adsorption, model, optimization, process integration, simulation, superstructure, water usage and treatment network


The paper addresses water recycling in process industry, inter alia, the issues of mathematical models’ complexity problem in the “process integration”-based structural optimization of sustainable water usage and treatment networks. The nature of addressing structural optimization problems requires iteratively querying individual process models, which are incorporated as objective functions and constraints within the optimization model, throughout the process of finding a solution, therefore the goal was to explore the intricacy of mentioned models. Within the framework of the research, the impact of complexity of water network constituent parts models on the optimization performance was investigated by Monte Carlo method for one step of the optimization procedure, as well as for the optimization procedure as a whole. Units’ models in form of algebraic equations (for direct equation calculation case), algebraic equations (for root search), ordinary differential equations (for Cauchy initial value problem with a case of two differential equations), ordinary differential equations (for boundary value problem), and partial differential equations (for two spatial variables) were examined with an analysis of their applicability for optimization purposes. The justification for employing both deterministic "counter-current mass transfer" models and statistical polynomial "input-output" steady-state algebraic models were established for addressing the specific problems under investigation. As the case study, special polynomial model was constructed based on the experimental design / response surface methodology and the dynamics simulation results on adsorption wastewater treatment within the packed bed column filled with activated carbon. Central composite rotatable design was formulated and subsequently executed using computational experimentation methods for the parametric identification of a nonlinear polynomial model. The evaluation confirmed that the constructed model exhibits satisfactory predictive accuracy.


Batstone, D. J. Mathematical modelling of anaerobic reactors treating domestic wastewater: rational criteria for model use. Reviews in Environmental Science and Bio/Technology, 2006, 5 (1), 57–71.

Boyko, T. V. Modeling of ion exchange systems of industrial wastewater treatment: dissertation abstract. (Kyiv, 1993). [In Ukrainian].

Deng, Ch.; Jiang, W.; Zhou, W.; Feng, X. New superstructure-based optimization of property-based industrial water system. Journal of Cleaner Production, 2018, 189, 878–886.

Diehl, S.; Farås, S. Control of an ideal activated sludge process in wastewater treatment via an ODE–PDE model. Journal of Process Control. 2013, 23 (3), 359–381.

El-Halwagi, M. Sustainable Design Through Process Integration: 2nd Edition. (Amsterdam-Oxford-Cambridge: Elsevier. 2017).

Fan, X.-Y.; Klemeš, J. J.; Jia, X.; Liu, Zh.-Y. An iterative method for design of total water networks with multiple contaminants. Journal of Cleaner Production, 2019, 240, 118098.

García, A.; Betancourt, F. Conservative mathematical model and numerical simulation of batch gravity settling with coalescence of liquid-liquid dispersions. Chem. Enging. Sci., 2019, 207, 1214–1229.

Givlyud, A. M.; Humnytskyi, Y. M.; Ruda, M. V. Purification of wastewater from the dairy industry by the adsorption method: monograph. («Mizhnarodna naukova hilʹdiya», 2022). URL: [In Ukrainian].

Huang, C.; Chang, C.; Ling, H.; Chang, C. A mathematical programming model for water usage and treatment network design. Ind Eng Chem Res., 1999, 38 (7), 2666–2679.

Jung, D.; Lee, S.; Hwang, H. Optimization difficulty indicator and testing framework for water distribution network complexity. Water, 2019, 11 (10), 2132.

Kyrychenko, K. C.; Sabliy, L. A. Calculation of optimum parameters of biological wastewater treatment based on the GPS-X program. 17th International Scientific and Practical Conference "Biotechnology of the 21st Century", 2023, 228–230. [In Ukrainian].

Ochando-Pulido, J. M.; Vellido-Pérez, J. A.; González-Hernández, R.; Martínez-Férez, A. Optimization and modeling of two-phase olive-oil washing wastewater integral treatment and phenolic compounds recovery by novel weak-base ion exchange resins. Separation and Purification Technology, 2020, 117084.

Ortega, G.; Rovira, A. A fast and accurate methodology for the calculation of the shading and blocking efficiency in central receiver systems. Renewable Energy. 2020, 154, 58–70.

Poplewski, G.; Foo, D. C. Y. An extended corner point method for the synthesis of flexible water network. Process Safety and Environmental Protection, 2020, 148 (2), 210–224.

Savelski, M. J.; Bagajewicz, M. J. On the necessary conditions of optimality of water utilization systems in process plants with multiple contaminants. Chem Eng Sci., 2003, 23–24, 5349–5362.

Shakhnovskij, A.; Jeżowski, J.; Kvitka, A. et al. Optymalizacja sieci wody procesowej przy zastosowaniu programowania matematycznego. Inżynieria Chemiczna i Procesowa, 2004, 25 (3), 1607-1612.

Shakhnovsky, A.; Kvitka, O. Design of sustainable industrial water networks: 1. Genesis of the systematic methods. Water and water purification technologies. Scientific and technical news, 2019, 24 (1), 34–44.

Takama, N.; Kiriyama, T.; Shiroko, K.; Umeda, T. Optimal water allocation in a petroleum refinery. Computers & Chemical Engineering, 1980, 4 (4), 251–258.

Vlasyuk, A. P.; Zhukovsky, V. V.; Zhukovska, N. A. Mathematical and computer modeling of mass transfer during filtration of salt solutions in porous and nanoporous media: monograph (Rivne: NUVHP. 2022). [In Ukrainian].

Wałczyk, K.; Poplewski, G.; Jeżowski, J. et. al. Optimization of water network with models of non-mass transfer processes. Chem. Proc. Enging., 2007, 28, 515–525.

Wang, Y.-P.; Smith, R. Wastewater Minimization. Chem. Eng. Sci., 1994, 49 (7), 981–1006.

Yang, F.; Fan, X.-Y.; Jia, X.; Klemeš, J. J., Liu, Zh.-Y. An iterative design approach for water networks with multiple regeneration units. Journal of Cleaner Production, 2020, 271, 122483.

Zheng, T.; Du, Z.; Cao, H.; Jiang, J. et. al. Development of a novel mobile industrial-scale fluidized adsorption process for emergency treatment of water polluted by aniline: CFD simulation and experiments. Advanced Powder Technology, 2016, 27 (4), 1576–1587.