mathrm{s}\mathrm{t}}^{\prime }$ is the new membrane area.

The objective function of the simultaneous design and operation optimization problem includes the minimization of the total process time, the minimization of the energy consumption for the separation, the maximization of the degree of separation, and the penalty applied to the control effort defined as the rate of change for the control actions E _Est and Q. In addition, the level of the circulation rate, Q, is explicitly included in the objective function as it affects the process economics and performance. Regarding process design cost, the optimization of the membrane area increase/decrease factor is imposed. The overall objective function of the simultaneous design and operation optimization problem is the following

$$\min F(x,u)=W_A*m_A-W_{\mathrm{Deg}}(1-\frac{C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(t_f)}{C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(t_0)})+W_t{\int }_0^{t_{\mathrm{f}}}\mathbb{d}t+W_E*E_{Est}*m_A*A_{\mathrm{s}\mathrm{t}}{\int }_{t_0}^{t_{\mathrm{f}}}\frac{j_{\mathrm{s}\mathrm{t}}}{V_{\mathrm{d}\mathrm{i}\mathrm{l}}}\mathbb{d}\mathrm{t}+{\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{\Delta }{u}^T{W}_R\mathrm{\Delta }u\mathbb{d}t+W_q*\sum _{i=1}^{\mathrm{N}\mathrm{E}}{Q}_i$$ 17

where W _q and W _A are additional weight matrices to those of eq used to normalize the different goals of the objective function.

The user may run the optimization problem for different combinations of the weighting factors to investigate the entire span of possible optimal solutions. Each combination of weights leads to a different combination of optimal operating profiles and design values. In order to select the designs that exhibit optimum trade-offs, a multicriteria assessment methodology is employed.

3.3. Multiobjective Problem Formulation

Observing the format of the considered objective functions both in the case of the online optimization problem (eq ) and in the case of the simultaneous dynamic operating and design optimization (eq ) shows that the final result is inextricably linked to the final choice of weight matrices. There is therefore a need for a systematic study and selection of the different weight cases. In this work, we use 52 different combinations, which lead to 52 optimum results, using eq as the objective function. The weights serve to generate different realizations for each one of the terms of eq , and hence, each optimum result corresponds to a different value for the key parameters that are included in these terms. The best way for investigating the trade-offs among these parameters is to perform a multicriteria analysis with them as objective functions.

Considering that all cases under study (computed for all the 52 weight values in eq ) are part of a set G, the proposed multicriteria problem takes the following form

$$\underset{G}{\min }{t}_{\mathrm{f}},E_{\mathrm{d}\mathrm{e}\mathrm{s}},E_{\mathrm{E}\mathrm{s}\mathrm{t},\max },{m\text{\_}A,Q}_{\max }$$ 18

$$\underset{G}{\max }{D}_{\mathrm{s}\mathrm{e}\mathrm{p}}$$ 19

The objective functions of formulations () and () represent specific terms of eq . Regarding the time-varying terms, objective t _f represents the ED batch duration, which must be minimized (third term of eq ), and objective E _des represents the energy consumption for the system, which must also be minimized (the fourth term of eq ). Objective E _Est,max is the maximal value of the voltage profile in the ED process, which indicates the capacity of the electric voltage source for the separation and is associated with the process plant economics. Objective m_A represents the first term of eq as it is desirable to minimize the change of the surface area, and objective Q _max represents the last term of eq as it is desirable to minimize the maximum flow in the system and term. Finally, objective D _sep represents the second term of eq as it is desirable to maximize the separation efficiency.

While this formulation will generate an adequate Pareto front for the calculated cases, the goal in this work is to draw comprehensive conclusions about the differences between the performance indicators. Therefore, the expressions shown in () and () are transformed into an overall index in order to then create the Pareto front between the overall index and each performance index. In this way, it can be observed how the overall performance (i.e., from all indicators at the same time) of each individual process case changes against each indicator separately. For the set of computed cases G and the set of indices Pr = {t _f ,E _des,E _Est,m_A,Q _max,D _sep}, a global index J _m is proposed, which merges the properties into a unified criterion that satisfies the objectives chosen and described in () and () (i.e., the simultaneous minimization and maximization of the corresponding properties), as follows

$$\underset{mϵG}{\min }{J}_{\mathrm{m}}=\sum _{l\in P_r}{a}_{l,m}·x_{l,m}^{*}$$ 20

where $x_{l,m}^{*}$ represents the considered scaled index on P _r for each instance of design m (i.e., for each set of optimum values for the parameters in Pr that corresponds to one of the 53 weight combinations). a _l,m represents a unit coefficient that is positive for properties to be minimized and negative for those to be maximized. Based on eq , the selection of design cases with increased performance translates into the minimization of the index J _m . Scaling gives equal importance to each index used in eq . In the present study, the above is carried out with the standardization method of Zarogiannis et al.

$$x_{l,m}^{*}=\frac{x_{l,m}-{\mu }_l^G}{{\mathrm{s}\mathrm{t}}_l^G}$$ 21

where x _l,m represents the initial value of the performance index and ${\mu }_l^G$ and ${\mathrm{s}\mathrm{t}}_l^G$ represent the mean and standard deviation of the index under consideration, respectively, calculated over the entire set of instances of process G. The solved multicriteria problem involves determining the Pareto indexes by creating a Pareto front per index l ∈ Pr, taking the total index J _m , for each m ∈ G, against each index represented by the values of x _l,m . The problem is formulated as follows

$$\begin{array}{l}\underset{m\in G}{\min J_m}\\ \underset{l\in \mathrm{Pr}}{\min }\text{or}\underset{l\in \mathrm{Pr}}{\max l}\end{array}$$ 22

For example, for the first set of optimum values for the parameters in Pr (that resulted from solving the optimization problem with eq for the first combination of weights), it holds that

$$x_{1,1}^{*}=t_{f,1}^{*},x_{2,1}^{*}=E_{\mathrm{d}\mathrm{e}\mathrm{s},1}^{*},x_{3,1}^{*}=E_{\mathrm{E}\mathrm{s}\mathrm{t},1}^{*},x_{4,1}^{*}={m\text{\_}A}_1^{*},x_{5,1}^{*}=Q_{\max ,1}^{*},x_{6,1}^{*}=D_{\mathrm{s}\mathrm{e}\mathrm{p},1}^{*}$$

hence

$$J_1=t_{\mathrm{f},1}^{*}+E_{\mathrm{d}\mathrm{e}\mathrm{s},1}^{*}+E_{\mathrm{E}\mathrm{s}\mathrm{t},1}^{*}+{m\text{\_}A}_1^{*}+Q_{\max ,1}^{*}-D_{\mathrm{s}\mathrm{e}\mathrm{p},1}^{*}$$ 23

In this case, J ₁ will be used, together with each one of index l ∈ Pr (e.g., t _f, E _des, and so forth).

4. Implementation and Discussion of Results

4.1. Online Operation Optimization

The operation optimization of a batch ED system for the recovery of sulfuric anions is performed. The process system parameters and initial conditions are taken from Voutetaki et al. and Latinis et al. and are shown in Table . Model parameters are listed in Table .

1. Initial Values of Main Variables and Parameters.

Variable	U st (V)	Q (L/s)	C conc (mol/m3)	C dil (mol/m3)	C conc (mol/m3)	C dil (mol/m3)	V tank (m3)	V tank (m3)
Initial values	20	0.042	20.4	20.4	20.4	20.4	0.02	0.02

A1. List of Parameters.

Parameters	Units	Values
A	m2	0.038
D AEM	m2 s–1	24.2 × 10–11
D CEM	m2 s–1	24.2 × 10–11
l AEM	m	0.1 × 10–3
l CEM	m	0.1 × 10–3
L	m	5 × 10–4
N	-	68
V comp	m–3	0.19 × 10–3
ϕ	-	0.92
F	C mol–1	96,485
R	J mol–1 K–1	8.314
z	-	2
E el	V	1.5
V conc	L	20
V dil	L	20
r AEM	Ω cm2	3.984
r CEM	Ω cm2	15

The goal of achieving a minimum final diluate concentration is introduced in the form of a constraint, so as not to affect the form of the objective function in (eq ), as well as the significance of the other goals, and was selected based on the usual values attained by the experimental results

$$C_{\mathrm{N}\mathrm{E},c_n}^{\mathrm{d}\mathrm{i}\mathrm{l},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}\le 600\mathrm{m}\mathrm{g}/\mathrm{l}$$ 24

The OCFE method uses ten equally sized finite elements, NE = 10, four internal collocation points, and one interpolation point at the beginning of each finite element. Therefore, fifth-order Lagrange interpolation polynomials are used for the approximation of the state variables within each finite element since the polynomial degree equals the number of collocation points plus one. − The size of each finite element coincides with the time duration of the control interval. This selection provides high-order accuracy without unnecessary computational cost. As shown by Rohman and Aziz, moderate collocation orders and element numbers already yield accurate dynamic profiles, and piecewise-constant control parametrization performs better than linear or second-order continuous representations. Therefore, fifth-order Lagrange interpolation polynomials are used for the approximation of the state variables within each finite element. The size of each finite element coincides with the time duration of the control interval.

To ensure implementable input trajectories, we imposed rate-of-change constraints on the manipulated variables (inequality constraints), ±25% per update for stack voltage and ±50% for flow rate from their nominal values (E _Est = 20 V and Q ^const = 3.08 × 10^–6 L/s), within our dynamic optimization. Such Δu bounds are standard in optimal control formulations solved by direct collocation or multiple shooting and are implemented as simple inequality constraints on control moves or their time derivatives. − In ED, rapid voltage increases can drive transient polarization and water splitting. Operation is commonly kept below the limiting current density (LCD), typically 70–80% of LCD. , Additionally, stack design can limit practicable flow because narrow channels incur a substantial pressure drop (1–2 bar at 6–12 cm s^–1). In parallel, vendor guidance defines electrical and hydraulic envelopes: the GE E-Cell MK-3 manual lists 400 V DC as the stack maximum and, at start up, instructs operators to slowly increase current to 2 A per stack, while the PCCell ED cell manual requires zero TMP and Δp ≤ 0.5 bar per cell (with a maximum of 30 V per cell). These provide equipment-based reasons to limit single-step changes in both voltage and flow. , The selected bounds therefore stabilize the computed trajectories and respect electrical and hydraulic constraints while preserving a wide feasible operating space.

The volumetric flow rate value corresponds to the treatment of 0.2 m³ effluent solution on an hourly basis. The “Optimizer” block of the closed loop problem determines the control actions. The weight values of the objective function remain constant for the online optimization problem during the batch duration.

During the ED process, the phenomenon of ion polarization is very common and it develops near the surface of the exchange membranes. Ion polarization develops films of ions at the sides of the membranes that inhibit the transfer of ions through the membranes, separating the dilute and concentrated compartments. In addition, the treated solution may contain impurities that are not intended to be collected but are excreted. These impurities settle on the surface of the exchange membranes. Therefore, the effective size of the final active surface of the membranes reduces and can be considered to be a disturbance for the system. In the present study, the correction term A _corr was used to describe the effect of ion membrane polarization on the value of A, namely, the membrane area of a cell pair. So, a new variable is introduced as follows

$$A_{\mathrm{f}}=A*A_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$$ 25

The value A _corr was chosen to be equal to A _corr = 1 – c(t), essentially expressing the change in the active surface A of the membranes. Consistent with the extensive evidence that fouling/scaling is the dominant cause of ED performance deterioration, via reduced conducting surface fraction and increased stack resistance, we modeled a deliberately severe loss of active area to stress-test the controller. − We chose the value c(0) = 0.5, which increases linearly with time, finally reaching the value c(t _f) = 0.65, to include the cases of severe disturbances (from 50% reduction to 65% reduction of the original value, see Figure ). This range overlaps with reported surface coverage levels (45%) and conservatively brackets unmodeled codisturbances whose first-order effects also appear as added resistance or reduced conductive area. − Therefore, the negative disturbance scenario being studied forces the algorithm to follow more drastic control actions to reach the desired separation goals. A broader scenario study (explicit temperature, pH, and feed composition variations and their interactions) is recommended as future work, but focusing on the empirically dominant foulant-driven area loss keeps the present analysis targeted and industrially relevant. , The presence of the disturbance may cause the system to deviate from its initial set specification in eq . To enable a feasible ED process system, the final constraint level for the diluate tank is adapted to the corresponding final value achieved in the optimal open loop case

$$C_{\mathrm{N}\mathrm{E},c_n}^{\mathrm{d}\mathrm{i}\mathrm{l},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}\le C_{\mathrm{N}\mathrm{E},c_n}^{\mathrm{d}\mathrm{i}\mathrm{l},\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k},\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}\text{\_}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$$ 26

5.

Time variation of the percentage reduction of the active membrane area.

The optimal control actions are then calculated from the solution of the problem in formulation (). The “Estimator” block calculates the estimate of the disturbance. The disturbance is estimated from the measurement data so that the optimization-based control is more effective.

The results of operation optimization under the influence of membrane performance degradation due to ion polarization are summarized in Figure . The optimal operating profile for the circulation flow rate and the applied voltage at the beginning of the batch is calculated (Figure ; square markers). The voltage profile increases gradually as the concentration of the sulfuric anions drops, and the concentration driving forces are diminished. The model predicts that a batch duration of 13.5 min is sufficient to meet the purity specifications. The deviation obtained between the measured and predicted values for the concentration of the treated solution after 3 min is then used to update the disturbance estimate, which is subsequently used to update the optimal profiles for the control vector (Figure ; circle markers). Due to the degradation of the membrane, the flow rate and the voltage profile are increased. Additionally, a total batch time of approximately 18 min is predicted. Successive model updates every 3 min provide renewed optimal profiles for the control variables. Saturation of both control variables is observed as the disturbance severely affected the ability of the ED process to meet the purity specifications. The overall implemented profile for the control actions is shown in Figure .

6.

(a) Updated time optimal overall flow rate trajectories and (b) updated time optimal applied voltage trajectories.

7.

(a) Uncorrected and updated control profile for the overall flow rate. (b) Uncorrected and updated control profile for overall applied voltage.

Figure depicts the concentration profiles in the concentrate and dilute tanks for different scenarios. The solid curves indicate the achieved concentration profiles under the influence of the disturbance and the implementation of the operation optimization. Clearly, the ED process system manages to meet the purity specification for the diluate tank C _final = 304 mL/L (Table ). The operation optimization with disturbance state updates enables an increase of 34.9% in the overall degree of separation but with a 3-fold increase in the energy consumption. The energy increase is explained by the severity of the disturbance and the attempt of the optimizer to counterbalance energy consumption with product purity. The batch duration is increased by only 15.5% (1 min, 26 s). The dashed–dotted curves (Figure ) show the concentration profiles without the implementation of a real-time update strategy of the control variables profiles. The ED process would clearly miss the target for the diluate tank concentration, reaching a dilute tank concentration of C _final = 755 mg/L (Table ), much higher than the limit of 600 mg/L. The ideal case without the disturbance would have resulted in the dotted profiles, which are drawn for comparison purposes. The behavior of the disturbance estimates along the duration of the batch is provided in Figure . The variability of the disturbance estimates reduces as more measurements become available, and they eventually reach the actual disturbance values.

8.

Initial, uncorrected, and updated concentrate and diluate tank concentrations.

2. Results of Initial, Uncorrected, and Updated Cases.

	t f (min)	C dil (t f) (mg/L)	C conc (t f) (mg/L)	E des (kWh/m3)	D sep (%)
initial selected optimal case	13.5	519	3481	2.1	74
uncorrected profile with disturbance	13.5	755	3245	2.1	63
updated profile with disturbance	15.6	304	3696	9.5	85

9.

Actual (predicted) values and calculated estimates for the disturbance.

Our results align with reported benefits of time-varying operation while addressing acid recovery objectives. In our disturbance scenario, updating the control profiles increased the achieved degree of separation by 34.9% while meeting the diluate purity target, with only a 15.5% increase in the batch time. As expected under severe disturbances, this came at a substantially higher energy consumption (3-fold vs the initial selected case) because the optimizer traded energy to enforce product quality constraints. The key advantage of our approach is that the optimization-based disturbance estimator identifies the magnitude of the unknown disturbance during the batch and continuously corrects the voltage and flow schedules. As more measurements arrive, the estimator variance shrinks and converges to the actual disturbance (Figure ). This online estimation is what prevents target violations observed with an uncorrected profile, which missed the diluate specification limit at 600 mg L^–1.

These findings are consistent with the literature on time-variant operation. In batch ED desalination, feed-forward voltage control increased production by up to 37% over constant voltage by maintaining high current without crossing limiting conditions. Time-variant voltage–flow cocontrol delivered up to 45% higher production than steady operation at the same stack size. Operating-regime studies further indicate energy reductions for constant-current and constant-entropy-generation relative to constant-voltage, and two-stage ED can reduce power by up to 29%. In renewable direct-drive contexts, cascade PID flow-commanded current control achieved about 94% photovoltaic utilization with >99.6–99.8% lower battery than literature medians, while real-time model-based EDR attained 77% direct PV use with −92% battery and −22% levelized cost of water, illustrating the value of closed-loop tracking for productivity and cost. ,

Within this context, our dynamic-optimization policy improves robustness to unknown disturbances by combining trajectory optimization with in-run disturbance estimation and rate-limited actuation. Relative to our constant-operation baseline, it delivers +34.9% separation with a +15.5% batch-time penalty at the expense of higher energy consumption, a trade-off pattern that matches the magnitudes and directions reported across prior time-variant control studies while uniquely preserving the diluate specification under disturbance.

4.2. Simultaneous Design and Operation Optimization

The design optimization of a specific industrial-scale unit with an average amount of Q = 45 m³/d liquid wastewater produced is performed. The characteristics of the wastewater in (SO₄ ) before treatment are C _conc (t ₀) = 3000 – 8000 mg/L (30.61–81.63 mol/m³), whereas the after-treatment requirement is C _dil (t _f ) <800 mg/L (8.16 mol/m³). The volume of the wastewater storage tank is V = 10 m³, so it can be deduced that the tank filling time is t _f = 5.3 h. Based on the tank filling time, the upper limit of the process time is chosen for the optimization problem, defined as $0\le t_{\mathrm{f}}^{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\text{\_}\mathrm{u}\mathrm{p}}\le 5$ h. The concentrate and diluate tanks have a capacity of V _tank = V _tank = 5 m³ _. The objective function is shown in eq . The design variable denoting the total membrane area, m_A, is the only design decision variable. The bounds for the control variables are as follows

$$0\le E_{Est}\le 360\mathrm{V}$$ 27

$$0\le Q\le 10000\mathrm{L}/\mathrm{h}$$ 28

$$0\le m\text{\_}A\le 20$$ 29

There are no bounds for the state variables. The OCFE model employs ten finite elements with four interior collocation points within each element. The initial values for the concentrations are set as follows

$$C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}(0)=C_{\mathrm{d}\mathrm{i}\mathrm{l}}(0)=C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(0)=C_{\mathrm{d}\mathrm{i}\mathrm{l}}^{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{k}}(0)=51.02\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^3$$ 30

The nominal values for the current voltage and fluid flow rate are E _Est = 50 V and Q ^const = 7.72 × 10^–8 m³/s (1 m³/h). The weighting matrix for the variation of the control variables is diagonal with the nonzero elements derived from the nominal operating values for the control variables as follows

$$W_R=\left[\begin{array}{lll}\frac{1}{{50}^2}& 0& 0\\ 0& \frac{1}{{\left(\frac{0.5}{3600}\right)}^2}& 0\\ 0& 0& 0\end{array}\right]$$ 31

The values of the remaining weight matrices are chosen to normalize the objectives to be comparable and to cover the entire area of the Pareto optimal front.

The design and operation optimization problem is solved for a set of combinations of weighting factors in the objective function of eq . Table in the Appendix shows the optimization results for the entire set of weighting factors. The Pareto fronts between the overall objective function, J, and each of the individual terms in the objective function are provided in Figure . In this way, trade-offs among the competing terms in the objective function can be assessed. Additionally, the shape of the Pareto fronts can be investigated, and the sensitivity of each term can be evaluated.

A2. Optimization Results for Different Combinations of Weights in the Objective Function for the Industrial Unit Simulations.

Case No.	Final Time (t f) [h]	C final $\left[\frac{\mathrm{m}\mathrm{o}\mathrm{l}}{{\mathrm{m}}^3}\right]$	Energy Consumption (E des ) $\left[\frac{\mathrm{k}\mathrm{W}\mathrm{h}}{{\mathrm{m}}^3}\right]$	Degree of Separation (D sep )	m_A	Maximum Flow Rate Q max $\left[\frac{{\mathrm{m}}^3}{\mathrm{h}}\right]$	Maximum Voltage (E Est) [V]	Performance Index J
1	3.00	7.970	3.38	0.8438	20.00	0.00266	64.2	0.8
2	2.52	7.983	6.46	0.8435	12.11	0.00210	153.8	0.4
3	2.22	7.991	8.49	0.8434	7.79	0.00223	239.3	2.5
4	2.49	7.988	6.82	0.8434	11.85	0.00197	190.7	1.1
5	1.93	8.016	4.17	0.8429	20.00	0.00249	92.3	1.0
6	3.07	7.972	4.10	0.8437	14.66	0.00185	115.4	–1.0
7	2.48	7.964	6.03	0.8439	7.13	0.00237	208.0	0.5
8	3.52	7.960	10.00	0.9025	16.59	0.00194	114.2	0.1
9	3.24	7.949	11.54	0.9026	12.43	0.00194	183.5	0.5
10	3.31	7.947	11.03	0.9027	12.97	0.00194	172.5	0.2
11	2.63	8.003	4.51	0.8431	17.70	0.00198	90.3	–0.1
12	2.78	7.932	4.75	0.8445	8.80	0.00253	164.9	–0.9
13	3.17	7.915	4.11	0.8449	18.51	0.00232	117.0	–0.9
14	2.66	7.936	5.06	0.8445	7.50	0.00260	185.7	–0.4
15	3.71	7.958	9.35	0.9025	17.93	0.00194	103.6	0.1
16	3.60	7.933	9.74	0.9028	15.35	0.00194	148.0	–0.4
17	2.34	7.999	4.25	0.8432	15.39	0.00249	87.4	0.1
18	3.72	7.949	9.35	0.9026	18.18	0.00194	110.0	0.0
19	3.08	7.995	4.12	0.8433	15.23	0.00183	86.1	–0.7
20	2.68	7.932	4.97	0.8445	7.68	0.00263	182.0	–0.5
21	3.72	7.932	9.45	0.9028	18.27	0.00194	128.7	–0.2
22	3.75	7.958	9.32	0.9025	19.18	0.00194	100.0	0.4
23	3.75	7.960	9.21	0.9025	19.04	0.00194	98.9	0.3
24	2.96	7.936	4.19	0.8444	12.51	0.00246	115.5	–1.2
25	1.88	8.015	4.35	0.8429	20.00	0.00272	85.0	1.5
26	2.74	7.927	4.65	0.8446	8.28	0.00265	169.7	–0.9
27	2.57	7.943	5.47	0.8443	7.02	0.00264	186.5	–0.1
28	2.56	7.942	5.55	0.8443	6.93	0.00265	194.0	0.0
29	3.56	7.946	9.79	0.9027	14.98	0.00194	132.6	–0.4
30	3.20	7.951	11.87	0.9026	12.49	0.00194	187.5	0.7
31	3.21	7.950	11.77	0.9026	12.45	0.00194	186.4	0.6
32	2.67	7.943	4.93	0.8443	7.55	0.00264	159.1	–0.6
33	2.48	7.944	6.01	0.8443	6.71	0.00264	221.1	0.7
34	2.41	7.949	6.38	0.8442	6.64	0.00264	233.7	1.2
35	2.57	7.943	5.48	0.8443	7.01	0.00264	189.4	–0.1
36	3.19	7.945	4.49	0.8443	20.00	0.00199	117.4	–0.2
37	3.19	7.930	4.52	0.8446	20.00	0.00212	120.2	–0.3
38	3.11	7.945	4.52	0.8443	16.71	0.00206	129.3	–0.5
39	3.17	7.947	4.53	0.8442	20.00	0.00198	117.8	–0.2
40	2.87	7.918	4.08	0.8448	10.14	0.00264	145.3	–1.4
41	3.13	7.965	4.34	0.8439	16.81	0.00185	119.9	–0.6
42	3.19	7.961	3.65	0.8440	19.62	0.00230	73.7	0.0
43	1.44	8.028	4.88	0.8426	20.00	0.00278	121.7	2.4
44	3.19	7.945	4.49	0.8443	20.00	0.00199	117.2	–0.2
45	2.51	7.942	5.87	0.8443	6.77	0.00264	215.8	0.5
46	2.48	7.944	6.01	0.8443	6.71	0.00264	221.0	0.7
47	2.57	7.942	5.48	0.8443	7.01	0.00264	189.8	–0.1
48	2.71	7.965	4.83	0.8439	8.00	0.00236	140.9	–0.9
49	2.68	7.951	4.83	0.8442	7.68	0.00264	142.4	–0.7
50	2.68	7.951	4.85	0.8442	7.65	0.00264	142.9	–0.7
51	2.75	7.929	4.57	0.8446	8.45	0.00264	160.6	–1.0
52	2.78	7.932	4.75	0.8445	8.80	0.00253	164.9	–0.9

10.

Pareto optimal fronts between (a) Degree of Separation and Overall Performance Index (J), (b) Energy Consumption and Overall Performance Index (J), (c) Maximum Desirable Fluid Flow rate and Overall Performance Index (J), (d) Area Growth Factor and Overall Performance Index (J), (e) Final time and Overall Performance Index (J), and (f) Maximum Desired Operating Voltage and Overall Performance Index (J).

In order to confirm that the 52 selected weight combinations provide adequate and unbiased coverage of the design space, we performed a statistical analysis of the resulting solutions using normalized boxplots (Figure ). These 52 combinations were obtained via a systematic enumeration of discrete weight vectors, in the spirit of exhaustive enumeration approaches reported in the literature (e.g., Prousalis et al.), to ensure that all relevant trade-off directions were explored. The degree of separation was excluded from this analysis since this was the only imposed requirement across all cases: all solutions had to meet the minimum recovery specification defined by regulations, and therefore, it does not serve as a discriminating objective in the Pareto analysis.

11.

Normalized boxplots of the 52 solutions across the main objectives. Each box shows the interquartile range (25–75%), the horizontal line marks the median, whiskers extend to the nonoutlier minima and maxima, and any points beyond whiskers indicate outliers. The wide distributions confirm that the selected weight combinations span the entire spectrum of feasible values.

The boxplots clearly indicate that the solutions span the entire normalized range (0–1) of each objective, with broad interquartile ranges that confirm good coverage of intermediate compromise points. Of particular importance are the Maximum Desirable Fluid Flow Rate, Area Growth Factor, and Maximum Desired Operating Voltage as these represent the design-related decision variables that ultimately define the capacity and specifications of the new system. The wide distributions observed for these three objectives demonstrate that the chosen weight combinations successfully generated solutions ranging from low-capacity/low-cost designs to high-capacity/high-demand designs. This shows that the resulting Pareto fronts capture the full set of meaningful trade-offs among design and operational goals.

The frequency chart of the appearance for each case in the Pareto fronts is presented in Figure . Case 40 appears in the Pareto optimal front for each objective term and overall objective function pair, whereas several other cases appear more than once in the Pareto fronts.

12.

Frequency chart of occurrence of studied cases by the Pareto chart.

For Case 40, the control actions and corresponding concentrations are presented in Figure . Both the recirculation flow rate and the applied voltage increase gradually over the duration of the batch. This is behavior observed also in the small-scale unit of Section accounting for the gradual decrease in the concentration driving forces in the ED compartments. The maximum fluid flow rate value (Q) is 2.64 L/s, and the maximum current voltage value (E _Est) is 145.3 V. The value of the membrane surface area growth coefficient is calculated to be m_A = 10.14, leading to a membrane exchange surface area per cell pair of $A_{\mathrm{s}\mathrm{t}}^{\prime }$ = 0.3853 m². The final diluate tank concentration is C _final = 7.918 mol/m³, with an achieved separation degree of D _sep = 84.47%. The system fulfills the concentration specification requirements in 2.87 h < 5 h. The energy consumption for the process is evaluated at 4.1 kWh/m³. The first selected case exhibits optimal characteristics in terms of many of the criteria. However, some objectives have the potential for significant improvement. Therefore, the area that achieves lower values in the maximum current voltage required, in the energy consumed, and in the area growth factor needs to be investigated.

13.

Case 40: (a) optimal control profiles, imposed voltage (left-blue), fluid flow rate (right-red) along the duration of the batch, and (b) concentrate and diluate tanks and compartments concentration profiles.

For Case 35, the control actions and the concentrate and diluate tank concentration profiles are listed in Figure . The corresponding operational characteristics of Figure a indicate as optimal design characteristics a maximum fluid flow rate value (Q) of 2.64 L/s and a maximum current voltage value (E _Est) of 189.4 V. The area growth factor is calculated to be m_A = 7.01, and the membrane area is $A_{\mathrm{s}\mathrm{t}}^{\prime }$ = 0.266 m². The final diluate tank concentration is C _final = 7.943 mol/m³, resulting in a degree of separation equal to D _sep = 84.43%. The duration of the process is 2.57 h < 5 h, and the energy consumption required is estimated to be 5.5 kWh/m³. Case 35 therefore uses a smaller area growth factor; therefore, overall, it requires a smaller membrane area than Case 40 in a shorter process time.

14.

Case 35: (a) process control actions, imposed voltage (left-blue), fluid flow rate (right-red) against time, and (b) concentrate and diluate tanks and compartments concentrations against process time.

For Case 1, the control actions and concentration profiles are listed in Figure . The optimum characteristics include a maximum fluid flow rate value (Q) of 2.66 L/s and a maximum current voltage value (E _Est ) of 64.2 V. The area growth factor was found to be m_A = 20, the maximum available for the algorithm, and the exchange area per membrane cell pair $A_{\mathrm{s}\mathrm{t}}^{\prime }$ = 0.76 m². The final dilution tank concentration value is calculated to be C _final = 7.97 mol/m³ and the degree of separation is D _sep = 84.38%. The current unit requires 3 h, which is again less than the constraint set at 5 h. The energy consumption required for the ED process is estimated to be 3.4 kWh/m³. Compared to Cases 40 and 35, Case 1 attains a lower (maximum) operating voltage value.

15.

Case 1: (a) process control actions, imposed voltage (left-blue), fluid flow rate (right-red) against time, and (b) concentrate and diluate tanks and compartments concentrations against process time.

The results of all of the characteristics of the selected cases are summarized in Table . It appears that all cases are within the limits of final concentration and process time that were set, so they are acceptable. More specifically, Case 40 comparatively manages to achieve a low final concentration value in the diluate tank and, thus, a greater final separation, with low energy consumption at the same time. Case 35 uses a small area growth factor of 7.01 at a short final process time of 2.57 h. As an undesirable feature, this case requires a large allowable voltage and a large power consumption, 38% higher than that of Case 40. Case 1 uses the maximum allowable area growth factor of 20 but achieves the smallest desired voltage value of 64.2 V. Another positive characteristic is the low energy consumption of 3.38 kWh/m³, 21% and 62% lower than the other selected cases.

3. Aggregate Results of Selected Industrial-Scale Unit Cases.

case No.	t f (h)	C dil (t f) (mol/m3)	C conc (t f) (mol/m3)	E des (kWh/m3)	exchange membrane area $A_{\mathrm{s}\mathrm{t}}^{\prime }$ (m2)	maximum flow rate Q (L/s)	maximum voltage E Est (V)
40	2.87	7.918	94.12	4.08	0.385	2.64	145.3
35	2.57	7.943	94.1	5.48	0.266	2.64	189.4
1	3.00	7.970	94.07	3.38	0.76	2.66	64.2

The published batch-ED work provides scaling theory and pilot demonstrations, but we did not find intermediate-area experimental validation bridging our lab unit (A = 0.038 m²) to A = 0.76 m² for sulfate/sulfuric systems. Accordingly, we verified the model and control strategy on the unit-scale ED using validated sulfuric acid parameters and then explored designs by varying the membrane area growth factor mA, explicitly reporting intermediate in silico areas (e.g., A = 0.266 and 0.385 m² in Table ) before the largest case. This pathway follows batch-ED scaling guidance (area–time product, Aτ ≈ const.) and is consistent with the current practice in the field; note also that pilot-scale batch ED has been demonstrated for sulfate/lead at 5 m² effective area. ,,

Potential scale-up challenges and how they affect optimal policies include the following: As area grows, (i) flow maldistribution reduces the limiting current density (LCD), so current density caps/penalties are warranted in optimization. (ii) Fouling/scaling at higher residence times and in more complex manifolds increases resistance and can push the optimum toward higher energy or longer batch time. , (iii) Pressure drop and pump work become non-negligible, reshaping the energy–time Pareto front. (iv) Shunt currents and Joule heating lower current efficiency and alter transport coefficients; conservative current density limits and temperature-aware properties mitigate this. In our formulation, we constrain voltage and flow and penalize Umax and Qmax to keep trajectories within realistic regimes and to avoid aggressive actuation, i.e., smooth, equipment-friendly control actions aligned with batch-ED operation theory. ,

Overall, it is observed that each one of the three selected cases exhibits trade-offs among the employed objective functions. Case 1 achieves the lowest energy consumption and the lowest operating voltage, whereas Case 35 achieves the shortest final process time and the smallest total membrane area growth factor, and finally, Case 40 achieves the smallest final dilute tank concentration. In this study, we deliberately focused on operating parameters and energy consumption without including costs related to maintenance, membrane replacement, or capital depreciation so as to avoid extending the scope into a full technoeconomic analysis or life cycle assessment (LCA). We advise that future work should complement the present results with a detailed LCA and technoeconomic evaluation to support end-users and engineers in selecting the most suitable case for industrial application.

5. Conclusions

The present work studied the effect of disturbances in the membranes in a batch electrodialysis system with recirculation. An online dynamic optimization system was developed to alleviate the effects of disturbances on the satisfaction of the effluent quality specifications. The system achieved optimal performance of the electrodialysis unit through control of the operating conditions. The successive optimization enables optimal adjustments periodically of the control variables to accommodate the set of specifications in the final concentrations of the diluate and concentrate tanks. The applied disturbance implicates the reduction of the active membranes’ surface by more than half (50–65%). The control actions of the iterative optimization algorithm manage to satisfy the final state requirement, slightly increasing the processing time and achieving an even lower final concentration in the diluate tank. To compensate for the disturbance, the system increases gradually the energy consumption, though maintaining a good balance among the competing control objectives.

Glossary

Nomenclature

A

membrane area (m²)

A _corr

coefficient of change in active membrane area A

A _f

membrane area under the disturbance (m²)

A _st

membrane area per cell pair (m²)

AEM

anion exchange membrane

C

concentration (mol/m³)

C ^CEM/AEM

membrane surface concentration (mol/m³)

c

change in active membrane area A

c _i

internal collocation points

c _u

vector of constraints of control variables

const_Est

constraint of power source voltage (V)

const_q

constraint of flow rate (m³/s)

CEM

cation exchange membrane

D

diffusion coefficient (m² s^–1)

E _AEM

AEM potential (V)

E _CEM

CEM potential (V)

E _des

electrical energy consumption (kWh/m³)

E _el

electrode potential difference (V)

E _stack

power source voltage (V)

F

Faraday constant (C mol^–1)

f

set of differential and algebraic equations related to mass balances

g

set of inequality constraints of differential and algebraic equations

h

set of equality constraints of differential and algebraic equations

I _stack

stack current (A)

J

objective function

j _st

current density per cell pair (A/m²)

L

membrane thickness (m)

M

mass matrix

N

number of cell pairs

NC

number of collocation points

NE

number of finite elements

NFE

number of finite elements used in the iterative disturbance estimation loop

Q

flow rate (m³/s)

W _x

weight matrix for state variables

W _R

weight matrix for control variables

r

residual equation of the approximation solution

W1

weight for power voltage

W2

weight for flow rate

R _AEM

AEM Ohmic resistance (Ω)

R _CEM

CEM Ohmic resistance (Ω)

R _conc

concentrate compartment resistance (Ω)

R _dil

dilute compartment resistance (Ω)

t

time (sec)

u

vector of manipulated or control variables

U _st

applied potential difference

V _comp

volume compartment

W _t

weight matrix for time

W _E

weight matrix for energy consumption

W _Deg

weight matrix for degree of separation

x

vector of state variables

z

ion charge

Glossary

Superscripts/subscripts

comp

compartment

conc

concentrate

dil

dilute

f

final value

i

finite element

j

collocation point

L

lower bound

model

state and control values calculated from the model

process

state and control values calculated from the plant

tank

tank

U

upper bound

0

initial value

Glossary

Greek letters

δ

the Dirac delta

δ_kj

the Kronecker delta

Δζ

length of time interval

Λ

Lagrange function

λ

Lagrange multiplier vector

Σ

number of the time domain elements

τ

dimensionless time

φ

current efficiency

φ_i

Lagrange polynomials

Appendix

The appendix tables show the model parameters and optimization results for the entire set of weighting factors.

The open access publishing of this article is financially supported by HEAL-Link.

The authors declare no competing financial interest.