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Adsorption kinetic models: Physical meanings, applications, and solving methods
Journal Pre-proof Adsorption kinetic models: Physical meanings, applications, and solving methods Jianlong Wang, Xuan Guo
PII:
S0304-3894(20)30144-8
DOI:
https://doi.org/10.1016/j.jhazmat.2020.122156
Reference:
HAZMAT 122156
To appear in:
Journal of Hazardous Materials
Received Date:
20 November 2019
Revised Date:
20 January 2020
Accepted Date:
20 January 2020
Please cite this article as: Wang J, Guo X, Adsorption kinetic models: Physical meanings, applications, and solving methods, Journal of Hazardous Materials (2020), doi: https://doi.org/10.1016/j.jhazmat.2020.122156
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Adsorption kinetic models: Physical meanings, applications, and solving methods Jianlong Wang 1,2 *, Xuan Guo 1 1 Collaborative Innovation Center for Advanced Nuclear Energy Technology, INET, Tsinghua University, Beijing 100084, P. R. China
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2 Beijing Key Laboratory of Radioactive Waste Treatment, Tsinghua University, Beijing 100084, P. R. China
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∗ Corresponding author
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Full post address:
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Tel.: +86 10 62784843
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Energy Science Building, INET, Tsinghua University, Beijing 100084, P. R. China
Fax: +86 10 62771150
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E-mail address: [email protected]
∗ Corresponding author. Tel.: +86 10 62784843; fax: +86 10 62771150. E-mail address: [email protected] 1
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Graphical abstract
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Highlights
Adsorption empirical kinetic models and mass transfer kinetic models were
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reviewed.
The physical meanings and applications of 16 adsorption kinetic models were analyzed.
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The model validity evaluation equations were summarized based on literature. A user interface for solving kinetic models was developed based on Excel software.
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Abstract 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
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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
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physical meanings, applications, as well as the solving methods of 16 adsorption kinetic
models. Firstly, the mathematical derivations, physical meanings and applications of
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the adsorption reaction models, the empirical models, the diffusion models, and the
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models for adsorption onto active sites were analyzed and discussed in detail. Secondly, the model validity evaluation equations were summarized based on literature. Thirdly,
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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.
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Key words: Adsorption; kinetic model; physical meaning; solving method
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Contents Abstract .................................................................................................................. 2 1. Introduction ........................................................................................................ 9 2. Adsorption reaction models and empirical models .......................................... 12 2.1 Pseudo-first-order (PFO) model ............................................................12 2.2 Pseudo-second-order (PSO) model ........................................................17 2.3 Mixed-order (MO) model ......................................................................22 2.5 Elovich model ........................................................................................24
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2.4 Ritchie’s equation and pseudo-nth-order (PNO) model .........................26
3. Diffusional models ........................................................................................... 30 3.1 External diffusion models ......................................................................30
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3.2 Internal diffusion models .......................................................................37 3.3 Pore volume and surface diffusion (PVSD) model................................41
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4. Models for adsorption onto active sites (AAS)................................................ 42 4.1 Langmuir kinetics model .......................................................................42
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4.2 Phenomenological AAS model ..............................................................43 5. Model validity evaluation ................................................................................ 44
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6. Solving methods for adsorption kinetic models............................................... 46 7. Concluding remarks and perspectives ............................................................. 50 Acknowledgments................................................................................................52
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References ............................................................................................................ 53
4
Nomenclature Modified pineapple crown leaf
4-CP
4-chlorophenol
a
Initial adsorption rate constant of the Elovich model (mg g-1 h-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)
Cet Cf CIP
-p re
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Equilibrium adsorbate concentration (mg·L-1) Equilibrium adsorbate concentration at the surface of the adsorbent (mg·L-1) Adsorbate concentration in the liquid film (mg·L-1) Ciprofloxacin
Chitosan coated polyacrylonitrile nanofibrous mat
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CPNM
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Ce
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(M) PCL
Ct
Adsorbate concentration at time t (mg·L-1)
Ctr
Adsorbate concentration at distant r, (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
5
Intraparticle diffusion coefficient (mm2·h-1)
DFC
Diclofenac
Dl
Diffusional constant
Dp
Effective pore volume diffusion coefficient (cm2·s-1)
Ds
Surface diffusion coefficient (cm2·s-1)
DY12
Direct yellow 12
E33
Bayoxide®E33
FTMA-MT
Surfactant-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)
ka
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&S kint KL
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External mass transfer coefficient (cm·s-1) Mass transfer coefficient between the bulk liquid and the surface of the adsorbent (cm·h-1) Internal mass transfer rate constant (h-1) Langmuir constant (L·mg-1) Mass transfer coefficient (cm·h-1)
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kM&W
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kF
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Def
kn
nth order rate constant (gn-1∙mg1-n∙min-1)
ks
External mass transfer coefficient (h-1)
KW&M
Intraparticle diffusion coefficient (mg·g·h-1/2)
m
Adsorbent mass (g)
MB
Methylene blue
6
Methyl blue
MG
Malachite green
MO1
Methyl orange
MOFs
UiO-66-type metal-organic frameworks
ms
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)
qe qet qexp
-p re
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Calculated adsorption capacity (mg·L-1) Adsorption capacity at equilibrium (mg·L-1) Equilibrium adsorption capacity in the pores of the adsorbent (mg·g-1) Experimental adsorption capacity (mg·L-1) Maximum adsorption capacity (mg·g-1)
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qm
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qcal
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MB1
qmax
Langmuir constant (mg·g-1)
qmean
Average value of the experimental adsorption capacity (mg·L-1)
qt
Adsorbed amount of the adsorbate at time t (mg·L-1)
r
Distant of the adsorbate and the surface of the particle (cm)
R
Rate coefficient (h-1)
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r0
Radius of the adsorbent particle (cm)
RB5G
Reactive blue dye 5G
RhB
Rhodamine B
ri
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
Sp
External surface area per mass of adsorbent (cm2·g-1)
t
Adsorption time (h)
tm
Modeling time
te
Adsorption equilibrium time
TA
Tannic acid
V
Solution volume (L)
α
nth order rate constant (h-1)
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εp θ ρ
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Particle porosity
Void fraction of the adsorbent (g·cm-3) Occupied fraction of the active sites (0 ≤ θ ≤ 1) Density of the adsorbent particle (g·cm-3) Apparent density of the adsorbent particle (g·mL-1)
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ρp χ2
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Correlation coefficient
-p
R2
Chi-square
8
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., 2018). The adsorption kinetic study provides
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information of the adsorption rate, the performance of the adsorbent used, and the mass transfer mechanisms. Knowing the adsorption kinetic is essential for the design of the adsorption systems. The adsorption mass transfer kinetic includes three steps, as shown
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in Fig. 1. The first step is the external diffusion. In this step, the adsorbate transfers
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through the liquid film around the adsorbent. The concentrations difference between
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the bulk solution and the surface of the adsorbent are the driving force of the external diffusion. The second step is the internal diffusion. The internal diffusion describes the
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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. Adsorbate
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Liquid film
Mass transfer steps
1 1: External diffusion
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2: Internal diffusion
Pores Active sites
3: Adsorption on active sites Adsorbent
Fig. 1 Adsorption mass transfer steps. 9
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 mixedorder (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 adsorption kinetic process.
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However, some problems are existed in the applications of these kinetic models. The first one is that the most applied PFO and PSO models are empirical models and lack
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of specific physical meanings. We cannot investigate the mass transfer mechanisms by these empirical kinetic models. Therefore, the physical meanings of the empirical
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kinetic models should be established. The second one is that the differential kinetic
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models, such as the phenomenological external/internal and adsorption in active sites models have specific physical meanings, but the solving methods are complicated. Few
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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
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models. The third one is that in some published papers, the kinetic models are applied or solved in inappropriate manner. For example, the Frusawa and Smith (F&S) model can only be used for modeling the adsorption process that the isotherm is linear (Frusawa and Smith, 1973). However, it has been applied for the adsorption process that the isotherm is Langmuir-type (Özer et al., 2005; Wang et al., 2008).
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In addition, the linear regression method is the most widely applied method in the calculations of the model parameters, owing to its simplicity (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 variables. This process could introduce propagate errors to the independent/dependent variables, and could cause inaccurate estimations of the parameters (El-Khaiary et al.,
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2010; Ho, 2006; Kumar and Sivanesan, 2006; Guo and Wang, 2019c). The nonlinear
method can provide consistent and accurate estimations for model parameters (El-
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Khaiary et al., 2010). It is of great significance to provide the nonlinear solving method for the adsorption kinetic models.
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Based on the above, the physical meanings, applications, and solving methods of
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16 adsorption kinetic models were thoroughly studied in this paper. The objectives of this review paper were: (1) to describe the mathematical
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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 summarize the applications of the adsorption kinetic models based
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on literature; (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.
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2. Adsorption reaction models and empirical models 2.1 Pseudo-first-order (PFO) model The PFO model was firstly proposed by Lagergren (1898). The differential form of the PFO model is described by Eq. (1) (Lagergren, 1898): 𝑑𝑞𝑡 𝑑𝑡
= 𝑘1 (𝑞𝑒 − 𝑞𝑡 )
(1)
Integrating Eq. (1) for the conditions of q0=0 yields:
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𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝑘1 𝑡 )
(2)
The linearized form of the PFO model is presented as follows. ln(𝑞𝑒 − 𝑞𝑡 ) = ln 𝑞𝑒 − 𝑘1 𝑡
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(3)
Eq. (3) has been frequently used to fit the kinetics data and to calculate the
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parameters qe and k1, by plotting ln(qe-qt) vs. t (Ma et al., 2018; Ersan et al., 2019;
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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
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linearization process may cause inaccurate estimations of the parameters (El-Khaiary et al., 2010; Ho, 2006; Kumar and Sivanesan, 2006). The nonlinear method, which can provide accurate estimations for model parameters, is provided in the following section.
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The PFO parameter qe is the equilibrium adsorption amount estimated 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 represented by the linear model (Eq. (4)). 𝑞𝑒 = 𝐾𝐶𝑒
(4)
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The PFO parameter k1 is frequently used to describe how fast the adsorption equilibrium is achieved (Plazinski et al., 2009). However, as shown in Eq. (1), the adsorption rate dqt/dt is related to both k1 and (qe-qt). Small value of k1 and 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 of k1. PFO rate = 𝑘1 (𝑞𝑒 − 𝑞𝑡 )
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(5)
The PFO model used to be considered as empirical model for a long time. Azizian
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(2004) deduced the PFO model from the Langmuir kinetics model (Eq. (6)) (Langmuir, 1918), and analyzed the physical meanings of this model. 𝑑𝑡
= 𝑘𝑎 𝐶𝑡 (1 − 𝜃) − 𝑘𝑑 𝜃
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𝑑𝜃
(7)
𝑚𝑞𝑚 𝑉
𝜃
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𝐶𝑡 = 𝐶0 − 𝛽𝜃 = 𝐶0 −
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Ct can be given by Eq. (7):
(6)
Substitution of Eq. (7) into Eq. (6) yields: 𝑑𝜃 𝑑𝑡
= 𝑘𝑎 (𝐶0 −
𝑚𝑞𝑚 𝑉
𝜃) (1 − 𝜃) − 𝑘𝑑 𝜃
(8)
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Effort has been made to study the physical meanings and the applications 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. 𝑑𝜃 𝑑𝑡
= 𝑘𝑎 𝐶0 (1 − 𝜃) − 𝑘𝑑 𝜃
(9)
Liu and Shen (2008) also concluded that the simplification of the Langmuir
13
kinetics model to the PFO and PSO models was C0-dependent. Zhang (2019) suggested that the conditions for the PFO model were: (1) C0/β approached to zero; (2) C0/β approached to infinite; and (3) kaβ/kd approached to zero. Our previous research (Guo and Wang, 2019a) concluded that mqm/V was approximately a constant for a given adsorption process, therefore when a few active sites were occupied compared with C0, mqmθ/V could be neglect. Three conditions could satisfy this hypothesis, as presented
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in Fig. 2.
The first one is that the value of C0 is high. Table 1 summarizes the applications
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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
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BBLS-CA, resorcinol onto CTAB/NaOH/flyash composites, Cu (II) onto RSP and
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SMSP could be better fitted by the PFO model than the PSO model at high C0 (Darwish et al., 2019; Khan et al., 2019; Ho and Mckay, 1998; Stähelin et al., 2018; Agarwal and
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Rani, 2017; Sari et al., 2017; Gunasundari and Kumar, 2017). Siyal et al. (2017) investigated the adsorption of anionic surfactant onto fly ash based geopolymer, concluded that when C0 increased from 100 mg·L-1 to 880 mg·L-1, the values of the
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correlation coefficient (R2) of the PFO model increased from 0.0035 to 0.937. Ma et al. (2008) reported that the values of R2 of the PFO model increased from 0.7556 to 0.9587, with the increase of C0 of Congo Red from 100 mg·L-1 to 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 to7.49×10-6 mol·L-1, the values of R2 of the PFO model increased from
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0.9704 to 0.9886 (Azizian e al., 2009). The second one is that the adsorption process is in the initial stage. Hu et al. (2018) found that when t approached 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
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external diffusion or the internal diffusion is the rate controlling step. The adsorption of metals ions and hydrophilic compounds onto microplastics could be represented by
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the PFO model (Guo et al., 2019a; Turner and Holmes, 2015). One possible reason is that microplastics are hydrophobic compounds, and the diffusion of hydrophilic
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compounds to the surface of the hydrophobic microplastics is difficult. Therefore, the
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external/internal diffusion is the rate limiting step. However, the adsorption of hydrophobic organic compounds (such as lubrication oil and polybrominated diphenyl
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ethers) onto microplastics 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 adsorption onto active sites
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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 in Table 1, the majority of literature modeled the whole adsorption process instead of modeling the initial stage of the adsorption. Therefore, the conditions
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for the PFO model are at high initial concentration of adsorbate and the adsorption is not controlled by the adsorption in active sites. In some cases, the PFO model could
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represent the external/internal diffusion.
Fig. 2 Physical meanings of the PFO and PSO models
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Table 1
Adsorbate
tm/t
te
C0 (mg·L-1)
Reference
BB69
0–250 min/250 min
250 min*
500
Ho and Mckay, 1998
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Applications of the PFO model
AB25
0–250 min /250 min
–
200
MO1
0–1440 min/ 1440 min
540 min
200
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
Adsorbent Peat Peat NiO NPs
16
Darwish et al., 2019
Resorcinol
0–24 h/ 24 h
3 h*
200
Agarwal and Rani, 2017
RSP
Cu(II)
0–150 min/150 min
110 min*
300
Gunasundari and Kumar,
SMSP
Cu (II)
0–150 min/150 min
60 min*
500
2017
Peat
Pb(II)
0–20 min/90 min
90 min*
504
Ho and Mckay, 1999
Peat
Pb(II)
0–20 min/90 min
30 min*
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*
5×10–3
Turner and Holmes, 2015
PE
Cu(Ⅱ)
0–168 h/168 h
30 h*
PE
Ni(Ⅱ)
0–168 h/168 h
30 h*
-p
*: Obtained from the figures in references.
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CTAB/NaOH/flyash
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2.2 Pseudo-second-order (PSO) model
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-: Not mentioned.
The PSO model (Eq. (10)) was firstly applied to model the adsorption of lead onto
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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. 𝑑𝑞𝑡
= 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2
(10)
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𝑑𝑡
The integrated PSO model is described as following:
𝑞𝑡 =
𝑞𝑒2 𝑘2 𝑡
(11)
1+𝑞𝑒 𝑘2 𝑡
In order to calculate the model parameters, the nonlinear PSO model is always transformed to the linear form (Eq. (12)). 𝑡 𝑞𝑡
=
1 𝑘2 𝑞𝑒2
+
𝑡
(12)
𝑞𝑒
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The linearization of the PSO model changes the weight of qt, and introduces propagated errors, which leads to the inaccurate calculations of the model parameters (El-Khaiary et al., 2010; Ho, 2006; 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 constant k1, the PSO rate constant k2 is also used to describe the rate of adsorption equilibrium (Plazinski et al., 2009). But the adsorption
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rate dqt/dt is related to both k2 and (qe-qt)2. Thus, it is more precise to calculate the PSO rate by Eq. (13). PSO rate = 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2
-p
(13)
The physical meanings of the PSO model have been investigated. Azizian (2004)
re
found that when C0 was low, the Langmuir kinetics model (Eq. (8)) could be simplified
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to the PSO model. Liu (2008) suggested that the PSO model was related to the vacant active sites. Zhang (2019) reported that the conditions for the PFO model were C0/β
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approached to 1 and kaβ/kd approached to infinite. Marczewski (2010a) found that the conditions for the PSO model were the equilibrium surface coverage fraction θe approached to 1 and C0/β approached to 1. Our previous study (Guo and Wang, 2019a)
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concluded that the PSO model could represent three conditions (Fig. 2). The first one is that the value of C0 is low. Sabarinathan et al. (2019) studied the
adsorption of MB onto molecular polyoxometalate, found that the values of R2 of the PSO model decreased from 0.8979 to 0.5905 with the increase of C0 from 140 mg·L-1 to 300 mg·L-1. The applications of the PSO model in literature are summarized in Table
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2. As shown in Table 2, at low initial concentrations of adsorbate, the adsorption kinetics 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 are better modeled by the PSO model (Ho and Mckay, 1998; Darwish et al., 2019; Khan et al., 2019; Stähelin et
Table 2 Applications of the PSO model
C0 (mg·L-1)
Adsorbate
tm/t
Peat
BB69
0–250 min /250 min
Peat
AB25
0–250 min /250 min
NiO NPs
MO1
0–1440 min/ 1440 min
540 min
50
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
0–24 h/ 24 h
3 h*
50
Agarwal
Resorcinol
30 min*
50
–
20
re
lP
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CTAB/NaOH/flyash
-p
Adsorbent
RSP
te
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al., 2018; Sari et al., 2017; Agarwal and Rani, 2017; Gunasundari and Kumar, 2017).
Reference
Ho and Mckay, 1998
Darwish et al., 2019
and
Rani,
2017
Cu(II)
0–150 min/150 min
100 min*
50
Gunasundari
Cu(II)
0–150 min/150 min
50 min*
50
Kumar, 2017
Pb(II)
20–90 min/90 min
90 min*
504
Ho and Mckay, 1999
Pb(II)
20–90 min/90 min
30 min*
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*
–
Hu et al., 2017
PS
lubrication oil
0–48 h/48 h
12 h*
–
PE
BDE-99
0-24 h/24 h
30 min
0.5
Peat Peat
Jo
SMSP
19
Xu et al., 2019
and
Al-modified biochar
Nitrate
0–24 h/24 h
24 h
50
Al-modified biochar
Phosphate
0–24 h/24 h
24 h
50
Modified clay
MO1
0–4 h/4 h
2h
200
Gamoudi and Srasra,
Modified clay
IC
0–4 h/4 h
1h
200
2019.
Modified clay
PR
0–4 h/4 h
1h
200
Ag(Ⅰ)
0–150 min/150 min
20 min
122
H. Wang et al., 2018
Cr(VI)
0–180 min/180 min
120 min*
20 and 30
Gogoi et al., 2018
Cr(III)
0–180 min/180 min
120 min*
20 and 30
Modified chitosan
Re(VII)
0–24 h
–
20
Lou et al., 2018
FTMA-MT
Phenol
0–21 h
7h
–
Li et al., 2018
Modified hydrochar
Pb(II)
0–360 min/360 min
240 min*
10
Xia et al., 2019
Modified-chitosan
Cr(VI)
0–150 min/150 min
P-CSs
MB
0–300 min/300 min
Modified mesoporous
Steroid estrogens
0–90 min/90 min
Modified
titanate
Yin et al., 2018
40–100
Islam, 2019
140 min
30–90
Tang et al., 2019
30 min
2
Gao et al., 2019
0–300 min/300 min
150 min
600–1000
Li et al., 2019
0–300 min/300 min
30 min
40–90
modified
Cd(II)
ur na
hydrochar
MB
lP
silica Maleylated
-p 90 min
re
(M) PCL
ro of
nanotubes
Modified bentonite
RhB
0–70 min/70 min
40 min
100
Modified bentonite
AR1
0–70 min/70 min
90 min
100
Cobalt
MG
0–70 min/70 min
40 min
100
Amiri et al., 2017
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
ferrite
silica
magnetic
Jo
nanocomposite Modified
Algerian
Huang et al., 2017
geomaterial
20
Modified
Cu(II)
0–90 min/90 min
50 min
200
chitosan/CoFe2O4
Pb(II)
0–90 min/90 min
50 min
200
Modified
Sr(II)
0–10 h/10 h
10 h
100
Cheng et al., 2019
U(VI)
0–180 min/180 min
180 min
480
Zhuang et al., 2018
DFC
0–500 min/500 min
200 min
80
Zhuang et al., 2019
As(III)
0–540min/540 min
300 min
10 and 30
Zhang et al., 2016
E33
Phosphate
0–294 h/294 h
294 h*
140
E33/Mn
Phosphate
0–294 h/294 h
294 h*
140
E33/Ag I
Phosphate
0–294 h/294 h
294 h*
140
E33/Ag II
Phosphate
0–294 h/294 h
NiO NPs
MO1
0–1440 min/ 1440 min
BBLS-CA
MB
0–360 min/360 min
Bacterial
Fan et al., 2017
cellulose membrane Magnetic
amidoxime
functionalized chitosan MOFs Cerium
modified
-p
ro of
chitosan
Lalley et al., 2016
140
540 min
200
Darwish et al., 2019
500
Sari et al., 2017
re
294 h*
90 min
-: Not mentioned.
lP
*: Obtained from the figures in references.
ur na
The second one is at the final stage of the adsorption process. As shown in Table 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
Jo
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%. 21
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 of R2 of the PSO model was 0.990, while for hydrochar, this value was 0.945 (Xia et al., 2019). As listed in Table 2, the adsorption of pollutants on modified materials, such as Al-modified biochar, modified clay, modified titanate nanotubes, modified pineapple crown leaf, modified chitosan, surfactant-modified
ro of
montmorillonite, modified hydrochar, H3PO4 modified corn stalks, modified
mesoporous silica, modified bentonite, cobalt ferrite silica magnetic nanocomposite,
-p
modified Algerian geomaterial, modified wheat bran, modified hydrogel, modified
bacterial cellulose membrane, magnetic amidoxime functionalized chitosan and MOFs
re
could be better modeled by the PSO model (Yin et al., 2018; Gamoudi and Srasra, 2019.;
lP
H. Wang et al., 2018; Gogoi et al., 2018; Lou et al., 2018; Li et al., 2018, 2019; Xia et al., 2019; Islam, 2019; Tang et al., 2019; Gao et al., 2019; Huang et al., 2017; Amiri et
ur na
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 kinetics is dominated by the adsorption onto
Jo
active site.
2.3 Mixed-order (MO) model The mixed-order (MO) model has the following form (Guo and Wang, 2019a):
𝑑𝑞𝑡 𝑑𝑡
= 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) + 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )2
(14)
The PFO and PSO rate of the MO model can be calculated by Eqs. (15)–(16).
22
PFO rate' = 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) (15) PSO rate' = 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )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
ro of
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
-p
controlling step is the diffusion or the adsorption; and (3) arbitrary initial adsorbate concentration in solution (Guo and Wang, 2019a).
re
Eq. (14) is similar in the form to the hybrid-order’ (HO) model (Eq. (17)) provided
by Marczewski (2010b). 𝑑𝑡 𝑑𝐹 𝑑𝑡
= 𝑘L1 (𝜃𝑒 − 𝜃𝑡 ) + 𝑘L2 (𝜃𝑒 − 𝜃𝑡 )2
(17)
= 𝑘1𝑎 (1 − 𝐹𝐿 ) + 𝑘2𝑎 𝑎eq (1 − 𝐹𝐿 )2
(18)
ur na
𝑑𝜃𝑡
lP
in Liu and Shen (2008), and the mixed 1, 2-order rate equation (MOE) (Eq. (18)) derived
Where kL1, kL2, k1a, aeq, and k2a are coefficients.
Jo
As described in Eq. (17), the HO model is the relationship between θt and t, 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 parameters. Therefore, the MO model presented by Guo and Wang (2019a) is
23
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
ro 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
-p
in the following section. 2.5 Elovich model
re
The basic assumptions of the Elovich model were (1) the activation energy
lP
increased with adsorption time and (2) the surface of the adsorbent was heterogeneous. The Elovich model is an empirical model without definite physical meanings. It is
ur na
commonly used to model the chemisorption of gas onto solid. The Elovich model has been described by Eq. (19) (Elovich and Larinov, 1962): 𝑑𝑞𝑡 𝑑𝑡
= 𝑎𝑒 −𝑏𝑞𝑡
(19)
Jo
Integrating Eq. (19) for the condition of q0 = 0 yields: 1
𝑞𝑡 = ln(1 + 𝑎𝑏𝑡)
(20)
𝑏
Eq. (20) is a nonlinear equation, which can be solved by the nonlinear least square
regression method (plot qt versus t) or by the linear method (plot qt versus ln(1+abt)). However, the nonlinear method is more complex than the linear method. And the linear
24
method needs to give the appropriate initial values of ab, which is difficult for the researchers (Ho, 2006). Chien and Clayton (1980) simplified Eq. (20) with the assumption of abt >> 1: 1
1
1
𝑏
𝑏
𝑏
𝑞𝑡 = ln(𝑎𝑏𝑡) = ln(𝑎𝑏) + ln(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 represent the adsorption of liquid-solid
ro of
systems. The applications of the Elovich model are summarized in Table 3. Eq. (21) is
the most frequently applied form of the Elovich model, which has been successfully
-p
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
re
parameters (Lin et al., 2018; Brito et al., 2018; Kalhor et al., 2018). However, the
lP
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
ur na
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 calculate the Elovich rate: Elovich rate = 𝑎𝑒 −𝑏𝑞𝑡
Jo
(22)
Table 3
Applications of the Elovich model Adsorbent
Adsorbate
Elovich model
Ref.
Alfisol
Tetracycline
Eq. (21)
Bao et al., 2010
25
Ultisol
Tetracycline
Eq. (21)
Contaminant barrier minerals
Pb(Ⅱ)
Eq. (21)
Contaminant barrier minerals
Cd(Ⅱ)
Eq. (21)
β-cyclodextrin-based adsorbent
Cu(Ⅱ)
Eq. (21)
Huang et al., 2012
Synthesized TiO2 nanoparticles
Hg(II)
Eq. (21)
Dou et al, 2011
TiO2/montmorillonite
Hg(II)
Eq. (21)
Surfactant-Modified Natural Zeolites
Benzene
Eq. (21)
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, 2014
Gl-PZSNTs-Fe3O4
MB
Eq. (21)
Wang et al., 2019
Honeycomb-like porous activated carbon
Cu(II)
Modified eggshell membrane
MO1
Modified eggshell membrane
MB1
Anodized iron oxide nanoflakes
Phosphate
Inyang et al., 2016
-p
ro of
Seifi et al., 2011
Mondal and Majumder, 2019
Eq. (21)
Candido et al., 2019
re
Eq. (21)
Afridi et al., 2019
DY12
Eq. (21)
Alinejad-Mir et al., 2018
BB69
Eq. (21)
Magdy and Altaher, 2018
Cr(VI)
Eq. (20)
Lin et al., 2018
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
Iron oxide-gelatin nanoadsorbent Cement kiln dust
Carbons
Jo
Modified rice straw
lP
Eq. (21)
ur na
Eq. (21)
2.4 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 26
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): 𝑑𝜃 𝑑𝑡
= 𝛼(1 − 𝜃)𝑛
(23)
After integration with the initial condition of q0 = 0, Ritchie’s equation becomes: For n=1, 𝜃 = 1 − 𝑒 −𝛼𝑡
(24)
ro of
For n≠1, 1
𝜃 = 1 − [1 + (𝑛 − 1)𝛼𝑡]1−𝑛
(25)
-p
θ can be described by the ratio of qt and q∞ (q∞ can be obtained from the adsorption isotherm model (q∞ = f(Ce)), Eq. (24) and Eq. (25) becomes:
re
For n=1,
For n≠1,
lP
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝛼𝑡 )
(26)
1
ur na
𝑞𝑡 = 𝑞∞ − 𝑞∞ [1 + (𝑛 − 1)𝛼𝑡]1−𝑛
(27)
When n=2, rearrangement of Eq. (27) yields: 𝑞𝑡 =
𝛼𝑞∞ 𝑡
(28)
1+𝛼𝑡
Jo
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 in Table 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
27
the RSO model to the adsorption data of CO2 onto piperazine-modified activated alumina. B. 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. 𝑑𝑞𝑡 𝑑𝑡
= 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛
(29)
ro of
For n = 1, the PNO model equals to the PFO model. After integration with the initial condition of q0 = 0 and n ≠ 1, the PNO model becomes: (𝑞𝑒 − 𝑞𝑡 )1-n = (n − 1)𝑘𝑛 𝑡 + 𝑞𝑒 1−n
-p
(30)
𝑞𝑡 = 𝑞𝑒 (1 −
1 (1+(n−1)𝑞𝑒
n−1 𝑘
𝑛 𝑡)
)
1/(n−1) )
re
Rearrangement of Eq. (30) yields:
(31)
lP
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
ur na
equation without specific physical meanings. The applications of the PNO model are summarized in Table 4. Caroni et al. (2009) indicated that values of n of tetracycline adsorption on chitosan particles increased with the increase of the initial concentrations
Jo
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 decreased to 0.979 at high C0 (6 mg·L-1). Leyva-Ramos et al. (2010) suggested that the values of n were not correlated with the experimental conditions, such as pH, C0, impeller speeds, and so on. The nonlinear form of the PNO model cannot be written as the linear form.
28
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 proposed for the calculation of the PNO rate: PNOrate = 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛
ro of
(32)
-p
Table 4 Applications of the Ritchie’s equation and the PNO model Adsorbent
Adsorbate
C0 (mg·L-1)
Bone char
Cd(Ⅱ) CO2
n
Ref.
224-560
RSO
2
Cheung et al., 2001
-
RSO
2
Fashi et al., 2018
50
Ritchie’s equation
-
B. Wang et al., 2018
MB
50
Ritchie’s equation
-
MB
50
Ritchie’s equation
-
Alkanes
-
RSO
2
Phthalonitrile compound
Alcohols
-
RSO
2
Phthalonitrile compound
Chlorinated
-
RSO
2
Macroporous ion-exchange resins
Al(Ⅲ)
77.5
PNO
1.17-1.47
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
activated
Alumina
Ball-milled biochar
MB
ur na
Calcium alginate
Ball-milled biochar encapsulated in calcium-alginate beads
lP
Piperazine-modified
Jo
Phthalonitrile compound
re
Model
29
Günay et al., 2018
Nekouei et al., 2019
Macroporous ion-exchange resins
Pb(Ⅱ)
183
PNO
0.998-1.36
Pure açaí biochar
MB
50
PNO
1.65
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, 2018
Chitosan spheres
MO1
-
PNO
2-2.5
Morais, 2008
Chitosan particles
Tetracycline
15*
PNO
4.5*
Caroni et al., 2009
Chitosan particles
Tetracycline
55*
PNO
16*
Montmorillonite-carbon hybrids
17β-Estradiol
6
PNO
0.979
Montmorillonite-carbon hybrids
17β-Estradiol
0.2
PNO
Bone char
fluoride
5.44
PNO
Bone char
fluoride
8.35
PNO
Carbons prepared from betel trunk
Phenol
-
Carbons prepared from betel trunk
MB
-
Carbons prepared from betel trunk
Basic brown 1
-
Carbons prepared from betel trunk
Acid
ro of
-p
1.3
Leyva-Ramos, 2010
2.0
PNO
1.50
PNO
3.51
re
-
lP
Carbons prepared from betel trunk
74
Liu et al., 2019
2.276
PNO
3.06
PNO
1.77
2,4-dichloropenol
-
PNO
1.63
ur na
(AB74)
blue
Pessôa et al., 2019
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
Tseng et al., 2014
Izadi et al., 2017
Jo
*: Obtained from the figures in references. -: Not mentioned.
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. Several equations have been 30
developed to model the external mass transfer process.
3.1.1 Boyd’s external diffusion equation
Boyd et al. (1947) deduced a kinetic model to describe the diffusion of adsorbate through a bounding liquid film: 𝑑𝑞𝑡 𝑑𝑡
= 4𝜋𝑟0 2 𝐷𝑙 (𝜕𝐶𝑓 /𝜕𝑟)𝑟=𝑟
(33)
0
ro of
Assuming that the concentration gradient is linear, Eq. (33) is simplified to Eq. (34) (Boyd et al., 1947): 𝑑𝑞𝑡 𝑑𝑡
= 𝑅(𝑞∞ − 𝑞𝑡 )
(34)
re
by the adsorption isotherm model (q∞ = f(Ce)).
-p
Where q∞ = 4πr03Clk/3 (k is the distribution coefficient). q∞ can also be obtained
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝑅𝑡 )
lP
Integration of Eq. (34) at initial condition of q0=0 yields (Boyd et al., 1947): (35)
ur na
This equation is similar with the PFO model. Ahmed and Theydan (2013) studied the adsorption kinetics of metronidazole onto microporous activated carbon using the linear form of Boyd’s equation (Eq. (36)), the results showed that the values of R2 of
Jo
Boyd’s equation ranged from 0.9471 to 0.9885. ln( 1 − 𝑞𝑡 /𝑞∞ ) = −𝑅𝑡 + 𝐴
(36)
Table 5 Applications of the diffusional models
31
Diffusional
Models name
Adsorbent
Adsorbate
Reference
External
Boyd’s
Microporous activated carbon
Metronidazole
Ahmed and Theydan, 2013
diffusion
diffusion equation
models
F&S model
Inactive
Azo dye
Wang et al., 2008
Özer et al., 2005
models external
CMC
immobilized
Aspergillus fumigatus beads Enteromorpha prolifera
AR337
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
Azo dye
Wang et al., 2008
CMC
-p
ro of
F&S model
immobilized
re
Aspergillus fumigatus beads Enteromorpha prolifera
AR337
M&W model
Enteromorpha prolifera
AB324
M&W model
Bone char
Cu (Ⅱ)
M&W model
Bone char
Cd(Ⅱ)
ur na
lP
M&W model
M&W model
Bone char
Zn(Ⅱ)
M&W model
Clinoptilolite rich mineral
Cr(VI)
M&W model
Bacteria loaded clinoptilolite rich
Cr(VI)
Özer et al., 2005
Choy et al., 2004
Erdoğan and Ulku, 2012
Jo
mineral
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
32
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)
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
Boyd’s intraparticle
Mansonia wood sawdust
Pb(II)
Ofomaja, 2010
diffusion
diffusion model
models
Boyd’s intraparticle
Dolomite
Cr(VI)
Albadarin et al., 2012
ro of
-p
diffusion model Boyd’s intraparticle
PMMA
Phenol
Starchy adsorbents
Okewale et al., 2013
MG
Sartape et al., 2017
Pre-treated bentonite clay
Ag(Ⅰ)
Cantuaria et al., 2016
Mesoporous activated carbon
Cd(Ⅱ)
Tan et al., 2016
Verde-lodo bentonite
Ag(Ⅰ)
Freitas et al., 2017
Verde-lodo bentonite
Cu(II)
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Wood apple shell
ur na
diffusion model Boyd’s intraparticle
Al-Muhtaseb et al., 2011
Water
diffusion model Boyd’s intraparticle
re
diffusion model Boyd’s intraparticle
Suzaki et al., 2017
diffusion model
Boyd’s intraparticle diffusion model
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Boyd’s intraparticle diffusion model Boyd’s intraparticle diffusion model Boyd’s intraparticle
Amine-functionalized
diffusion model
mesoporous alumina
ordered
33
MB
Gan et al., 2015
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)
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
IMT model
Dowex Optipore SD-2
-p
Boyd’s intraparticle
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)
diffusion model
ro of
AB 25
ur na
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re
RB5G
IMT model
Fixed-bed columns
Cu(II)
IMT model
Fixed-bed columns
Ni(II)
IMT model
Fixed-bed columns
Zn(II)
IMT model
Microplastics
SMX
Guo et al., 2019b
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
Jo
PVSD model
Apiratikul and Pavasant, 2008
34
Suzaki et al., 2017
PVSD model
Biochar
Ibuprofen
Ocampo-Perez et al., 2019
3.1.2 Frusawa and Smith (F&S) model
Frusawa and Smith (1973) developed an adsorption rate equation: 𝐶𝑡 𝐶0
=
1 1+𝑚𝑠 𝐾
+
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚 𝐾
𝑠 − 𝑚 𝐾 𝑘F&S 𝑆𝑡 𝑠
(37)
The value of kF&SS is frequently used to describe the external diffusional process
ro of
(Özer et al., 2005, Panday et al., 1986) The F&S model assumes that the external diffusion is the slowest step, the
intraparticle diffusion is negligible, and the isotherm is linear (Eq. (4)) (Frusawa and
-p
Smith, 1973). Note that Eq. (37) cannot be used for modeling the adsorption condition
re
that the isotherm is not linear.
The applications of the F&S model are summarized in Table 5. Unfortunately,
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most studies applied Eq. (37) to their adsorption kinetics data when the isotherm is
ur na
described by the Langmuir isotherm model (Özer et al., 2005; Wang et al., 2008; Öztürk and Malkoc, 2014; Fu et al., 2012, 2015). In order to establish the relationship between qt and t, Eq. (38) is brought into Eq. (37), and the result is presented in Eq. (39): 𝑉
(𝐶0 − 𝐶𝑡 ) =
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𝑞𝑡 = 𝑞𝑡 =
𝑚
𝐶0
𝑚𝑠
(1 −
1
1+𝑚𝑠 𝐾
(𝐶0 −𝐶𝑡 )
(38)
𝑚𝑠
−
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚 𝐾
𝑠 − 𝑚 𝐾 𝑘F&S 𝑆𝑡 𝑠 )
(39)
Eq. (39) can be solved by the nonlinear regression method, which is provided in the following section.
35
3.1.3 Mathews and Weber (M&W) model
The Mathews and Weber (M&W) model is presented as follows (Mathews and Weber, 1977). 𝑘𝑀&𝑊 =
𝑟0 𝜌(1−𝜀) 𝑙𝑛(𝐶0 /𝐶𝑡 ) 3𝑚𝑠
(40)
𝑡
The coefficient item 3ms/(r0(1-)) can be replaced by the outer surface coefficient
𝐶𝑡 𝐶0
ro of
S (cm-1). Rearrangement of Eq. (40) yields: = 𝑒 −𝑘M&W 𝑆𝑡
(41)
The value of kM&WS is calculated to describe the external diffusion. Eq. (41) is the
-p
most frequently used form of the M&W model. As summarized in Table 5, the M&W
re
model has been used to describe the adsorption kinetics data of dyes and metals onto biosorbent and mineral (Wang et al., 2008, Choy et al., 2004; Erdoğan and Ulku, 2012;
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Dávila-Guzmán et al., 2013; Šljivić et al., 2011). Combining Eq. (38) and Eq. (41) yields: 𝐶0
(1−e−𝑘M&W 𝑆𝑡 )
ur na
𝑞𝑡 =
𝑚𝑠
(42)
Eq. (42) can be solved by the nonlinear regression method.
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3.1.4 Phenomenological external mass transfer (EMT) model
The EMT model (Eq. (43)) assumes that the film diffusion is the slowest step, and
the equilibrium is obtained on the surface of the adsorbent (Ruthven, 1984; Hines and Maddox, 1985). The driving force of the external diffusion is the concentration gradient of adsorbent in the liquid film.
36
𝑑𝑞𝑡 𝑑𝑡
=
𝑘𝑠 𝜌
(𝐶𝑡 − 𝐶et ) = 𝑘ext (𝐶𝑡 − 𝐶et )
(43)
Then the equilibrium phenomenon is described by the adsorption isotherm model. The isotherm model is presented as qet = f(Cet). Then Cet = f-1(qet). For example, if the Langmuir model can adequately represent the adsorption isotherm, the equilibrium phenomenon is described as following: 𝑞𝑡 =
𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶et
(44)
1+𝐾𝐿 𝐶et
𝑑𝑞𝑡 𝑑𝑡
= 𝑘ext (𝐶0 −
𝑚𝑞𝑡 𝑉
−
𝑞𝑡 𝑞max 𝐾𝐿 −𝑞𝑡 𝐾𝐿
ro of
Substitution of Eqs. (38) and (44) into Eq. (43) yields: )
(45)
-p
The initial conditions of the EMT model are q0 = 0 and Cet(0) = 0. The EMT model can be solved by the Runge-Kutta method. The solving method is provided in the
re
following section.
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As shown in Table 5, the adsorption processes of dyes onto Dowex Optipore SD2 and SB biomass, as well as metals ions onto argassum horneri and fixed-bed columns
ur na
have been modeled by the EMT model. The equilibrium phenomenon was described by the Langmuir (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017; Sausen et al., 2018), BET (Scheufele et al., 2016), Linear (Guo et al., 2019b), and Freundlich (Hu
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et al., 2019; Guo et al., 2019b) isotherms. 3.2 Internal diffusion models The internal diffusion models assume that the diffusion of adsorbate within
adsorbent is the slowest step. The diffusion of adsorbate in the liquid film around the adsorbent and the adsorption onto the active sites are instantaneous. Here we reviewed
37
three most used internal diffusion models, i.e., the Boyd’s intraparticle diffusion model, the Weber and Morris (W&M) model, and the phenomenological internal mass transfer (IMT) model.
3.2.1 Boyd’s intraparticle diffusion model
Boyd et al. (1947) reported an intraparticle diffusion model: 6 𝜋2
∑∞ 𝑛=1
1 𝑛2
𝑒 −𝑛
2 𝐵𝑡
(46)
ro of
𝐹=1 −
Where F = qt/q∞, q∞ (mg·g-1) is obtained by the adsorption isotherm model (q∞ = f(Ce)) (Yao and Chen, 2017).
-p
As the value of F up to 0.05, Eq. (46) is simplified to Eq. (47) (Boyd et al., 1947): (47)
re
𝐹 = 1.08√𝐵𝑡
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Reichenberg (1953) suggested that for the values of F ranged from 0 to 0.85, Eq. (46) could be rearranged to:
Or
6 𝜋3/2
3
√𝐵𝑡 − 𝜋2 (𝐵𝑡)
ur na
𝐹=
𝐵𝑡 = 2𝜋 −
𝜋2 𝐹 3
− 2𝜋 (1 −
𝜋𝐹 1/2 3
)
(48 a)
(48 b)
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When the values of F range from 0.86 to 1, Eq. (46) is transformed to Eq. (49) (Reichenberg, 1953): 𝐹 =1−
𝜋2 6
𝑒 −𝐵𝑡
(49 a)
Or 𝐵𝑡 = −ln
𝜋2 6
(1 − 𝐹)
(49 b)
If the experimental data obey the Boyd’s intraparticle diffusion model, B is a 38
constant. In other words, if the plot of Bt vs t is a straight line and passes through (0, 0), the adsorption process is controlled by the intraparticle diffusion. Please note that in the applications of Boyd’s intraparticle diffusion model, the term Bt is the product of B and t. In the past decades, Boyd’s intraparticle diffusion equation has been used to represent the internal mass transfer processes (Table 5). For example, Ofomaja (2010)
ro of
studied the adsorption of Pb(II) onto mansonia wood sawdust using the Boyd’s model.
Albadarin et al. (2012) demonstrated that the values of R2 of Boyd’s model for the
-p
adsorption of Cr(Ⅵ) (C0 = 10-50 mg·L-1) onto dolomite ranged from 0.899 to 0.951.
Al-Muhtaseb et al. (2011) concluded that the plot of Bt vs t didn’t pass through (0, 0),
re
which indicated that the adsorption of phenol onto PMMA was not controlled by the
lP
intraparticle diffusion.
ur na
3.2.2 Weber and Morris (W&M) model
Weber and Morris (1963) deduced a model to describe the intraparticle diffusion process. The W&M model is presented as following: 𝑞𝑡 = 𝑘W&M 𝑡 1/2
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(50)
By plotting qt vs. t, the parameter kW&M can be estimated. The plot of qt vs t1/2 is a
straight line and passes through (0, 0) when the intraparticle diffusion is the controlling process. Otherwise, the adsorption is controlled by multiple processes. Because W&M model is simple and convenient, it has been widely applied to find out the rate controlling step, such as phenol onto PMMA (Al-Muhtaseb et al., 2011), 39
TA, MB, phenol, and 4-CP on activated carbons (Wu et al., 2009), Cu(II), Cd(II), and Pb(II) onto Caulerpa lentillifera (Apiratikul and Pavasant, 2008), Dibenzothiophene, MB and AB 25 onto activated carbons (Danmaliki and Saleh, 2016; Archin et al., 2019) (Table 5).
3.2.3 Phenomenological internal mass transfer (IMT) model
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The internal mass transfer phenomenon can be described by the Fick’s law (Crank, 1979): 𝜕𝑞𝑡 (𝑟𝑖 ,𝑡) 𝜕𝑡
=
𝐷ef 𝜕 𝑟𝑖 2 𝜕𝑟𝑖
(𝑟𝑖 2
𝜕𝑞𝑡 (𝑟𝑖 ,𝑡) 𝜕𝑟𝑖
)
(51)
-p
Eq. (51) can be simplified to the LDF model (Glueckauf and Coates,1947; Yang,
𝜕𝑞𝑡
= 𝑘𝑖𝑛𝑡 (𝑞et − 𝑞𝑡 )
(52)
lP
𝜕𝑡
re
1987):
The IMT model assumes that the internal diffusion is the slowest process, and the
ur na
equilibrium is obtained at the liquid-solid interface. The adsorption isotherm is used to describe the equilibrium phenomenon. For example, if the Langmuir model can describe the adsorption equilibrium data, qet is calculated as following: 𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶𝑡
(53)
1+𝐾𝐿 𝐶𝑡
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𝑞et =
Substitution of Eqs. (38) and (53) into Eq. (52) yields:
𝜕𝑞𝑡 𝜕𝑡
= 𝑘𝑖𝑛𝑡 (
𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶𝑡
1+𝐾𝐿 (𝐶0 −𝑚𝑞𝑡 /𝑉)
− 𝑞𝑡 )
(54)
Eq. (54) can be solved by the Runge-Kutta method with the initial condition of q0 = 0. The applications of the IMT model are shown in Table 5. The equilibrium phenomenon in the liquid-solid interface was described by the BET (Scheufele et al., 40
2016), Freundlich (Hu et al., 2019), and Langmuir isotherms (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017). 3.3 Pore volume and surface diffusion (PVSD) model The basic assumptions of the PVSD model are that (1) the adsorbent particle is spherical, (2) the convective mass transport in the pores is negligible, (3) the solution is
ro of
homogeneous, and (4) the adsorption on active sites is simultaneous. The PVSD model is described by Eqs. (55)-(60) (Leyva-Ramos and Geankoplis, 1985): 𝑉
𝑑𝐶𝑡
= −𝑚𝑆𝑝 𝑘𝐹 (𝐶𝑡 − 𝐶tr |r=r0 )
𝑑𝑡
(55)
𝑡 = 0, 𝐶𝑡 = 𝐶0 𝜕𝐶tr 𝜕𝑡
+ 𝜌𝑝
𝜕𝑞𝑡 𝜕𝑡
=
1 𝜕 𝑟 2 𝜕𝑟
(𝑟 2 (𝐷𝑝
𝜕𝐶tr 𝜕𝑟
+ 𝜌𝑝 𝐷𝑠
𝜕𝑟
𝐷𝑝
|𝑟=0 = 0 𝜕𝐶tr 𝜕𝑟
|𝑟=𝑟0 + 𝜌𝑝 𝐷𝑠
𝜕𝑞𝑡
))
lP
𝜕𝐶tr
𝜕𝑟
re
𝑡 = 0,0 ≤ 𝑟 ≤ 𝑟0 , 𝐶tr = 0
𝜕𝑞𝑡
-p
𝜀𝑝
(56)
| 𝜕𝑟 𝑟=𝑟0
= 𝑘𝐹 (𝐶𝑡 − 𝐶tr )|𝑟=𝑟0
(57) (58) (59) (60)
ur na
The equilibrium phenomenon is described by the adsorption isotherm: qet = f(Cet). The PVSD model is simplified to the pore volume diffusion (PVD) model when Ds = 0 and Dp ≠ 0 or to the surface diffusion (SD) model when Dp = 0 and Ds ≠ 0.
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Detailed numerical solution of the PVSD model is given by Souza et al. (2017a). In recent years, the PVSD model has been used to describe the diffusional adsorption of CV and MG onto bentonite clay (Souza et al., 2017a, 2019), pyridine and metronidazole onto granular activated carbon (Ocampo-Perez et al., 2010, DíazBlancas et al., 2018), acetaminophen onto activated carbon pellets (Ocampo-Perez et
41
al., 2017), and ibuprofen onto biochar (Ocampo-Perez et al., 2019) (Table 5).
4. Models for adsorption onto active sites (AAS) The models for AAS assume that the adsorption onto active site is the slowest step, and the diffusion process is negligible. Langmuir kinetics model and Phenomenological AAS model, as well as the Ritchie’s equation reviewed in the previous section can
ro of
describe the AAS phenomenon. 4.1 Langmuir kinetics model
The Langmuir kinetics model can be described by Eq. (6) (Langmuir, 1918), as
-p
shown in the previous section. θ can be calculated by qt/qe, and then Eq. (6) is
𝑑𝑞𝑡 𝑑𝑡
= 𝑘𝑎 𝐶𝑡 (𝑞𝑒 − 𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡
re
transformed to Eq. (61):
(61)
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At adsorption equilibrium, Eq. (61) is reduced to the Langmuir isotherm model (Langmuir, 1918). Eq. (61) need to be solved by the Runge-Kutta method with the
ur na
initial condition of q0 = 0 based on MATLAB or other programming software, which makes it difficult to apply. Efforts have been made to find the analytical solution of the Langmuir kinetics. Liu and Shen (2008) reported that the Langmuir kinetics could be
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solved by Eq. (17). The model coefficients kL1 and kL2 are calculated as following: 𝑘L1 = √𝑘𝑎 2 (𝐶0 − 𝑞𝑒 𝑚𝑠 )2 +2𝑘𝑎 𝑘𝑑 (𝐶0 + 𝑞𝑒 𝑚𝑠 ) + 𝑘𝑑
(62)
𝑘L2 = 𝑘𝑎 𝑞𝑒 𝑚𝑠
(63)
Shen et al. (2009) developed a simplified Langmuir kinetics model using a simple geometric approach. Tien and Ramarao (2014) reported a transformed Langmuir rate
42
equation related to the PFO and PSO models. For the applications of the Langmuir kinetics model, Al-Jabari (2016) studied the adsorption kinetics of Cr(III) onto mineral particles using the Langmuir kinetics. Marczewski et al. (2013) investigated the adsorption and desorption kinetics of benzene on carbons using the integrated Langmuir kinetics model. Marczewski (2011) extended the application of Langmuir kinetics in dilute solutions.
ro of
The solving methods reported by the previous studies are also complex and may lead to difficulties in parameters estimations. In the following section, a convenient
-p
method is provided. In addition, Eq. (64) is recommended to calculate the Langmuir adsorption rate.
re
Langmuir rate = 𝑘𝑎 𝐶𝑡 (𝑞𝑒 − 𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡
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4.2 Phenomenological AAS model
(64)
The Phenomenological AAS mode, which is developed based on the Langmuir
𝑑𝑞𝑡 𝑑𝑡
ur na
kinetics model (Langmuir, 1918; Thomas, 1944), can be described by Eq. (65): = 𝑘𝑎 𝐶𝑡 (𝑞max −𝑞𝑡 ) −
𝑘𝑎
𝐾𝐿
𝑞𝑡
(65)
Substitution of Eq. (38) into Eq. (65) yields: 𝑑𝑞𝑡
= 𝑘𝑎 (𝐶0 −
Jo
𝑑𝑡
𝑚𝑞𝑡 𝑉
) (𝑞max − 𝑞𝑡 ) −
𝑘𝑎 𝐾𝐿
𝑞𝑡
(66)
Eq. (66) can be solved by the Runge-Kutta method with the initial condition of q0
= 0. The AAS model has been adopted to study the adsorption mechanism (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017; Hu et al., 2019; Guo et al., 2019b; Sausen et al., 2018).
43
5. Model validity evaluation In the past decade, various statistical parameters, such as R2, adjR2, 2, SSE, MSE, and HYBRID were used to evaluate the performance of the kinetic models (Eqs. (67)(72)). ∑(𝑞mean −𝑞cal )2
(67)
∑(𝑞cal −𝑞mean )2 +∑(𝑞cal −𝑞exp )2
𝐴𝑑𝑗𝑅2 = 1 − (1 − 𝑅2 ) 𝜒2 = ∑
(𝑁𝑒𝑥𝑝 −1)
(68)
(𝑁𝑒𝑥𝑝 −𝑁𝑝𝑎𝑟𝑎 −1)
(𝑞𝑒𝑥𝑝 −𝑞cal )2 )
ro of
𝑅2 =
(69)
𝑞cal 2
𝑆𝑆𝐸 = ∑(𝑞exp − 𝑞cal )2 1 𝑁𝑒𝑥𝑝
𝐻𝑌𝐵𝑅𝐼𝐷 =
∑(𝑞exp − 𝑞cal )2 100 𝑁𝑒𝑥𝑝 −𝑁𝑝𝑎𝑟𝑎
∑
(71)
-p
𝑀𝑆𝐸 =
(70)
𝑞𝑒𝑥𝑝 −𝑞cal 𝑞𝑒𝑥𝑝
(72)
re
Here we summarized the application of the statistical parameters based on
lP
literature (Guo and Wang, 2019a; Blanco et al., 2017; Hu et al., 2019; Suzaki et al., 2017; Wang et al., 2008; Ma et al., 2018; Ersan et al., 2019; Shang et al., 2019; Delgado
ur na
et al., 2019; Darwish et al., 2019; Sabarinathan et al., 2019; Khan et al., 2019; Gamoudi and Srasra, 2019; Stähelin et al., 2018; Sari et al., 2017; Gunasundari and Kumar, 2017; Lou et al., 2019; Guo et al., 2019a, 2019b; Yin et al., 2018; H. Wang et al., 2018; Gogoi
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et al., 2018; Xia et al., 2019; Islam, 2019; Tang et al., 2019; Gao et al., 2019; Ahmed and Theydan, 2013; Öztürk and Malkoc, 2014; Fu et al., 2012, 2015; Panday et al., 1986; Choy et al., 2004; Erdoğan and Ulku, 2012; Dávila-Guzmán et al., 2013; Šljivić et al., 2011; Sausen et al., 2018; Ofomaja, 2010; Albadarin et al., 2012; Al-Muhtaseb et al., 2011; Sartape et al., 2017; Cantuaria et al., 2016; Tan et al., 2016; Freitas et al.,
44
2017; Danmaliki and Saleh, 2016; Archin et al., 2019; Tsibranska and Hristova, 2011; Álvarez-Gutiérrez et al., 2017; Sreńscek-Nazzal et al., 2015; AkankshaKalra et al., 2019; Riahi et al., 2017; Saadi et al., 2015; Devi and Murugappan, 2016; Tanzifi et al., 2018; Dastkhoon et al., 2017; Karimi et al., 2016; Karri and Sahu, 2018; Siyal et al., 2019), as shown in Fig. 3. Fig. 3 indicated that R2 and adjR2 are two of the most frequently calculated parameters. Over 60% of the literature has compared the values
ro of
of R2 and adjR2 to evaluate the kinetic models. The values of χ2, SSE, and HYBRID are
also calculated in some studies. The proportions of the above statistical parameters are
-p
11%, 11%, and 10%, respectively. Few studies adopted MSE for the evaluation of their models. However, the values of R2 of different kinetic models always have small
re
difference. For example, Delgado et al. (2019) reported that the values of R2 of the PFO
lP
and PSO models were 0.998 and 0.999, respectively. Burakova et al. (2018) found that the values of R2 of the PFO and the W&M models were 0.88 and 0.899, respectively.
ur na
Therefore, we recommend calculating more statistical parameters to evaluate the fitting
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results thoroughly and rigorously.
45
Fig. 3 Statistical parameters of the adsorption kinetic models
6. Solving methods for adsorption kinetic models The adsorption kinetic models reviewed in this paper are summarized in Table 6. In this paper, a convenient UI for solving the kinetic models was developed based on Excel (see in Supplementary Information). The Runge-Kutta method is employed to
ro of
solve the differential equations (such as the MO model and the Langmuir kinetics model), and the solver add-in is applied for the calculation of the model parameters.
The statistical parameters (R2, adjR2, 2, SSE, MSE, and HYBRID) are calculated based
-p
on Eqs. (67)-(72). Readers can download the UI attached in Supplementary Information
Table 6
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List of the adsorption kinetic models
re
and input their experimental data, then the fitting results will show in the sheet of Excel.
Differential equation
PFO model
𝑑𝑞𝑡 = 𝑘1 (𝑞𝑒 − 𝑞𝑡 ) 𝑑𝑡
PSO model
𝑑𝑞𝑡 = 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2 𝑑𝑡
MO model
𝑑𝑞𝑡 = 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) + 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )2 𝑑𝑡
-
𝑑𝜃 = 𝜕(1 − 𝜃)𝑛 𝑑𝑡
𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝜕𝑡 ) (n=1)
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Models
equation PNO model
References
𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝑘1𝑡 )
Lagergren, 1898
𝑞𝑡 =
𝑞𝑒2 𝑘2 𝑡 1 + 𝑞𝑒 𝑘2 𝑡
Ho et al., 1996 Guo and Wang, 2019a Ritchie, 1977 1
𝑞𝑡 = 𝑞𝑒 − 𝑞𝑒 [1 + (𝑛-1)𝜕𝑡]1−𝑛 (n≠1)
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Ritchie’s
Integrated form or nonlinear form
𝑑𝑞𝑡 = 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛 𝑑𝑡
𝑞𝑡 = 𝑞𝑒 (1 −
1
)
Özer, 2007
(1+(n-1)𝑞𝑒 n-1 𝑘𝑛 𝑡)1/(n-1) )
(n ≠ 1) Elovich model Boyd’s external diffusion
𝑑𝑞𝑡 = 𝑎𝑒 −𝑏𝑞𝑡 𝑑𝑡
𝑞𝑡 =
1 ln(1 + 𝑎𝑏𝑡) 𝑏
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝑅𝑡 )
𝑑𝑞𝑡 = 𝑅(𝑞∞ − 𝑞𝑡 ) 𝑑𝑡 46
Elovich
and
Larinov, 1962 Boyd et al., 1947
equation F&S model
-
𝐶𝑡 𝐶0
=
1 1+𝑚𝑠 𝐾
+
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚𝑠 𝐾 𝑘 𝑆𝑡 𝑚𝑠 𝐾 F&S
Frusawa
−
and
Smith, 1973
(Linear isotherm) M&W model
𝑑𝑞𝑡 𝑘𝑠 = (𝐶𝑡 − 𝐶et ) = 𝑘ext (𝐶𝑡 − 𝐶et ) 𝑑𝑡 𝜌
EMT model
𝐶𝑡 = 𝑒 −𝑘M&W𝑆𝑡 𝐶0
Mathews
-
Ruthven,
and
Weber, 1977 1984;
Hines
and
Maddox, 1985 Boyd’s
-
𝐵𝑡 = 2𝜋 −
𝜋2𝐹
F ≤ 0.85)
diffusion model
𝐵𝑡 = −ln W&M model
− 2𝜋 (1 −
𝜋2 6
3
)
(0 ≤
Boyd et al., 1947
(1 − 𝐹) (0.86 ≤ F ≤ 1)
𝑞𝑡 = 𝑘W&M 𝑡 1/2
-
𝜋𝐹 1/2
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intraparticle
3
Weber
and
𝜕𝑞𝑡 = 𝑘𝑖𝑛𝑡 (𝑞et − 𝑞𝑡 ) 𝜕𝑡
𝑉
𝑑𝐶𝑡 = −𝑚𝑆𝑝 𝑘𝐹 (𝐶𝑡 − 𝐶tr |r=r0 ) 𝑑𝑡
𝑡 = 0, 𝐶𝑡 = 𝐶0
-
Leyva-Ramos
Coates,
and 1947;
Yang, 1987
and Geankoplis, 1985
𝜕𝐶tr 𝜕𝑞𝑡 1 𝜕 𝜕𝐶tr 𝜕𝑞𝑡 + 𝜌𝑝 = 2 (𝑟 2 (𝐷𝑝 + 𝜌𝑝 𝐷𝑠 )) 𝜕𝑡 𝜕𝑡 𝑟 𝜕𝑟 𝜕𝑟 𝜕𝑟
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𝜀𝑝
Glueckauf
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PVSD model
-
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IMT model
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Morris, 1963
𝑡 = 0,0 ≤ 𝑟 ≤ 𝑟0 , 𝐶tr = 0 𝜕𝐶tr | =0 𝜕𝑟 𝑟=0 𝐷𝑝
𝑑𝑞𝑡 = 𝑘𝑎 𝐶𝑡 (𝑞𝑒 −𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡 𝑑𝑡
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Langmuir
𝜕𝐶tr 𝜕𝑞𝑡 |𝑟=𝑟0 + 𝜌𝑝 𝐷𝑠 | = 𝑘𝐹 (𝐶𝑡 − 𝐶tr )|𝑟=𝑟0 𝜕𝑟 𝜕𝑟 𝑟=𝑟0
kinetics model
-
Langmuir, 1918
-
Langmuir, 1918;
𝑑𝜃 = 𝑘𝑎 𝐶𝑡 (1 − 𝜃) − 𝑘𝑑 𝜃 𝑑𝑡
AAS model
𝑑𝑞𝑡 𝑚𝑞𝑡 𝑘𝑎 = 𝑘𝑎 (𝐶0 − ) (𝑞max − 𝑞𝑡 ) − 𝑞𝑡 𝑑𝑡 𝑉 𝐾𝐿
Thomas, 1944
Detailed instructions for this UI are explained as follows. (1) Download the UI in Supplementary Information. 47
(2) Open the UI, activate Solver Add-in. Then press Alt + F11 to get Visual Basic Editor, go to the Tools menu and select References from the drop-down menu. Select “Visual Basic For Applications”, “Microsoft Excel 16.0 Object Library”, “OLE Automation”, “Microsoft Office 16.0 Object Library”, “Microsoft Forms 2.0 Object Library”, “Solver”, and “Ref Edit Control”. (3) The model selection interface is shown in the Excel (Fig. 4 (left)). Readers can
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choose one kinetic model, and click the “OK” button. (If the model selection interface is not displayed, please enable the edit view.)
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(4) After click the “OK” button, the data input window is shown in the Excel (Fig. 4 (right)). Please input the adsorption time t and the experimental adsorption capacity
re
qt, as well as other data based on the kinetic models used, and click the “Fit” button.
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Please note that each t and qt should be separated by a comma. (Fig. 4 (right)) (5) The results are shown in “a1” sheet (Fig. (5)). If this UI fails to solve your
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kinetic model, please close the Excel and re-open it, and then try another initial value of the parameter. You may have to try several times to get the proper initial value of the parameters.
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(6) After solving one kinetic model, please record the parameters of the models and export the figure of the fitting results (Fig. 5) in a new Excel. Then close the UI and re-open it. You can choose another kinetic model for calculation.
48
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Fig. 4 The interface of the UI (left) and the data input window (right)
Fig. 5 Fitting results of the adsorption kinetic model.
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Several points should be noticed when using the UI for calculation: (1) In the data input window, if the background of the input position is yellow (Fig.
4 (right)), this value is related to the experimental data or the data calculated by the isotherms. These values will not change in the solving process. (2) For the Ritchie’s equation, the value of n is assumed to be larger than 1, because calculation errors are generated when 0 < n < 1. 49
(3) Eqs. (39) and (42) are used as the equations of the F&S and M&W models. (4) For the EMT, IMT, and AAS models, the Langmuir isotherm is adopted to describe the adsorption equilibrium phenomenon. (5) Please note that the PVSD and the Boyd’s intraparticle diffusion models are not included in this UI. Because the PVSD model is complicated and cannot be solved by the Excel software. And the Boyd’s intraparticle diffusion model is a piecewise
ro of
function, which can be simply solved by the linear regression method.
(6) The full names of the abbreviation of the kinetic models in this UI can be seen
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7. Concluding remarks and perspectives
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in the nomenclature table in this paper.
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This review aimed to study the physical meanings, applications, and solving methods of 16 adsorption kinetic models, including 6 adsorption reaction models and
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empirical models, 8 diffusional models, and 2 models for adsorption onto active sites. For the adsorption reaction models and empirical adsorption kinetic models, the most widely used PFO model and PSO model can be deduced by the Langmuir kinetics.
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The theoretical analysis and the review on literature indicated that the PFO model represented the conditions of (1) at high initial concentrations of adsorbate; (2) at the initial stage of adsorption, and (3) few active sites exist in the adsorbent material. In some cases, the PFO model can describe the diffusional kinetics process. The PSO model could represent the conditions of (1) at high initial concentrations of adsorbate; (2) at the final stage of adsorption, and (3) the adsorbent is abundant with active sites. 50
The PSO model reveals the mechanism of adsorption in active sites in most cases. The MO kinetics can describe the whole adsorption processes. The empirical Elovich and the PNO models are applied in some studies, but they are lacking of specific physical meanings and cannot offer important information about the mass transfer mechanisms. For the mass transfer models, the external diffusion process can be described by the Boyd’s external diffusion equation, F&S, M&W, and phenomenological EMT
ro of
models. These models have specific physical meanings (external diffusion), and they
are more suitable to model the external mass transfer processes. The Boyd’s
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intraparticle diffusion model, M&W, and phenomenological IMT models can represent
the intraparticle diffusion process. For the adsorption in active sites process, the
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Langmuir kinetics, Ritchie’s equation and the phenomenological AAS models can
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describe this process. These models can give the mass transfer mechanisms more accurately. The chosen of the kinetic models should be based on the mass transfer
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processes.
In addition, the statistical parameters R2, adjR2, 2, SSE, MSE, and HYBRID are widely used to evaluate the performance of the kinetic models. In which R2 and adjR2
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are the most frequently applied statistical parameters. To rigorously evaluate the fitting results of the kinetic models, we recommend calculating all the above statistical parameters.
The adsorption kinetic models include the nonlinear kinetic models (such as the nonlinear PFO, PSO models and the Elovich model) and the differential kinetic models
51
(such as the MO model, the phenomenological EMT, IMT, and AAS models, and the Langmuir kinetics model). The solving methods for these models are required. In this paper, an UI for solving the kinetic models is developed based on Excel solver add-in and the Runge-Kutta method, which can be used conveniently to solve the models reported in this review. In recent years, some other novel kinetic models, such as the Largitte double step
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model (Largitte and Pasquier, 2016), the film-pore mass transfer (FPMT) model (Guo
and Wang, 2019d), Gaulke’s unified kinetic model (Gaulke et al., 2016), and the fractal-
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like adsorption kinetic models (El Boundati et al., 2019) have also been proposed.
Although these novel models are not frequently used by researchers, they are expected
Declaration of interests
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to help to investigate the adsorption mechanisms, and to model the adsorption systems.
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Adsorption kinetic models: Physical meanings, applications, and solving methods
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Jianlong Wang 1,2 *, Xuan Guo 1
Acknowledgments The research was supported by the National Key Research and Development Program (2016YFC1402507) and the Program for Changjiang Scholars and Innovative Research Team in University (IRT-13026).
52
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PII:
S0304-3894(20)30144-8
DOI:
https://doi.org/10.1016/j.jhazmat.2020.122156
Reference:
HAZMAT 122156
To appear in:
Journal of Hazardous Materials
Received Date:
20 November 2019
Revised Date:
20 January 2020
Accepted Date:
20 January 2020
Please cite this article as: Wang J, Guo X, Adsorption kinetic models: Physical meanings, applications, and solving methods, Journal of Hazardous Materials (2020), doi: https://doi.org/10.1016/j.jhazmat.2020.122156
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Adsorption kinetic models: Physical meanings, applications, and solving methods Jianlong Wang 1,2 *, Xuan Guo 1 1 Collaborative Innovation Center for Advanced Nuclear Energy Technology, INET, Tsinghua University, Beijing 100084, P. R. China
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2 Beijing Key Laboratory of Radioactive Waste Treatment, Tsinghua University, Beijing 100084, P. R. China
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∗ Corresponding author
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Full post address:
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Tel.: +86 10 62784843
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Energy Science Building, INET, Tsinghua University, Beijing 100084, P. R. China
Fax: +86 10 62771150
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E-mail address: [email protected]
∗ Corresponding author. Tel.: +86 10 62784843; fax: +86 10 62771150. E-mail address: [email protected] 1
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Graphical abstract
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Highlights
Adsorption empirical kinetic models and mass transfer kinetic models were
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reviewed.
The physical meanings and applications of 16 adsorption kinetic models were analyzed.
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The model validity evaluation equations were summarized based on literature. A user interface for solving kinetic models was developed based on Excel software.
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Abstract 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
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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
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physical meanings, applications, as well as the solving methods of 16 adsorption kinetic
models. Firstly, the mathematical derivations, physical meanings and applications of
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the adsorption reaction models, the empirical models, the diffusion models, and the
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models for adsorption onto active sites were analyzed and discussed in detail. Secondly, the model validity evaluation equations were summarized based on literature. Thirdly,
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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.
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Key words: Adsorption; kinetic model; physical meaning; solving method
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Contents Abstract .................................................................................................................. 2 1. Introduction ........................................................................................................ 9 2. Adsorption reaction models and empirical models .......................................... 12 2.1 Pseudo-first-order (PFO) model ............................................................12 2.2 Pseudo-second-order (PSO) model ........................................................17 2.3 Mixed-order (MO) model ......................................................................22 2.5 Elovich model ........................................................................................24
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2.4 Ritchie’s equation and pseudo-nth-order (PNO) model .........................26
3. Diffusional models ........................................................................................... 30 3.1 External diffusion models ......................................................................30
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3.2 Internal diffusion models .......................................................................37 3.3 Pore volume and surface diffusion (PVSD) model................................41
re
4. Models for adsorption onto active sites (AAS)................................................ 42 4.1 Langmuir kinetics model .......................................................................42
lP
4.2 Phenomenological AAS model ..............................................................43 5. Model validity evaluation ................................................................................ 44
ur na
6. Solving methods for adsorption kinetic models............................................... 46 7. Concluding remarks and perspectives ............................................................. 50 Acknowledgments................................................................................................52
Jo
References ............................................................................................................ 53
4
Nomenclature Modified pineapple crown leaf
4-CP
4-chlorophenol
a
Initial adsorption rate constant of the Elovich model (mg g-1 h-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)
Cet Cf CIP
-p re
lP
Equilibrium adsorbate concentration (mg·L-1) Equilibrium adsorbate concentration at the surface of the adsorbent (mg·L-1) Adsorbate concentration in the liquid film (mg·L-1) Ciprofloxacin
Chitosan coated polyacrylonitrile nanofibrous mat
Jo
CPNM
ur na
Ce
ro of
(M) PCL
Ct
Adsorbate concentration at time t (mg·L-1)
Ctr
Adsorbate concentration at distant r, (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
5
Intraparticle diffusion coefficient (mm2·h-1)
DFC
Diclofenac
Dl
Diffusional constant
Dp
Effective pore volume diffusion coefficient (cm2·s-1)
Ds
Surface diffusion coefficient (cm2·s-1)
DY12
Direct yellow 12
E33
Bayoxide®E33
FTMA-MT
Surfactant-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)
ka
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&S kint KL
-p
re
lP
External mass transfer coefficient (cm·s-1) Mass transfer coefficient between the bulk liquid and the surface of the adsorbent (cm·h-1) Internal mass transfer rate constant (h-1) Langmuir constant (L·mg-1) Mass transfer coefficient (cm·h-1)
Jo
kM&W
ur na
kF
ro of
Def
kn
nth order rate constant (gn-1∙mg1-n∙min-1)
ks
External mass transfer coefficient (h-1)
KW&M
Intraparticle diffusion coefficient (mg·g·h-1/2)
m
Adsorbent mass (g)
MB
Methylene blue
6
Methyl blue
MG
Malachite green
MO1
Methyl orange
MOFs
UiO-66-type metal-organic frameworks
ms
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)
qe qet qexp
-p re
lP
Calculated adsorption capacity (mg·L-1) Adsorption capacity at equilibrium (mg·L-1) Equilibrium adsorption capacity in the pores of the adsorbent (mg·g-1) Experimental adsorption capacity (mg·L-1) Maximum adsorption capacity (mg·g-1)
Jo
qm
ur na
qcal
ro of
MB1
qmax
Langmuir constant (mg·g-1)
qmean
Average value of the experimental adsorption capacity (mg·L-1)
qt
Adsorbed amount of the adsorbate at time t (mg·L-1)
r
Distant of the adsorbate and the surface of the particle (cm)
R
Rate coefficient (h-1)
7
r0
Radius of the adsorbent particle (cm)
RB5G
Reactive blue dye 5G
RhB
Rhodamine B
ri
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
Sp
External surface area per mass of adsorbent (cm2·g-1)
t
Adsorption time (h)
tm
Modeling time
te
Adsorption equilibrium time
TA
Tannic acid
V
Solution volume (L)
α
nth order rate constant (h-1)
ur na
εp θ ρ
re
lP
Particle porosity
Void fraction of the adsorbent (g·cm-3) Occupied fraction of the active sites (0 ≤ θ ≤ 1) Density of the adsorbent particle (g·cm-3) Apparent density of the adsorbent particle (g·mL-1)
Jo
ρp χ2
ro of
Correlation coefficient
-p
R2
Chi-square
8
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., 2018). The adsorption kinetic study provides
ro of
information of the adsorption rate, the performance of the adsorbent used, and the mass transfer mechanisms. Knowing the adsorption kinetic is essential for the design of the adsorption systems. The adsorption mass transfer kinetic includes three steps, as shown
-p
in Fig. 1. The first step is the external diffusion. In this step, the adsorbate transfers
re
through the liquid film around the adsorbent. The concentrations difference between
lP
the bulk solution and the surface of the adsorbent are the driving force of the external diffusion. The second step is the internal diffusion. The internal diffusion describes the
ur na
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. Adsorbate
Jo
Liquid film
Mass transfer steps
1 1: External diffusion
32
2: Internal diffusion
Pores Active sites
3: Adsorption on active sites Adsorbent
Fig. 1 Adsorption mass transfer steps. 9
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 mixedorder (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 adsorption kinetic process.
ro of
However, some problems are existed in the applications of these kinetic models. The first one is that the most applied PFO and PSO models are empirical models and lack
-p
of specific physical meanings. We cannot investigate the mass transfer mechanisms by these empirical kinetic models. Therefore, the physical meanings of the empirical
re
kinetic models should be established. The second one is that the differential kinetic
lP
models, such as the phenomenological external/internal and adsorption in active sites models have specific physical meanings, but the solving methods are complicated. Few
ur na
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
Jo
models. The third one is that in some published papers, the kinetic models are applied or solved in inappropriate manner. For example, the Frusawa and Smith (F&S) model can only be used for modeling the adsorption process that the isotherm is linear (Frusawa and Smith, 1973). However, it has been applied for the adsorption process that the isotherm is Langmuir-type (Özer et al., 2005; Wang et al., 2008).
10
In addition, the linear regression method is the most widely applied method in the calculations of the model parameters, owing to its simplicity (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 variables. This process could introduce propagate errors to the independent/dependent variables, and could cause inaccurate estimations of the parameters (El-Khaiary et al.,
ro of
2010; Ho, 2006; Kumar and Sivanesan, 2006; Guo and Wang, 2019c). The nonlinear
method can provide consistent and accurate estimations for model parameters (El-
-p
Khaiary et al., 2010). It is of great significance to provide the nonlinear solving method for the adsorption kinetic models.
re
Based on the above, the physical meanings, applications, and solving methods of
lP
16 adsorption kinetic models were thoroughly studied in this paper. The objectives of this review paper were: (1) to describe the mathematical
ur na
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 summarize the applications of the adsorption kinetic models based
Jo
on literature; (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.
11
2. Adsorption reaction models and empirical models 2.1 Pseudo-first-order (PFO) model The PFO model was firstly proposed by Lagergren (1898). The differential form of the PFO model is described by Eq. (1) (Lagergren, 1898): 𝑑𝑞𝑡 𝑑𝑡
= 𝑘1 (𝑞𝑒 − 𝑞𝑡 )
(1)
Integrating Eq. (1) for the conditions of q0=0 yields:
ro of
𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝑘1 𝑡 )
(2)
The linearized form of the PFO model is presented as follows. ln(𝑞𝑒 − 𝑞𝑡 ) = ln 𝑞𝑒 − 𝑘1 𝑡
-p
(3)
Eq. (3) has been frequently used to fit the kinetics data and to calculate the
re
parameters qe and k1, by plotting ln(qe-qt) vs. t (Ma et al., 2018; Ersan et al., 2019;
lP
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
ur na
linearization process may cause inaccurate estimations of the parameters (El-Khaiary et al., 2010; Ho, 2006; Kumar and Sivanesan, 2006). The nonlinear method, which can provide accurate estimations for model parameters, is provided in the following section.
Jo
The PFO parameter qe is the equilibrium adsorption amount estimated 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 represented by the linear model (Eq. (4)). 𝑞𝑒 = 𝐾𝐶𝑒
(4)
12
The PFO parameter k1 is frequently used to describe how fast the adsorption equilibrium is achieved (Plazinski et al., 2009). However, as shown in Eq. (1), the adsorption rate dqt/dt is related to both k1 and (qe-qt). Small value of k1 and 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 of k1. PFO rate = 𝑘1 (𝑞𝑒 − 𝑞𝑡 )
ro of
(5)
The PFO model used to be considered as empirical model for a long time. Azizian
-p
(2004) deduced the PFO model from the Langmuir kinetics model (Eq. (6)) (Langmuir, 1918), and analyzed the physical meanings of this model. 𝑑𝑡
= 𝑘𝑎 𝐶𝑡 (1 − 𝜃) − 𝑘𝑑 𝜃
re
𝑑𝜃
(7)
𝑚𝑞𝑚 𝑉
𝜃
ur na
𝐶𝑡 = 𝐶0 − 𝛽𝜃 = 𝐶0 −
lP
Ct can be given by Eq. (7):
(6)
Substitution of Eq. (7) into Eq. (6) yields: 𝑑𝜃 𝑑𝑡
= 𝑘𝑎 (𝐶0 −
𝑚𝑞𝑚 𝑉
𝜃) (1 − 𝜃) − 𝑘𝑑 𝜃
(8)
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Effort has been made to study the physical meanings and the applications 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. 𝑑𝜃 𝑑𝑡
= 𝑘𝑎 𝐶0 (1 − 𝜃) − 𝑘𝑑 𝜃
(9)
Liu and Shen (2008) also concluded that the simplification of the Langmuir
13
kinetics model to the PFO and PSO models was C0-dependent. Zhang (2019) suggested that the conditions for the PFO model were: (1) C0/β approached to zero; (2) C0/β approached to infinite; and (3) kaβ/kd approached to zero. Our previous research (Guo and Wang, 2019a) concluded that mqm/V was approximately a constant for a given adsorption process, therefore when a few active sites were occupied compared with C0, mqmθ/V could be neglect. Three conditions could satisfy this hypothesis, as presented
ro of
in Fig. 2.
The first one is that the value of C0 is high. Table 1 summarizes the applications
-p
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
re
BBLS-CA, resorcinol onto CTAB/NaOH/flyash composites, Cu (II) onto RSP and
lP
SMSP could be better fitted by the PFO model than the PSO model at high C0 (Darwish et al., 2019; Khan et al., 2019; Ho and Mckay, 1998; Stähelin et al., 2018; Agarwal and
ur na
Rani, 2017; Sari et al., 2017; Gunasundari and Kumar, 2017). Siyal et al. (2017) investigated the adsorption of anionic surfactant onto fly ash based geopolymer, concluded that when C0 increased from 100 mg·L-1 to 880 mg·L-1, the values of the
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correlation coefficient (R2) of the PFO model increased from 0.0035 to 0.937. Ma et al. (2008) reported that the values of R2 of the PFO model increased from 0.7556 to 0.9587, with the increase of C0 of Congo Red from 100 mg·L-1 to 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 to7.49×10-6 mol·L-1, the values of R2 of the PFO model increased from
14
0.9704 to 0.9886 (Azizian e al., 2009). The second one is that the adsorption process is in the initial stage. Hu et al. (2018) found that when t approached 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
ro of
external diffusion or the internal diffusion is the rate controlling step. The adsorption of metals ions and hydrophilic compounds onto microplastics could be represented by
-p
the PFO model (Guo et al., 2019a; Turner and Holmes, 2015). One possible reason is that microplastics are hydrophobic compounds, and the diffusion of hydrophilic
re
compounds to the surface of the hydrophobic microplastics is difficult. Therefore, the
lP
external/internal diffusion is the rate limiting step. However, the adsorption of hydrophobic organic compounds (such as lubrication oil and polybrominated diphenyl
ur na
ethers) onto microplastics 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 adsorption onto active sites
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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 in Table 1, the majority of literature modeled the whole adsorption process instead of modeling the initial stage of the adsorption. Therefore, the conditions
15
for the PFO model are at high initial concentration of adsorbate and the adsorption is not controlled by the adsorption in active sites. In some cases, the PFO model could
lP
re
-p
ro of
represent the external/internal diffusion.
Fig. 2 Physical meanings of the PFO and PSO models
ur na
Table 1
Adsorbate
tm/t
te
C0 (mg·L-1)
Reference
BB69
0–250 min/250 min
250 min*
500
Ho and Mckay, 1998
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Applications of the PFO model
AB25
0–250 min /250 min
–
200
MO1
0–1440 min/ 1440 min
540 min
200
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
Adsorbent Peat Peat NiO NPs
16
Darwish et al., 2019
Resorcinol
0–24 h/ 24 h
3 h*
200
Agarwal and Rani, 2017
RSP
Cu(II)
0–150 min/150 min
110 min*
300
Gunasundari and Kumar,
SMSP
Cu (II)
0–150 min/150 min
60 min*
500
2017
Peat
Pb(II)
0–20 min/90 min
90 min*
504
Ho and Mckay, 1999
Peat
Pb(II)
0–20 min/90 min
30 min*
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*
5×10–3
Turner and Holmes, 2015
PE
Cu(Ⅱ)
0–168 h/168 h
30 h*
PE
Ni(Ⅱ)
0–168 h/168 h
30 h*
-p
*: Obtained from the figures in references.
ro of
CTAB/NaOH/flyash
lP
2.2 Pseudo-second-order (PSO) model
re
-: Not mentioned.
The PSO model (Eq. (10)) was firstly applied to model the adsorption of lead onto
ur na
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. 𝑑𝑞𝑡
= 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2
(10)
Jo
𝑑𝑡
The integrated PSO model is described as following:
𝑞𝑡 =
𝑞𝑒2 𝑘2 𝑡
(11)
1+𝑞𝑒 𝑘2 𝑡
In order to calculate the model parameters, the nonlinear PSO model is always transformed to the linear form (Eq. (12)). 𝑡 𝑞𝑡
=
1 𝑘2 𝑞𝑒2
+
𝑡
(12)
𝑞𝑒
17
The linearization of the PSO model changes the weight of qt, and introduces propagated errors, which leads to the inaccurate calculations of the model parameters (El-Khaiary et al., 2010; Ho, 2006; 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 constant k1, the PSO rate constant k2 is also used to describe the rate of adsorption equilibrium (Plazinski et al., 2009). But the adsorption
ro of
rate dqt/dt is related to both k2 and (qe-qt)2. Thus, it is more precise to calculate the PSO rate by Eq. (13). PSO rate = 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2
-p
(13)
The physical meanings of the PSO model have been investigated. Azizian (2004)
re
found that when C0 was low, the Langmuir kinetics model (Eq. (8)) could be simplified
lP
to the PSO model. Liu (2008) suggested that the PSO model was related to the vacant active sites. Zhang (2019) reported that the conditions for the PFO model were C0/β
ur na
approached to 1 and kaβ/kd approached to infinite. Marczewski (2010a) found that the conditions for the PSO model were the equilibrium surface coverage fraction θe approached to 1 and C0/β approached to 1. Our previous study (Guo and Wang, 2019a)
Jo
concluded that the PSO model could represent three conditions (Fig. 2). The first one is that the value of C0 is low. Sabarinathan et al. (2019) studied the
adsorption of MB onto molecular polyoxometalate, found that the values of R2 of the PSO model decreased from 0.8979 to 0.5905 with the increase of C0 from 140 mg·L-1 to 300 mg·L-1. The applications of the PSO model in literature are summarized in Table
18
2. As shown in Table 2, at low initial concentrations of adsorbate, the adsorption kinetics 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 are better modeled by the PSO model (Ho and Mckay, 1998; Darwish et al., 2019; Khan et al., 2019; Stähelin et
Table 2 Applications of the PSO model
C0 (mg·L-1)
Adsorbate
tm/t
Peat
BB69
0–250 min /250 min
Peat
AB25
0–250 min /250 min
NiO NPs
MO1
0–1440 min/ 1440 min
540 min
50
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
0–24 h/ 24 h
3 h*
50
Agarwal
Resorcinol
30 min*
50
–
20
re
lP
ur na
CTAB/NaOH/flyash
-p
Adsorbent
RSP
te
ro of
al., 2018; Sari et al., 2017; Agarwal and Rani, 2017; Gunasundari and Kumar, 2017).
Reference
Ho and Mckay, 1998
Darwish et al., 2019
and
Rani,
2017
Cu(II)
0–150 min/150 min
100 min*
50
Gunasundari
Cu(II)
0–150 min/150 min
50 min*
50
Kumar, 2017
Pb(II)
20–90 min/90 min
90 min*
504
Ho and Mckay, 1999
Pb(II)
20–90 min/90 min
30 min*
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*
–
Hu et al., 2017
PS
lubrication oil
0–48 h/48 h
12 h*
–
PE
BDE-99
0-24 h/24 h
30 min
0.5
Peat Peat
Jo
SMSP
19
Xu et al., 2019
and
Al-modified biochar
Nitrate
0–24 h/24 h
24 h
50
Al-modified biochar
Phosphate
0–24 h/24 h
24 h
50
Modified clay
MO1
0–4 h/4 h
2h
200
Gamoudi and Srasra,
Modified clay
IC
0–4 h/4 h
1h
200
2019.
Modified clay
PR
0–4 h/4 h
1h
200
Ag(Ⅰ)
0–150 min/150 min
20 min
122
H. Wang et al., 2018
Cr(VI)
0–180 min/180 min
120 min*
20 and 30
Gogoi et al., 2018
Cr(III)
0–180 min/180 min
120 min*
20 and 30
Modified chitosan
Re(VII)
0–24 h
–
20
Lou et al., 2018
FTMA-MT
Phenol
0–21 h
7h
–
Li et al., 2018
Modified hydrochar
Pb(II)
0–360 min/360 min
240 min*
10
Xia et al., 2019
Modified-chitosan
Cr(VI)
0–150 min/150 min
P-CSs
MB
0–300 min/300 min
Modified mesoporous
Steroid estrogens
0–90 min/90 min
Modified
titanate
Yin et al., 2018
40–100
Islam, 2019
140 min
30–90
Tang et al., 2019
30 min
2
Gao et al., 2019
0–300 min/300 min
150 min
600–1000
Li et al., 2019
0–300 min/300 min
30 min
40–90
modified
Cd(II)
ur na
hydrochar
MB
lP
silica Maleylated
-p 90 min
re
(M) PCL
ro of
nanotubes
Modified bentonite
RhB
0–70 min/70 min
40 min
100
Modified bentonite
AR1
0–70 min/70 min
90 min
100
Cobalt
MG
0–70 min/70 min
40 min
100
Amiri et al., 2017
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
ferrite
silica
magnetic
Jo
nanocomposite Modified
Algerian
Huang et al., 2017
geomaterial
20
Modified
Cu(II)
0–90 min/90 min
50 min
200
chitosan/CoFe2O4
Pb(II)
0–90 min/90 min
50 min
200
Modified
Sr(II)
0–10 h/10 h
10 h
100
Cheng et al., 2019
U(VI)
0–180 min/180 min
180 min
480
Zhuang et al., 2018
DFC
0–500 min/500 min
200 min
80
Zhuang et al., 2019
As(III)
0–540min/540 min
300 min
10 and 30
Zhang et al., 2016
E33
Phosphate
0–294 h/294 h
294 h*
140
E33/Mn
Phosphate
0–294 h/294 h
294 h*
140
E33/Ag I
Phosphate
0–294 h/294 h
294 h*
140
E33/Ag II
Phosphate
0–294 h/294 h
NiO NPs
MO1
0–1440 min/ 1440 min
BBLS-CA
MB
0–360 min/360 min
Bacterial
Fan et al., 2017
cellulose membrane Magnetic
amidoxime
functionalized chitosan MOFs Cerium
modified
-p
ro of
chitosan
Lalley et al., 2016
140
540 min
200
Darwish et al., 2019
500
Sari et al., 2017
re
294 h*
90 min
-: Not mentioned.
lP
*: Obtained from the figures in references.
ur na
The second one is at the final stage of the adsorption process. As shown in Table 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
Jo
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%. 21
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 of R2 of the PSO model was 0.990, while for hydrochar, this value was 0.945 (Xia et al., 2019). As listed in Table 2, the adsorption of pollutants on modified materials, such as Al-modified biochar, modified clay, modified titanate nanotubes, modified pineapple crown leaf, modified chitosan, surfactant-modified
ro of
montmorillonite, modified hydrochar, H3PO4 modified corn stalks, modified
mesoporous silica, modified bentonite, cobalt ferrite silica magnetic nanocomposite,
-p
modified Algerian geomaterial, modified wheat bran, modified hydrogel, modified
bacterial cellulose membrane, magnetic amidoxime functionalized chitosan and MOFs
re
could be better modeled by the PSO model (Yin et al., 2018; Gamoudi and Srasra, 2019.;
lP
H. Wang et al., 2018; Gogoi et al., 2018; Lou et al., 2018; Li et al., 2018, 2019; Xia et al., 2019; Islam, 2019; Tang et al., 2019; Gao et al., 2019; Huang et al., 2017; Amiri et
ur na
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 kinetics is dominated by the adsorption onto
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active site.
2.3 Mixed-order (MO) model The mixed-order (MO) model has the following form (Guo and Wang, 2019a):
𝑑𝑞𝑡 𝑑𝑡
= 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) + 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )2
(14)
The PFO and PSO rate of the MO model can be calculated by Eqs. (15)–(16).
22
PFO rate' = 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) (15) PSO rate' = 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )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
ro of
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
-p
controlling step is the diffusion or the adsorption; and (3) arbitrary initial adsorbate concentration in solution (Guo and Wang, 2019a).
re
Eq. (14) is similar in the form to the hybrid-order’ (HO) model (Eq. (17)) provided
by Marczewski (2010b). 𝑑𝑡 𝑑𝐹 𝑑𝑡
= 𝑘L1 (𝜃𝑒 − 𝜃𝑡 ) + 𝑘L2 (𝜃𝑒 − 𝜃𝑡 )2
(17)
= 𝑘1𝑎 (1 − 𝐹𝐿 ) + 𝑘2𝑎 𝑎eq (1 − 𝐹𝐿 )2
(18)
ur na
𝑑𝜃𝑡
lP
in Liu and Shen (2008), and the mixed 1, 2-order rate equation (MOE) (Eq. (18)) derived
Where kL1, kL2, k1a, aeq, and k2a are coefficients.
Jo
As described in Eq. (17), the HO model is the relationship between θt and t, 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 parameters. Therefore, the MO model presented by Guo and Wang (2019a) is
23
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
ro 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
-p
in the following section. 2.5 Elovich model
re
The basic assumptions of the Elovich model were (1) the activation energy
lP
increased with adsorption time and (2) the surface of the adsorbent was heterogeneous. The Elovich model is an empirical model without definite physical meanings. It is
ur na
commonly used to model the chemisorption of gas onto solid. The Elovich model has been described by Eq. (19) (Elovich and Larinov, 1962): 𝑑𝑞𝑡 𝑑𝑡
= 𝑎𝑒 −𝑏𝑞𝑡
(19)
Jo
Integrating Eq. (19) for the condition of q0 = 0 yields: 1
𝑞𝑡 = ln(1 + 𝑎𝑏𝑡)
(20)
𝑏
Eq. (20) is a nonlinear equation, which can be solved by the nonlinear least square
regression method (plot qt versus t) or by the linear method (plot qt versus ln(1+abt)). However, the nonlinear method is more complex than the linear method. And the linear
24
method needs to give the appropriate initial values of ab, which is difficult for the researchers (Ho, 2006). Chien and Clayton (1980) simplified Eq. (20) with the assumption of abt >> 1: 1
1
1
𝑏
𝑏
𝑏
𝑞𝑡 = ln(𝑎𝑏𝑡) = ln(𝑎𝑏) + ln(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 represent the adsorption of liquid-solid
ro of
systems. The applications of the Elovich model are summarized in Table 3. Eq. (21) is
the most frequently applied form of the Elovich model, which has been successfully
-p
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
re
parameters (Lin et al., 2018; Brito et al., 2018; Kalhor et al., 2018). However, the
lP
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
ur na
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 calculate the Elovich rate: Elovich rate = 𝑎𝑒 −𝑏𝑞𝑡
Jo
(22)
Table 3
Applications of the Elovich model Adsorbent
Adsorbate
Elovich model
Ref.
Alfisol
Tetracycline
Eq. (21)
Bao et al., 2010
25
Ultisol
Tetracycline
Eq. (21)
Contaminant barrier minerals
Pb(Ⅱ)
Eq. (21)
Contaminant barrier minerals
Cd(Ⅱ)
Eq. (21)
β-cyclodextrin-based adsorbent
Cu(Ⅱ)
Eq. (21)
Huang et al., 2012
Synthesized TiO2 nanoparticles
Hg(II)
Eq. (21)
Dou et al, 2011
TiO2/montmorillonite
Hg(II)
Eq. (21)
Surfactant-Modified Natural Zeolites
Benzene
Eq. (21)
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, 2014
Gl-PZSNTs-Fe3O4
MB
Eq. (21)
Wang et al., 2019
Honeycomb-like porous activated carbon
Cu(II)
Modified eggshell membrane
MO1
Modified eggshell membrane
MB1
Anodized iron oxide nanoflakes
Phosphate
Inyang et al., 2016
-p
ro of
Seifi et al., 2011
Mondal and Majumder, 2019
Eq. (21)
Candido et al., 2019
re
Eq. (21)
Afridi et al., 2019
DY12
Eq. (21)
Alinejad-Mir et al., 2018
BB69
Eq. (21)
Magdy and Altaher, 2018
Cr(VI)
Eq. (20)
Lin et al., 2018
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
Iron oxide-gelatin nanoadsorbent Cement kiln dust
Carbons
Jo
Modified rice straw
lP
Eq. (21)
ur na
Eq. (21)
2.4 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 26
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): 𝑑𝜃 𝑑𝑡
= 𝛼(1 − 𝜃)𝑛
(23)
After integration with the initial condition of q0 = 0, Ritchie’s equation becomes: For n=1, 𝜃 = 1 − 𝑒 −𝛼𝑡
(24)
ro of
For n≠1, 1
𝜃 = 1 − [1 + (𝑛 − 1)𝛼𝑡]1−𝑛
(25)
-p
θ can be described by the ratio of qt and q∞ (q∞ can be obtained from the adsorption isotherm model (q∞ = f(Ce)), Eq. (24) and Eq. (25) becomes:
re
For n=1,
For n≠1,
lP
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝛼𝑡 )
(26)
1
ur na
𝑞𝑡 = 𝑞∞ − 𝑞∞ [1 + (𝑛 − 1)𝛼𝑡]1−𝑛
(27)
When n=2, rearrangement of Eq. (27) yields: 𝑞𝑡 =
𝛼𝑞∞ 𝑡
(28)
1+𝛼𝑡
Jo
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 in Table 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
27
the RSO model to the adsorption data of CO2 onto piperazine-modified activated alumina. B. 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. 𝑑𝑞𝑡 𝑑𝑡
= 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛
(29)
ro of
For n = 1, the PNO model equals to the PFO model. After integration with the initial condition of q0 = 0 and n ≠ 1, the PNO model becomes: (𝑞𝑒 − 𝑞𝑡 )1-n = (n − 1)𝑘𝑛 𝑡 + 𝑞𝑒 1−n
-p
(30)
𝑞𝑡 = 𝑞𝑒 (1 −
1 (1+(n−1)𝑞𝑒
n−1 𝑘
𝑛 𝑡)
)
1/(n−1) )
re
Rearrangement of Eq. (30) yields:
(31)
lP
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
ur na
equation without specific physical meanings. The applications of the PNO model are summarized in Table 4. Caroni et al. (2009) indicated that values of n of tetracycline adsorption on chitosan particles increased with the increase of the initial concentrations
Jo
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 decreased to 0.979 at high C0 (6 mg·L-1). Leyva-Ramos et al. (2010) suggested that the values of n were not correlated with the experimental conditions, such as pH, C0, impeller speeds, and so on. The nonlinear form of the PNO model cannot be written as the linear form.
28
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 proposed for the calculation of the PNO rate: PNOrate = 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛
ro of
(32)
-p
Table 4 Applications of the Ritchie’s equation and the PNO model Adsorbent
Adsorbate
C0 (mg·L-1)
Bone char
Cd(Ⅱ) CO2
n
Ref.
224-560
RSO
2
Cheung et al., 2001
-
RSO
2
Fashi et al., 2018
50
Ritchie’s equation
-
B. Wang et al., 2018
MB
50
Ritchie’s equation
-
MB
50
Ritchie’s equation
-
Alkanes
-
RSO
2
Phthalonitrile compound
Alcohols
-
RSO
2
Phthalonitrile compound
Chlorinated
-
RSO
2
Macroporous ion-exchange resins
Al(Ⅲ)
77.5
PNO
1.17-1.47
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
activated
Alumina
Ball-milled biochar
MB
ur na
Calcium alginate
Ball-milled biochar encapsulated in calcium-alginate beads
lP
Piperazine-modified
Jo
Phthalonitrile compound
re
Model
29
Günay et al., 2018
Nekouei et al., 2019
Macroporous ion-exchange resins
Pb(Ⅱ)
183
PNO
0.998-1.36
Pure açaí biochar
MB
50
PNO
1.65
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, 2018
Chitosan spheres
MO1
-
PNO
2-2.5
Morais, 2008
Chitosan particles
Tetracycline
15*
PNO
4.5*
Caroni et al., 2009
Chitosan particles
Tetracycline
55*
PNO
16*
Montmorillonite-carbon hybrids
17β-Estradiol
6
PNO
0.979
Montmorillonite-carbon hybrids
17β-Estradiol
0.2
PNO
Bone char
fluoride
5.44
PNO
Bone char
fluoride
8.35
PNO
Carbons prepared from betel trunk
Phenol
-
Carbons prepared from betel trunk
MB
-
Carbons prepared from betel trunk
Basic brown 1
-
Carbons prepared from betel trunk
Acid
ro of
-p
1.3
Leyva-Ramos, 2010
2.0
PNO
1.50
PNO
3.51
re
-
lP
Carbons prepared from betel trunk
74
Liu et al., 2019
2.276
PNO
3.06
PNO
1.77
2,4-dichloropenol
-
PNO
1.63
ur na
(AB74)
blue
Pessôa et al., 2019
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
Tseng et al., 2014
Izadi et al., 2017
Jo
*: Obtained from the figures in references. -: Not mentioned.
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. Several equations have been 30
developed to model the external mass transfer process.
3.1.1 Boyd’s external diffusion equation
Boyd et al. (1947) deduced a kinetic model to describe the diffusion of adsorbate through a bounding liquid film: 𝑑𝑞𝑡 𝑑𝑡
= 4𝜋𝑟0 2 𝐷𝑙 (𝜕𝐶𝑓 /𝜕𝑟)𝑟=𝑟
(33)
0
ro of
Assuming that the concentration gradient is linear, Eq. (33) is simplified to Eq. (34) (Boyd et al., 1947): 𝑑𝑞𝑡 𝑑𝑡
= 𝑅(𝑞∞ − 𝑞𝑡 )
(34)
re
by the adsorption isotherm model (q∞ = f(Ce)).
-p
Where q∞ = 4πr03Clk/3 (k is the distribution coefficient). q∞ can also be obtained
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝑅𝑡 )
lP
Integration of Eq. (34) at initial condition of q0=0 yields (Boyd et al., 1947): (35)
ur na
This equation is similar with the PFO model. Ahmed and Theydan (2013) studied the adsorption kinetics of metronidazole onto microporous activated carbon using the linear form of Boyd’s equation (Eq. (36)), the results showed that the values of R2 of
Jo
Boyd’s equation ranged from 0.9471 to 0.9885. ln( 1 − 𝑞𝑡 /𝑞∞ ) = −𝑅𝑡 + 𝐴
(36)
Table 5 Applications of the diffusional models
31
Diffusional
Models name
Adsorbent
Adsorbate
Reference
External
Boyd’s
Microporous activated carbon
Metronidazole
Ahmed and Theydan, 2013
diffusion
diffusion equation
models
F&S model
Inactive
Azo dye
Wang et al., 2008
Özer et al., 2005
models external
CMC
immobilized
Aspergillus fumigatus beads Enteromorpha prolifera
AR337
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
Azo dye
Wang et al., 2008
CMC
-p
ro of
F&S model
immobilized
re
Aspergillus fumigatus beads Enteromorpha prolifera
AR337
M&W model
Enteromorpha prolifera
AB324
M&W model
Bone char
Cu (Ⅱ)
M&W model
Bone char
Cd(Ⅱ)
ur na
lP
M&W model
M&W model
Bone char
Zn(Ⅱ)
M&W model
Clinoptilolite rich mineral
Cr(VI)
M&W model
Bacteria loaded clinoptilolite rich
Cr(VI)
Özer et al., 2005
Choy et al., 2004
Erdoğan and Ulku, 2012
Jo
mineral
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
32
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)
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
Boyd’s intraparticle
Mansonia wood sawdust
Pb(II)
Ofomaja, 2010
diffusion
diffusion model
models
Boyd’s intraparticle
Dolomite
Cr(VI)
Albadarin et al., 2012
ro of
-p
diffusion model Boyd’s intraparticle
PMMA
Phenol
Starchy adsorbents
Okewale et al., 2013
MG
Sartape et al., 2017
Pre-treated bentonite clay
Ag(Ⅰ)
Cantuaria et al., 2016
Mesoporous activated carbon
Cd(Ⅱ)
Tan et al., 2016
Verde-lodo bentonite
Ag(Ⅰ)
Freitas et al., 2017
Verde-lodo bentonite
Cu(II)
lP
Wood apple shell
ur na
diffusion model Boyd’s intraparticle
Al-Muhtaseb et al., 2011
Water
diffusion model Boyd’s intraparticle
re
diffusion model Boyd’s intraparticle
Suzaki et al., 2017
diffusion model
Boyd’s intraparticle diffusion model
Jo
Boyd’s intraparticle diffusion model Boyd’s intraparticle diffusion model Boyd’s intraparticle
Amine-functionalized
diffusion model
mesoporous alumina
ordered
33
MB
Gan et al., 2015
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)
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
IMT model
Dowex Optipore SD-2
-p
Boyd’s intraparticle
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)
diffusion model
ro of
AB 25
ur na
lP
re
RB5G
IMT model
Fixed-bed columns
Cu(II)
IMT model
Fixed-bed columns
Ni(II)
IMT model
Fixed-bed columns
Zn(II)
IMT model
Microplastics
SMX
Guo et al., 2019b
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
Jo
PVSD model
Apiratikul and Pavasant, 2008
34
Suzaki et al., 2017
PVSD model
Biochar
Ibuprofen
Ocampo-Perez et al., 2019
3.1.2 Frusawa and Smith (F&S) model
Frusawa and Smith (1973) developed an adsorption rate equation: 𝐶𝑡 𝐶0
=
1 1+𝑚𝑠 𝐾
+
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚 𝐾
𝑠 − 𝑚 𝐾 𝑘F&S 𝑆𝑡 𝑠
(37)
The value of kF&SS is frequently used to describe the external diffusional process
ro of
(Özer et al., 2005, Panday et al., 1986) The F&S model assumes that the external diffusion is the slowest step, the
intraparticle diffusion is negligible, and the isotherm is linear (Eq. (4)) (Frusawa and
-p
Smith, 1973). Note that Eq. (37) cannot be used for modeling the adsorption condition
re
that the isotherm is not linear.
The applications of the F&S model are summarized in Table 5. Unfortunately,
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most studies applied Eq. (37) to their adsorption kinetics data when the isotherm is
ur na
described by the Langmuir isotherm model (Özer et al., 2005; Wang et al., 2008; Öztürk and Malkoc, 2014; Fu et al., 2012, 2015). In order to establish the relationship between qt and t, Eq. (38) is brought into Eq. (37), and the result is presented in Eq. (39): 𝑉
(𝐶0 − 𝐶𝑡 ) =
Jo
𝑞𝑡 = 𝑞𝑡 =
𝑚
𝐶0
𝑚𝑠
(1 −
1
1+𝑚𝑠 𝐾
(𝐶0 −𝐶𝑡 )
(38)
𝑚𝑠
−
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚 𝐾
𝑠 − 𝑚 𝐾 𝑘F&S 𝑆𝑡 𝑠 )
(39)
Eq. (39) can be solved by the nonlinear regression method, which is provided in the following section.
35
3.1.3 Mathews and Weber (M&W) model
The Mathews and Weber (M&W) model is presented as follows (Mathews and Weber, 1977). 𝑘𝑀&𝑊 =
𝑟0 𝜌(1−𝜀) 𝑙𝑛(𝐶0 /𝐶𝑡 ) 3𝑚𝑠
(40)
𝑡
The coefficient item 3ms/(r0(1-)) can be replaced by the outer surface coefficient
𝐶𝑡 𝐶0
ro of
S (cm-1). Rearrangement of Eq. (40) yields: = 𝑒 −𝑘M&W 𝑆𝑡
(41)
The value of kM&WS is calculated to describe the external diffusion. Eq. (41) is the
-p
most frequently used form of the M&W model. As summarized in Table 5, the M&W
re
model has been used to describe the adsorption kinetics data of dyes and metals onto biosorbent and mineral (Wang et al., 2008, Choy et al., 2004; Erdoğan and Ulku, 2012;
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Dávila-Guzmán et al., 2013; Šljivić et al., 2011). Combining Eq. (38) and Eq. (41) yields: 𝐶0
(1−e−𝑘M&W 𝑆𝑡 )
ur na
𝑞𝑡 =
𝑚𝑠
(42)
Eq. (42) can be solved by the nonlinear regression method.
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3.1.4 Phenomenological external mass transfer (EMT) model
The EMT model (Eq. (43)) assumes that the film diffusion is the slowest step, and
the equilibrium is obtained on the surface of the adsorbent (Ruthven, 1984; Hines and Maddox, 1985). The driving force of the external diffusion is the concentration gradient of adsorbent in the liquid film.
36
𝑑𝑞𝑡 𝑑𝑡
=
𝑘𝑠 𝜌
(𝐶𝑡 − 𝐶et ) = 𝑘ext (𝐶𝑡 − 𝐶et )
(43)
Then the equilibrium phenomenon is described by the adsorption isotherm model. The isotherm model is presented as qet = f(Cet). Then Cet = f-1(qet). For example, if the Langmuir model can adequately represent the adsorption isotherm, the equilibrium phenomenon is described as following: 𝑞𝑡 =
𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶et
(44)
1+𝐾𝐿 𝐶et
𝑑𝑞𝑡 𝑑𝑡
= 𝑘ext (𝐶0 −
𝑚𝑞𝑡 𝑉
−
𝑞𝑡 𝑞max 𝐾𝐿 −𝑞𝑡 𝐾𝐿
ro of
Substitution of Eqs. (38) and (44) into Eq. (43) yields: )
(45)
-p
The initial conditions of the EMT model are q0 = 0 and Cet(0) = 0. The EMT model can be solved by the Runge-Kutta method. The solving method is provided in the
re
following section.
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As shown in Table 5, the adsorption processes of dyes onto Dowex Optipore SD2 and SB biomass, as well as metals ions onto argassum horneri and fixed-bed columns
ur na
have been modeled by the EMT model. The equilibrium phenomenon was described by the Langmuir (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017; Sausen et al., 2018), BET (Scheufele et al., 2016), Linear (Guo et al., 2019b), and Freundlich (Hu
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et al., 2019; Guo et al., 2019b) isotherms. 3.2 Internal diffusion models The internal diffusion models assume that the diffusion of adsorbate within
adsorbent is the slowest step. The diffusion of adsorbate in the liquid film around the adsorbent and the adsorption onto the active sites are instantaneous. Here we reviewed
37
three most used internal diffusion models, i.e., the Boyd’s intraparticle diffusion model, the Weber and Morris (W&M) model, and the phenomenological internal mass transfer (IMT) model.
3.2.1 Boyd’s intraparticle diffusion model
Boyd et al. (1947) reported an intraparticle diffusion model: 6 𝜋2
∑∞ 𝑛=1
1 𝑛2
𝑒 −𝑛
2 𝐵𝑡
(46)
ro of
𝐹=1 −
Where F = qt/q∞, q∞ (mg·g-1) is obtained by the adsorption isotherm model (q∞ = f(Ce)) (Yao and Chen, 2017).
-p
As the value of F up to 0.05, Eq. (46) is simplified to Eq. (47) (Boyd et al., 1947): (47)
re
𝐹 = 1.08√𝐵𝑡
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Reichenberg (1953) suggested that for the values of F ranged from 0 to 0.85, Eq. (46) could be rearranged to:
Or
6 𝜋3/2
3
√𝐵𝑡 − 𝜋2 (𝐵𝑡)
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𝐹=
𝐵𝑡 = 2𝜋 −
𝜋2 𝐹 3
− 2𝜋 (1 −
𝜋𝐹 1/2 3
)
(48 a)
(48 b)
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When the values of F range from 0.86 to 1, Eq. (46) is transformed to Eq. (49) (Reichenberg, 1953): 𝐹 =1−
𝜋2 6
𝑒 −𝐵𝑡
(49 a)
Or 𝐵𝑡 = −ln
𝜋2 6
(1 − 𝐹)
(49 b)
If the experimental data obey the Boyd’s intraparticle diffusion model, B is a 38
constant. In other words, if the plot of Bt vs t is a straight line and passes through (0, 0), the adsorption process is controlled by the intraparticle diffusion. Please note that in the applications of Boyd’s intraparticle diffusion model, the term Bt is the product of B and t. In the past decades, Boyd’s intraparticle diffusion equation has been used to represent the internal mass transfer processes (Table 5). For example, Ofomaja (2010)
ro of
studied the adsorption of Pb(II) onto mansonia wood sawdust using the Boyd’s model.
Albadarin et al. (2012) demonstrated that the values of R2 of Boyd’s model for the
-p
adsorption of Cr(Ⅵ) (C0 = 10-50 mg·L-1) onto dolomite ranged from 0.899 to 0.951.
Al-Muhtaseb et al. (2011) concluded that the plot of Bt vs t didn’t pass through (0, 0),
re
which indicated that the adsorption of phenol onto PMMA was not controlled by the
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intraparticle diffusion.
ur na
3.2.2 Weber and Morris (W&M) model
Weber and Morris (1963) deduced a model to describe the intraparticle diffusion process. The W&M model is presented as following: 𝑞𝑡 = 𝑘W&M 𝑡 1/2
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(50)
By plotting qt vs. t, the parameter kW&M can be estimated. The plot of qt vs t1/2 is a
straight line and passes through (0, 0) when the intraparticle diffusion is the controlling process. Otherwise, the adsorption is controlled by multiple processes. Because W&M model is simple and convenient, it has been widely applied to find out the rate controlling step, such as phenol onto PMMA (Al-Muhtaseb et al., 2011), 39
TA, MB, phenol, and 4-CP on activated carbons (Wu et al., 2009), Cu(II), Cd(II), and Pb(II) onto Caulerpa lentillifera (Apiratikul and Pavasant, 2008), Dibenzothiophene, MB and AB 25 onto activated carbons (Danmaliki and Saleh, 2016; Archin et al., 2019) (Table 5).
3.2.3 Phenomenological internal mass transfer (IMT) model
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The internal mass transfer phenomenon can be described by the Fick’s law (Crank, 1979): 𝜕𝑞𝑡 (𝑟𝑖 ,𝑡) 𝜕𝑡
=
𝐷ef 𝜕 𝑟𝑖 2 𝜕𝑟𝑖
(𝑟𝑖 2
𝜕𝑞𝑡 (𝑟𝑖 ,𝑡) 𝜕𝑟𝑖
)
(51)
-p
Eq. (51) can be simplified to the LDF model (Glueckauf and Coates,1947; Yang,
𝜕𝑞𝑡
= 𝑘𝑖𝑛𝑡 (𝑞et − 𝑞𝑡 )
(52)
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𝜕𝑡
re
1987):
The IMT model assumes that the internal diffusion is the slowest process, and the
ur na
equilibrium is obtained at the liquid-solid interface. The adsorption isotherm is used to describe the equilibrium phenomenon. For example, if the Langmuir model can describe the adsorption equilibrium data, qet is calculated as following: 𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶𝑡
(53)
1+𝐾𝐿 𝐶𝑡
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𝑞et =
Substitution of Eqs. (38) and (53) into Eq. (52) yields:
𝜕𝑞𝑡 𝜕𝑡
= 𝑘𝑖𝑛𝑡 (
𝑞𝑚𝑎𝑥 𝐾𝐿 𝐶𝑡
1+𝐾𝐿 (𝐶0 −𝑚𝑞𝑡 /𝑉)
− 𝑞𝑡 )
(54)
Eq. (54) can be solved by the Runge-Kutta method with the initial condition of q0 = 0. The applications of the IMT model are shown in Table 5. The equilibrium phenomenon in the liquid-solid interface was described by the BET (Scheufele et al., 40
2016), Freundlich (Hu et al., 2019), and Langmuir isotherms (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017). 3.3 Pore volume and surface diffusion (PVSD) model The basic assumptions of the PVSD model are that (1) the adsorbent particle is spherical, (2) the convective mass transport in the pores is negligible, (3) the solution is
ro of
homogeneous, and (4) the adsorption on active sites is simultaneous. The PVSD model is described by Eqs. (55)-(60) (Leyva-Ramos and Geankoplis, 1985): 𝑉
𝑑𝐶𝑡
= −𝑚𝑆𝑝 𝑘𝐹 (𝐶𝑡 − 𝐶tr |r=r0 )
𝑑𝑡
(55)
𝑡 = 0, 𝐶𝑡 = 𝐶0 𝜕𝐶tr 𝜕𝑡
+ 𝜌𝑝
𝜕𝑞𝑡 𝜕𝑡
=
1 𝜕 𝑟 2 𝜕𝑟
(𝑟 2 (𝐷𝑝
𝜕𝐶tr 𝜕𝑟
+ 𝜌𝑝 𝐷𝑠
𝜕𝑟
𝐷𝑝
|𝑟=0 = 0 𝜕𝐶tr 𝜕𝑟
|𝑟=𝑟0 + 𝜌𝑝 𝐷𝑠
𝜕𝑞𝑡
))
lP
𝜕𝐶tr
𝜕𝑟
re
𝑡 = 0,0 ≤ 𝑟 ≤ 𝑟0 , 𝐶tr = 0
𝜕𝑞𝑡
-p
𝜀𝑝
(56)
| 𝜕𝑟 𝑟=𝑟0
= 𝑘𝐹 (𝐶𝑡 − 𝐶tr )|𝑟=𝑟0
(57) (58) (59) (60)
ur na
The equilibrium phenomenon is described by the adsorption isotherm: qet = f(Cet). The PVSD model is simplified to the pore volume diffusion (PVD) model when Ds = 0 and Dp ≠ 0 or to the surface diffusion (SD) model when Dp = 0 and Ds ≠ 0.
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Detailed numerical solution of the PVSD model is given by Souza et al. (2017a). In recent years, the PVSD model has been used to describe the diffusional adsorption of CV and MG onto bentonite clay (Souza et al., 2017a, 2019), pyridine and metronidazole onto granular activated carbon (Ocampo-Perez et al., 2010, DíazBlancas et al., 2018), acetaminophen onto activated carbon pellets (Ocampo-Perez et
41
al., 2017), and ibuprofen onto biochar (Ocampo-Perez et al., 2019) (Table 5).
4. Models for adsorption onto active sites (AAS) The models for AAS assume that the adsorption onto active site is the slowest step, and the diffusion process is negligible. Langmuir kinetics model and Phenomenological AAS model, as well as the Ritchie’s equation reviewed in the previous section can
ro of
describe the AAS phenomenon. 4.1 Langmuir kinetics model
The Langmuir kinetics model can be described by Eq. (6) (Langmuir, 1918), as
-p
shown in the previous section. θ can be calculated by qt/qe, and then Eq. (6) is
𝑑𝑞𝑡 𝑑𝑡
= 𝑘𝑎 𝐶𝑡 (𝑞𝑒 − 𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡
re
transformed to Eq. (61):
(61)
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At adsorption equilibrium, Eq. (61) is reduced to the Langmuir isotherm model (Langmuir, 1918). Eq. (61) need to be solved by the Runge-Kutta method with the
ur na
initial condition of q0 = 0 based on MATLAB or other programming software, which makes it difficult to apply. Efforts have been made to find the analytical solution of the Langmuir kinetics. Liu and Shen (2008) reported that the Langmuir kinetics could be
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solved by Eq. (17). The model coefficients kL1 and kL2 are calculated as following: 𝑘L1 = √𝑘𝑎 2 (𝐶0 − 𝑞𝑒 𝑚𝑠 )2 +2𝑘𝑎 𝑘𝑑 (𝐶0 + 𝑞𝑒 𝑚𝑠 ) + 𝑘𝑑
(62)
𝑘L2 = 𝑘𝑎 𝑞𝑒 𝑚𝑠
(63)
Shen et al. (2009) developed a simplified Langmuir kinetics model using a simple geometric approach. Tien and Ramarao (2014) reported a transformed Langmuir rate
42
equation related to the PFO and PSO models. For the applications of the Langmuir kinetics model, Al-Jabari (2016) studied the adsorption kinetics of Cr(III) onto mineral particles using the Langmuir kinetics. Marczewski et al. (2013) investigated the adsorption and desorption kinetics of benzene on carbons using the integrated Langmuir kinetics model. Marczewski (2011) extended the application of Langmuir kinetics in dilute solutions.
ro of
The solving methods reported by the previous studies are also complex and may lead to difficulties in parameters estimations. In the following section, a convenient
-p
method is provided. In addition, Eq. (64) is recommended to calculate the Langmuir adsorption rate.
re
Langmuir rate = 𝑘𝑎 𝐶𝑡 (𝑞𝑒 − 𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡
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4.2 Phenomenological AAS model
(64)
The Phenomenological AAS mode, which is developed based on the Langmuir
𝑑𝑞𝑡 𝑑𝑡
ur na
kinetics model (Langmuir, 1918; Thomas, 1944), can be described by Eq. (65): = 𝑘𝑎 𝐶𝑡 (𝑞max −𝑞𝑡 ) −
𝑘𝑎
𝐾𝐿
𝑞𝑡
(65)
Substitution of Eq. (38) into Eq. (65) yields: 𝑑𝑞𝑡
= 𝑘𝑎 (𝐶0 −
Jo
𝑑𝑡
𝑚𝑞𝑡 𝑉
) (𝑞max − 𝑞𝑡 ) −
𝑘𝑎 𝐾𝐿
𝑞𝑡
(66)
Eq. (66) can be solved by the Runge-Kutta method with the initial condition of q0
= 0. The AAS model has been adopted to study the adsorption mechanism (Blanco et al., 2017; Marin et al., 2014; Suzaki et al., 2017; Hu et al., 2019; Guo et al., 2019b; Sausen et al., 2018).
43
5. Model validity evaluation In the past decade, various statistical parameters, such as R2, adjR2, 2, SSE, MSE, and HYBRID were used to evaluate the performance of the kinetic models (Eqs. (67)(72)). ∑(𝑞mean −𝑞cal )2
(67)
∑(𝑞cal −𝑞mean )2 +∑(𝑞cal −𝑞exp )2
𝐴𝑑𝑗𝑅2 = 1 − (1 − 𝑅2 ) 𝜒2 = ∑
(𝑁𝑒𝑥𝑝 −1)
(68)
(𝑁𝑒𝑥𝑝 −𝑁𝑝𝑎𝑟𝑎 −1)
(𝑞𝑒𝑥𝑝 −𝑞cal )2 )
ro of
𝑅2 =
(69)
𝑞cal 2
𝑆𝑆𝐸 = ∑(𝑞exp − 𝑞cal )2 1 𝑁𝑒𝑥𝑝
𝐻𝑌𝐵𝑅𝐼𝐷 =
∑(𝑞exp − 𝑞cal )2 100 𝑁𝑒𝑥𝑝 −𝑁𝑝𝑎𝑟𝑎
∑
(71)
-p
𝑀𝑆𝐸 =
(70)
𝑞𝑒𝑥𝑝 −𝑞cal 𝑞𝑒𝑥𝑝
(72)
re
Here we summarized the application of the statistical parameters based on
lP
literature (Guo and Wang, 2019a; Blanco et al., 2017; Hu et al., 2019; Suzaki et al., 2017; Wang et al., 2008; Ma et al., 2018; Ersan et al., 2019; Shang et al., 2019; Delgado
ur na
et al., 2019; Darwish et al., 2019; Sabarinathan et al., 2019; Khan et al., 2019; Gamoudi and Srasra, 2019; Stähelin et al., 2018; Sari et al., 2017; Gunasundari and Kumar, 2017; Lou et al., 2019; Guo et al., 2019a, 2019b; Yin et al., 2018; H. Wang et al., 2018; Gogoi
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et al., 2018; Xia et al., 2019; Islam, 2019; Tang et al., 2019; Gao et al., 2019; Ahmed and Theydan, 2013; Öztürk and Malkoc, 2014; Fu et al., 2012, 2015; Panday et al., 1986; Choy et al., 2004; Erdoğan and Ulku, 2012; Dávila-Guzmán et al., 2013; Šljivić et al., 2011; Sausen et al., 2018; Ofomaja, 2010; Albadarin et al., 2012; Al-Muhtaseb et al., 2011; Sartape et al., 2017; Cantuaria et al., 2016; Tan et al., 2016; Freitas et al.,
44
2017; Danmaliki and Saleh, 2016; Archin et al., 2019; Tsibranska and Hristova, 2011; Álvarez-Gutiérrez et al., 2017; Sreńscek-Nazzal et al., 2015; AkankshaKalra et al., 2019; Riahi et al., 2017; Saadi et al., 2015; Devi and Murugappan, 2016; Tanzifi et al., 2018; Dastkhoon et al., 2017; Karimi et al., 2016; Karri and Sahu, 2018; Siyal et al., 2019), as shown in Fig. 3. Fig. 3 indicated that R2 and adjR2 are two of the most frequently calculated parameters. Over 60% of the literature has compared the values
ro of
of R2 and adjR2 to evaluate the kinetic models. The values of χ2, SSE, and HYBRID are
also calculated in some studies. The proportions of the above statistical parameters are
-p
11%, 11%, and 10%, respectively. Few studies adopted MSE for the evaluation of their models. However, the values of R2 of different kinetic models always have small
re
difference. For example, Delgado et al. (2019) reported that the values of R2 of the PFO
lP
and PSO models were 0.998 and 0.999, respectively. Burakova et al. (2018) found that the values of R2 of the PFO and the W&M models were 0.88 and 0.899, respectively.
ur na
Therefore, we recommend calculating more statistical parameters to evaluate the fitting
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results thoroughly and rigorously.
45
Fig. 3 Statistical parameters of the adsorption kinetic models
6. Solving methods for adsorption kinetic models The adsorption kinetic models reviewed in this paper are summarized in Table 6. In this paper, a convenient UI for solving the kinetic models was developed based on Excel (see in Supplementary Information). The Runge-Kutta method is employed to
ro of
solve the differential equations (such as the MO model and the Langmuir kinetics model), and the solver add-in is applied for the calculation of the model parameters.
The statistical parameters (R2, adjR2, 2, SSE, MSE, and HYBRID) are calculated based
-p
on Eqs. (67)-(72). Readers can download the UI attached in Supplementary Information
Table 6
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List of the adsorption kinetic models
re
and input their experimental data, then the fitting results will show in the sheet of Excel.
Differential equation
PFO model
𝑑𝑞𝑡 = 𝑘1 (𝑞𝑒 − 𝑞𝑡 ) 𝑑𝑡
PSO model
𝑑𝑞𝑡 = 𝑘2 (𝑞𝑒 − 𝑞𝑡 )2 𝑑𝑡
MO model
𝑑𝑞𝑡 = 𝑘1’ (𝑞𝑒 − 𝑞𝑡 ) + 𝑘2’ (𝑞𝑒 − 𝑞𝑡 )2 𝑑𝑡
-
𝑑𝜃 = 𝜕(1 − 𝜃)𝑛 𝑑𝑡
𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝜕𝑡 ) (n=1)
ur na
Models
equation PNO model
References
𝑞𝑡 = 𝑞𝑒 (1 − 𝑒 −𝑘1𝑡 )
Lagergren, 1898
𝑞𝑡 =
𝑞𝑒2 𝑘2 𝑡 1 + 𝑞𝑒 𝑘2 𝑡
Ho et al., 1996 Guo and Wang, 2019a Ritchie, 1977 1
𝑞𝑡 = 𝑞𝑒 − 𝑞𝑒 [1 + (𝑛-1)𝜕𝑡]1−𝑛 (n≠1)
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Ritchie’s
Integrated form or nonlinear form
𝑑𝑞𝑡 = 𝑘𝑛 (𝑞𝑒 − 𝑞𝑡 )𝑛 𝑑𝑡
𝑞𝑡 = 𝑞𝑒 (1 −
1
)
Özer, 2007
(1+(n-1)𝑞𝑒 n-1 𝑘𝑛 𝑡)1/(n-1) )
(n ≠ 1) Elovich model Boyd’s external diffusion
𝑑𝑞𝑡 = 𝑎𝑒 −𝑏𝑞𝑡 𝑑𝑡
𝑞𝑡 =
1 ln(1 + 𝑎𝑏𝑡) 𝑏
𝑞𝑡 = 𝑞∞ (1 − 𝑒 −𝑅𝑡 )
𝑑𝑞𝑡 = 𝑅(𝑞∞ − 𝑞𝑡 ) 𝑑𝑡 46
Elovich
and
Larinov, 1962 Boyd et al., 1947
equation F&S model
-
𝐶𝑡 𝐶0
=
1 1+𝑚𝑠 𝐾
+
𝑚𝑠 𝐾 1+𝑚𝑠 𝐾
𝑒
1+𝑚𝑠 𝐾 𝑘 𝑆𝑡 𝑚𝑠 𝐾 F&S
Frusawa
−
and
Smith, 1973
(Linear isotherm) M&W model
𝑑𝑞𝑡 𝑘𝑠 = (𝐶𝑡 − 𝐶et ) = 𝑘ext (𝐶𝑡 − 𝐶et ) 𝑑𝑡 𝜌
EMT model
𝐶𝑡 = 𝑒 −𝑘M&W𝑆𝑡 𝐶0
Mathews
-
Ruthven,
and
Weber, 1977 1984;
Hines
and
Maddox, 1985 Boyd’s
-
𝐵𝑡 = 2𝜋 −
𝜋2𝐹
F ≤ 0.85)
diffusion model
𝐵𝑡 = −ln W&M model
− 2𝜋 (1 −
𝜋2 6
3
)
(0 ≤
Boyd et al., 1947
(1 − 𝐹) (0.86 ≤ F ≤ 1)
𝑞𝑡 = 𝑘W&M 𝑡 1/2
-
𝜋𝐹 1/2
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intraparticle
3
Weber
and
𝜕𝑞𝑡 = 𝑘𝑖𝑛𝑡 (𝑞et − 𝑞𝑡 ) 𝜕𝑡
𝑉
𝑑𝐶𝑡 = −𝑚𝑆𝑝 𝑘𝐹 (𝐶𝑡 − 𝐶tr |r=r0 ) 𝑑𝑡
𝑡 = 0, 𝐶𝑡 = 𝐶0
-
Leyva-Ramos
Coates,
and 1947;
Yang, 1987
and Geankoplis, 1985
𝜕𝐶tr 𝜕𝑞𝑡 1 𝜕 𝜕𝐶tr 𝜕𝑞𝑡 + 𝜌𝑝 = 2 (𝑟 2 (𝐷𝑝 + 𝜌𝑝 𝐷𝑠 )) 𝜕𝑡 𝜕𝑡 𝑟 𝜕𝑟 𝜕𝑟 𝜕𝑟
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𝜀𝑝
Glueckauf
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PVSD model
-
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IMT model
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Morris, 1963
𝑡 = 0,0 ≤ 𝑟 ≤ 𝑟0 , 𝐶tr = 0 𝜕𝐶tr | =0 𝜕𝑟 𝑟=0 𝐷𝑝
𝑑𝑞𝑡 = 𝑘𝑎 𝐶𝑡 (𝑞𝑒 −𝑞𝑡 ) − 𝑘𝑑 𝑞𝑡 𝑑𝑡
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Langmuir
𝜕𝐶tr 𝜕𝑞𝑡 |𝑟=𝑟0 + 𝜌𝑝 𝐷𝑠 | = 𝑘𝐹 (𝐶𝑡 − 𝐶tr )|𝑟=𝑟0 𝜕𝑟 𝜕𝑟 𝑟=𝑟0
kinetics model
-
Langmuir, 1918
-
Langmuir, 1918;
𝑑𝜃 = 𝑘𝑎 𝐶𝑡 (1 − 𝜃) − 𝑘𝑑 𝜃 𝑑𝑡
AAS model
𝑑𝑞𝑡 𝑚𝑞𝑡 𝑘𝑎 = 𝑘𝑎 (𝐶0 − ) (𝑞max − 𝑞𝑡 ) − 𝑞𝑡 𝑑𝑡 𝑉 𝐾𝐿
Thomas, 1944
Detailed instructions for this UI are explained as follows. (1) Download the UI in Supplementary Information. 47
(2) Open the UI, activate Solver Add-in. Then press Alt + F11 to get Visual Basic Editor, go to the Tools menu and select References from the drop-down menu. Select “Visual Basic For Applications”, “Microsoft Excel 16.0 Object Library”, “OLE Automation”, “Microsoft Office 16.0 Object Library”, “Microsoft Forms 2.0 Object Library”, “Solver”, and “Ref Edit Control”. (3) The model selection interface is shown in the Excel (Fig. 4 (left)). Readers can
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choose one kinetic model, and click the “OK” button. (If the model selection interface is not displayed, please enable the edit view.)
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(4) After click the “OK” button, the data input window is shown in the Excel (Fig. 4 (right)). Please input the adsorption time t and the experimental adsorption capacity
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qt, as well as other data based on the kinetic models used, and click the “Fit” button.
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Please note that each t and qt should be separated by a comma. (Fig. 4 (right)) (5) The results are shown in “a1” sheet (Fig. (5)). If this UI fails to solve your
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kinetic model, please close the Excel and re-open it, and then try another initial value of the parameter. You may have to try several times to get the proper initial value of the parameters.
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(6) After solving one kinetic model, please record the parameters of the models and export the figure of the fitting results (Fig. 5) in a new Excel. Then close the UI and re-open it. You can choose another kinetic model for calculation.
48
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Fig. 4 The interface of the UI (left) and the data input window (right)
Fig. 5 Fitting results of the adsorption kinetic model.
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Several points should be noticed when using the UI for calculation: (1) In the data input window, if the background of the input position is yellow (Fig.
4 (right)), this value is related to the experimental data or the data calculated by the isotherms. These values will not change in the solving process. (2) For the Ritchie’s equation, the value of n is assumed to be larger than 1, because calculation errors are generated when 0 < n < 1. 49
(3) Eqs. (39) and (42) are used as the equations of the F&S and M&W models. (4) For the EMT, IMT, and AAS models, the Langmuir isotherm is adopted to describe the adsorption equilibrium phenomenon. (5) Please note that the PVSD and the Boyd’s intraparticle diffusion models are not included in this UI. Because the PVSD model is complicated and cannot be solved by the Excel software. And the Boyd’s intraparticle diffusion model is a piecewise
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function, which can be simply solved by the linear regression method.
(6) The full names of the abbreviation of the kinetic models in this UI can be seen
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7. Concluding remarks and perspectives
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in the nomenclature table in this paper.
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This review aimed to study the physical meanings, applications, and solving methods of 16 adsorption kinetic models, including 6 adsorption reaction models and
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empirical models, 8 diffusional models, and 2 models for adsorption onto active sites. For the adsorption reaction models and empirical adsorption kinetic models, the most widely used PFO model and PSO model can be deduced by the Langmuir kinetics.
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The theoretical analysis and the review on literature indicated that the PFO model represented the conditions of (1) at high initial concentrations of adsorbate; (2) at the initial stage of adsorption, and (3) few active sites exist in the adsorbent material. In some cases, the PFO model can describe the diffusional kinetics process. The PSO model could represent the conditions of (1) at high initial concentrations of adsorbate; (2) at the final stage of adsorption, and (3) the adsorbent is abundant with active sites. 50
The PSO model reveals the mechanism of adsorption in active sites in most cases. The MO kinetics can describe the whole adsorption processes. The empirical Elovich and the PNO models are applied in some studies, but they are lacking of specific physical meanings and cannot offer important information about the mass transfer mechanisms. For the mass transfer models, the external diffusion process can be described by the Boyd’s external diffusion equation, F&S, M&W, and phenomenological EMT
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models. These models have specific physical meanings (external diffusion), and they
are more suitable to model the external mass transfer processes. The Boyd’s
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intraparticle diffusion model, M&W, and phenomenological IMT models can represent
the intraparticle diffusion process. For the adsorption in active sites process, the
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Langmuir kinetics, Ritchie’s equation and the phenomenological AAS models can
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describe this process. These models can give the mass transfer mechanisms more accurately. The chosen of the kinetic models should be based on the mass transfer
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processes.
In addition, the statistical parameters R2, adjR2, 2, SSE, MSE, and HYBRID are widely used to evaluate the performance of the kinetic models. In which R2 and adjR2
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are the most frequently applied statistical parameters. To rigorously evaluate the fitting results of the kinetic models, we recommend calculating all the above statistical parameters.
The adsorption kinetic models include the nonlinear kinetic models (such as the nonlinear PFO, PSO models and the Elovich model) and the differential kinetic models
51
(such as the MO model, the phenomenological EMT, IMT, and AAS models, and the Langmuir kinetics model). The solving methods for these models are required. In this paper, an UI for solving the kinetic models is developed based on Excel solver add-in and the Runge-Kutta method, which can be used conveniently to solve the models reported in this review. In recent years, some other novel kinetic models, such as the Largitte double step
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model (Largitte and Pasquier, 2016), the film-pore mass transfer (FPMT) model (Guo
and Wang, 2019d), Gaulke’s unified kinetic model (Gaulke et al., 2016), and the fractal-
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like adsorption kinetic models (El Boundati et al., 2019) have also been proposed.
Although these novel models are not frequently used by researchers, they are expected
Declaration of interests
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to help to investigate the adsorption mechanisms, and to model the adsorption systems.
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Adsorption kinetic models: Physical meanings, applications, and solving methods
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Jianlong Wang 1,2 *, Xuan Guo 1
Acknowledgments The research was supported by the National Key Research and Development Program (2016YFC1402507) and the Program for Changjiang Scholars and Innovative Research Team in University (IRT-13026).
52
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