Optimal Control of Chromate Removal via Enhanced Modeling Using the Method of Moments
Publication: Journal of Environmental Engineering
Volume 150, Issue 5
Abstract
Single-use anion-exchange resins can reduce hazardous chromates to safe levels in drinking water. However, since most process control strategies monitor effluent concentrations, detection of any chromate leakage leads to premature resin replacement. Furthermore, variations in the inlet chromate concentration and other process conditions make process control a challenging step. In this work, we capture the uncertainty of the process conditions by applying the Ito process of Brownian motion with a drift into a stochastic optimal control strategy. The ion-exchange process is modeled using the method of moments, which helps capture the process dynamics, later formulated into mathematical objectives representing desired chromate removal. We then solved our developed models as an optimal control problem via Pontryagin’s maximum principle. The objectives enabled a successful control via flow rate adjustments leading to higher chromate extraction. Such an approach maximizes the capacity of the resin and column efficiency to remove toxic compounds from water while capturing deviations in the process conditions.
Practical Applications
When dealing with highly toxic compounds like chromium, it is critical that its concentration in drinking water is kept at a low, safe level. As single-use ion-exchange resins are used to extract the hazardous chemical, changes in inlet concentrations can lead to premature leakage. Hence, an optimal control strategy is needed for the purification system while monitoring the inlet concentration rather than the outlet concentration to avoid a process control delay. For a successful optimization, predicting the output concentration based on the inlet conditions becomes necessary to maximize the performance of the extraction process. In this work, predictive modeling while capturing the uncertainties of the system maximized the chromate removal in less time than running the process at a constant flow rate. The results demonstrate that changing the flow rate with time is an improved strategy to achieve such performance. The flow rate change is a unique approach to an industry that designs its processes around a constant flow rate and reacts too late when system deviations have already occurred. Therefore, applying the approach described in this work will maximize the utilization of the purification process resulting in less waste produced and safer drinking water.
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Data Availability Statement
All data, models, and code generated or used during the study appear in the published article.
Acknowledgments
The authors thank the Department of Chemical Engineering at Rowan University for their assistance in acquiring computational licenses and resources used in this study. Special thanks to colleagues, Swapana Jerpoth and Emmanuel Aboagye, for their paper review and feedback. The authors also acknowledge Purolite for their support.
References
Abbasi, S., and U. M. Diwekar. 2013. “Characterization and stochastic modeling of uncertainties in the biodiesel production.” Clean Technol. Environ. Policy 16 (Jan): 79–94. https://doi.org/10.1007/s10098-013-0596-4.
Artstein, Z. 2011. “Pontryagin maximum principle revisited with feedbacks.” Eur. J. Control 17 (1): 46–54. https://doi.org/10.3166/ejc.17.46-54.
Balan, C., I. Volf, and D. Bilba. 2013. “Chromium (VI) removal from aqueous solutions by Purolite base anion-exchange resins with gel structure.” Chem. Ind. Chem. Eng. Q. 19 (4): 615–628. https://doi.org/10.2298/CICEQ120531095B.
Benavides, P. T., and U. Diwekar. 2012. “Optimal control of biodiesel production in a batch reactor.” Fuel 94 (Apr): 218–226. https://doi.org/10.1016/j.fuel.2011.08.035.
Benavides, P. T., and U. Diwekar. 2013. “Studying various optimal control problems in biodiesel production in a batch reactor under uncertainty.” Fuel 103 (Jan): 585–592. https://doi.org/10.1016/j.fuel.2012.06.089.
Biswas, S., and U. Mishra. 2015. “Continuous fixed-bed column study and adsorption modeling: Removal of lead ion from aqueous solution by charcoal originated from chemical carbonization of rubber wood sawdust.” J. Chem. 2015 (Jan): 1–9. https://doi.org/10.1155/2015/907379.
Boscain, U., and B. Piccoli. 2005. “A short introduction to optimal control.” In Contrôle Non Linéaire et Applications: Cours Donnés à l’école d’été Du Cimpa de l’Université de Tlemcen/Sari Tewfit, 19–66. Palaiseau, France: École Polytechnique.
Brereton, T. 2014. Stochastic simulation of processes, fields and structures, 108–121. Ulm, Germany: Ulm Univ.
Briskot, T., et al. 2019. “Prediction uncertainty assessment of chromatography models using Bayesian inference.” J. Chromatogr. A 1587 (Feb): 101–110. https://doi.org/10.1016/j.chroma.2018.11.076.
Brito, F., et al. 1997. “Equilibria of Chromate(VI) species in acid medium and Ab initio studies of these species.” Polyhedron 16 (21): 3835–3846. https://doi.org/10.1016/S0277-5387(97)00128-9.
Carta, G., and A. Jungbauer. 2010. “Effects of dispersion and adsorption kinetics on column performance.” In Protein chromatography, 237–276. New York: Wiley.
Charola, S., R. Yadav, P. Das, and S. Maiti. 2018. “Fixed-bed adsorption of reactive orange 84 dye onto activated carbon prepared from empty cotton flower agro-waste.” Sustainable Environ. Res. 28 (6): 298–308. https://doi.org/10.1016/j.serj.2018.09.003.
Chiang, A. C. 1992. “Optimum control: The maximum principle.” In Elements of dynamic optimization, 167–177. New York: McGraw-Hill.
Corder, G. D., and P. L. Lee. 1986. “Feedforward control of a wastewater plant.” Water Res. 20 (3): 301–309. https://doi.org/10.1016/0043-1354(86)90077-1.
Costa, M., and C. B. Klein. 2006. “Toxicity and carcinogenicity of chromium compounds in humans.” Crit. Rev. Toxicol. 36 (2): 155–163. https://doi.org/10.1080/10408440500534032.
de Dardel, F., and T. V. Arden. 2008. “Ion exchangers.” In Ullmann’s encyclopedia of industrial chemistry. New York: Wiley.
Diwekar, U. 2008. “Optimal control and dynamic optimization.” In Introduction to applied optimization, 215–277. Berlin: Springer.
Ferreira, M. G. S., M. L. Zheludkevich, and J. Tedim. 2011. “9—Advanced protective coatings for aeronautical applications.” In Nanocoatings and ultra-thin films, edited by A. Salam, H. Makhlouf, and I. Tiginyanu, 235–279. Sawston, UK: Woodhead.
Fogler, H. S. 2016. “Residence time distributions of chemical reactors.” In Elements of chemical reaction engineering, 767–806. Boston: Prentice Hall.
Gallego, P. T. B. 2013. Optimal control of batch production of biodiesel fuel under uncertainty. Urbana, IL: Univ. of Illinois.
Ghanem, F., S. S. Jerpoth, and K. M. Yenkie. 2022. “Improved models for chromate removal using ion exchangers in drinking water applications.” J. Environ. Eng. 148 (5): 04022015. https://doi.org/10.1061/(ASCE)EE.1943-7870.0001997.
Goltz, M. N., and P. V. Roberts. 1987. “Using the method of moments to analyze three-dimensional diffusion-limited solute transport from temporal and spatial perspectives.” Water Resour. Res. 23 (8): 1575–1585. https://doi.org/10.1029/WR023i008p01575.
Hamdaoui, O. 2009. “Removal of Copper(II) from aqueous phase by Purolite C100-MB cation exchange resin in fixed bed columns: Modeling.” J. Hazardous Mater. 161 (2–3): 737–746. https://doi.org/10.1016/j.jhazmat.2008.04.016.
Hamill, P. 2018. Lagrangians and hamiltonians. 4th ed. Cambridge, UK: Cambridge University Press.
Harmand, J., C. Lobry, A. Rapaport, and T. Sari. 2019. Optimal control in bioprocesses: Pontryagin’s maximum principle in practice. 1st ed. New York: Wiley.
Hutcheson, J. 2006. “Ultrapure water: Systems for microelectronics.” Filtr. Sep. 43 (5): 22–25. https://doi.org/10.1016/S0015-1882(06)70888-6.
Jawitz, J. W. 2004. “Moments of truncated continuous univariate distributions.” Adv. Water Resour. 27 (3): 269–281. https://doi.org/10.1016/j.advwatres.2003.12.002.
Kabir, G. 2008. “Removal of chromate in trace concentration using ion exchange from tannery wastewater.” Int. J. Environ. Res. 2 (4): 377–384. https://doi.org/10.22059/IJER.2010.218.
Kalaruban, M., et al. 2016. “Removing nitrate from water using iron-modified Dowex 21K XLT ion exchange resin: Batch and fluidised-bed adsorption studies.” Sep. Purif. Technol. 158 (Jan): 62–70. https://doi.org/10.1016/j.seppur.2015.12.022.
Kao, P. C. K. 2019. “Brownian motion and other diffusion processes.” In An introduction to stochastic processes, 373–420. Urbana, IL: Dover.
Lei, Z., C. Li, and B. Chen. 2003. “Extractive distillation: A review.” Sep. Purif. Rev. 32 (2): 121–213. https://doi.org/10.1081/SPM-120026627.
Li, X., et al. 2016. “Chromium removal from strong base anion exchange waste brines.” J. Am. Water Works Assoc. 108 (4): 247–255. https://doi.org/10.5942/jawwa.2016.108.0049.
Li, X. 2016. “Meeting the New California MCL for Hexavalent chromium with strong base anion exchange resin.” Am. Water Works Assoc. 108 (9): 474–481. https://doi.org/10.5942/jawwa.2016.108.0112.
Lin, S. H., and C. D. Kiang. 2003. “Chromic acid recovery from waste acid solution by an ion exchange process: Equilibrium and column ion exchange modeling.” Chem. Eng. J. 92 (1–3): 193–199. https://doi.org/10.1016/S1385-8947(02)00140-7.
Luo, J., O. A. Cirpka, and P. K. Kitanidis. 2006. “Temporal-moment matching for truncated breakthrough curves for step or step-pulse injection.” Adv. Water Resour. 29 (9): 1306–1333. https://doi.org/10.1016/j.advwatres.2005.10.005.
Millar, G. J., S. J. Couperthwaite, M. de Bruyn, and C. W. Leung. 2015. “Ion exchange treatment of saline solutions using Lanxess S108H strong acid cation resin.” Chem. Eng. J. 280 (Nov): 525–535. https://doi.org/10.1016/j.cej.2015.06.008.
Mustafa, Y. A., and S. E. Ebrahim. 2010. “Utilization of Thomas model to predict the breakthrough curves for adsorption and ion exchange.” J. Eng. 16 (4): 6206–6223.
Nagaki, M., R. D. Hughes, J. Y. N. Lau, and R. Williams. 1991. “Removal of endotoxin and cytokines by adsorbents and the effect of plasma protein binding.” Int. J. Artif. Organs 14 (1): 43–50. https://doi.org/10.1177/039139889101400109.
Purolite Website. 2021. “Purolite A600E.” Accessed April 14, 2021. https://www.purolite.com/product/a600e.
Recepoğlu, Y. K., et al. 2018. “Packed bed column dynamic study for boron removal from geothermal brine by a chelating fiber and breakthrough curve analysis by using mathematical models.” Desalination 437 (Jul): 1–6. https://doi.org/10.1016/j.desal.2018.02.022.
Rodriguez-Gonzalez, P. T., V. Rico-Ramirez, R. Rico-Martinez, and U. M. Diwekar. 2019. “A new approach to solving stochastic optimal control problems.” Mathematics 7 (1207): 13. https://doi.org/10.3390/math7121207.
Sharma, D. 2022. “Simulated moving bed technology: Overview and use in biorefineries.” In Biorefineries—Selected processes, 1–21. London: IntechOpen.
Shastri, Y., and U. Diwekar. 2006. “An optimal control and options theory approach to forecasting and managing sustainable systems.” In Proc., Int. Congress on Environmental Modeling and Software. Provo, UT: Brigham Young Univ.
Staby, A., et al. 2007. “Comparison of chromatographic ion-exchange resins.” J. Chromatogr. A 1164 (1–2): 82–94. https://doi.org/10.1016/S0021-9673(00)00780-9.
Szabados, T. 2010. “An elementary introduction to the wiener process and stochastic integrals.” Preprint, submitted February 21, 2021. http://arxiv.org/abs/1008.1510.
Yenkie, K. M., and U. Diwekar. 2012. “Stochastic optimal control of seeded batch crystallizer applying the ITO process.” Indus. Eng. Chem. Res. 52 (1): 108–122. https://doi.org/10.1021/ie300491v.
Yenkie, K. M., and U. M. Diwekar. 2018. “The ‘no sampling parameter estimation (NSPE)’ algorithm for stochastic differential equations.” Chem. Eng. Res. Des. 129 (Jan): 376–383. https://doi.org/10.1016/j.cherd.2017.11.018.
Yenkie, K. M., U. M. Diwekar, and A. A. Linninger. 2016. “Simulation-free estimation of reaction propensities in cellular reactions and gene signaling networks.” Comput. Chem. Eng. 87 (Apr): 154–163. https://doi.org/10.1016/j.compchemeng.2016.01.010.
Yu, C., A. W. Warrick, and M. H. Conklin. 1999. “A moment method for analyzing breakthrough curves of step inputs.” Water Resour. Res. 35 (11): 3567–3572. https://doi.org/10.1029/1999WR900225.
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© 2024 American Society of Civil Engineers.
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Received: May 31, 2023
Accepted: Dec 26, 2023
Published online: Mar 5, 2024
Published in print: May 1, 2024
Discussion open until: Aug 5, 2024
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