Isotherm investigation for the sorption of fluoride onto BioF: comparison of linear and nonlinear regression method
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Abstract
A comparison of the linear and nonlinear regression method in selecting the optimum isotherm among three most commonly used adsorption isotherms (Langmuir, Freundlich, and Redlich–Peterson) was made to the experimental data of fluoride (F) sorption onto BioF at a solution temperature of 30 ± 1 °C. The coefficient of correlation (\(r^{2}\)) was used to select the best theoretical isotherm among the investigated ones. A total of four Langmuir linear equations were discussed and out of which linear form of most popular Langmuir1 and Langmuir2 showed the higher coefficient of determination (0.976 and 0.989) as compared to other Langmuir linear equations. Freundlich and Redlich–Peterson isotherms showed a better fit to the experimental data in linear leastsquare method, while in nonlinear method Redlich–Peterson isotherm equations showed the best fit to the tested data set. The present study showed that the nonlinear method could be a better way to obtain the isotherm parameters and represent the most suitable isotherm. Redlich–Peterson isotherm was found to be the best representative (\(r^{2}\) = 0.999) for this sorption system. It is also observed that the values of \(\beta\) are not close to unity, which means the isotherms are approaching the Freundlich but not the Langmuir isotherm.
Keywords
Sorption BioF Fluoride Equilibrium isotherm Linear regression Nonlinear regressionAbbreviations
 F
Fluoride
 BioF
Biofilter
 r^{2}
Coefficient of correlation
 NaF
Sodium fluoride
 mg
Milligramme
 g/L
Gramme per litre
 mg/L
Milligramme per litre
 °C
Degree centigrade
 HDPE
Highdensity polyethylene
 K_{L}
Adsorption equilibrium constant
 Q_{m}
Maximum adsorption capacity
 q_{e}
Adsorbate adsorbed onto adsorbent
 C_{e}
Equilibrium liquidphase concentration
 K_{F}
Freundlich constant
 1/n
Heterogeneity factor
 K_{R}
Redlich–Peterson constant
 a_{R}
Redlich–Peterson constant
 β
Redlich–Peterson constant
 q_{m}
Equilibrium capacity
Introduction
Among the various methods used for defluoridation of drinking water, the adsorption process has been widely used because of its simplicity, affordability, easy operation, and satisfactory results (Liu et al. 2010; Deng et al. 2011; Bhatnagar et al. 2011; Xiang et al. 2014). Adsorption process includes the selective transfer of solute components onto the surface or the bulk of solid adsorbent materials in the aqueous phase (Kumar and Sivanesan 2007). The effectiveness of an adsorbent is estimated on the basis of its uptake capacity, adsorption rate, mechanical strength, possibility of regeneration, and reuse options (Tang and Zhang 2016). Among these, adsorbent capacity is the most important parameter which plays a vital role in overall process of adsorption (Oh and Park 2002; Gong et al. 2005). The uptake capacity and adsorption performance are usually determined on the basis of equilibrium experiments and sorption isotherms describing the interaction of pollutant with the adsorbent material (Brdar et al. 2012). The equilibrium studies are also very important in optimizing the design parameters for any adsorption system which provide sufficient information about physicochemical data to evaluate the adsorption process as a unit operation (LeyvaRamos et al. 2010). The distribution of a solute between solid adsorbent and liquid phase is also a measure of the position of equilibrium. Therefore, equilibrium data should be accurately fit into different isotherm models to find a suitable one that can be used to design the process (Khaled et al. 2009).
Among the various tested isotherms for the defluoridation of drinking water, Langmuir, Freundlich, and Redlich–Peterson isotherms are frequently used over a wide concentration range of solute and sorbent to describe the adsorption equilibrium for water and wastewater treatment applications (Ho et al. 2005; Ho 2006a, b; Kumar and Sivanesan 2007).
The conventional approach for parameter evaluation of nonlinear forms of aforementioned isotherms involves linearization of the expressions through transformation, followed by the linear regression method. The main disadvantage of the linear regression technique, that limit its use, includes estimation of only two variables in an empirical equation; whereas, nonlinear optimization provides a more complex, yet mathematically rigorous method to determine the isotherm parameter values (Pal et al. 2013).
Linear regression analysis has been frequently employed in accessing the quality of fits and adsorption performance for fluoride removal from aqueous solutions (Fan et al. 2003; Onyango et al. 2004; Kamble et al. 2009; Foo and Hameed 2010). The fitting validity of different models was tested in linearized forms using coefficient of determination. Some other methods, recently were reported to predict the optimum isotherm, which includes correlation coefficient, the sum of errors squared, a hybrid error function, Spearman’s correlation coefficient, Standard deviation of relative errors, coefficient of nondetermination Marquardt’s percent standard deviation, the average relative error, and the sum of absolute errors (Kumar et al. 2008; Foo and Hameed 2010). Currently, nonlinear regression method is observed to be the best way in selecting the optimum isotherm, but very limited published literature is available for such type of adsorption systems, i.e., BioF–F system. This method of nonlinear regression involves the step of minimizing the error distribution between the experimental equilibrium data and predicted isotherm (Krishni et al. 2014).
Having these considerations in mind, a comparison of linear least squares method and nonlinear regression method was discussed in present study using the experimental adsorption data of F onto BioF. The three widely used isotherms (Langmuir, Freundlich, and Redlich–Peterson) were investigated to discuss this issue. A trialanderror optimization method was used for the nonlinear regression using the solver addin function, Microsoft Excel, Microsoft Corporation (Kumar and Sivanesan 2007; Krishni et al. 2014). In order to solve the nonlinear equations of applied isotherms, an efficient Microsoft’s addin software xlstat is used in this study (ShahmohammadiKalalagh and Babazadeh 2014; Kausar et al. 2014). This study was done in continuation of our previous work (Yadav et al. 2014, 2015) to examine the most suitable isotherm for BioF and F system using linear and nonlinear equations.
Experimental programmes
Materials and methods
All chemicals used throughout this study were of analytical grade and purchased mainly from Merck India Limited. The stock solution of fluoride was prepared by dissolving the appropriate amount (221 mg) of anhydrous NaF in 1 L of doubledistilled water to a concentration of 100 mg/L. The test working solutions of F were prepared by successive dilution with doubledistilled water. All the batch adsorption studies were undertaken using BioF adsorbent, manufactured by HES Water Engineers (I) Pvt. Ltd. (a joint venture company of water engineers, Australia). More details about this adsorbent are given in our previous studies (Yadav et al. 2014, 2015).
Batch equilibrium adsorption experiments were conducted to investigate the adsorption behaviour of BioF at a constant dose of 10 g/L and varying concentrations of fluoride. All the adsorption experiments were carried out at room temperature of 30 ± 1 °C. To study the various process parameters, a series of conical flasks having test solution and adsorbent, was then shaken at a constant speed of 90 rpm in an orbital shaker with thermostatic control (Remi, India). At the end of the required contact time (when equilibrium was achieved), flasks were removed from the shaker and allowed to stand for 5 min for the adsorbent to settle down. After the fluoride adsorption equilibrium studies, the treated and untreated samples were filtered through Whatman filter paper No. 42 and stored in HDPE bottles for the further analysis of the residual F using an ion meter (Thermo Scientific Orion 5Star ion meter).
Isotherm models
Selected adsorption isotherms and their linear forms with corresponding plots
Isotherm model (nonlinear form)  Linear form  Plot  

Langmuir 1  \(q_{\text{e }} = \frac{{Q_{\text{m}} K_{\text{L}} C_{\text{e}} }}{{1 + K_{\text{L}} C_{\text{e}} }}\)  \({\raise0.7ex\hbox{${C_{\text{e}} }$} \!\mathord{\left/ {\vphantom {{C_{\text{e}} } {q_{\text{e}} }}}\right.\kern0pt} \!\lower0.7ex\hbox{${q_{\text{e}} }$}} = \frac{1}{{q_{\text{m}} }}C_{\text{e}} + \frac{1}{{K_{\text{L}} q_{\text{m}} }}\)  \({\raise0.7ex\hbox{${C_{\text{e}} }$} \!\mathord{\left/ {\vphantom {{C_{\text{e}} } {q_{\text{e}} }}}\right.\kern0pt} \!\lower0.7ex\hbox{${q_{\text{e}} }$}}{\text{ vs }}C_{\text{e}}\) 
Langmuir 2  \(\frac{1}{{q_{\text{e}} }} = \left( {\frac{1}{{K_{\text{L}} q_{\text{m}} }}} \right)\frac{1}{{C_{\text{e}} }} + \frac{1}{{q_{\text{m}} }}\)  \(\frac{1}{{q_{\text{e}} }} \,{\text{vs }} \frac{1}{{C_{\text{e}} }}\)  
Langmuir 3  \(q_{\text{e}} = q_{\text{m}}  \left( {\frac{1}{{K_{\text{L}} }}} \right)\frac{{q_{\text{e}} }}{{C_{\text{e}} }}\)  \(q_{\text{e}} \,{\text{vs }} \frac{{q_{\text{e}} }}{{C_{\text{e}} }}\)  
Langmuir 4  \(\frac{{q_{\text{e}} }}{{C_{\text{e}} }} = K_{\text{L}} q_{\text{m}}  K_{\text{L}} q_{\text{e}}\)  \(\frac{{q_{\text{e}} }}{{C_{\text{e}} }} \,{\text{vs }} q_{\text{e}}\)  
Freundlich  \(q_{\text{e }} = K_{\text{F}} C_{\text{e}}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 n}}\right.\kern0pt} \!\lower0.7ex\hbox{$n$}}}}\)  \(\log q_{\text{e}} = \log K_{\text{F}} + \frac{1}{n}\log C_{\text{e}}\)  \(\log q_{\text{e}} {\text{ vs }}\log C_{\text{e}}\) 
Redlich–Peterson  \(q_{\text{e}} = \frac{{K_{\text{R}} C_{\text{e}} }}{{1 + a_{\text{R}} C_{\text{e}}^{\beta } }}\)  \(\ln \left( {K_{\text{R}} \frac{{C_{\text{e}} }}{{q_{\text{e}} }}  1} \right) = \beta \ln \left( {C_{\text{e}} } \right) + \ln \left( {a_{\text{R}} } \right)\)  \(\ln \left( {K_{\text{R}} \frac{{C_{\text{e}} }}{{q_{\text{e}} }}  1} \right) \,{\text{vs }} { \ln }\left( {C_{\text{e}} } \right)\) 
Results and discussion
Linear regression method
Isotherm parameters obtained using the linear and nonlinear method
Langmuir isotherm  Freundlich isotherm  Redlich–Peterson isotherm  

\(Q_{\text{m}}\)  \(K_{\text{L}}\)  \(R^{2}\)  \(K_{\text{F}}\)  \(n\)  \(R^{2}\)  \(K_{\text{R}}\)  \(a_{\text{R}}\)  \(\beta\)  \(R^{2}\)  
Linear 1  0.39  2.7  0.976  0.31  1.79  0.999  18.8  59.94  0.45  0.999 
Linear 2  0.31  4.01  0.989  
Linear 3  0.34  3.48  0.903  
Linear 4  0.36  3.14  0.903  
Nonlinear  0.531  0.246  0.969  0.631  1.697  0.969  20.12  64.77  0.464  0.999 
Nonlinear regression method
Comparative account of linear and nonlinear regression method
In this study, it is important to mention here that predicting the optimum isotherm by using only linear method is not appropriate (Yadav et al. 2014), as different forms of one single Langmuir equation may be applicable for a particular adsorption system. Subsequently, the results produced by these four equations may differ significantly as shown in Table 2. The probable reason behind unlike outcomes of different linearized forms of one equation may be the variation in derived error functions. Moreover, the error distribution may vary depending on the way of linearization. The same has been evidenced for the equilibrium sorption data of present adsorption system. However, another possible reason behind the variable results may be the different axial settings, which alter the result of linear regression and influence the determination process. Thus, it can be concluded that it is more appropriate to use nonlinear method to estimate the parameters of an isotherm or a rate equation (Kumar and Sivanesan 2007). Also, nonlinear method had an advantage that the error distribution does not get altered as in case of linear technique, because all the equilibrium parameters are fixed on the same axis.
Conclusions

The equilibrium sorption data of F onto BioF sorbent is explained using the linear and nonlinear forms of Langmuir, Freundlich and Redlich–Peterson isotherms.

The comparison of linear and nonlinear regression method shows that nonlinear regression is more reliable as compared to linear regression method for the prediction of bestfit isotherm as well as parameter determination for the adsorption of F onto BioF.

The values of \(r^{2}\) of Langmuir1 and Langmuir2 presented in this study are close to those of the nonlinear form of Langmuir isotherm, while Langmuir3 and Langmuir4 showed almost similar \(r^{2}\) values. The values of \(r^{2}\) of linear forms of Freundlich isotherm were observed to be different in comparison to the value of nonlinear form of the Redlich–Peterson isotherm.

Redlich–Peterson isotherm was found to be the bestfit among the investigated isotherms suggesting that the isotherms are approaching the Freundlich, but not the Langmuir isotherm on basis of \(\beta\) value which is not close to unity.
Notes
Acknowledgements
The authors are thankful to Dr. A.B. Gupta for his suggestions and support in carrying out experimental work.
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