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Statistical techniques for the optimization of cesium removal from aqueous solutions onto iron-based nanoparticle-zeolite composites


Statistical optimization of performance determining factors is essential for the development of a cesium removal system from aqueous solutions. Therefore, factorial experimental design and multiple regression techniques were employed to assess the primary and interaction effects of the pH, initial concentration, and contact time in the cesium removal process using nanoscale zero-valent iron-zeolite (nZVI-Z) and nano-Fe/Cu-zeolite (nFe/Cu-Z) as an adsorbent. The optimum region of cesium removal was identified by constructing a contour plot. The study revealed that initial concentration was the most significant factor followed by contact time. The study also suggested that maximum cesium removal occurred at pH, initial concentration, and contact time of 6, 200 mg/L, and 30 min, respectively. Moreover, the statistically significant interaction effect was observed between contact time and initial concentration. The experimental data were also fitted with Tόth, Langmuir, Dubinin-Astakhov (D-A), Freundlich, and Hill models and found that the Tόth model fitted better compared with the other four models based on Akaike information criterion (AIC) and root-mean-square deviation (RMSD). The findings of this paper can undoubtedly contribute to constructing the optimum statistical process of removing hazardous pollutants from the water, which significantly impacts on human health and the environment.

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Data availability

The datasets used and analyzed during the current work are available from the corresponding author on reasonable request.


A :

Level of pH (−)

a 0 :

Lower level of pH (−)

a 1 :

Higher level of pH (−)


Interaction effect between pH (−) and initial concentration (mg/L)


Interaction effect between pH (−) and contact time (min)


Interaction effect among pH (−), initial concentration (mg/L), and contact time (min)

B :

Initial concentration (mg/L)

b 0 :

Lower level of initial concentration (mg/L)

b 0t :

Equilibrium constant of Tόth model (L/mg)

b 1 :

Higher level of initial concentration (mg/L)


Interaction effect between initial concentration (mg/L) and contact time (min)

C :

Contact time (min)

c 0 :

Lower level of contact time (min)

c 1 :

Higher level of contact time (min)

C e :

Equilibrium concentration (mg/L)

E :

Adsorption characteristic energy (J/mol)

K D :

Hill constant (−)

K f :

Freundlich constant ((mg/g) (mg/L)− (1/n))

K L :

Langmuir constant (L/mg)

n :

Heterogeneity parameter (−)

n H :

Binding interaction coefficient of the Hill model (−)

N :

Number of observations (−)

Q :

Isosteric heat of adsorption (J/mol)

q e :

Instantaneous uptake (mg/g)

q m :

Maximum uptake (mg/g)

R :

Universal gas constant (J/(mol/K))


Akaike information criterion

Adj MSS:

Adjusted mean sum of squares

Adj SS:

Adjusted sum of squares


Analysis of variance




Initial concentration

Cr (VI):

Hexavalent chromium






Degrees of freedom




Value of F-statistic


Nanoscale zero-valent iron


Zero-valent iron nanoparticle-zeolite

P value:

Probability value


Root-mean-square deviation

R 2 (adj):

Adjusted R2

SE coef.:

Standard error of coefficient




Value of student T-statistic

W cal :

Calculated cesium removal using model (mg/g)

W exp :

Experimental cesium removal (mg/g)


poly-γ-glutamic acid

α :

Level of significance (−)

ρ c :

Density of cesium (mg/L)

τ :

Average removal of cesium (mg/g)

Κ 0 :

Overall mean (mg/g)

Κ i :

Regression coefficients (−)


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Authors and Affiliations



Md. Matiar Rahman: conceptualization, formal analysis, methodology, software, writing-original draft. Shamal Chandra Karmaker: data collection, formal analysis, investigation. Animesh Pal: investigation, methodology, writing-review & editing. Osama Eljamal: investigation, writing-review & editing. Bidyut Baran Saha: conceptualization, investigation, writing-review & editing, supervision.

Corresponding author

Correspondence to Bidyut Baran Saha.

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The authors declare that they have no conflict of interest.

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Responsible Editor: Tito Roberto Cadaval Jr

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Rahman, M.M., Karmaker, S.C., Pal, A. et al. Statistical techniques for the optimization of cesium removal from aqueous solutions onto iron-based nanoparticle-zeolite composites. Environ Sci Pollut Res 28, 12918–12931 (2021).

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  • Cesium removal
  • Composites
  • Factorial experiment
  • Interaction effect
  • Statistical techniques