Physical Meanings, Applications, and Solving Methods

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Journal of Hazardous Materials

journal homepage:www.elsevier.com/locate/jhazmat

Review

Adsorption kinetic models: Physical meanings, applications, and solving methods

Jianlong Wang

^a,b,

*, Xuan Guo

^a

aCollaborative Innovation Center for Advanced Nuclear Energy Technology, INET, Tsinghua University, Beijing 100084, PR China

bBeijing Key Laboratory of Radioactive Waste Treatment, Tsinghua University, Beijing 100084, PR China

G R A P H I C A L A B S T R A C T

A R T I C L E I N F O Editor: Daniel C.W. Tsang Keywords:

Adsorption Kinetic model Physical meaning Solving method

A B S T R A C T

Adsorption technology has been widely applied in water and wastewater treatment, due to its low cost and high efficiency. The adsorption kinetic models have been used to evaluate the performance of the adsorbent and to investigate the adsorption mass transfer mechanisms. However, the physical meanings and the solving methods of the kinetic models have not been well established. The proper interpretation of the physical meanings and the standard solving methods for the adsorption kinetic models are very important for the applications of the kinetic models. This paper mainly focused on the physical meanings, applications, as well as the solving methods of 16 adsorption kinetic models. Firstly, the mathematical derivations, physical meanings and applications of the adsorption reaction models, the empirical models, the diffusion models, and the models for adsorption onto active sites were analyzed and discussed in detail. Secondly, the model validity evaluation equations were summarized based on literature. Thirdly, a convenient user interface (UI) for solving the kinetic models was developed based on Excel software and provided in supplementary information, which is helpful for readers to simulate the adsorption kinetic process.

1. Introduction

Adsorption is a mass transfer process that pollutants from the liquid phase to the solid adsorbent. Adsorption is one of the most widely used

technologies in water and waste water treatment, because it has many advantages, such as simple design, low price, easy maintenance, and high efficiency (Wang and Chen, 2009,2014;Wang and Zhuang, 2017, 2019;Wang et al., 2018a,b,c). The adsorption kinetic study provides

https://doi.org/10.1016/j.jhazmat.2020.122156

Received 20 November 2019; Received in revised form 20 January 2020; Accepted 20 January 2020

⁎Corresponding author at: Energy Science Building, INET, Tsinghua University, Beijing 100084, PR China.

E-mail addresses:wangjl@mail.tsinghua.edu.cn,wangjl@tsinghua.edu.cn(J. Wang).

Available online 25 January 2020

0304-3894/ © 2020 Elsevier B.V. All rights reserved.

T

Nomenclature

(M) PCL Modified pineapple crown leaf 4-CP 4-chlorophenol

a Initial adsorption rate constant of the Elovich model (mg g⁻¹h⁻¹)

A Constant of the Boyd’s equation AB113 Acid Blue 113

AB25 Acid blue 25 AB434 Acid Blue 324

AdjR² Adjust correlation coefficient AR1 Acid red 1

AR337 Acid Red 337

b Desorption rate constant of the Elovich model (g mg⁻¹) B Boy’s coefficient (h⁻¹)

BB69 Basic blue 69

BBLS-CA Sludge-based carbonaceous adsorbent BDE-99 Polybrominated diphenyl ethers-99 BY2 Basic Yellow 2

C0 Initial adsorbate concentration (mg·L⁻¹) C_e Equilibrium adsorbate concentration (mg·L⁻¹)

Cet Equilibrium adsorbate concentration at the surface of the adsorbent (mg·L⁻¹)

C_f Adsorbate concentration in the liquid film (mg·L⁻¹) CIP Ciprofloxacin

CPNM Chitosan coated polyacrylonitrile nanofibrous mat Ct Adsorbate concentration at timet(mg·L⁻¹) Ctr Adsorbate concentration at distantr, (mg·L⁻¹)

Ctr|r 0=Adsorbate concentration at the center of the adsorbent (mg·L⁻¹)

Ctr|r r0=Adsorbate concentration at the external surface of the adsorbent (mg·L⁻¹)

CV Crystal violet

D_ef Intraparticle diffusion coefficient (mm²·h⁻¹) DFC Diclofenac

D^l Diffusional constant

D_p Effective pore volume diffusion coefficient (cm²·s⁻¹) Ds Surface diffusion coefficient (cm²·s⁻¹)

DY12 Direct yellow 12 E33 Bayoxide®E33

FTMA-MTSurfactant-modified montmorillonite HYBRID Hybrid fractional error function IC Indigo carmine

K Partition coefficient of the linear model (L·g⁻¹) k1 Pseudo-first-order rate constant (h⁻¹)

k1’ First-order rate constant of the mixed-order model (h⁻¹) k₂ Pseudo-second-order rate constant (g·mg⁻¹·h⁻¹) k2’ Second-order rate constant of the mixed-order model

(g·mg⁻¹·h⁻¹)

k_a Adsorption rate constant (L·mg⁻¹·h⁻¹) kd Desorption rate constant (h⁻¹)

kext Universal external mass transfer coefficient (L·g⁻¹·h⁻¹) kF External mass transfer coefficient (cm·s⁻¹)

kF&S Mass transfer coefficient between the bulk liquid and the surface of the adsorbent (cm·h⁻¹)

kint Internal mass transfer rate constant (h⁻¹) KL Langmuir constant (L·mg⁻¹)

kM&W Mass transfer coefficient (cm·h⁻¹) kn nth order rate constant (gⁿ⁻¹∙mg^1-n∙min^-1) k_s External mass transfer coefficient (h⁻¹)

KW&M Intraparticle diffusion coefficient (mg·g·h^−1/2) m Adsorbent mass (g)

MB Methylene blue MB1 Methyl blue MG Malachite green MO1 Methyl orange

MOFs UiO-66-type metal-organic frameworks

m_s Mass of adsorbent per unit volume of solution (g·L⁻¹) MSE Mean sum of squares error

MWB Modified wheat bran

n Number of active sites occupied by an adsorbate ion/

molecule.

Nexp Number of the data points Npara Number of the model parameters

PA Polyamide

P-CSs H3PO4– modified corn stalks PE Polyethylene

PMMA Poly(methyl methacrylate) PR Phenol red

PS Polystyrene PVC Polyvinyl chloride

q∞ Equilibrium adsorption capacity at infinite time (mg·g⁻¹) qcal Calculated adsorption capacity (mg·L⁻¹)

q_e Adsorption capacity at equilibrium (mg·L⁻¹)

qet Equilibrium adsorption capacity in the pores of the ad- sorbent (mg·g⁻¹)

qexp Experimental adsorption capacity (mg·L⁻¹) qm Maximum adsorption capacity (mg·g⁻¹) qmax Langmuir constant (mg·g⁻¹)

qmean Average value of the experimental adsorption capacity (mg·L⁻¹)

qt Adsorbed amount of the adsorbate at timet(mg·L⁻¹) r Distant of the adsorbate and the surface of the particle R (cm)Rate coefficient (h⁻¹)

R² Correlation coefficient

r₀ Radius of the adsorbent particle (cm) RB5G Reactive blue dye 5G

RhB Rhodamine B

r_i Radial position of the adsorbate (mm) RSP Raw Spirulina platensis

SSE Residual sum of squares error

S Outer surface of adsorbent per unit volume (cm⁻¹) SMSP Sulphuric acid modified SP

SMT Sulfamethazine SMX Sulfamethoxazole

S_p External surface area per mass of adsorbent (cm²·g⁻¹) t Adsorption time (h)

tm Modeling time

t_e Adsorption equilibrium time TA Tannic acid

V Solution volume (L)

α nth order rate constant (h⁻¹) ε Particle porosity

ε_p Void fraction of the adsorbent (g·cm⁻³)

θ Occupied fraction of the active sites (0 ≤θ≤ 1) ρ Density of the adsorbent particle (g·cm⁻³)

ρp Apparent density of the adsorbent particle (g·mL⁻¹) χ² Chi-square

information of the adsorption rate, the performance of the adsorbent used, and the mass transfer mechanisms. Knowing the adsorption ki- netic is essential for the design of the adsorption systems. The adsorp- tion mass transfer kinetic includes three steps, as shown inFig. 1. The first step is the external diffusion. In this step, the adsorbate transfers through the liquid film around the adsorbent. The concentrations dif- ference between the bulk solution and the surface of the adsorbent are the driving force of the external diffusion. The second step is the in- ternal diffusion. The internal diffusion describes the diffusion of the adsorbate in the pores of the adsorbent. The third step is the adsorption of the adsorbate in the active sites of the adsorbent.

Various adsorption kinetic models, such as the pseudo-first-order (PFO) model (Lagergren, 1898), the pseudo-second-order (PSO) model (Ho et al., 1996), the mixed-order (MO) model (Guo and Wang, 2019a), the Ritchie’s equation (Ritchie, 1977), the Elovich model (Elovich and Larinov, 1962), and the phenomenological mass transfer models (Blanco et al., 2017;Marin et al., 2014;Scheufele et al., 2016;Hu et al., 2019; Suzaki et al., 2017) have been developed to describe the ad- sorption kinetic process. However, some problems are existed in the applications of these kinetic models. The first one is that the most ap- plied PFO and PSO models are empirical models and lack of specific physical meanings. We cannot investigate the mass transfer mechan- isms by these empirical kinetic models. Therefore, the physical mean- ings of the empirical kinetic models should be established. The second one is that the differential kinetic models, such as the phenomen- ological external/internal and adsorption in active sites models have specific physical meanings, but the solving methods are complicated.

Few studies have investigated the mass transfer processes by these models (Blanco et al., 2017;Marin et al., 2014;Scheufele et al., 2016;

Hu et al., 2019; Suzaki et al., 2017; Guo and Wang, 2019b). The complex solving methods hinder the applications of these models. The third one is that in some published papers, the kinetic models are ap- plied or solved in inappropriate manner. For example, the Frusawa and Smith (F&S) model can only be used for modeling the adsorption pro- cess that the isotherm is linear (Frusawa and Smith, 1973). However, it has been applied for the adsorption process that the isotherm is Lang- muir-type (Özer et al., 2005;Wang et al., 2008).

In addition, the linear regression method is the most widely applied method in the calculations of the model parameters, owing to its sim- plicity (Ma et al., 2018;Ersan et al., 2019;Shang et al., 2019;Delgado et al., 2019;Darwish et al., 2019;Sabarinathan et al., 2019). However, the linearization process changed the independent/dependent vari- ables. This process could introduce propagate errors to the in- dependent/dependent variables, and could cause inaccurate estima- tions of the parameters (El-Khaiary et al., 2010;Ho, 2006a,b;Kumar and Sivanesan, 2006;Guo and Wang, 2019c). The nonlinear method can provide consistent and accurate estimations for model parameters

(El-Khaiary et al., 2010). It is of great significance to provide the nonlinear solving method for the adsorption kinetic models.

Based on the above, the physical meanings, applications, and sol- ving methods of 16 adsorption kinetic models were thoroughly studied in this paper.

The objectives of this review paper were: (1) to describe the mathematical derivations and to analyze the physical meanings of the adsorption reaction kinetic models, the empirical models, the diffusion models, and the models for adsorption onto active sites; (2) to sum- marize the applications of the adsorption kinetic models based on lit- erature; (3) to review the model validity evaluation equations; and (4) to develop a convenient user interface (UI) for solving the kinetic models based on Excel.

2. Adsorption reaction models and empirical models 2.1. Pseudo-first-order (PFO) model

The PFO model was firstly proposed byLagergren (1898). The dif- ferential form of the PFO model is described by Eq. (1)(Lagergren, 1898):

dq =

dt^t k q1( _e q_t) (1)

Integrating Eq.(1)for the conditions ofq0= 0 yields:

=

q_t q_e(1 e ^{k t1}) (2)

The linearized form of the PFO model is presented as follows.

=

q q q k t

ln( _{e t}) ln _{e 1} (3)

Eq.(3)has been frequently used to fit the kinetics data and to cal- culate the parametersqeandk1, by plotting ln(qe-qt) vs.t(Ma et al., 2018; Ersan et al., 2019; Shang et al., 2019; Delgado et al., 2019;

Darwish et al., 2019; Sabarinathan et al., 2019; Khan et al., 2019;

Agarwal and Rani, 2017; Gamoudi and Srasra, 2019). However, the linearization process may cause inaccurate estimations of the para- meters (El-Khaiary et al., 2010; Ho, 2006a,b; Kumar and Sivanesan, 2006). The nonlinear method, which can provide accurate estimations for model parameters, is provided in the following section.

The PFO parameterq_eis the equilibrium adsorption amount esti- mated by the PFO model.Rodrigues and Silva (2016)reported that the PFO model was theoretically consistent and equaled to the linear driving force (LDF) model, when the adsorption isotherm could be re- presented by the linear model (Eq.(4)).

=

q_e KCe (4)

The PFO parameterk₁is frequently used to describe how fast the adsorption equilibrium is achieved (Plazinski et al., 2009). However, as Fig. 1.Adsorption mass transfer steps.

shown in Eq.(1), the adsorption rate dqt/dtis related to bothk1and (qe- qt). Small value ofk1and big value of (qe-qt) could be obtained when the adsorption is slow. Therefore, it is more precise to calculate the PFO rate by Eq.(5), instead of describing the adsorption rate by comparing the values ofk1.

=k q q

PFO rate 1( _{e t}) (5)

The PFO model used to be considered as empirical model for a long time. Azizian (2004)deduced the PFO model from the Langmuir ki- netics model (Eq. (6)) (Langmuir, 1918), and analyzed the physical meanings of this model.

d =

dt k Ca t(1 ) kd (6)

Ctcan be given by Eq.(7):

= =

C C C mq

t Vm

0 0 (7)

Substitution of Eq.(7)into Eq.(6)yields:

d =

dt k C mq

V (1 ) k

a m

d

0 (8)

Effort has been made to study the physical meanings and the ap- plications of the PFO model. Azizian (2004) concluded that when C0> >mqmθ/V, Eq.(8)was transformed to Eq.(9), which could be simplified to the PFO model.

d =

dt k Ca 0(1 ) kd (9)

Liu and Shen (2008)also concluded that the simplification of the Langmuir kinetics model to the PFO and PSO models wasC0-dependent.

Zhang (2019)suggested that the conditions for the PFO model were: (1) C0/βapproached to zero; (2) C0/βapproached to infinite; and (3)kaβ/

kdapproached to zero. Our previous research (Guo and Wang, 2019a) concluded that mqm/Vwas approximately a constant for a given ad- sorption process, therefore when a few active sites were occupied compared withC0,mqmθ/Vcould be neglect. Three conditions could satisfy this hypothesis, as presented inFig. 2.

The first one is that the value ofC₀is high.Table 1summarizes the applications of the PFO model. The adsorption of BB69 and AB25 onto peat, MO1 onto NiO NPs and CuO NPs, Cr(VI) onto iron electrodes,

benzene onto activated carbon, MB onto BBLS-CA, resorcinol onto CTAB/NaOH/flyash composites, Cu (II) onto RSP and SMSP could be better fitted by the PFO model than the PSO model at highC0(Darwish et al., 2019;Khan et al., 2019;Ho and Mckay, 1998;Stähelin et al., 2018; Agarwal and Rani, 2017; Sari et al., 2017; Gunasundari and Kumar, 2017).Siyal et al. (2019)investigated the adsorption of anionic surfactant onto fly ash based geopolymer, concluded that whenC0in- creased from 100 mg L⁻¹to 880 mg L⁻¹, the values of the correlation coefficient (R²) of the PFO model increased from 0.0035 to 0.937.Ma et al. (2018)reported that the values ofR²of the PFO model increased from 0.7556 to 0.9587, with the increase ofC0of Congo Red from 100 mg L⁻¹to 800 mg L⁻¹. In the reported of MV onto granular activated carbon, it was concluded that by increasing C₀ from 2.77 × 10⁻⁶ to7.49 × 10⁻⁶mol·L⁻¹, the values ofR²of the PFO model increased from 0.9704 to 0.9886 (Azizian et al., 2009).

The second one is that the adsorption process is in the initial stage.

Hu et al. (2018)found that whentapproached to zero, the PSO model approximated to the PFO model. Ho and Mckay (1999)studied the adsorption of Pb(II) onto peat, found that the PFO model adequately represented the kinetics data for the first 20 min (Table 1).

The third one is that the adsorbent material has a few active sites. In this sense, the external diffusion or the internal diffusion is the rate controlling step. The adsorption of metals ions and hydrophilic com- pounds onto microplastics could be represented by the PFO model (Guo et al., 2019a;Turner and Holmes, 2015). One possible reason is that microplastics are hydrophobic compounds, and the diffusion of hy- drophilic compounds to the surface of the hydrophobic microplastics is difficult. Therefore, the external/internal diffusion is the rate limiting step. However, the adsorption of hydrophobic organic compounds (such as lubrication oil and polybrominated diphenyl ethers) onto mi- croplastics could be better described by the PSO model (Hu et al., 2017;

Xu et al., 2019). The diffusion of hydrophobic compounds to the surface of microplastics is easier than the hydrophilic compounds. The ad- sorption onto active sites may be the rate controlling step. Therefore, the PFO model represents the condition that a few active sites exist in the adsorbent material or a few adsorbate ions/molecules can interact with the active sites.

As summarized inTable 1, the majority of literature modeled the whole adsorption process instead of modeling the initial stage of the adsorption. Therefore, the conditions for the PFO model are at high

Fig. 2.Physical meanings of the PFO and PSO models.

initial concentration of adsorbate and the adsorption is not controlled by the adsorption in active sites. In some cases, the PFO model could represent the external/internal diffusion.

2.2. Pseudo-second-order (PSO) model

The PSO model (Eq.(10)) was firstly applied to model the adsorp- tion of lead onto peat (Ho et al., 1996). Then the PSO model was widely adopted to describe the adsorption processes. Most published papers used the PSO model to predict the adsorption experimental data and to calculate the adsorption rate constants.

dq =

dt^t k q₂( _e q_t)² (10)

The integrated PSO model is described as following:

= + q q k t

q k t

t 1 e

e 2 2

2 (11)

In order to calculate the model parameters, the nonlinear PSO model is always transformed to the linear form (Eq.(12)).

= +

t q k q

t q 1

t 2 e2 e

(12) The linearization of the PSO model changes the weight ofq_t, and introduces propagated errors, which leads to the inaccurate calculations of the model parameters (El-Khaiary et al., 2010;Ho, 2006a,b;Kumar and Sivanesan, 2006;Guo and Wang, 2019c). The nonlinear method for solving the PSO model is provided in the following section.

Similar with the PFO rate constantk1, the PSO rate constantk2is also used to describe the rate of adsorption equilibrium (Plazinski et al., 2009). But the adsorption rate dqt/dtis related to bothk2and (qe-qt)². Thus, it is more precise to calculate the PSO rate by Eq.(13).

=k q q

PSO rate ₂( _{e t})² (13)

The physical meanings of the PSO model have been investigated.

Azizian (2004) found that when C0 was low, the Langmuir kinetics model (Eq.(8)) could be simplified to the PSO model.Liu (2008)sug- gested that the PSO model was related to the vacant active sites.Zhang (2019)reported that the conditions for the PFO model were C0/βap- proached to 1 andkaβ/kdapproached to infinite.Marczewski (2010a) found that the conditions for the PSO model were the equilibrium surface coverage fractionθ_eapproached to 1 and C₀/βapproached to 1.

Our previous study (Guo and Wang, 2019a) concluded that the PSO

model could represent three conditions (Fig. 2).

The first one is that the value ofC0is low.Sabarinathan et al. (2019) studied the adsorption of MB onto molecular polyoxometalate, found that the values ofR²of the PSO model decreased from 0.8979 to 0.5905 with the increase ofC₀from 140 mg L⁻¹to 300 mg L⁻¹. The appli- cations of the PSO model in literature are summarized inTable 2. As shown inTable 2, at low initial concentrations of adsorbate, the ad- sorption kinetics of BB69 and AB25 onto peat, MO₁onto NiO NPs and CuO NPs, Cr(VI) onto iron electrodes, benzene onto activated carbon, MB onto BBLS-CA, resorcinol onto CTAB/NaOH/flyash composites, Cu (II) onto RSP and SMSP are better modeled by the PSO model (Ho and Mckay, 1998;Darwish et al., 2019;Khan et al., 2019;Stähelin et al., 2018; Sari et al., 2017; Agarwal and Rani, 2017; Gunasundari and Kumar, 2017).

The second one is at the final stage of the adsorption process. As shown inTable 2, the PSO model could better represent the adsorption of Pb(II) onto peat for 20–90 min than the PFO model (Ho and Mckay, 1999). However, this phenomenon may extend the applications of the PSO model, because the adsorption kinetics data reported in most published papers are from the initial to the final (equilibrium) stage of the adsorption process (Tables 1 and 2).Simonin (2016)demonstrated that the kinetics data closed to or at equilibrium produced bias and unfairly promoted the PSO model. Simonin (2016) suggested using kinetics data for which the fractional uptake was lower than 85 %.

The third one is that the adsorbent material is abundant with active sites. In the reported of Pb(Ⅱ) onto hydrochar and modified hydrochar, it was observed that for modified hydrochar, the value ofR²of the PSO model was 0.990, while for hydrochar, this value was 0.945 (Xia et al., 2019). As listed inTable 2, the adsorption of pollutants on modified materials, such as Al-modified biochar, modified clay, modified titanate nanotubes, modified pineapple crown leaf, modified chitosan, surfac- tant-modified montmorillonite, modified hydrochar, H3PO4modified corn stalks, modified mesoporous silica, modified bentonite, cobalt ferrite silica magnetic nanocomposite, modified Algerian geomaterial, modified wheat bran, modified hydrogel, modified bacterial cellulose membrane, magnetic amidoxime functionalized chitosan and MOFs could be better modeled by the PSO model (Yin et al., 2018;Gamoudi and Srasra, 2019;Wang et al., 2018b;Gogoi et al., 2018;Lou et al., 2018;Li et al., 2018,2019;Xia et al., 2019;Islam et al., 2019;Tang et al., 2019;Gao et al., 2019;Huang et al., 2017;Amiri et al., 2017;

Bentaleb et al., 2017;Zhang et al., 2017; Zhuang et al., 2017,2018, 2019;Fan et al., 2017; Cheng et al., 2019). In general, the modified materials are abundant with active sites. Therefore, the adsorption Table 1

Applications of the PFO model.

Adsorbent Adsorbate tm/t te C0(mg·L⁻¹) Reference

Peat BB69 0–250 min/250 min 250 min^a 500 Ho and Mckay, 1998

Peat AB25 0–250 min/250 min – 200

NiO NPs MO1 0–1440 min/1440 min 540 min 200 Darwish et al., 2019

CuO NPs MO1 0–1440 min/1440 min 540 min 200

Iron electrodes Cr(VI) 0–120 min/120 min – 100 Khan et al., 2019

Activated carbon Benzene 0–120 min/120 min 70 min 110 Stähelin et al., 2018

BBLS-CA MB 0–360 min/360 min 90 min 500 Sari et al., 2017

CTAB/NaOH/flyash Resorcinol 0–24 h/24 h 3 h^a 200 Agarwal and Rani, 2017

RSP Cu(II) 0–150 min/150 min 110 min^a 300 Gunasundari and Kumar, 2017

SMSP Cu (II) 0–150 min/150 min 60 min^a 500

Peat Pb(II) 0–20 min/90 min 90 min^a 504 Ho and Mckay, 1999

Peat Pb(II) 0–20 min/90 min 30 min^a 209

PA SMT 0–24 h/24 h 16 h 2 Guo et al., 2019a

PE SMT 0–24 h/24 h 16 h

PVC SMT 0–24 h/24 h 16 h

PE Ag(Ⅰ) 0–168 h/168 h 30 h^a 5 × 10^–3 Turner and Holmes, 2015

PE Cu(Ⅱ) 0–168 h/168 h 30 h^a

PE Ni(Ⅱ) 0–168 h/168 h 30 h^a

-:Not mentioned.

a Obtained from the figures in references.

kinetics is dominated by the adsorption onto active site.

2.3. Mixed-order (MO) model

The mixed-order (MO) model has the following form (Guo and Wang, 2019a):

= +

dq

dt^t k q₁^’( _e q_t) k q₂^’( _e q_t)² (14) The PFO and PSO rate of the MO model can be calculated by Eqs.

(15) and (16).

=k q q

PFO rate' ₁^’( _{e t}) (15)

=k q q

PSO rate' ₂^’( _{e t})² (16)

In most cases, the PFO rate and PSO rate describe the diffusion step

and the step of adsorption on active sites, respectively (Guo and Wang, 2019a). In addition, the MO model represents the overall adsorption process. The following conditions satisfy the assumption of the MO model: (1) arbitrary stage of the adsorption; (2) the rate controlling step is the diffusion or the adsorption; and (3) arbitrary initial adsorbate concentration in solution (Guo and Wang, 2019a).

Eq.(14)is similar in the form to the hybrid-order’ (HO) model (Eq.

(17)) provided inLiu and Shen (2008), and the mixed 1, 2-order rate equation (MOE) (Eq.(18)) derived byMarczewski (2010b).

= +

d

dt^t kL1(e t) kL2( e t)2

(17)

= +

dF

dt k1a(1 FL) k a2a eq(1 FL)2

(18) Wherek_L1,k_L2,k_1a,a_eq, andk_2aare coefficients.

As described in Eq.(17), the HO model is the relationship between Table 2

Applications of the PSO model.

Adsorbent Adsorbate tm/t te C0(mg·L⁻¹) Reference

Peat BB69 0–250 min/250 min 30 min^a 50 Ho and Mckay, 1998

Peat AB25 0–250 min/250 min – 20

NiO NPs MO1 0–1440 min/1440 min 540 min 50 Darwish et al., 2019

CuO NPs MO1 0–1440 min/1440 min 540 min 50

Iron electrodes Cr(VI) 0–120 min/120 min – 60 Khan et al., 2019

Activated carbon Benzene 0–120 min/120 min 70 min 69 Stähelin et al., 2018

BBLS-CA MB 0–360 min/360 min 90 min 100 Sari et al., 2017

CTAB/NaOH/flyash Resorcinol 0–24 h/24 h 3 h^a 50 Agarwal and Rani, 2017

RSP Cu(II) 0–150 min/150 min 100 min^a 50 Gunasundari and Kumar, 2017

SMSP Cu(II) 0–150 min/150 min 50 min^a 50

Peat Pb(II) 20–90 min/90 min 90 min^a 504 Ho and Mckay, 1999

Peat Pb(II) 20–90 min/90 min 30 min^a 209

CPNM AB113 0–120 h/120 h 120 h 100 Lou et al., 2019

PE lubrication oil 0–48 h/48 h 12 h^a – Hu et al., 2017

PS lubrication oil 0–48 h/48 h 12 h^a –

PE BDE-99 0–24 h/24 h 30 min 0.5 Xu et al., 2019

Al-modified biochar Nitrate 0–24 h/24 h 24 h 50 Yin et al., 2018

Al-modified biochar Phosphate 0–24 h/24 h 24 h 50

Modified clay MO1 0–4 h/4 h 2 h 200 Gamoudi and Srasra, 2019.

Modified clay IC 0–4 h/4 h 1 h 200

Modified clay PR 0–4 h/4 h 1 h 200

Modified titanate nanotubes Ag(Ⅰ) 0–150 min/150 min 20 min 122 Wang et al., 2018b

(M) PCL Cr(VI) 0–180 min/180 min 120 min^a 20 and 30 Gogoi et al., 2018

Cr(III) 0–180 min/180 min 120 min^a 20 and 30

Modified chitosan Re(VII) 0–24 h – 20 Lou et al., 2018

FTMA-MT Phenol 0–21 h 7 h – Li et al., 2018

Modified hydrochar Pb(II) 0–360 min/360 min 240 min^a 10 Xia et al., 2019

Modified-chitosan Cr(VI) 0–150 min/150 min 90 min 40–100 Islam et al., 2019

P-CSs MB 0–300 min/300 min 140 min 30–90 Tang et al., 2019

Modified mesoporous silica Steroid estrogens 0–90 min/90 min 30 min 2 Gao et al., 2019

Maleylated modified hydrochar MB 0–300 min/300 min 150 min 600–1000 Li et al., 2019

Cd(II) 0–300 min/300 min 30 min 40–90

Modified bentonite RhB 0–70 min/70 min 40 min 100 Huang et al., 2017

Modified bentonite AR1 0–70 min/70 min 90 min 100

Cobalt ferrite silica magnetic nanocomposite MG 0–70 min/70 min 40 min 100 Amiri et al., 2017

Modified Algerian geomaterial 2,4-dichlorophenol 0–200 min/200 min 60 min 81.5–244.5 Bentaleb et al., 2017

MWB Anionic azo dyes 0–12 h/12 h 11 h 50 Zhang et al., 2017

Modified hydrogel Tetracycline 0–24 h/24 h 24 h 300 Zhuang et al., 2017

Modified hydrogel CIP 0–24 h/24 h 24 h 300

Modified chitosan/CoFe2O4 Cu(II) 0–90 min/90 min 50 min 200 Fan et al., 2017

Pb(II) 0–90 min/90 min 50 min 200

Modified Bacterial cellulose membrane Sr(II) 0–10 h/10 h 10 h 100 Cheng et al., 2019

Magnetic amidoxime functionalized chitosan U(VI) 0–180 min/180 min 180 min 480 Zhuang et al., 2018

MOFs DFC 0–500 min/500 min 200 min 80 Zhuang et al., 2019

Cerium modified chitosan As(III) 0–540 min/540 min 300 min 10 and 30 Zhang et al., 2016

E33 Phosphate 0–294 h/294 h 294 h^a 140 Lalley et al., 2016

E33/Mn Phosphate 0–294 h/294 h 294 h^a 140

E33/Ag I Phosphate 0–294 h/294 h 294 h^a 140

E33/Ag II Phosphate 0–294 h/294 h 294 h^a 140

NiO NPs MO1 0–1440 min/1440 min 540 min 200 Darwish et al., 2019

BBLS-CA MB 0–360 min/360 min 90 min 500 Sari et al., 2017

-: Not mentioned.

a Obtained from the figures in references.

θtandt, which makes it difficult to directly apply the HO model to the kinetics experimental data. The MOE was simplified according to some assumptions, such as k2aaeq/(k1a+k2aaeq) < 1 (Marczewski, 2010a), leading to the lower precision in the estimations of the model para- meters. Therefore, the MO model presented byGuo and Wang (2019a) is recommended to model the kinetics processes.

The MO model has been successfully used to describe the adsorption of SMT onto MOFs, and SMX, SMT, and CEP-C onto microplastics (Guo and Wang, 2019b;Zhuang et al., 2020). The MO model is a differential equation, which can be solved by the Runge-Kutta method. Although the solving method of the MO model based on MATLAB is provided in our previous study (Guo and Wang, 2019a), the application of the program in MATLAB is still complicated. Therefore, we developed an UI in Excel software, which is easily to be used for solving the MO model.

The details are shown in the following section.

2.4. Elovich model

The basic assumptions of the Elovich model were (1) the activation energy increased with adsorption time and (2) the surface of the ad- sorbent was heterogeneous. The Elovich model is an empirical model without definite physical meanings. It is commonly used to model the chemisorption of gas onto solid. The Elovich model has been described by Eq.(19)(Elovich and Larinov, 1962):

dq =

dt^t ae ^bqt (19)

Integrating Eq.(19)for the condition ofq0= 0 yields:

= +

q_t b1ln(1 abt)

(20) Eq.(20)is a nonlinear equation, which can be solved by the non- linear least square regression method (plotq_tversus t) or by the linear method (plotqtversus ln(1+abt)). However, the nonlinear method is more complex than the linear method. And the linear method needs to give the appropriate initial values ofab, which is difficult for the re- searchers (Ho, 2006a,b).Chien and Clayton (1980)simplified Eq.(20) with the assumption ofabt> > 1:

= = +

q b abt

b ab b t 1ln( ) 1ln( ) 1ln( )

t (21)

Plotting qt versus ln(t) can solve Eq. (21) by the linear regression method. In the past decade, the Elovich model has been used to re- present the adsorption of liquid-solid systems. The applications of the Elovich model are summarized in Table 3. Eq.(21) is the most fre- quently applied form of the Elovich model, which has been successfully used for modeling the adsorption of metals ions and organic pollutants on adsorbents. Only a few published papers adopted Eq.(20)for the estimations of the Elovich parameters (Lin et al., 2018; Brito et al., 2018; Kalhor et al., 2018). However, the assumption for Eq. (21), namely, abt> > 1, may reduce the accuracy of the Elovich model.

Thus, although the application of Eq.(21)is simple and convenient, it is not recommend using Eq.(21)to model the kinetics data.

In the following section, we provided the nonlinear method for Eq.

(20)based on Excel UI. In addition, Eq.(22)is recommended to cal- culate the Elovich rate:

=ae

Elovich rate ^bqt (22)

2.5. Ritchie’s equation and pseudo-nth-order (PNO) model

Ritchie’s equation was firstly proposed for modeling the adsorption kinetic data of gases on solids (Ritchie, 1977). The physical meaning of Ritchie’s equation is that the adsorption is dominated by the adsorption in active sites. And one adsorbate ion/molecule can occupy n active sites. The desorption process is not considered in this model. Ritchie’s equation is presented by Eq.(23)(Ritchie, 1977):

d =

dt (1 )ⁿ (23)

After integration with the initial condition of q₀ = 0, Ritchie’s equation becomes:

For n = 1,

=1 e ^t (24)

For n ≠ 1,

=1 [1+(n 1) ]t 1n

1 (25)

θcan be described by the ratio ofqtandq∞(q∞can be obtained from the adsorption isotherm model (q∞=f(Ce)), Eqs.(24) and(25) be- comes:

Table 3

Applications of the Elovich model.

Adsorbent Adsorbate Elovich model Ref.

Alfisol Tetracycline Eq.(21) Bao et al., 2010

Ultisol Tetracycline Eq.(21)

Contaminant barrier minerals Pb(Ⅱ) Eq.(21) Inyang et al., 2016

Contaminant barrier minerals Cd(Ⅱ) Eq.(21)

β-cyclodextrin-based adsorbent Cu(Ⅱ) Eq.(21) Huang et al., 2012

Synthesized TiO2nanoparticles Hg(II) Eq.(21) Dou et al., 2011

TiO2/montmorillonite Hg(II) Eq.(21)

Surfactant-Modified Natural Zeolites Benzene Eq.(21) Seifi et al., 2011

Surfactant-Modified Natural Zeolites Toluene Eq.(21)

Surfactant-Modified Natural Zeolites Ethylbenzene Eq.(21)

Surfactant-Modified Natural Zeolites Xylenes Eq.(21)

Desert soils Inositol hexaphosphate Eq.(21) Fuentes et al., 2014

Gl-PZSNTs-Fe3O4 MB Eq.(21) Wang et al., 2019

Honeycomb-like porous activated carbon Cu(II) Eq.(21) Mondal and Majumder, 2019

Modified eggshell membrane MO1 Eq.(21) Candido et al., 2019

Modified eggshell membrane MB1 Eq.(21)

Anodized iron oxide nanoflakes Phosphate Eq.(21) Afridi et al., 2019

Iron oxide-gelatin nanoadsorbent DY12 Eq.(21) Alinejad-Mir et al., 2018

Cement kiln dust BB69 Eq.(21) Magdy and Altaher, 2018

Modified rice straw Cr(VI) Eq.(20) Lin et al., 2018

Carbons Dianix®royal blue CC Eq.(20) Brito et al., 2018

Amino functionalized silica nano hollow sphere Imidacloprid pesticide Eq.(20) Kalhor et al., 2018

Raw pomegranate peel Cu(II) Eq.(21) Ben-Ali et al., 2017

Alfalfa white proteins concentrate Polyphenols Eq.(21) Frdaous et al., 2017

For n = 1,

=

q_t q (1 e ^t) (26)

For n ≠ 1,

= +

q_t q q [1 (n 1) ]t 1n

1 (27)

When n = 2, rearrangement of Eq.(27)yields:

= +

q q t

t

t 1 (28)

Eq.(28)is the Ritchie’s second-order (RSO) equation. Note that the RSO equation is different with the nonlinear form of the PSO model (Eq.

(11)). The applications of Ritchie’s equation are summarized inTable 4.

The RSO model is frequently used in the adsorption of gas or liquid onto solid.Cheung et al. (2001)studied the adsorption kinetics of cadmium onto bone char by using the RSO model.Fashi et al. (2018)applied the RSO model to the adsorption data of CO2 onto piperazine-modified activated alumina. Wang et al. (2018) demonstrated that Ritchie’s equation could best fit the kinetics data of MB onto calcium alginate, ball-milled biochar, and their composites.

Özer (2007) developed the pseudo-nth-order (PNO) model (Eq.

(29)) for modeling the adsorption kinetics data of Pb(II) on sulphuric acid-treated wheat bran.

dq =

dt^t k qn( e qt)n

(29) For n = 1, the PNO model equals to the PFO model. After in- tegration with the initial condition ofq0= 0 and n ≠ 1, the PNO model becomes:

= +

q q k t q

( _{e t})1-n ( n 1) n e1 n (30)

Rearrangement of Eq.(30)yields:

= +

q q

1 1 k t

(1 (n 1) )

t e

q n1/(n 1)

en 1 (31)

The PNO model mainly represents the adsorption processes that the order factor is between 1 and 2 or larger than 2. Unfortunately, the PNO model is an empirical equation without specific physical meanings. The applications of the PNO model are summarized inTable 4.Caroni et al.

(2009)indicated that values of n of tetracycline adsorption on chitosan particles increased with the increase of the initial concentrations of tetracycline.Liu et al. (2019)reported that the value of n was 2.276 at low initial concentration of 17β-Estradiol (0.2 mg L⁻¹), while it de- creased to 0.979 at highC₀(6 mg L⁻¹).Leyva-Ramos et al. (2010) suggested that the values of n were not correlated with the experi- mental conditions, such as pH,C0, impeller speeds, and so on.

The nonlinear form of the PNO model cannot be written as the linear form. Therefore, the solution of the PNO model is more complex than the PFO and PSO models.Özer (2007)solved the PNO model using the nonlinear regression method by Statistica 6.0 software.Tseng et al.

(2014) used commercial SigmaPlot 11 software to solve the PNO model. In the following section, a convenient method for solving the Ritchie’s equation and the PNO model was provided. Eq.(32)is pro- posed for the calculation of the PNO rate:

=k q q

PNOrate n( e t)n (32)

3. Diffusional models 3.1. External diffusion models

The external diffusion models assume that the diffusion of adsorbate in a bounding liquid film around the adsorbent is the slowest step.

Table 4

Applications of the Ritchie’s equation and the PNO model.

Adsorbent Adsorbate C0(mg L⁻¹) Model n Ref.

Bone char Cd(Ⅱ) 224–560 RSO 2 Cheung et al., 2001

Piperazine-modified activated Alumina CO2 – RSO 2 Fashi et al., 2018

Calcium alginate MB 50 Ritchie’s equation – Wang et al., 2018a

Ball-milled biochar MB 50 Ritchie’s equation –

Ball-milled biochar encapsulated in calcium-alginate beads MB 50 Ritchie’s equation –

Phthalonitrile compound Alkanes – RSO 2 Günay et al., 2018

Phthalonitrile compound Alcohols – RSO 2

Phthalonitrile compound Chlorinated – RSO 2

Macroporous ion-exchange resins Al(Ⅲ) 77.5 PNO 1.17–1.47 Nekouei et al., 2019

Macroporous ion-exchange resins Cu(Ⅱ) 7653 PNO 0.94–1.37

Macroporous ion-exchange resins Zn(Ⅱ) 425.5 PNO 0.38–1.38

Macroporous ion-exchange resins Ni(Ⅱ) 85.5 PNO 1.25–1.73

Macroporous ion-exchange resins Pb(Ⅱ) 183 PNO 0.998–1.36

Pure açaí biochar MB 50 PNO 1.65 Pessôa et al., 2019

NaOH-açaí biochar MB 50 PNO 2.26

Sulphuric acid-treated wheat bran Pb(II) 100 PNO 1.717–1.929 Özer, 2007

Molecularly imprinted polymers Quinoline 20 PNO 0.98–2.85 Saavedra et al., 2018

Chitosan spheres MO1 – PNO 2–2.5 Morais et al., 2008

Chitosan particles Tetracycline 15^a PNO 4.5^a Caroni et al., 2009

Chitosan particles Tetracycline 55^a PNO 16^a

Montmorillonite-carbon hybrids 17β-Estradiol 6 PNO 0.979 Liu et al., 2019

Montmorillonite-carbon hybrids 17β-Estradiol 0.2 PNO 2.276

Bone char fluoride 5.44 PNO 1.3 Leyva-Ramos et al., 2010

Bone char fluoride 8.35 PNO 2.0

Carbons prepared from betel trunk Phenol – PNO 1.50 Tseng et al., 2014

Carbons prepared from betel trunk MB – PNO 3.51

Carbons prepared from betel trunk Basic brown 1 – PNO 3.06

Carbons prepared from betel trunk Acid blue 74 (AB74) – PNO 1.77

Carbons prepared from betel trunk 2,4-dichloropenol – PNO 1.63

Carbons prepared from betel trunk 4-chloropenol – PNO 1.28

Carbons prepared from betel trunk 4-Cresol – PNO 1.40

Resin Iron ions 3000 PNO 1.92 Izadi et al., 2017

-: Not mentioned.

a Obtained from the figures in references.

Table 5

Applications of the diffusional models.

Diffusional models Models name Adsorbent Adsorbate Reference

External diffusion models Boyd’s external diffusion equation Microporous activated carbon Metronidazole Ahmed and Theydan, 2013 F&S model Inactive CMC immobilized Aspergillus fumigatus beads Azo dye Wang et al., 2008

F&S model Enteromorpha prolifera AR337 Özer et al., 2005

F&S model Enteromorpha prolifera AB324

F&S model Natural untreated clay BY2 Öztürk and Malkoc, 2014

F&S model Bamboo Charcoal Organic materials Fu et al., 2012

F&S model Bamboo Charcoal Organic matter Fu et al., 2015

F&S model Wollastonite Cu(Ⅱ) Panday et al., 1986

M&W model Inactive CMC immobilized Aspergillus fumigatus beads Azo dye Wang et al., 2008

M&W model Enteromorpha prolifera AR337 Özer et al., 2005

M&W model Enteromorpha prolifera AB324

M&W model Bone char Cu (Ⅱ) Choy et al., 2004

M&W model Bone char Cd(Ⅱ)

M&W model Bone char Zn(Ⅱ)

M&W model Clinoptilolite rich mineral Cr(VI) Erdoğan and Ulku, 2012

M&W model Bacteria loaded clinoptilolite rich mineral Cr(VI)

M&W model Spent coffee ground Cu (Ⅱ) Dávila-Guzmán et al., 2013

M&W model Zeolite Cu (Ⅱ) Šljivić et al., 2011

M&W model Clay Cu (Ⅱ)

M&W model Diatomite Cu (Ⅱ)

EMT model Dowex Optipore SD-2 RB5G Blanco et al., 2017

EMT model Dowex Optipore SD-2 RB5G Marin et al., 2014

EMT model SB biomass RB5G Scheufele et al., 2016

EMT model Sargassum horneri Cs(Ⅰ) Hu et al., 2019

EMT model Sargassum horneri Sr(II)

EMT model Fixed-bed columns Cu(II) Suzaki et al., 2017

EMT model Fixed-bed columns Ni(II)

EMT model Fixed-bed columns Zn(II)

EMT model Microplastics SMX Guo et al., 2019b

EMT model Fixed-bed CIP Sausen et al., 2018

Internal diffusion models Boyd’s intraparticle diffusion model Mansonia wood sawdust Pb(II) Ofomaja, 2010

Boyd’s intraparticle diffusion model Dolomite Cr(VI) Albadarin et al., 2012

Boyd’s intraparticle diffusion model PMMA Phenol Al-Muhtaseb et al., 2011

Boyd’s intraparticle diffusion model Starchy adsorbents Water Okewale et al., 2013

Boyd’s intraparticle diffusion model Wood apple shell MG Sartape et al., 2017

Boyd’s intraparticle diffusion model Pre-treated bentonite clay Ag(Ⅰ) Cantuaria et al., 2016 Boyd’s intraparticle diffusion model Mesoporous activated carbon Cd(Ⅱ) Tan et al., 2016 Boyd’s intraparticle diffusion model Verde-lodo bentonite Ag(Ⅰ) Freitas et al., 2017 Boyd’s intraparticle diffusion model Verde-lodo bentonite Cu(II)

Boyd’s intraparticle diffusion model Amine-functionalized ordered mesoporous alumina MB Gan et al., 2015

Boyd’s intraparticle diffusion model Sand Neutral Red Rauf et al., 2007

W&M model PMMA Phenol Al-Muhtaseb et al., 2011

W&M model Activated carbons MB Wu et al., 2009

W&M model Activated carbons TA

W&M model Activated carbons Phenol

W&M model Activated carbons 4-CP

W&M model Caulerpa lentillifera Cu(II) Apiratikul and Pavasant, 2008

W&M model Caulerpa lentillifera Cd(II)

W&M model Caulerpa lentillifera Pb(II)

W&M model Activated carbon Dibenzothiophene Danmaliki and Saleh, 2016

W&M model Activated carbon MB Archin et al., 2019

W&M model Activated carbon AB 25

IMT model Dowex Optipore SD-2 RB5G Blanco et al., 2017

IMT model Dowex Optipore SD-2 RB5G Marin et al., 2014

IMT model SB biomass RB5G Scheufele et al., 2016

IMT model Sargassum horneri Cs(Ⅰ) Hu et al., 2019

IMT model Sargassum horneri Sr(II)

IMT model Fixed-bed columns Cu(II) Suzaki et al., 2017

IMT model Fixed-bed columns Ni(II)

IMT model Fixed-bed columns Zn(II)

IMT model Microplastics SMX Guo et al., 2019b

PVSD model PVSD model Bentonite clay MG Souza et al., 2017a

PVSD model Bentonite clay MG Souza et al., 2017b

PVSD model Bentonite clay CV Souza et al., 2019

PVSD model Granular activated carbon Pyridine Ocampo-Perez et al., 2010

PVSD model Granular activated carbon Metronidazole Díaz-Blancas et al., 2018

PVSD model Activated carbon pellets Acetaminophen Ocampo-Perez et al., 2017

PVSD model Biochar Ibuprofen Ocampo-Perez et al., 2019