Abstract

Adsorption has become a competitive method in the field of wastewater and air treatment. Adsorption kinetics is one of the main factors that must be understood before the applicability of any adsorbent. In every adsorption process, linear or nonlinear analysis of the kinetics is applied. The goodness of fit index (coefficient of correlation or sum of squares) is applied to access the best model. The usage of linear or non-linear from of the adsorption kinetics has an impact on the distribution of error function. Almost in every adsorption study, linear forms have been used to conclude the best kinetic model that influence the adsorption mechanism which might be an error. Therefore, this review highlights the mistakes in the usage of linear and non-linear models. The applicability of the adsorption kinetics in wastewater treatment is also illuminated.

Keywords: goodness of fit, error function, Boyd model, diffusion model, dyes, ion adsorption, emerging contaminants

#### 1. Introduction

Although water covers 70% of the world's surface, the availability of fresh water for animal consumption is becoming scarce. This is attributed to the improved standards that have resulted in rapid usage of pollutant infused substances such as pesticides, heavy metals, anions, pharmaceuticals, phenolic, personal care products, dyes, and hydrocarbons. Different methods have been developed to remove such substances from the wastewater, these include; biological, precipitation, membrane technology, electrochemical, and adsorption. Currently, the latter has received a considerable attention because it is cost and energy effective, easy to design and operate. Numerous adsorbents (carbon, clay, polymer, activates sludge, and zeolite) have been developed to remove solute form wastewater [1–3]. These adsorbents have large surface area and pore volume, thermal stability, with low acid/base reactivity. This makes them suitable to remove a wide range of organic and inorganic dissolved pollutants from wastewater and air.

The utmost parameter to consider while designing the adsorption system is adsorption kinetics; kinetics determine the rate at which the adsorption occurs. Kinetics are influenced by the surface complexity of the adsorbent, solute concentration and flow. Pseudo-First-order (PFO), Pseudo-Second-order (PSO), Elovich, and Intra-particle (IP) model are some of the kinetics that foretells the adsorbentadsorbate interaction. The first two models have been widely applied in almost every sorption process. The suitability of any model depends on the error level correlation coefficient (R<sup>2</sup> ) or Sum of Squared Errors (SSE). To study adsorption kinetics, the linear forms have been applied; a linear form of PSO has been favoured over PFO model for the last 2 decades.

validity of the model. PFO is varied under Henry regime adsorption, and under high sorbent dosage [4]. To account for experimental differences during the sorption of

> qe qt

qe � qt

(3)

is proportional

), the

(6)

(7)

(8)

<sup>¼</sup> lnqe � <sup>k</sup>1<sup>t</sup> (4)

<sup>2</sup> (5)

<sup>t</sup> (9)

=qt vs. t. Even

<sup>e</sup> � k2qeqt (10)

against t gives a better fit of dye adsorption compared

dyes on activated carbon, PFO is modified as follows [5];

qt qe

and the linear form

qe

þ ln qe � qt

DOI: http://dx.doi.org/10.5772/intechopen.80495

2.2 Pseudo second order (PSO) model

The plot of qt

need to be tested.

curvilinear form of PSO.

linearized form of PSO is;

189

dqt dt <sup>¼</sup> <sup>k</sup><sup>1</sup>

Modelling of Adsorption Kinetic Processes—Errors,Theory and Application

þ ln qe � qt

of solute on the surface of the adsorbent—the driving force qe � qt

dqt

Eq. (6) has been rearranged in different forms, Eq. (7)–(10).

t qt

1 qt

qt <sup>t</sup> <sup>¼</sup> <sup>k</sup>2q<sup>2</sup>

Therefore, the PSO constants can be determined from a graph of <sup>t</sup>

to Lagergren PFO. Different models for adsorption of various solutes have been developed, but the applicability and consistence of their linear and non-linear form

PSO model assumes that the rate of adsorption of solute is proportional to the available sites on the adsorbent. And the reaction rate is dependent on the amount

to the number of active sites available on the adsorbent [4, 6]. Eq. (5) shows the

dt <sup>¼</sup> <sup>k</sup><sup>2</sup> qe � qt

k<sup>2</sup> is PSO rate constant. Applying the integral limits for t (0, t) and qt (0, qt

qt <sup>¼</sup> <sup>t</sup> 1 k2q<sup>2</sup> e þ t qe

<sup>¼</sup> <sup>1</sup> k2q<sup>2</sup> e

<sup>¼</sup> <sup>1</sup> k2q<sup>2</sup> e 1

qt <sup>¼</sup> qe � <sup>1</sup>

though the PSO model may be affected by pH, dose amount, particle size, and temperature—the model assesses the impact of observable rate parameters.

þ t qe

t þ 1 qe

k2qe qt

The approach used to linearize the curvilinear function of PSO determines the distribution of the error function of the same kinetic model. Among the linearized equations of PSO, Eq. (7) yields better fitting results compared to other forms.

The linearization of a function involves assumptions—any change in the assumption means a change in the error parameter. This leads to biasness in the data producing vast outcomes which violate the variance and normality assumption of the standard least squares. For example, the linearization of PSO has resulted in more than five different forms of PSO model. Using the non-linear form of the function means distributing the error on the entire curve. Generally, analysis of PFO and PSO is done by plotting ln qt � qe and <sup>t</sup>=qt against time, respectively. As the system approaches equilibrium, t =qt ≈ <sup>t</sup> =qe produces a perfect fitting for PSO. However, at equilibrium, ln qe � qt becomes enormously large reducing the fitting index. In addition, R2 has been compared using different scales of various kinetics which is statically wrong. Therefore, this work highlights the gaps in the linearization of adsorption kinetics, and the suitability of each kinetic model towards the removal of dyes, organics, metals, and anions from solution.

### 2. Adsorption kinetics

Adsorption is the process by which solute molecules attach to the surface of an adsorbent. The adsorption process is done in batch or column setup. Adsorption kinetics is a curve (or line) that describes the rate of retention or release of a solute from an aqueous environment to solid-phase interface at a given adsorbents dose, temperature, flow rate and pH. During adsorption two main processes are involved; physical (physisorption) or chemical (chemisorption). Physical adsorption is as a result of weak forces of attraction (van der Waals), while chemisorption involves the formation of a strong bond between the solute and the adsorbent that involves the transfer of electrons.

#### 2.1 Pseudo first order model (PFO)

Also known as Lagergren model, PFO describes the adsorption of solute onto adsorbent following the first order mechanism;

$$\frac{dq\_t}{dt} = k\_1(q\_\epsilon - q\_t) \tag{1}$$

where qt is adsorbate adsorbed onto adsorbent at time t (mg/g), qe is equilibrium adsorption capacity (mg/g), and k<sup>1</sup> is rate constant per min. The integral of Eq. (1) from t ¼ 0 to t ¼ t and qt ¼ 0 and qt ¼ qt yields a linear expression of PFO, Eq. (2).

$$
\ln \left( q\_e - q\_t \right) = \ln q\_e - k\_1 t \tag{2}
$$

The value of k<sup>1</sup> is determined by plotting ln qe � qt vs. t. Albeit some studies have found k<sup>1</sup> to increase with initial solute concentration ð Þ C<sup>0</sup> or independent of C0, the rate constant is always inversely proportional to the initial concentration of the solute. This is because a longer time is required for a large initial solute concentration. The controlling mechanism is affected by experimental conditions, thus the Modelling of Adsorption Kinetic Processes—Errors,Theory and Application DOI: http://dx.doi.org/10.5772/intechopen.80495

validity of the model. PFO is varied under Henry regime adsorption, and under high sorbent dosage [4]. To account for experimental differences during the sorption of dyes on activated carbon, PFO is modified as follows [5];

$$\frac{dq\_t}{dt} = k\_1 \frac{q\_\epsilon}{q\_t} \left(q\_\epsilon - q\_t\right) \tag{3}$$

and the linear form

and Intra-particle (IP) model are some of the kinetics that foretells the adsorbentadsorbate interaction. The first two models have been widely applied in almost every sorption process. The suitability of any model depends on the error level—

kinetics, the linear forms have been applied; a linear form of PSO has been favoured

The linearization of a function involves assumptions—any change in the assumption means a change in the error parameter. This leads to biasness in the data producing vast outcomes which violate the variance and normality assumption of the standard least squares. For example, the linearization of PSO has resulted in more than five different forms of PSO model. Using the non-linear form of the function means distributing the error on the entire curve. Generally, analysis of

> =qt ≈ <sup>t</sup>

removal of dyes, organics, metals, and anions from solution.

index. In addition, R2 has been compared using different scales of various kinetics which is statically wrong. Therefore, this work highlights the gaps in the linearization of adsorption kinetics, and the suitability of each kinetic model towards the

Adsorption is the process by which solute molecules attach to the surface of an adsorbent. The adsorption process is done in batch or column setup. Adsorption kinetics is a curve (or line) that describes the rate of retention or release of a solute from an aqueous environment to solid-phase interface at a given adsorbents dose, temperature, flow rate and pH. During adsorption two main processes are involved; physical (physisorption) or chemical (chemisorption). Physical adsorption is as a result of weak forces of attraction (van der Waals), while chemisorption involves the formation of a strong bond between the solute and the adsorbent that involves

Also known as Lagergren model, PFO describes the adsorption of solute onto

dt <sup>¼</sup> <sup>k</sup><sup>1</sup> qe � qt

have found k<sup>1</sup> to increase with initial solute concentration ð Þ C<sup>0</sup> or independent of C0, the rate constant is always inversely proportional to the initial concentration of the solute. This is because a longer time is required for a large initial solute concentration. The controlling mechanism is affected by experimental conditions, thus the

where qt is adsorbate adsorbed onto adsorbent at time t (mg/g), qe is equilibrium adsorption capacity (mg/g), and k<sup>1</sup> is rate constant per min. The integral of Eq. (1) from t ¼ 0 to t ¼ t and qt ¼ 0 and qt ¼ qt yields a linear expression of PFO, Eq. (2).

dqt

ln qe � qt

The value of k<sup>1</sup> is determined by plotting ln qe � qt

) or Sum of Squared Errors (SSE). To study adsorption

and <sup>t</sup>=qt against time, respectively. As

becomes enormously large reducing the fitting

=qe produces a perfect fitting for PSO.

(1)

vs. t. Albeit some studies

<sup>¼</sup> lnqe � <sup>k</sup>1<sup>t</sup> (2)

correlation coefficient (R<sup>2</sup>

over PFO model for the last 2 decades.

Advanced Sorption Process Applications

PFO and PSO is done by plotting ln qt � qe

the system approaches equilibrium, t

However, at equilibrium, ln qe � qt

2. Adsorption kinetics

the transfer of electrons.

188

2.1 Pseudo first order model (PFO)

adsorbent following the first order mechanism;

$$\frac{q\_t}{q\_\varepsilon} + \ln(q\_\varepsilon - q\_t) = \ln q\_\varepsilon - k\_1 t \tag{4}$$

The plot of qt qe þ ln qe � qt against t gives a better fit of dye adsorption compared to Lagergren PFO. Different models for adsorption of various solutes have been developed, but the applicability and consistence of their linear and non-linear form need to be tested.

#### 2.2 Pseudo second order (PSO) model

PSO model assumes that the rate of adsorption of solute is proportional to the available sites on the adsorbent. And the reaction rate is dependent on the amount of solute on the surface of the adsorbent—the driving force qe � qt is proportional to the number of active sites available on the adsorbent [4, 6]. Eq. (5) shows the curvilinear form of PSO.

$$k\frac{dq\_t}{dt} = k\_2(q\_\varepsilon - q\_t)^2\tag{5}$$

k<sup>2</sup> is PSO rate constant. Applying the integral limits for t (0, t) and qt (0, qt ), the linearized form of PSO is;

$$q\_t = \frac{t}{\frac{1}{k\_2 q\_e^2} + \frac{t}{q\_e}}\tag{6}$$

Eq. (6) has been rearranged in different forms, Eq. (7)–(10).

$$\frac{t}{q\_t} = \left[\frac{1}{k\_2 q\_e^2}\right] + \frac{t}{q\_e} \tag{7}$$

$$\frac{1}{q\_t} = \left[\frac{1}{k\_2 q\_e^2}\right] \frac{1}{t} + \frac{1}{q\_e} \tag{8}$$

$$q\_t = q\_\epsilon - \left[\frac{1}{k\_2 q\_\epsilon}\right] \frac{q\_t}{t} \tag{9}$$

$$\frac{q\_t}{t} = k\_2 q\_e^2 - k\_2 q\_e q\_t \tag{10}$$

The approach used to linearize the curvilinear function of PSO determines the distribution of the error function of the same kinetic model. Among the linearized equations of PSO, Eq. (7) yields better fitting results compared to other forms. Therefore, the PSO constants can be determined from a graph of <sup>t</sup> =qt vs. t. Even though the PSO model may be affected by pH, dose amount, particle size, and temperature—the model assesses the impact of observable rate parameters.

PSO can be used to determine the initial solute uptake and adsorption capacity of an adsorbent. Within the last 2 decades, PSO fits the experiment better and it has been concluded that the adsorption mechanism is chemisorption in nature, involving the transfer of electrons between the adsorbate and adsorbent—this conclusion is wrong. Adsorption mechanism cannot be based on simple fitting of PSO model.

When the solute concentration is low, Eq. (7) explains the adsorption mechanism more than any other kinetic model; however, at high initial concentration, PFO model is favoured [7]. This is because at low C<sup>0</sup> the value of ln qe � qt increases exponentially increasing the error function—which is the reverse for high C0. Although the applicable of linear forms has improved, they may be misleading in developing kinetic systems. For example, the R2 of the linear PSO model during the adsorption of methylene green 5 onto activated was above 0.99; however, using non-linear form, the fit index was below 0.70 [6]. Both PSO and PFO do not explain the diffusion of solute into the adsorbent; therefore, before any conclusions are made about adsorption mechanism, diffusion models should be investigated.

#### 2.3 Elovich model

To further understand the chemisorption nature of adsorption, Elovich model (developed by Zeldowitsch) is applied. This model helps to predict the mass and surface diffusion, activation and deactivation energy of a system. Although the model was initially applied in gaseous systems, its applicability in wastewater processes has been redeemed meaningful. The model assumes that the rate of adsorption of solute decreases exponentially as the amount of adsorbed solute increase.

$$\frac{dq\_t}{dt} = a \exp^{-\beta q\_t} \tag{11}$$

qt ¼ Kp

Modelling of Adsorption Kinetic Processes—Errors,Theory and Application

DOI: http://dx.doi.org/10.5772/intechopen.80495

diffusion controls the adsorption process. However, on many occasions, the plot does not pass through the origin and it gives multiple linear sections; these sections corresponds to different mechanisms that control the adsorption process. There are four main mechanisms that describe the transfer of solute from a solution to the adsorbent. The first is called mass transfer (bulk movement) of solute particles as soon as the adsorbent is dropped into the solution. This process is too fast, thus it is not considered during the design of kinetic systems. The second mechanism is called film diffusion; it involves the slow movement of solutes from the boundary layer to the adsorbent's surface. When the solute reach the surface of the adsorbent, they move to the pores of the adsorbent—third mechanism. The final mechanism involves rapid adsorptive attachment of the solute on the active sites of the pores; being a rapid process, it is not considered during engineering design of kinetics [6]. If the system is characterised by poor mixing, small solute size, and low concentration, film diffusion becomes the rate controlling step; otherwise, IP diffusion con-

Misrepresentation of diffusion model: Couple of papers have assessed IP model using a straight of Eq. (14); however, in reality, pore diffusion is a slow process making Eq. (14) to follow a curvilinear trend. When the segment analysis is applied, the values of Kp and C differ enormously. The segments can be got by

To understand if film diffusion is the rate controlling step, Boyd developed a single-resistance model that can be used to assess this effect. Boyd assumes that the boundary layer surrounding the adsorbent has a greater effect on the diffusion of

> 1 n2

<sup>F</sup> <sup>¼</sup> qt q∞

It is hard to estimate appropriate values of Bt with Eq. (15); Bt can be calculated

The graph of Bt vs t helps to predict the rate limiting step. If the graph approximates y ¼ mx þ 0:0 line, the rate limiting step is intra-particle diffusion, otherwise film diffusion model governs the process. Treatment of water was estimated to

� �exp �n<sup>2</sup>

Bt

<sup>3</sup> � <sup>2</sup><sup>π</sup> <sup>1</sup> � <sup>π</sup><sup>F</sup>

0:86 ≤ F ≤ 1: Bt ¼ �0:4977 � ln 1ð Þ � F (18)

3 � �<sup>1</sup>=<sup>2</sup>

� � (15)

) to solute adsorbed at infinite

(16)

(17)

visual or application of regression on different points.

solute [3]. To determine this effect, Eq. (15) is applied.

F is the fraction of solute adsorbed at time t (qt

<sup>F</sup> <sup>¼</sup> <sup>1</sup> � <sup>6</sup>

using the integrated Fourier transform of Eqs. (17) and (18).

<sup>0</sup> <sup>≤</sup> <sup>F</sup> <sup>≤</sup> <sup>0</sup>:85: Bt <sup>¼</sup> <sup>2</sup><sup>π</sup> � <sup>π</sup><sup>2</sup><sup>F</sup>

<sup>π</sup><sup>2</sup> <sup>∑</sup> ∞ 1

time (q∞) (t<sup>∞</sup> > 24h for better results). Bt is a mathematical function of F.

The plot of qt vs ffiffi

trols the process.

2.4.1 Boyd model

191

ffiffi

Kp is a rate constant mg/g.min0.5, and C is boundary layer thickness. The values of C determines the boundary layer effect—higher values, the greater the effect.

<sup>t</sup> <sup>p</sup> , gives a linear function. If the line passes through the origin, IP

<sup>t</sup> <sup>p</sup> <sup>þ</sup> <sup>C</sup> (14)

As qt <sup>≈</sup> 0, dqt dt ≈ α which is the initial adsorption rate (mg/g.min), and β is desorption constant. Integrating and applying the limits for t (0, t) and qt 0; qt , the Elovich model can be linearized as;

$$q\_t = \frac{1}{\beta} \ln \left[ t + \frac{1}{a\beta} \right] - \frac{1}{\beta} \ln(a\beta) \tag{12}$$

As the system approaches equilibrium t ≫ <sup>1</sup> αβ, thus Eq. (12) becomes;

$$q\_t = \frac{\mathbf{1}}{\beta} \ln[\alpha \beta] + \frac{\mathbf{1}}{\beta} \text{Int} \tag{13}$$

The graph of qt vs t helps to determine the nature of adsorption on the heterogeneous surface of the adsorbent, whether chemisorption or not. A number of solutes have been reported to follow Elovich kinetics model [8, 9].

#### 2.4 Intra-particle diffusion (IP) model

IP model has been widely applied to examine the rate limiting step during adsorption. The adsorption of solute in a solution involves mass transfer of adsorbate (film diffusion), surface diffusion, and pore diffusion. Film diffusion is an independent step, whereas surface and pore diffusion may occur simultaneously. IP is studied by examining Weber and Morris (1963) model, Eq. (14).

Modelling of Adsorption Kinetic Processes—Errors,Theory and Application DOI: http://dx.doi.org/10.5772/intechopen.80495

$$q\_t = \mathcal{K}\_p \sqrt{t} + \mathcal{C} \tag{14}$$

Kp is a rate constant mg/g.min0.5, and C is boundary layer thickness. The values of C determines the boundary layer effect—higher values, the greater the effect. The plot of qt vs ffiffi <sup>t</sup> <sup>p</sup> , gives a linear function. If the line passes through the origin, IP diffusion controls the adsorption process. However, on many occasions, the plot does not pass through the origin and it gives multiple linear sections; these sections corresponds to different mechanisms that control the adsorption process. There are four main mechanisms that describe the transfer of solute from a solution to the adsorbent. The first is called mass transfer (bulk movement) of solute particles as soon as the adsorbent is dropped into the solution. This process is too fast, thus it is not considered during the design of kinetic systems. The second mechanism is called film diffusion; it involves the slow movement of solutes from the boundary layer to the adsorbent's surface. When the solute reach the surface of the adsorbent, they move to the pores of the adsorbent—third mechanism. The final mechanism involves rapid adsorptive attachment of the solute on the active sites of the pores; being a rapid process, it is not considered during engineering design of kinetics [6]. If the system is characterised by poor mixing, small solute size, and low concentration, film diffusion becomes the rate controlling step; otherwise, IP diffusion controls the process.

Misrepresentation of diffusion model: Couple of papers have assessed IP model using a straight of Eq. (14); however, in reality, pore diffusion is a slow process making Eq. (14) to follow a curvilinear trend. When the segment analysis is applied, the values of Kp and C differ enormously. The segments can be got by visual or application of regression on different points.

#### 2.4.1 Boyd model

PSO can be used to determine the initial solute uptake and adsorption capacity of an adsorbent. Within the last 2 decades, PSO fits the experiment better and it has been concluded that the adsorption mechanism is chemisorption in nature, involving the transfer of electrons between the adsorbate and adsorbent—this conclusion is wrong. Adsorption mechanism cannot be based on simple fitting of PSO model. When the solute concentration is low, Eq. (7) explains the adsorption mechanism more than any other kinetic model; however, at high initial concentration, PFO model is favoured [7]. This is because at low C<sup>0</sup> the value of ln qe � qt

increases exponentially increasing the error function—which is the reverse for high C0. Although the applicable of linear forms has improved, they may be misleading in developing kinetic systems. For example, the R2 of the linear PSO model during the adsorption of methylene green 5 onto activated was above 0.99; however, using non-linear form, the fit index was below 0.70 [6]. Both PSO and PFO do not explain the diffusion of solute into the adsorbent; therefore, before any conclusions are made about adsorption mechanism, diffusion models should be investigated.

To further understand the chemisorption nature of adsorption, Elovich model (developed by Zeldowitsch) is applied. This model helps to predict the mass and surface diffusion, activation and deactivation energy of a system. Although the model was initially applied in gaseous systems, its applicability in wastewater processes has been redeemed meaningful. The model assumes that the rate of adsorption of solute decreases exponentially as the amount of adsorbed solute increase.

dt ≈ α which is the initial adsorption rate (mg/g.min), and β is

� 1 β

β

dt <sup>¼</sup> <sup>α</sup> exp �βqt (11)

αβ, thus Eq. (12) becomes;

lnð Þ αβ (12)

lnt (13)

dqt

qt <sup>¼</sup> <sup>1</sup> β ln t þ

have been reported to follow Elovich kinetics model [8, 9].

qt <sup>¼</sup> <sup>1</sup> β

IP is studied by examining Weber and Morris (1963) model, Eq. (14).

As the system approaches equilibrium t ≫ <sup>1</sup>

desorption constant. Integrating and applying the limits for t (0, t) and qt 0; qt

1 αβ 

ln½ �þ αβ <sup>1</sup>

The graph of qt vs t helps to determine the nature of adsorption on the heterogeneous surface of the adsorbent, whether chemisorption or not. A number of solutes

IP model has been widely applied to examine the rate limiting step during adsorption. The adsorption of solute in a solution involves mass transfer of adsorbate (film diffusion), surface diffusion, and pore diffusion. Film diffusion is an independent step, whereas surface and pore diffusion may occur simultaneously.

2.3 Elovich model

Advanced Sorption Process Applications

As qt <sup>≈</sup> 0, dqt

190

the Elovich model can be linearized as;

2.4 Intra-particle diffusion (IP) model

,

To understand if film diffusion is the rate controlling step, Boyd developed a single-resistance model that can be used to assess this effect. Boyd assumes that the boundary layer surrounding the adsorbent has a greater effect on the diffusion of solute [3]. To determine this effect, Eq. (15) is applied.

$$F = 1 - \frac{6}{\pi^2} \sum\_{1}^{\infty} \left( \frac{1}{n^2} \right) \exp\left( -n^2 B\_t \right) \tag{15}$$

F is the fraction of solute adsorbed at time t (qt ) to solute adsorbed at infinite time (q∞) (t<sup>∞</sup> > 24h for better results). Bt is a mathematical function of F.

$$F = \frac{q\_t}{q\_{\infty}}\tag{16}$$

It is hard to estimate appropriate values of Bt with Eq. (15); Bt can be calculated using the integrated Fourier transform of Eqs. (17) and (18).

$$0 \le F \le 0.85 \colon B\_t = 2\pi - \frac{\pi^2 F}{3} - 2\pi \left(1 - \frac{\pi F}{3}\right)^{1/2} \tag{17}$$

$$0.86 \le F \le 1 \colon B\_t = -0.4977 - \ln(1 - F) \tag{18}$$

The graph of Bt vs t helps to predict the rate limiting step. If the graph approximates y ¼ mx þ 0:0 line, the rate limiting step is intra-particle diffusion, otherwise film diffusion model governs the process. Treatment of water was estimated to

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.

3.4 Hybrid fractional error function

DOI: http://dx.doi.org/10.5772/intechopen.80495

3.5 Sum of normalised errors (SNE)

3.6 Misuse of fitting index

of the parameters.

4.1 Dyes

193

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.

Modelling of Adsorption Kinetic Processes—Errors,Theory and Application

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.

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

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

in both

the third form of PSO (Eq. (9)), the presence of dependent parameter qt

basing on R<sup>2</sup> can be misleading to the industry of adsorption mechanism.

Dyes are organic substances that cause a permanent or temporary change in colour of a material; they are resistant to detergents. Dyes are widely employed in leather, food, textile, paper, rubber, and plastic industries. When dyes are released in the hydrosphere, they can block sunlight penetration, thus affecting the marine life. In addition, they give unpleasant colour to water making it unsafe for human consumption. To reduce the impact of dyes on the ecosystem, adsorption method has been employed to remove dyes from wastewater. Different kinetic models have been employed to study the adsorption of dyes from solution, these include; PFO, PSO, Elvoich, and IP models. The suitability of any model depends on error functions. Table 2 summarises the non-linear adsorption kinetics of different adsorbent.

4. Linear and non-linear fitting application

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
