3.4 Hybrid fractional error function

follow IP model [10, 11]. However, in many studies film diffusion is the limiting step during the initial stages of the process followed by IP diffusion when particles reach the surface of the adsorbent [1, 12, 13]. There is barely any non-linear form of IP model. A couple of published papers have mispresented Boyd model. The values of Bt are obtained using Eq. (18) over the entire time scale [11, 12], this is wrong.

To determine the kinetic model that best describes the interaction between the adsorbent and solute, the goodness of fit is used. The coefficient of correlation (R<sup>2</sup>

This is one of the most used error function in determining the model. The main challenge of using SSE is that at higher concentration the squares of error increase.

Almost given in every adsorption study, R2 shows the degree of variability of dependent variable which is explained by all independent variables. It ranges from

Although not commonly used in adsorption kinetics analysis, this error function

) <sup>∑</sup><sup>n</sup>

qcal: calculated amount of adsorbate adsorbed onto adsorbent, qexp : experimental amount of adsorbate adsorbed, n:

<sup>i</sup>¼<sup>1</sup> qcal�qexp ½ �<sup>2</sup>

þ∑<sup>n</sup>

1 <sup>n</sup>�<sup>p</sup> <sup>∑</sup><sup>n</sup> i¼1

<sup>n</sup>�<sup>p</sup> <sup>∑</sup><sup>p</sup> i¼1 qexp �qcal qexp h i i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � r

<sup>i</sup>¼<sup>1</sup> qcal � qexp h i<sup>2</sup>

<sup>i</sup>¼<sup>1</sup> qcal�<sup>q</sup> exp ½ �<sup>2</sup>

qexp �qcal q exp � �<sup>2</sup> i

i

∑<sup>n</sup> <sup>i</sup>¼<sup>1</sup> qcal�qexp ½ �<sup>2</sup>

is a modification of geometric mean distribution. It is based on the number of

Error function Expression

Marquardt's percent standard deviation (MPSD) <sup>100</sup>

Sum-of-squared errors (SSE) <sup>∑</sup><sup>n</sup>

Hybrid fractional error function <sup>100</sup>

sum-of-squared errors (SSE), average relative error, Spearman's correlation coefficient, non-linear chi-square test, hybrid fractional error function, Marquardt's percent standard deviation, and standard deviation of relative errors are some of the error functions that have been employed to study model fit. Table 1 summaries the

),

3. Goodness of fit

Advanced Sorption Process Applications

error function [4, 14–17].

3.1 Sum of squared errors

3.2 Coefficient of correlation

This gives a good fit which is not the case always [15].

0 to 1, with values close to zero showing a perfect fit.

3.3 Marquardt's percent standard deviation

degrees of freedom of a system [16].

data points, p: number of parameters in each model.

Table 1.

192

Common error functions.

Coefficient of correlation (R2

Developed by Porter [17], the model was aimed to improve the applicability of SSE at a lower concentration. The error function is divided by the measured value.

## 3.5 Sum of normalised errors (SNE)

Different error functions yield different value of goodness fit—thus it may be difficult to select the best model fit. SNE provide a normalised value of the different error functions, making comparison very easy. SNE is done by dividing the error value of the different functions by the highest error for a given kinetic model.

### 3.6 Misuse of fitting index

The assessment of adsorption kinetics using error function has been misused in almost all adsorption papers. The problem arises when error function of linearized equations of non-linear functions are expended to determine the suitability of a model. In some linearized models, to reduce the error factor, log or square root transforms are applied if the error increases with the dependent factor. And if the error variance decrease with increasing dependent factor, then exponential or square alters are applied. However, the use of R<sup>2</sup> or SSE does not detect the biasness of the parameters.

The dependent variable in adsorption kinetic is not entirely linear over the given values of the independent variable. Eq. (7) shows the linearized form of PSO. The inverse of data weights <sup>1</sup> <sup>q</sup> <sup>=</sup> <sup>t</sup><sup>Þ</sup> and the presence of independent variable ð Þ<sup>t</sup> in both dependent and independent sides causes false correlation. The inversing of variables on both sides of Eq. (8) distorts the error distribution over the entire data. In the third form of PSO (Eq. (9)), the presence of dependent parameter qt in both the independent and dependent section leads to spurious correlation. While in Eq. (10), the presence of independent variable violates the least squares assumption [18]. R<sup>2</sup> is a very sensitive parameter that can cause spurious conclusions. R2 varies with the range of independent parameter—if the range is big, R<sup>2</sup> will be fit; and if the range is small, fit will be poor. Adding more data points decreases the degree of freedom of a system; this favours model fit. Therefore, making conclusions solely basing on R<sup>2</sup> can be misleading to the industry of adsorption mechanism.
