(59f) Automated Synthesis of Hybrid Models for Ionic Separations

Electrochemical separation is an efficient separation technology for various industries from water treatment to oil refineries[1, 2]. However, designing and synthesizing targeted processes is a challenge. Mathematical models aid in developing new technologies through simulation, sensitivity analysis, techno-economic analyses, and optimization. They provide valuable information and reduce the time and resources required for new designs. Combining data-driven and knowledge-based modeling approaches can further accelerate the development of new technologies[3]. Therefore, mathematical models are essential for electrochemical separation processes, offering a powerful tool for efficient and effective separations.

A specific modeling approach, called compositional modeling[4] has been shown to be effective in obtaining useful models in situations where extensive data and/or physical knowledge are not available (e.g., separation processes featuring new materials or configurations)[5, 6]. The modeling task is described as an optimization problem[7, 8] such that given the physical structure of the electrochemical separation system S, a set of statements about its behavior, a library of model fragments {m₁, ..., m_n} and a set of rules constraining their use, we seek the most useful, coherent scenario model ξ={m₁σ₁, ..., m_nσ_n} for answering a query (e.g., device’s performance), where σ_i is a binding parameter from the variables in m_i to objects in S and each instance m_iσ_i is unique. The resulting model can be used to generate synthetic data to train a Machine Learning-based surrogate model (MLBSM) on which Transfer Learning (TL) can be applied to improve the model performance[6].

In this work, a generalized hybrid modeling framework that exploits compositional modeling and Machine Learning to develop physics-aware models for electrochemical separations systems is proposed. The modeling approach aims to facilitate the synthesis of suitable mathematical models for electrochemical separations. The framework is showcased by modeling a representative electrochemical separation technology, namely electrodeionization (EDI) for water desalination.

In an EDI device, ionizable species are removed from a liquid matrix by applying an electric field. The device is typically arranged as a cell, with two electrodes on each side and a train of anion and cation exchange membranes (AEMs and CEMs respectively) separated by solution compartments. When a voltage is applied, electrically charged solutes are forced to move, that is, anions migrate towards the anode and cations towards the cathode. The ion exchange membranes serve as barriers such that anions move through AEMs but are retained by CEMs, while the opposite is true for cations. The result is the depletion of ionic species in the dilute solution compartment and the accumulation of ionic species in the concentrate solution compartment[1]. The migration of ionic species continues until an electrical equilibrium is reached.

The accumulation of charges over time as a consequence of their migration through the ion-exchange membranes resembles the behavior of an electric capacitor. Meanwhile, the restriction the ion-exchange membranes impose over the mobility of the charges is analogous to an electric resistance. These two observations can be exploited to approximate the device level behavior of an EDI device as an electric circuit. This representation, although simplistic, provides a basis for the synthesis of compositional models for electrochemical separation devices.

The compositional modeling approach is outlined in Figure 1 (A). The first step is to obtain an equivalent electric (EC) circuit that approximates the behavior of the device. The EC is then modeled as a first order linear differential equation. The solution to this ODE stands as the basis of the model. Nonlinearities are then added to the model in the form of corrections to the base model. The functional for each of the nonlinearities is chosen from a library of model fragments via optimization using genetic algorithms. Once the compositional model is obtained, synthetic data can be generated by simulating system responses to changes in the input variables.

Figure 3 (B-C) shows the experimental values and those predicted for the best model selected from the optimization step, along with the functional forms selected for the compositional model. Experimental data was taken from Palakkal et al. [9]. This model has 5 input variables, namely time ( ), voltage (V), total resistance (R), and the activity coefficients for the concentrate and the diluate, (γ_conc) and (γ_dil), respectively. For the case of the evolution of charge (concentration) over time, the predicted values are in good agreement with the observed experimental data. The predicted evolution of current on the other hand, although close to the experimental values, tends to be underestimated. The overall behavior of the system, however, is captured with a rather simple model.

After hyperparameter optimization, the selected architecture was an artificial neural network with 14 layers (5:11x64:32:3), with a batch size of 1450, and a learning rate of 0.003. A validation dataset (6 samples) from laboratory experiments was kept from the model during training and used for model evaluation only. For the TL step, hyperparameter optimization yielded learning rates of 4.17x10^-9, 1.0x10^-5, and 0.00016 for the input, hidden, and output layers respectively, and a batch size of 8. Given the small size of the experimental dataset available, training data for TL was obtained by fitting a polynomial to the experimental data and interpolating (oversampling). The resulting dataset contained a total of 24 samples. Figure 3(E) shows the performance of the best architecture and training scheme on training and testing data after TL.

References

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