Modification of breakthrough models in a continuous-flow fixed-bed column: Mathematical characteristics of breakthrough curves and rate profiles

Journal Pre-proofs Modification of breakthrough models in a continuous-flow fixed-bed column: Mathematical characteristics of breakthrough curves and rate profiles Qili Hu, Yanhua Xie, Zhenya Zhang PII: DOI: Reference:

S1383-5866(19)34315-1 https://doi.org/10.1016/j.seppur.2019.116399 SEPPUR 116399

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Separation and Purification Technology

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21 September 2019 3 December 2019 5 December 2019

Please cite this article as: Q. Hu, Y. Xie, Z. Zhang, Modification of breakthrough models in a continuous-flow fixed-bed column: Mathematical characteristics of breakthrough curves and rate profiles, Separation and Purification Technology (2019), doi: https://doi.org/10.1016/j.seppur.2019.116399

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Modification of breakthrough models in a continuous-flow fixed-bed column: Mathematical characteristics of breakthrough curves and rate profiles Qili Hua,b, Yanhua Xiea,*, Zhenya Zhangb,* a College b

of Environment and Ecology, Chengdu University of Technology, Chengdu 610059, China

Graduate School of Life and Environmental Sciences, University of Tsukuba, 1-1-1 Tennodai,

Tsukuba, Ibaraki 305-8572, Japan

* Corresponding author. E-mail: [email protected] (Y. Xie). E-mail: [email protected] (Z. Zhang).

1

Abstract In order to more completely describe mathematical characteristics of the breakthrough curve, this work defines the four parameters: Maximum specific breakthrough rate μmax, lag time λ, inflection point ti and half-operating time t50. The breakthrough models include the Bohart– Adams, Thomas, Yoon–Nelson, Clark, Wolborska and dose-response models. Attempts are made to address mathematical relationships between the breakthrough models, propose modified breakthrough models, investigate effects of model parameters on the breakthrough curve and rate profile and reveal their physical meanings. The fitting performance of the breakthrough models is verified by the adsorption of nitrate on the chitosan-Fe(III) composite. The results indicate that the model terms q0m/vc0 and a0x/uc0 are the operating time required to reach 50% breakthrough. The Clark model has the best fitting performance with high adjusted determination factor (Adj. R2 = 0.9976) and low reduced chi-squared value (χ2 = 2.70 × 10−4). In addition, an inconsistency concerning application of the Wolborska model is proposed to avoid this situation where it is repeated in subsequent publications. This work is expected to help readers better understand the breakthrough models and select the appropriate model to analyze the dynamic behaviors in a continuous-flow fixed-bed column. Keywords: Adsorption; Modification; Breakthrough model; Rate profile; Parameters

2

Nomenclature A

dimensionless Clark constant

a

dimensionless dose-response constant

a0

adsorption capacity (mg L−1)

b

dose-response constant (min−1)

c

effluent solute concentration (mg L−1)

c0

influent solute concentration (mg L−1)

cb

concentration of solute at the breakthrough time (mg L−1)

kBA

Bohart–Adams rate constant (mL min−1 mg−1)

kT

Thomas rate constant (mL min−1 mg−1)

kYN

Yoon–Nelson rate constant (min−1)

m

absorbent mass filled in the column (g)

n

Freundlich constant or number of data points

q0

saturation capacity (mg g−1)

r

Clark constant (min−1)

t

operating time (min)

t50

half-operating time (min)

tb

breakthrough time (min)

ti

inflection point (min)

ts

saturation time (min)

u

linear flow velocity (cm min−1)

v

flow rate (mL min−1) 3

x

bed height (cm)

βa

kinetic coefficient of the external mass transfer (min−1)

λ

lag time (min)

μmax

maximum specific breakthrough rate (min−1)

τ

operating time required to reach 50% breakthrough (min)

f

number of degrees of freedom

p

number of model parameters

y

c/c0

ýi

predicted value

yi

observed value

ωi

weighting coefficient (ωi = 1/yi)

R2

multiple coefficient of determination (R2 = 1 − RSS/TSS)

Adj. R2

adjusted determination factor

χ2

reduced chi-squared value

TSS

total sum of squares

RSS

residual sum of squares

4

1. Introduction Adsorption is one of the most widely used technologies for water reuse and wastewater treatment, involving the selective interaction of one or more compounds from aqueous solution on the solid surfaces [1]. The mass transfer processes in porous adsorbents include the four steps: Bulk transport, film diffusion, intraparticle diffusion and adsorptive attachment [2]. Adsorption has currently been an indispensable unit operation in industrial application including separation and purification, industrial catalysis and pollution control [3]. The importance of adsorption is growing due to the occurrence of emerging contaminants such as pesticides, industrial compounds, pharmaceuticals and personal care products [4,5]. The prevalent use of adsorption stems from its low cost, fast kinetics, low energy for regeneration, insensitivity to toxic substances and complete removal of pollutants even from dilute solution [6,7]. In the practical operation of full-scale adsorption process, a continuous-flow fixed-bed column is preferred [8]. In such a system, the adsorption of various solutes is a time- and distance-dependent process [9]. The mass transfer is favorable due to the existence of the concentration gradient, which is the driving force of the adsorption process [10]. A good understanding of the dynamic behavior of adsorption systems is required for process design and optimization [11]. The dynamic adsorption behaviors of the solutes in a fixed-bed column are represented by the effluent concentration-time profile, i.e. the breakthrough curve. The shape of this curve is determined by the shape of the equilibrium isotherm and is influenced by the individual transport processes in the column and adsorbents [8]. The most effective adsorption performance can be obtained when the shape of the breakthrough curve is as sharp as possible [12]. The breakthrough time and the saturation time can’t be exactly predicted by the breakthrough models due to the asymptotic form of the breakthrough curve [9]. Therefore, the operating times at c/c0 = 0.05 and c/c0 = 0.95 are often defined as the breakthrough time tb and the saturation time ts, respectively [13]. 5

The appropriate design of an adsorption process requires the development of a mathematical model that can describe the dynamic adsorption behaviors and predict the breakthrough curve in a fixed-bed column [11]. One approach has been to solve simultaneously the partial differential equations (PDEs) describing the mass and the heat balance in a fixed-bed column [14]. However, the simultaneous solution of PDEs for a given adsorption system, although more general and mathematically rigid, requires complicated numerical solutions. Thus, it is desirable to use the simplified models to satisfactorily predict the fixed-bed adsorption behaviors. Many attempts have been made to develop simplified breakthrough models to reduce computational time and facilitate optimization studies. Several fixed-bed adsorption models with explicit equations have been widely used to describe the breakthrough behaviors, including the Bohart–Adams, Thomas, Yoon–Nelson, Clark, Wolborska and dose-response models (see Table 1). These simplified models are primarily established according to the description of mass transfer within adsorption systems and they can describe the experimental data satisfactorily for most practical design purposes. However, in many publications with respect to the adsorption of various contaminants from aqueous solution, the authors mainly focus on the fitting performance of these simplified models and the determination of the parameters and errors. Some crucial problems are still to be addressed: (i) Mathematical relationships between breakthrough models; (ii) Physical meanings of some specific terms; (iii) Effects of model parameters on the breakthrough curve and rate profile; (iv) Complete description of the breakthrough curve; and (v) Establishment of the modified breakthrough models. The objectives of this work are to help readers further understand mathematical characteristics of the breakthrough curve and select the appropriate model to analyze the dynamic behaviors in a continuous-flow fixed-bed column. 2. Mathematical relationships between breakthrough models In order to select an appropriate breakthrough model to describe the dynamic behaviors in a fixed-bed 6

column, it is extremely necessary to reveal mathematical relationships between breakthrough models. In the previous studies [15-18], some researchers believe that the Bohart–Adams, Thomas and Yoon–Nelson models are separate and independent when the three models are used to correlate with the same set of the experimental data. In other words, the breakthrough curves are not coincident and error values are not equal. However, it can be clearly seen from Table 1 that the three models share the identical mathematical forms and their parameters are interchangeable: kYN = kBAc0 = kTc0 and τ = a0x/uc0 = q0m/vc0 (where c0, v, x, m and u are dependent of initial operating conditions). It follows that the Bohart–Adams, Thomas and Yoon– Nelson models are equivalent in mathematical nature. Consequently, the fitting curves provided by the three models are coincident and all error values are equal when the curve fitting is carried out. Conclusions can be drawn that the physical meanings of the terms q0m/vc0 and a0x/uc0 are the operating time required to reach 50% breakthrough and that kYN is numerically equal to a product of kBA (kT) and c0. Moreover, the change in certain parameter of q0m/vc0 or a0x/uc0 without adjusting q0m/vc0 or a0x/uc0 does not alter adsorption performance. The most prominent advantage of using q0m/vc0 or a0x/uc0 consists in the fact that one can readily see which group of the parameters affects adsorption performance rather than examining the effect of each parameter. The revelation of mathematical relationships between the Bohart–Adams, Thomas and Yoon–Nelson models can also reduce workload significantly instead of the complex curve fitting. It is worth noting that the Yoon–Nelson model does not require detailed data with respect to the characteristics of adsorbates, the type of an adsorbent and the physical properties of a fixed-bed column, indicating that the parameters kYN and τ may be regarded as lumped parameters that embed some physical processes and operating features. As shown in Fig. S1, the Yoon–Nelson model can be derived from a 1

logistic function y = 1 + exp( ― t) through stretching and translation transformations successively. Thus, the 7

fitting curve provided by the Yoon–Nelson model is centrosymmetric with respect to the point (τ, 0.5). From a mathematical perspective, the stretching transformation affects its degree of curvature, while the translation transformation simply influences the location of the curve. As a result, kT and τ can be regarded as the shape and location parameters, respectively. In addition, it can be clearly seen from Table 1 that the Clark model can reduce to the Yoon–Nelson model when n = 2 according to the following relationship: A = exp(kYNτ)

(1)

r = kYN

(2)

It follows that the Clark model can be seen as a generalized form of the Yoon–Nelson model. The use of the Clark model requires that an adsorption process should follow the Freundlich isotherm in a batch adsorption system [19]. If not, the parameter n is not appropriate to calculate A and r, probably resulting in distinct deviations. Thus, it is more reasonable to directly calculate the parameter n instead of the n value obtained from the batch reactor when the adsorption process does not follow the Freundlich isotherm. It is predicted that the Clark model may provide better fitting performance than the Yoon–Nelson model because the extra adjustable parameter n makes the curve fitting more flexible. Fig. 1 shows the transformation processes of the Wolborska model. It can be derived from an exponential function y = exp(t) by translation and stretching transformations successively. As a result, the Wolborska model does not represent a S-shaped curve and thereby fails to describe the breakthrough curve completely. It may be only applied to the region of low breakthrough concentration, in which the process kinetics is controlled by mass transfer and axial diffusion in the liquid phase [20]. The characteristic parameters βa and a0 does not precisely reflect an actual fixed-bed adsorption system, either. In our opinion, the use of the Wolborska model should be avoided for the modeling of the dynamic behaviors in a fixed-bed column.

8

3. Modification of breakthrough models Similar to the bacterial growth curve [21], the breakthrough curve can be also divided into three phases (Fig. 2): Lag, logarithmic and stationary phases. In the lag phase, the fresh adsorbents packed in a fixedbed column contain large numbers of available adsorption sites and thereby hold the adsorbate molecules consecutively, leading to the very low effluent concentration; In the logarithmic phase, as the adsorption progresses, the decrease in adsorption sites results in the rapid increase in the effluent concentration; In the stationary phase, the adsorbent particles approach the saturation state gradually and thus the effluent concentration tends to a stationary value. The three phases of the breakthrough curve can be described by three parameters: Lag time λ, maximum specific breakthrough rate μmax and influent concentration c0. Herein, μmax and λ are defined as the slope of the tangent line at the inflection point of the breakthrough curve and the t-axis intercept of this tangent line, respectively. The parameter μmax can reflect whether the adsorption is favorable and the parameter λ represents the required operating time before the breakthrough curve begins to rise. In general, it is difficult to accurately obtain μmax and λ if they are not estimated directly in the equation. Therefore, in order to effectively predict μmax and λ, all breakthrough models are rewritten by substituting model parameters and operating conditions with μmax and λ. To concisely report this work, the detail derivation processes refer to Supplemental material and the modified Bohart–Adams, Thomas, Yoon–Nelson, Clark and dose-response models are listed in Table 1. As mentioned above, the Wolborska model does not represent a sigmoidal curve. Thus, it has no corresponding modified model. It should be noted that the modified breakthrough models are derived by variable substitution. Therefore, the fitting curves are coincident between the Bohart–Adams, Thomas, Yoon–Nelson, Clark, dose-response models and the corresponding modified breakthrough models.

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4. Symmetry of breakthrough models In this work, the inflection point ti and half-operating time t50 are proposed to identify the symmetry of the breakthrough curve and more adequately describe the breakthrough curve. Herein, ti and t50 are defined as the operating time required to reach one point where the shape of the breakthrough curve is converted from the concave to the convex and 50% breakthrough (c/c0 = 0.5), respectively. The mathematical relationships between the defined four parameters and model parameters and operating conditions are listed in Table 2. The relative magnitude of ti and t50 can reflect the symmetry of the breakthrough curve. As mentioned above, the Bohart–Adams, Thomas and Yoon–Nelson models provide a symmetric breakthrough curve. Thus, ti and t50 are equal. For the Clark and dose-response models, the magnitude of t50 − ti can be given as:

(

)

n―1 1 t50 ― ti = ln n ― 1 r 2 ―1

t50 ― ti =

[

1 a

( )] a―1

1 1― b a+1

(3)

(4)

It is obvious that the positive or negative value of t50 − ti simply depends on the parameters n and a respectively. In the Clark model, the parameter n represents a Freundlich constant and the value of n lying in the range of 1–10 confirms the favorable condition for adsorption [22]. A plot of t50 − ti as a function of n is described in Fig. S2. It is found that 1 < n < 2, t50 > ti; n = 2, t50 = ti; and 2 < n < 10, t50 < ti. Thus, the Clark model can describe an asymmetric breakthrough curve when n ≠ 2, and the breakthrough curve becomes more asymmetric when the parameter n deviates from 2. In the dose-response model, the influence of the parameter a on the breakthrough curve is shown in Fig. 3. All breakthrough curves for different a values pass through one point (1/b, 0.5) and the shape of the curves only depends on the magnitude of a. It is observed that the dose-response model does not represent 10

a sigmoidal curve but exhibits a parabola-like curve when a ≤ 1. The sigmoidal curve occurs only at a > 1 and the degree of curvature becomes larger with the increase in a. It is observed from Fig. S3 that a > 1, t50 > ti. Therefore, the dose-response model provides an asymmetric breakthrough curve. The breakthrough curve becomes more symmetric with the increase in a since ti is closer to t50. 5. Breakthrough curve and rate profile The breakthrough curve and rate profile of the Yoon–Nelson model for different kYN are illustrated in Fig. 4. All of the breakthrough curves pass through one point (τ, 0.5) and the degree of curvature increases with the increase in kYN (become steeper). Each rate profile is a symmetric Gaussian distribution shape concerning the vertical line t = τ and its shape is only dependent of kYN. The height decreases and the width increases with the decrease in kYN. As mentioned above, τ is a location parameter and thereby simply affects the location of the breakthrough curve and rate profile. The effects of kT (kBA) and q0 (a0) on the breakthrough curve and rate profile are consistent with that of kYN and τ respectively because the Bohart– Adams, Thomas and Yoon–Nelson models are equivalent in mathematical nature. In the Clark model, one can readily see from Fig. 5 that the parameter n can affect the curvature and location of the breakthrough curve. It becomes steeper with the decrease in n. The rate profiles for different n exhibit three types: (i) A symmetric Gaussian distribution shape (n = 2); (ii) An asymmetric quasiGaussian distribution shape with a widened right-hand side (1 < n < 2); and (iii) An asymmetric quasiGaussian distribution shape with a widened left-hand side (2 < n < 10). As mentioned above, the Clark model can be converted to the Yoon–Nelson model. Thus, the effects of the parameters A and r on the breakthrough curve and rate profile are equivalent to that of kYN and τ, respectively. In the dose-response model, the effects of the parameters a and b on the breakthrough curve and rete profile are represented in Fig. 6. The breakthrough curve become steeper with the increase in a (a > 1). The 11

operating time required to reach 50% breakthrough is exactly equal to 1/b. Thus, the half-operating time t50 depends on the parameter b. The corresponding rate profile is an asymmetric quasi-Gaussian distribution shape with a widened right-hand side. The increase in a leads to the increased height and reduced width, while the location of the peak is shifted to the left side. It can be clearly seen from Fig. 6b and Fig. 6d that the breakthrough curves and rate profiles are shifted to the right side by a certain distance with the decrease in b. The degree of curvature for the breakthrough curves and maximum specific breakthrough rate μmax also decrease. The Bohart–Adams, Thomas, Yoon–Nelson, Clark and dose-response models represent different types of the breakthrough curves and rate profiles, which provide more flexible selection for the modeling of the fixed-bed adsorption data. 6. Application of breakthrough models 6.1. Fixed-bed adsorption In this work, the adsorption of nitrate on the chitosan-Fe(III) composite is used to evaluate the fitting performance of the breakthrough models. In order to concisely report this work, the preparation of the chitosan-Fe(III) composite refers to Supplementary material. The fixed-bed column was made of Pyrex glass tube (Inner diameter = 0.7 cm, length = 60 cm). The chitosan-Fe(III) beads (3 g) were immersed in deionized water for full swelling prior to use, which avoided adverse effects on the effective bed height, followed by addition of them to the fixed-bed column. The gauze was kept at the top and bottom of the column to stabilize the adsorbent and provide a uniform flow. The nitrate solution was fed into the column from the bottom by a peristaltic pump at a flow rate of 6.1 mL min−1. The nitrate solution was collected at the outlet at preset time intervals and measured by a UV/vis spectrophotometer (UV–1800, Shimadzu, Japan).

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6.2.

Evaluation of fitting performance

To be adequately modeled by the breakthrough models, the distribution of the experimental data must approach the shape of the breakthrough curve. In this work, the adjusted determination factor (Adj. R2) [23] and reduced chi-squared value (χ2) [24] are employed to evaluate the validity of the breakthrough models. The data fitting is carried out by an iterative nonlinear regression using Origin 9 software (OriginLab, USA).

( )

Adj. R2 = 1 ― (1 ― R2)

n―1 n―p

(5)

n

1 χ = ωi(yi ― ýi)2 f𝑖 = 1 2

∑

(6)

One can readily see from Fig. 7 that the fitting curves provided by the Bohart–Adams, Thomas and Yoon–Nelson models are coincident and the plot of the predicted versus observed values are also coincident accordingly, which prove that the three models represent the same breakthrough curve from experimental perspective. Therefore, as shown in Table 3, the Bohart–Adams, Thomas and Yoon–Nelson models provide the same parameters (umax = 3.86 × 10−3 min−1, λ = 33.9 min, t50 = 163.4 min, and ti = 163.4 min) and error values (Adj. R2 = 0.9878 and χ2 = 1.37 × 10−3). The model parameters and operating conditions conform to kYN = kBAc0 = kTc0 and τ = a0x/uc0 = q0m/vc0, which is perfectly consistent with the theoretical analysis. It is worth noting that the Clark model has best fitting performance among these models because the predicted values are also closer to the observed values, especially in the region of 0–6 h. As shown in Table 3, the Clark model has higher adjusted determination factor (Adj. R2 = 0.9976) and lower reduced chi-squared value (χ2 = 2.70 × 10−4). The order of the optimal fitting is: Clark > dose-response > Bohart–Adams = Thomas = Yoon–Nelson models and thus the adsorption of nitrate on the chitosan-Fe(III) composite follows the Clark model. As mentioned above, the breakthrough curve will be asymmetric when the parameter n ≠ 2 for the Clark model and the parameter a > 1 for the dose-response model. In this study, n = 1.0073 and a 13

= 2.47 indicate that the breakthrough curve is asymmetric and the rate profile will be a quasi-Gaussian distribution shape with a widened right-hand side. This asymmetric breakthrough curve may be ascribed to the fact that the intraparticle diffusion is the rate-controlling step and the adsorbent consists of two or more constituents of unequal reactivity [9,25]. Moreover, two different adsorption systems from literature are selected to further validate the theoretical results: Adsorption of methylene blue on porous silica microsphere [26] and adsorption of aniline on modified jute fiber [27]. Given that similar conclusions can be drawn for the two adsorption systems, in order to concisely report this work, the fitting results are put in Supplementary material. This work may help readers better select the optimum model to analyze the dynamic behaviors in a fixed-bed column. 7. Conclusions The Bohart–Adams, Thomas and Yoon–Nelson models represent the logistic function from mathematical perspective, which provide the same symmetric breakthrough curve. The Clark model is the generalized form of the above three breakthrough models and thus provides better fitting performance due to an extra adjustable parameter n. The dose-response model represents a sigmoidal curve only when the parameter a is more than unity (a > 1). The Wolborska model represents the exponential function in mathematical nature and does not provide a complete sigmoidal curve. The maximum specific breakthrough rate μmax, lag time λ, inflection point ti and half-operating time t50 can better reflect mathematical characteristics of the breakthrough curve and thereby contribute to selecting the optimum breakthrough model and gaining insights into the dynamic adsorption processes. The modified breakthrough models with μmax and λ may be important alternative methods for analysis of continuous-flow adsorption systems. The physical meanings of the terms q0m/vc0 and a0x/uc0 are the operating time required to reach 50% breakthrough. The model parameters are probably lumped constants that are related to the physical 14

processes and operating conditions. Acknowledgements The authors sincerely express their appreciation to Prof. Xiangjun Pei (Chengdu University of Technology, China). The first author acknowledges the support of the China Scholarship Council (No. 201606400063). References [1] V. Russo, M. Trifuoggi, M. Di Serio, R. Tesser, Fluid-solid adsorption in batch and continuous processing: A review and insights into modeling, Chem. Eng. Technol. 40 (2017) 799-820. [2] W.J. Weber Jr, Evolution of a technology, J. Environ. Eng. 110 (1984) 899-917. [3] J. Rouquerol, F. Rouquerol, P. Llewellyn, G. Maurin, K.S. Sing, Adsorption by Powders and Porous Solids: Principles, Methodology and Applications, Academic press, London, 2013. [4] J. Martín, M.D.M. Orta, S. Medina-Carrasco, J.L. Santos, I. Aparicio, E. Alonso, Removal of priority and emerging pollutants from aqueous media by adsorption onto synthetic organo-funtionalized highcharge swelling micas, Environ. Res. 164 (2018) 488-494. [5] K.L. Tan, B.H. Hameed, Insight into the adsorption kinetics models for the removal of contaminants from aqueous solutions, J. Taiwan Inst. Chem. Eng. 74 (2017) 25-48. [6] A.L. Yaumi, M.Z.A. Bakar, B.H. Hameed, Reusable nitrogen-doped mesoporous carbon adsorbent for carbon dioxide adsorption in fixed-bed, Energy 138 (2017) 776-784. [7] K.Y. Foo, B.H. Hameed, Insights into the modeling of adsorption isotherm systems, Chem. Eng. J. 156 (2010) 2-10. [8] G. Alberti, V. Amendola, M. Pesavento, R. Biesuz, Beyond the synthesis of novel solid phases: Review on modelling of sorption phenomena, Coordin. Chem. Rev. 256 (2012) 28-45. 15

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[18]T.A.H. Nguyen, H.H. Ngo, W.S. Guo, T.Q. Pham, F.M. Li, T.V. Nguyen, X.T. Bui, Adsorption of phosphate from aqueous solutions and sewage using zirconium loaded okara (ZLO): Fixed-bed column study, Sci. Total Environ. 523 (2015) 40-49. [19]R.M. Clark, Evaluating the cost and performance of field-scale granular activated carbon systems, Environ. Sci. Technol. 21 (1987) 573-580. [20]A. Wolborska, Adsorption on activated carbon of p-nitrophenol from aqueous solution, Water Res. 23 (1989) 85-91. [21]M. Zwietering, I. Jongenburger, F. Rombouts, K. Van't Riet, Modeling of the bacterial growth curve, Appl. Environ. Microbiol. 56 (1990) 1875-1881. [22]J. Preethi, S.M. Prabhu, S. Meenakshi, Effective adsorption of hexavalent chromium using biopolymer assisted oxyhydroxide materials from aqueous solution, React. Funct. Polym. 117 (2017) 16-24. [23]N.G. Hossein-Zadeh, Modeling the growth curve of Iranian Shall sheep using non-linear growth models, Small Ruminant Res. 130 (2015) 60-66. [24]S.N. do Carmo Ramos, A.L.P. Xavier, F.S. Teodoro, L.F. Gil, L.V.A. Gurgel, Removal of cobalt (II), copper (II), and nickel (II) ions from aqueous solutions using phthalate-functionalized sugarcane bagasse: mono-and multicomponent adsorption in batch mode, Ind. Crop. Prod. 79 (2016) 116-130. [25]G. Bohart, E. Adams, Some aspects of the behavior of charcoal with respect to chlorine, J. Am. Chem. Soc. 42 (1920) 523-544. [26]T. Ataei-Germi, A. Nematollahzadeh, Bimodal porous silica microspheres decorated with polydopamine nano-particles for the adsorption of methylene blue in fixed-bed columns, J. Colloid Interface Sci. 470 (2016) 172–182. [27]Q. Hu, P. Wang, J. Jiang, H. Pan, D.W. Gao, Column adsorption of aniline by a surface modified jute 17

fiber and its regeneration property, J. Environ. Chem. Eng. 4 (2016) 2243–2249.

18

Fig. 1. Transformation processes of the Wolborska model.

19

Fig. 2. Schematic diagram of the breakthrough curve and tangent line containing various parameters.

20

Fig. 3. Effect of the parameter a on the breakthrough curve for the dose-response model.

21

Fig. 4. Effects of the rate constant on (a) breakthrough curve and (b) rate profile for the Yoon–Nelson model (k1 > k2 > k3).

22

Fig. 5. Effects of the parameter n on (a) breakthrough curve and (b) rate profile for the Clark model (n1 > n2 > n3).

23

Fig. 6. Effects of the parameters a and b on breakthrough curve and rate profile for the dose-response model (a1 > a2 > a3 > 1 and b1 > b2 > b3).

24

Fig. 7. Adsorption of nitrate on the chitosan-Fe(III) composite: (a) Breakthrough curve and (b) predicted value versus observed value.

25

Table 1 Breakthrough models and the corresponding modified models. Name

Equation

Bohart–Adams

c = c0

Thomas

Yoon–Nelson

Clark

Dose-response

c = c0

Modified equation 1

[ ( )]

1 + exp kBAc0

a0x uc0

―t

1

[ (

1 + exp kTc0

q0m νc0

c 1 = c0 1 + exp[kYN(τ ― t)] c = c0

1 1 ― n )] 1

[1 + Aexp( ―rt

c 1 =1― c0 1 + (bt)a

)]

―t

c 1 = c0 1 + exp[4μmax(λ ― t) + 2]

c 1 = c0 1 + exp[4μmax(λ ― t) + 2] c 1 = c0 1 + exp[4μmax(λ ― t) + 2] c = c0

1

{

[

1 + (n ― 1) ⋅ exp μmax ⋅ n

n n ― 1(

c =1― c0

]}

1 n―1

λ ― t) + n 1

{[

2 t (λμmax + 1 + λμmax) ― 1 1+ λ (λμmax + 1 + λμmax)2 + 1

26

]

1+

1

(λμmax + 1 + λμmax)2

}

(λμmax + 1 + λμmax)2

Wolborska

(

)

βac0 βa c = exp t― x c0 a0 u

Non-existence

In the dose-response model, b = vc0/q0m.

27

Table 2 Summaries and calculations of the four parameters defined in this work. Models Bohart–Adams

μmax kBAc0

a0x

4

uc0

kTc0

q0m

4

νc0

Thomas

kYN

Yoon–Nelson Clark

λ

r⋅n

n ― n 1

―

a―1

Dose-response

b ⋅ (a ― 1) 4a

a

ti

2

a0x

a0x

kBAc0

uc0

uc0

2 kTc0

q0m

q0m

νc0

νc0

τ

τ

―

2

τ―

4 ―

―

t50

a+1

⋅ (a + 1)

a

kYN n―1

[( ) ]

1 ln r

+n

A

a+1

( )

1 a―1 b a+1

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a

n―1 ―1 1 2 ― ln r A

(

1 b

)

1 n―1 ― ln r A

( ) 1 a

( )

1 a―1 b a+1

Table 3 The parameters and errors obtained from different breakthrough models for nitrate adsorption on the chitosan-Fe(III) composite. Bohart– Adams Paramete rs

kBA (mL min− 1

mg−1 ) 3.09 × 10−4 μmax (min−1) λ (min) t50 (min) ti (min) χ2 Adj. R2

a0 (mg L−1)

2942. 3

Thomas kT (mL min− 1

mg−1 ) 3.09 × 10−4

q0 (mg g−1)

16607. 3

Yoon–Nelson

kYN (min−1 )

1.55 × 10−2

τ (min )

163. 4

doseresponse

Clark

A

2.7 8× 10−

r (min−1 )

n

a

b (min−1 )

1.09 × 10−2

1.007 3

2.4 7

6.74 × 10−3

2

3.86 × 10−3

3.86 × 10−3

3.86 × 10−3

4.01 × 10−3

4.92 × 10−3

33.9 163.4 163.4 1.37 × 10−3 0.9878

33.9 163.4 163.4 1.37 × 10−3 0.9878

33.9 163.4 163.4 1.37 × 10−3 0.9878

30.4 155.8 122.5 2.70 × 10−4 0.9976

44.2 148.4 104.6 3.32 × 10−4 0.9970

Operating conditions: c0 = 50 mg L−1; v = 6.1 mL min−1; x = 44 cm; u = 15.85 cm min−1; m = 3 g.

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Graphical abstract

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Highlights  The defined four parameters can more completely describe the breakthrough curve.  Mathematical relationships between the breakthrough models are revealed.  Physical meanings of some specific terms are explicated in detail.  Modified breakthrough models are developed using the parameters μmax and λ.  Effects of model parameters on breakthrough curve and rate profile are discussed.

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CRediT author statement

Qili Hu: Writing original draft, Conceptualization, Methodology, Validation Yanhua Xie: Data curation, Editing, Supervision Zhenya Zhang: Conceptualization, Methodology, Supervision

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Declaration of interests ■

The authors declare that they have no known competing financial interests or personal

relationships that could have appeared to influence the work reported in this paper. □

The authors declare the following financial interests/personal relationships which may be

considered as potential competing interests:

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