1 Introduction

Water pollution is recognized as one of the major problems in the world. It represents a serious threat to human health because of the uncontrolled industrial discharges, the intensive use of chemical fertilizers in agriculture, and the disorderly exploitation of water resources. Many minerals such as Ca²⁺, Mg²⁺, Na⁺, Cl⁻, and F⁻ are essential to the body; however, the excessive intake of these substances can cause harmful effects to human health and the environment [1]. In addition to arsenic and nitrate, fluoride ions (F⁻) have been classified as one of the most dangerous groundwater contaminants that lead to large-scale health problems by the World Health Organization [2]. Different technologies such as adsorption [3,4], oxidation [5], ion exchange [6], precipitation [7], electrodialysis [6], and reverse osmosis have been used for fluoride removal from water. Adsorption is reported as one of the most effective and economical techniques for water defluorination. The adsorption process is known to be simple and efficient in pollutant treatment in concentrated form at a low cost. In addition, it is relatively considered as an environmentally friendly process [7]. Many low-cost adsorbents have been used for fluoride removal from water [8,9], including activated alumina [10], activated carbon [11], kaolinite, bentonite, and montmorillonite [12], and so forth. The frequent use of clays is mainly because of their abundance and large reactive surface. In addition, clays can be preferred to synthetic adsorbent materials because of their relatively low costs [13].

Clay and clay mineral application in fluoride removal from water is well supported in the literature. In fact, considerable quantities of montmorillonite, kaolinite, or pyrophyllite clay display a huge potential for fluoride adsorption [14–16]. The positively charged surface of clay minerals provides an explanation for the high affinity uptake of the negatively charged ions at near neutral pH. The fluoride removal capacity of clay and clay minerals is impacted by different factors, such as pH and the medium's strength, and thermodynamic conditions [17,18]. Raw clay and clay minerals are rarely used and they are often used after physicochemical modifications such as metal oxide amendment, acid treatment, and thermal activation. Such modifications enhance stability at the optimal operating pH, active sites accessibility, adsorption capacity, and surface area [17,19]. Effective defluoridation with acid-activated kaolinite compared with raw kaolinite, where maximum sorption capacity of acid-activated clay ranged between 0.0450 and 0.0557 mg g⁻¹ at different temperatures, was reported in Ref. [20].

In the past decades, most articles on fluoride adsorption onto clay fields have interpreted the results with a pseudo-first-order or pseudo-second-order kinetic models [14,17,19,21,22]. Besides, Elovich kinetics model has been applied to chemisorption on the heterogeneous surface [4]. The traditional and routine fitting with pseudo-first-order and pseudo-second-order kinetic equations can sometimes give good fits with a coefficient of determination or correlation function (R²) values greater than 0.995 as this is the case here. However, we cannot be assured of an equation that does not give a correct time variation in the rate and veils the true evolution of the kinetics, whose rate is not necessarily uniform (pseudo-first-order), linear (pseudo-second-order). Nevertheless, it is well documented in the literature that the overall adsorption in a porous adsorbent must not only consider the adsorption rate on an active site but also the external mass transfer, surface diffusion, intraparticle diffusion, and pore volume diffusion [21].

The aims of this work were as follows: first, to study the fluoride adsorption kinetics onto activated clay; second, to assess the solution agitation speed, initial fluoride ion concentration, and adsorbent dosage effect on matter transfer rate, and eventually to find a simple mathematical model suitable for a better understanding of the physical phenomena involved in a more appropriate design, operation, optimization, and control of the industrial process separation. Simplified models of external diffusion are tested to describe the adsorption process. The external diffusion coefficient, k_f, is determined using different models and then correlated with the experimental variables.

2 Experimental section

3 Theoretical model

The solute adsorption mechanism on the surface of a solid goes through several successive elementary stages. Each of these stages can control the global phenomenon under given conditions. Fluoride adsorption requires (1) sorbate transport from the bulk of the solution to the exterior film surrounding the adsorbent; (2) sorbate movement across the external liquid film boundary layer to external surface sites; and (3) sorption of sorbate at internal surface sites. The last step is much faster than the first two and consequently its contribution to the overall resistance to the adsorption process is often neglected. Accordingly, the limiting step of the process can be of external type, internal type, or both.

Two resistance models are very complex because the transfer equations are coupled and the resolution of the equation system requires numerical resolution methods.

It is difficult to predict if one or both resistances (external or internal) limit the overall process without adsorption experiments.

Several researchers have studied the problem of external transport to define the conditions under which the hypotheses of this type of transfer and consequently the calculating method of the coefficient in the outer film, k_f, are valid [23,27]. In all these calculating methods k_f the resistance in the outer film is the limiting step at the beginning of adsorption.

In the present work, we faced the problem of determining the method of choice for calculating k_f and so retain the model that best describes the physical reality of the adsorption process. Therefore, four methods taken from the bibliography, namely, the following models of the initial slopes of Furusawa and Smith (Model I), Mathews and Weber (Model II) [23], Furusawa and Smith for a linear isotherm (Model III), and dimensional analysis of Harriott (Model IV) [23] were tested. Note that the first three methods are based on experimental adsorption data whereas the last is completely empirical.

4 Results and discussion

4.1 Equilibrium studies

The equilibrium biosorption data are analyzed using the Langmuir model as shown by the following equation:

qe=qmbCe1+bCe

(12)

where q_e is the biosorption capacity at equilibrium (mg g⁻¹), C_e is the fluoride ion concentration at equilibrium (mg L⁻¹), and q_m (mg g⁻¹) and b (L g⁻¹) are the Langmuir's constants. The essential characteristics of Langmuir isotherm can be expressed in terms of a dimensionless constant separation factor or equilibrium parameter R_L, which is defined by Weber [26].

RL=1(1+KLC0)

(13)

Fig. 1 shows the Langmuir isotherm plots of the experimental data for fluoride ion biosorption onto activated clay. The isotherm constants of the Langmuir model are shown in Table 2. The results indicate that the adsorption capacity of activated clay particles is high for the F⁻ 75.757 mg g⁻¹ at 25 °C. The R_L values dictate favorable adsorption for 0 < R_L < 1. On the basis of the R² = 0.995, there is an overall agreement that Langmuir model is the most suitable for designing industrial adsorption processes. The comparison of maximum adsorption capacity of F⁻ ions onto various adsorbents is presented in Table 3. It shows that the activated clay studied in this work has acceptable adsorption capacity.

Table 2

Langmuir isotherm constants for fluoride ions onto activated clay.

Table 2

Constants	qm (mg g−1)	b (L mg−1)	KL = b×qm (L g−1)	RL	R2
Fluoride	75.757	0.776	58.787	0.36	0.995

Table 3

Clay minerals used for fluoride removal from water.

Table 3

Clay mineral	Adsorption capacity (mg g−1)	Applied isotherm model	Reference
Magnesium incorporated bentonite	2.26	Langmuir	[19]
Fe3+ modified bentonite	2.91	Langmuir	[17]
Montmorillonite	1.32	Langmuir	[14]
Pyrophyllite	2.2	Langmuir	[15]

4.2 Kinetic studies – external mass transfer

4.2.1 Stirring speed effect

Fig. 2 shows that adsorption rate depends on the stirring speed, N, and that it is fast for t < 20 min, then it slows down. This phenomenon can be due to the increase in the stirring speed, which affects the resistance to the material transfer outside the clay particle [34]. Hereafter, for the calculation of k_f, we will detail the method of initial slopes for its simplicity. Fig. 3 shows the variation of ln k_f as a function of ln N. Relying on this variation, we found the following correlation between k_f and N: k_f = 1.077 × 10⁻⁴ N^0.639

4.2.2 Adsorbent dosage effect

Fig. 4 illustrates that for a contact time of 20 min, the adsorption kinetics goes up by increasing the mass of clay to 1.5 g. It also displays beyond 90 min, the material transfer rate tends to zero. As shown in Fig. 5, the correlation between k_f and m is

kf = 22.965 × 10−4 m0.55

This result indicates that the fluoride solute transport in the outer film is also controlled by the clay mass in solution. Increasing the mass of clay in solution helps to augment the liquid film transfer coefficient. Literature in general indicates an increased removal efficiency (RE) with adsorbent dose. Kim et al. [35] observed that an increase in pyrophyllite dose (from 0.05 to 2.0  L⁻¹) improves the RE (25–98.5%) at a fixed initial fluoride concentration. Similar observations have been reported by others [19,36] while carrying out fluoride removal studies with kaolinite and bentonite. It can be inferred from the literature that the increase in RE is mainly attributed to the enhancement number of active sites available for adsorption of fluoride ions.

4.2.3 Initial concentration effect

Fig. 6 shows that the adsorption is faster at its beginning than at its end. For instance, after 40 min, the adsorption slows down: this deceleration is probably because of the resistance to the material transfer inside clay particles. Toward high concentrations, the gap becomes tight and the adsorbed fraction is small. In addition, the time required to reach equilibrium varies with the initial fluoride concentration. As shown in Fig. 7, the correlation between k_f and C₀ is

kf = 36.52 × 10−4 C0−0.132

The negative value of p indicates that, for the fluoride/clay pair studied, the transfer coefficient in the outer film decreases with the initial fluoride ion concentration, in agreement with the observations of Mckay et al. [37]. Using magnesium incorporated bentonite and pyrophyllite [15,19] showed that increase in initial fluoride concentration leads to a decrease in RE. The authors explained the behavior in terms of exhausted binding capacity of bentonite or increased competition for the active adsorption sites at higher fluoride concentration.

Table 4 regroups the final and complete result of k_f values calculated by the different models. Table 5 involves a comparison between our results and those found in the bibliography. Thus, we remark the following:

• - Despite the diversity of studied pairs, the magnitude order of k_f is in the range of 10⁻⁵–10⁻⁴ m s⁻¹.
• - For all solute/adsorbent pairs, an increase in variation of k_f with m.
• - The effect of the initial solute concentration is negative: an effect contrary to that of the stirring speed.
• - The calculation of k_f by the adimensional analysis method does not provide the variation of this coefficient with the initial concentration and the mass of adsorbent in solution. Besides, the laws of the boundary layer theory state that k_f is proportional to the solute diffusion coefficient of the solute in the surrounding medium and the flow conditions around the solid particles. We made this variation with C₀ and m at the concentration gradient established at the beginning of the adsorption process. Despite this handicap by this method, the order of magnitude of k_f is always in the margin mentioned.
• - The difference between the values of k_f calculated by models II and III is very small. This result is attributed to the term m_s×K_L/(1 + m_s×K_L), which is almost equal to 1 in the case of the study couple. The use of Model I for the experimental determination of k_f can be suggested because of its simplicity.

Table 4

Results of the calculation of k_f by the different models (×10⁻⁵ m s⁻¹).

Table 4

Solute	Effect of model	I	II	III	IV
F−	N (rpm)	150	13.38	1.86	1.68	13.5
200	20.60	2.17	1.89	20.3
250	23.62	2.27	1.95	23.5
300	33.07	2.47	2.70	33.45
C0 (mg L−1)	10	10.89	3.10	2.52	33.45
30	10.23	2.49	2.00	33.45
50	8.00	2.10	1.19	33.45
70	7.66	1.50	1.65	33.45
m (g)	0.5	10.23	1.44	1.65	33.45
1	20.60	2.75	2.57	33.45
1.5	34.25	3.75	4.44	33.45
2	36.48	7.08	9.52	33.45

Table 5

External transfer coefficient k_f (m s⁻¹); comparison with the bibliography.

Table 5

Adsorbent	Field of validity	Solute	Variables	Model	Reference
N (rpm)	C0 (g L−1) or T (°C)	m (g)
Bone char	2 ≤ C0 ≤ 5 mmol/L 6.5 ≤ m ≤ 10.5 g 250 ≤ dp ≤ 100 μm	Cd2+		4.36 × 10−6 ≤ kf ≤ 7.30 × 10−6	5.47 × 10−6 ≤ kf ≤ 6.23 × 10−6	FS	[23]
Cu2+	6.14 × 10−4 ≤ kf ≤ 1.0 × 10−3	8.01 × 10−6 ≤ kf ≤ 8.99 × 10−6
Zn2+		4.75 × 10−4 ≤ kf ≤ 7.39 × 10−4	5.34 × 10−6 ≤ kf ≤ 6.40 × 10−6
Natural clay	207 ≤ N ≤ 684 rpm 0.05 ≤ m ≤ 0.25 g 5 ≤ C0 ≤ 30 mg L−1 0.45 ≤ dp ≤ 1.75	BM	kf = 2.73 × 10−5 N0.28	kf = 5440 × 10−5 C0−1.91	kf = 190 × 10−5 m1.83	FS	[32]
VM	kf =1.22 × 10−5 N0.19	kf = 11.63 × 10−5 C01.4	kf = 7.57 × 10−5 m0.28
RC	kf = 1.26 × 10−9 N1.81	kf = 17.1 × 10−5 C0−0.66	kf = 26 × 10−5 m0.88
China clay	30 < T < 50 °C	F−		2.716 < kf < 5.042 × 10−5		FS	[33]
Activated clay	150≤ N ≤ 300 rpm 0.5 ≤ m ≤ 1.5 g 30 ≤ C0 ≤ 90 mg L−1	F−	kf = 1.077 × 10−4 N0.639	kf = 36.52 × 10−4 C0−0.132	kf = 22.965 × 10−4 m0.55	FS	This study

5 Conclusions

In the present work, we studied the adsorption kinetics of fluoride onto activated clay to design an industrial adsorber. It was demonstrated in the first part that the optimal adsorption time is 30 min for a solution clay concentration of 1.5 g L⁻¹, at a minimum stirring speed of 300 rpm. Under these conditions, the pollutant elimination rate is greater than 90%. The transfer coefficient in the outer boundary layer was determined. This coefficient varied with the agitation speed, the initial fluoride ion concentration, and the adsorbent mass. Despite the diversity of the studied pairs, k_f magnitude order was in the range of 10⁻⁵–10⁻⁴ m s⁻¹.

Nomenclature

A_P

surface area of sorbent (m⁻¹)

equation constant

Langmuir isotherm constant (g L⁻¹)

liquid phase concentration at radius r (mg L⁻¹)

initial liquid phase concentration (mg L⁻¹)

equilibrium liquid phase concentration (mg L⁻¹)

liquid phase concentration at external sorbent surface (mg L⁻¹)

liquid phase concentration at time t (mg L⁻¹)

molecular diffusion coefficient of fluoride ions (m² s⁻¹)

diameter of sorbent (m)

Langmuir constant (L g⁻¹)

External mass transfer coefficient (m s⁻¹)

Equilibrium external mass transfer coefficient (m s⁻¹)

mass of dry sorbent (g)

molecular weight of the solvent (g mol⁻¹)

concentration of sorbent (g L⁻¹)

stirring speed (rpm)

equation constant

radius of concentration front of metal ions penetrating adsorbent (m)

contact time (s)

terminal velocity (m s⁻¹)

liquid phase volume (L)

molar volume (m³ mol⁻¹)

solvent-fluoride ion association factor (pour l'eau φ = 2.26)

dynamic viscosity of fluid (kg m⁻¹ s⁻¹)

kinematic viscosity (m² s⁻¹)

particle density (kg m⁻³)

solution density (kg m⁻³)