(434f) Chromate Control in an Ion Exchange Process Under Uncertainty

Problem (background/scope/motivation):

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 make corrective measures. Therefore, process engineers use predictive modeling[1] to make the appropriate decisions before experiencing premature breakthrough of such chemicals leading to inefficient use of the ion exchange resin capacity or the contamination of the water supply.

When the 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 the ion exchange process via inlet flow rate control based on inlet concentration.

Methods:

Optimal control is applied on a dynamic model, such as the Thomas model[2], via inlet flow rate control. This is accomplished via the application of the 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.

We used an improved version of the Thomas model[4], as the sigmoidal function predicting the characteristics of the chromate removal. The first 4 temporal moments[5] are formulated to represent the dynamic ion exchange process model in the form of Ordinary Differential Equations (ODEs), such as the zeroth moment or concentration, the first normalized moment ore residence time, the second central normalized moment or variance, and the third normalized moment or skewness,

The optimal control strategy to predict the operating policy of the ion exchange system is conducted via the Pontryagin’s Maximum Principle (PMP). While the moments represent the state variables describing the ion exchange process, PMP requires the introduction of adjoint variables which complement the state variables. The optimal control objective function in then rewritten as the Hamiltonian function which is a combination of state and adjoint variables[6]. The objective to maximize the chromate removal is achieved when the Hamiltonian function deviation with flow rate changes is minimized. The overall flowchart of the steps taken are summarized in the figure.

Results:

Both deterministic and stochastic optimal control were applied to the chromate removal system. While both control strategies achieved the desired results, the stochastic process was able to initiate flow rate changes earlier than the deterministic allowing for more gradual adjustments. The main reason for this difference is that the stochastic process considers the uncertainty of the measurements. Therefore, the stochastic optimal control becomes the weighted average of a multitude of deterministic optimal control problems and will mirror the results of a real life system.

Implications:

Since single use anion exchange resins are applied in the removal of chromate, it is critical to maximize the capacity of such resins to minimize the disposal waste. While using lower flow rate can maximize such capacity, it can also add unnecessary processing cost in case of low inlet chromate concentration. Therefore, the use of optimal control based on the inlet conditions becomes economically necessary for the efficient use of the anion exchange resin and to prevent the toxic compounds from contaminating our water supply. Such strategy can also be applied with various other contaminants such as lead, arsenic, PFAS, etc.

References:

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