(419g) Optimal Control of an Ion Exchange Process Under Uncertainty

Single-use anion exchange resins are standard tools to reduce hazardous chemicals such as chromates to low parts per billion concentrations in drinking water. Most processes monitor effluent concentrations when applying control strategies to their process. But when small leakages of chromates are detected, it is usually too late to take corrective measures. Therefore, process engineers use predictive modeling[1] to make the appropriate decisions before experiencing a premature breakthrough of such chemicals leading to inefficient use of the ion exchange resin capacity.

Objective:

When inlet chromate concentration increases, flow rate control is delayed until an increase of the effluent chromate concentration is detected. Therefore, better control is expected via monitoring influent concentrations to maximize the resin life. The objective of this work is to apply optimal control on a dynamic model, such as the Thomas model[2], via inlet flow rate control. This is accomplished via the application of Pontryagin’s maximum principle[3] through the following steps: (i) integrate the Thomas model with the method of moments to generate an ordinary differential equation model (ii) determine the dynamic influent flow rate profile to achieve maximum chromate removal, and (iii) capture uncertainty within the system using the Ito’s stochastic differential equation and (iv) apply stochastic optimal control to predict a robust operating policy.

Methodology:

An improved version of the Thomas model[4] is used, as the sigmoidal function predicting the characteristics of the chromate removal as seen in (eq. 1).

Thomas model = ψ = = (1)

where K_T - Thomas parameter derived from previous work, q_m – resin maximum capacity parameter derived from previous work, V – the volume of resin in the column, and Q – the flow rate into the column.

The first 3 temporal moments[5] are formulated to represent the model seen in (eq. 2) to (eq. 4).

y₁ = Zeroth moment = m₀ = = ψ = (2)

y₂ = First normalized moment = μ₁ = = = t_m = time till (3)

y₃ = Second central normalized moment = μ_2c = = = σ² = Variance (4)

where m_i == ith temporal moment. (5)

The moments represent the dynamic ion exchange process model in the form of Ordinary Differential Equations (ODEs).

The optimal control strategy to predict the operating policy of the ion exchange system is conducted via Pontryagin’s Maximum Principle (PMP). The moments represent the state variables describing the ion exchange process. PMP requires the introduction of adjoint variables that complement the state variables and the optimal control objective function is then rewritten as the Hamiltonian function which is a combination of state and adjoint variables[6] as represented in (eq 6).

H = (6)

where y_i –state variable, z_i –adjoint Ito process drift variable, w_i = adjoint Ito process variance variable, and

g_i – stochastic variance parameter[7].

The objective to maximize the chromate removal is achieved when the Hamiltonian function deviation with flow rate changes is minimized.

Summary:

Single-use anion exchange resins are used to remove hazardous chromates from drinking water applications. Predictive modeling is a useful tool to include optimal control based on the inlet concentration rather than the effluent concentration. The Pontryagin maximum principle, formulated within the Hamiltonian function[8], is used to maximize the resin loading capacity at a defined breakthrough point. The Hamiltonian is defined via the method of moments when representing the ion exchange system. The effectiveness of such control strategies is successful at maximizing the limited capacity of the ion exchange resin and minimizing toxic compounds from contaminating safe drinking water.

Keywords: Thomas Model, Hamiltonian system, Pontryagin maximum principle, stochastic optimal control, method of moments.

References:

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[6] K. M. Yenkie and U. Diwekar, “Stochastic Optimal Control of Seeded Batch Crystallizer Applying the Ito Process,” Ind. Eng. Chem. Res., p. 120604103933002, Jun. 2012, doi: 10.1021/ie300491v.

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[8] J. Harmand, C. Lobry, A. Rapaport, and T. Sari, Optimal Control in Bioprocesses: Pontryagin’s Maximum Principle in Practice, First., vol. 3. Wiley, 2019.