Abstract
Multicomponent adsorption processes are affected by both mixture and process variables viz. feed composition, pH, adsorbent dosage, and adsorbent type. Optimization of multicomponent adsorption processes with multiple objectives is challenging. It is important to accurately identify possible solutions and select the compromise solution that best satisfies the different objectives. Conventional algorithms, when applied to multicomponent adsorption, were found to identify the Pareto front less accurately, thereby necessitating the need for a reliable method. The steep portion of the Pareto front was especially not captured satisfactorily by the different conventional algorithms such as pattern search (PS), Non-dominated Sorting Genetic Algorithm (NSGA-II), and Epsilon-Constraint (EC). This portion assumes importance, if the compromise solution occurs in its vicinity. To address these challenges, a novel bi-objective optimization technique termed as elliptical method (EM) was developed and described in this work. It involves an exhaustive search, provides a well distributed Pareto front, and clearly delineates the steep region. After validating with benchmark problems, EM was applied to batch multi-component adsorption. The two objectives optimized simultaneously were adsorbent loading and percentage removal of the different solutes. The Pareto front and the compromise solution involving the best combination of the two objectives were significantly superior in the elliptical method when compared to those obtained from typical algorithms including epsilon-constraint (EC) method. The Pareto front was also well defined by the elliptical method without discontinuities near the extreme and steep regions. The number of points found by EM in the steeper region for the grade II adsorbent was 10 times greater than those found by the EC method while the PS and NSGA could not delineate this portion. The average time taken (considering both adsorbents) for EM per solution was 0.17 s which was at least 30.6% faster than the other methods. The compromise solution with the elliptical method was superior to the other methods. For instance, with grade II adsorbent, the compromise solution from the elliptical method suggested operating conditions that led to a total adsorbent loading and percentage removal of 333.4 mg/g and 56.0%. On the other hand, pattern search gave 324.1 mg/g and 56.5%, whereas the NSGA-II method gave 321.9 mg/g and 53.3%. For this adsorbent, elliptical method’s compromise solution was 50% and 20% closer in terms of the Euclidean distance to the utopia point than NSGA and PS methods, respectively. The elliptical method will facilitate reliable wastewater tertiary treatment taking into cognizance the utilization of the adsorbent as well as the percentage purity requirement.
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The datasets used and/or analyzed are available on reasonable request.
Abbreviations
- C0, ACT :
-
Initial concentration of acetaminophen (mg/L)
- C0, BTA :
-
Initial concentration of benzotriazole (mg/L)
- C0, CAF :
-
Initial concentration of caffeine (mg/L)
- \({C}_{0,total}\) :
-
Total initial concentration (mg/L)
- \({C}_{e,total}\) :
-
Total equilibrium concentration (mg/L)
- \({m}_{A}\) :
-
Mass of adsorbent (g)
- \(P{R}_{Total}\) :
-
Total percentage removal of solutes
- \(p{v}_{actual}\) :
-
Actual value of a process variable
- \(p{v}_{coded}\) :
-
Coded process variable
- \(p{v}_{high}\) :
-
Upper bound of a process variable
- \(p{v}_{low}\) :
-
Lower bound of a process variable
- \({q}_{Total}\) :
-
Total solute loading (mg/g)
- \({V}_{L}\) :
-
Volume of solution (L)
- \(\varphi ({\varvec{x}})\) :
-
Objective function in the first stage of the elliptical method
- \(\psi ({\varvec{x}})\) :
-
Objective function in the second stage of the elliptical method
- A:
-
Coded form of initial concentration of acetaminophen
- ACT:
-
Acetaminophen
- B:
-
Coded form of initial concentration of benzotriazole
- BTA:
-
Benzotriazole
- C:
-
Coded form of initial concentration of caffeine
- CAF:
-
Caffeine
- CS:
-
Compromise solution
- D:
-
Coded form of pH
- E:
-
Coded form of adsorbent dose
- EM:
-
Elliptical method
- NSGA:
-
Non-dominated Sorting Genetic Algorithm
- OS:
-
Overall spread
- PF:
-
Pareto front
- PS:
-
Pattern search
- PSO:
-
Particle Swarm Optimization
- U:
-
Utopia point
- VV:
-
Van Veldhuisen’s test functions
- ZDT:
-
Zitzler-Deb-Thiele’s test functions
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Hariharan Balamirtham—conceptualization, methodology, and original draft preparation; Bharathi Ganesan Retnam—experimental work and process modeling; Kannan Aravamudan—supervision of both experimental and computational works, co-authoring the manuscript, reviewing, and editing the drafts. The authors read and approved the final manuscript.
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Balamirtham, H., Retnam, B.G. & Aravamudan, K. Identifying steep pareto fronts in multicomponent adsorption using a novel elliptical method. Environ Sci Pollut Res 29, 80336–80352 (2022). https://doi.org/10.1007/s11356-022-21358-9
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DOI: https://doi.org/10.1007/s11356-022-21358-9