1.1 Basic concepts and theory of adsorption
1.1.7 Important aspects in isotherm modeling
The linear forms of the Langmuir and Freundlich equations are still frequently applied. The isotherm parameters can be easily determined from the slope and intercept of the linear plots. For three or four parameter models, linearization in ordinary manner is not longer possible and non-linear regression is applied in modeling. This is done by minimizing the error distribution between experimental and predicted adsorption isotherms. Different error functions can be used in minimization. Furthermore, important aspects in non-linear regression are the experimental data range and initial guess values. These are discussed below.
Error functions
Non-linear regression involves minimization of the error distribution between experimental and predicted adsorption isotherms usually by minimizing or maximizing the used error function depending on its definition. Table 2 shows the most commonly used error functions. Roughly, these can be divided into equations that account for the amount of parameters (p) and those that do not. Moreover, in most of the error functions the deviation between the experimental and simulated qe values is divided by the experimental qe to decrease weighting towards high concentration data [14].
Table 2. The list of error functions [14,27].
Error function Definition
The coefficient of determination,
correlation coefficient (R2)
( )
( ) ( )
The sum of the square of the errors
(ERRSQ)
∑ ( )
The hybrid error function (HYBRID)
( )
∑
= Marquardt’s percent standard deviation
(MPSD)
∑
The average relative error (ARE), Meanerror (ME)
∑
The sum of absolute errors (EABS)
i
n = number of data points, p = number of parameters of the isotherm equation, exp = experimental/measured, calc = calculated/estimated.
Several authors have introduced the comparison of different error functions. In the case of metal ion adsorption on peat the best correlation between simulated and experimental data was obtained using HYBRID and MPSD [14], of which the former was applicable for two parameter isotherms and the latter for three. HYBRID function was also appropriate in modeling of arsenic adsorption on iron oxide coated cement [46], basic dye adsorption by kudzu [47], and dye adsorption by
cationized starch-based material [48]. EABS function provided the best approximation for Cu(II) adsorption on chitosan [49] and the R2 for the basic red 9 adsorption on activated carbon [50]. For adsorption of methylene blue on activated carbon R2 function gave the closest fit to the experimental data for three parameter models and MPSD for two-parameter models [51].
Previous examples show that the applicability of different error functions is clearly case dependent.
Besides using error functions in fitting procedure, these functions are more often used in evaluation of goodness of fit [52]. The sum of normalized errors (SNE) is presented to provide a meaningful way to find the best fitting isotherm. This alone, however, should also not be used for selecting the optimum isotherm. In addition, the extent of which the theory behind the model and determined adsorbent properties converges should carefully be evaluated [51].
Data range
Limited discussion exists in the literature on how the data range affects isotherm fittings. After a literature survey, it can be stated that in over 80% of the studies fitting of isotherms and error analysis have been done using only eight or less experimental points. It is obvious that in these cases critical data might be missing from the isotherm curves. If high enough concentrations are not used in the experimental studies, the applied isotherm model may propose maximum adsorption capacity far from the true value. In addition, if all the experimental points lie in the initial part (linear range) or plateau of the isotherm curve the deviations between different isotherm models may not be observable. Therefore, proper experimental data should contain points from the concentration range that covers the whole isotherm. Even a wide experimental data range is used, however, minor changes in the data points may lead to notable changes in the fitting results [53].
Initial guess values
In most of the cases in literature, authors do not state how they have selected the initial values for non-linear fittings. The reason may be that a selection of the initial values has not affected the fitting results. Trial and error approach is supposed to be adequate [13]. Convergence difficulties, however, may arise when the initial guesses are set far from their true values [54]. To optimize the iteration procedure Ncibi [52] used the values obtained from linear fittings as initial
guesses for non-linear regressions. It is also reasonable to use the experimentally obtained qm as an initial guess whenever possible. Kinniburgh [13] stated that in the case of the BiLangmuir model, initial guesses of the other KBiL and qm should be taken from the fit of the simple Langmuir model. On the whole, to ensure the best fitting result, it is reasonable to test a few sets of initial guesses in non-linear regression.
Linear and non-linear fitting
The most commonly applied method in the fitting of models to the experimental data is the least squares regression combined with linearization of adsorption isotherms. Linear fitting of the two-parameter Langmuir and Freundlich isotherms is often used due to the simplicity of the method.
A lot of literature, however, states that a better way to obtain fitting parameters is non-linear regression [50-52,55-59]. This is attributed to the change of the error distribution caused by linearization [13]. Moreover, the Langmuir isotherm can be linearized in four different ways (Table 3) resulting in different parameter estimations [55,57,58].
Table 3. Linear forms of the Langmuir isotherm [55,57,58].
Isotherm Linear form Plot
Langmuir-lin1
When the non-linear form of the isotherm equation is fitted, the least squares problem can be solved either as a general unconstrained optimization problem (Microsoft Excel Solver) or an iterative algorithm such as the Gauss-Newton or Levenberg-Marquardt algorithm (Matlab, Origin, some special software) [60,61]. The benefit of the non-linear regression over the linear one is that it does not assume an equal error distribution for all the values of x [55]. Hence, non-linear fitting is generally a better choice for parameter estimation.
Lastly, it should be noted that the lack of the both linear and non-linear regression is an assumption that the independent variable (x) is error free. This assumption can lead to a wrong set of estimated parameters. To overcome this El-Khaiary [61] used orthogonal distance regression in modeling and found it to be a better method. Unfortunately, this procedure has not been widely applied.
Two-component systems
Besides the above discussion, a few special aspects need to be emphasized in the modeling of two-component systems. First, these systems are more complicated when the fitting as well as interpretation of the results can be difficult. The experimental data range should also include a large amount of points measured in different concentrations and different ratios of target components. This causes more experimental work. Therefore, some of the authors have fitted the two-component models to the experimental data by taking all the parameters from the one-component modeling results [42,44,62]. Despite this method being quite successful in some cases, it is important to note that equation parameters can be affected by the competing ions [63].
Moreover, when one-component parameters are not utilized, the two-component modeling involves the adjusting of 4 to 8 parameters. This can lead to a situation where experimental data is well fitted with the model, but the model does not conform the actual physical behavior of the system [3]. As in the case of one-component modeling different two-component isotherms can be compared using an error analysis. To justify the obtained results, however, the theory behind a model and experimentally observed properties of adsorption system should be carefully compared [51]. It is also reasonable to compare the results of one- and two-component modeling.