A Methodology to Estimate the Sorption Parameters from Batch and Column Tests: The Case Study of Methylene Blue Sorption onto Banana Peels

: In the present work, a methodology is presented where batch and ﬁxed-bed column tests of dye sorption onto granular biosorbents are analyzed with properly selected models to estimate the parameters required for the rational design of pilot-scale units. The sorption of methylene blue (MB) onto banana peels (BP) was investigated as a case study. To identify the mechanisms of MB sorption onto BP, the pore structure and surface of BP were characterized with mercury intrusion porosimetry (MIP), Fourier transform infrared spectroscopy (FTIR), and scanning electron microscopy (SEM). Batch tests were performed over the temperature range of 15–45 ◦ C, and three models (Langmuir, Freundlich, Langmuir–Freundlich) were ﬁtted to equilibrium and kinetic data for (i) estimating thermodynamic / kinetic parameters and (ii) choosing the model with the best goodness-of-ﬁt. Sorption tests on ﬁxed-bed columns were combined with a one-dimensional macroscopic convection / dispersion / sorption model to estimate the sorption parameters of BP beds. MB sorption onto BP was a purely exothermic ( ∆ H 0 ~ − 20 kJ / mol), reversible, and monolayer chemisorption with high activation energy for the desorption step (E d ~29 kJ / mol) and low activation energy for the adsorption step (E a ~9 kJ / mol). The Langmuir isotherm (K L = 141.9 m 3 / kg, T = 25 ◦ C) and Langmuir kinetic model (k d = 1.05 × 10 − 5 s − 1 ) provided the best ﬁtting to equilibrium and transient data of batch tests. The sorption capacity ~0.15–0.22 kg / kg and kinetic constant 0.3 × 10 − 5 s − 1 –4.0 × 10 − 5 s − 1 estimated from tests on BP beds were comparable to those obtained from batch tests.


Introduction
The presence of dyes in liquid industrial waste poses a risk to the ecosystem and human health [1][2][3]. Methylene blue (MB) is a cationic azo dye widely used in industry, and several diseases (e.g., hypertension, anemia) have been associated with its presence in the human body [4]. Adsorption is a simple and effective method of removing dyes from aqueous systems [5,6]. Among the adsorbents occasionally used to remove pollutants from wastewater, activated carbon is the most popular [7,8]. In recent years, emphasis has been placed on low-cost adsorbents made from agricultural waste as alternatives to commercial activated carbon [9]. Such materials, prepared from the peels and shells of fruits or leaves of trees, have been used to remove dyes from wastewater [10][11][12][13]. Special attention has been paid to olive-pomace [14,15] and oil waste mills [16] for the removal of dyes from wastewater. Still, it is worth mentioning the capacity of olive pomace/chitosan composites to adsorb antibiotics from wastewater [17].
(100 and 200 mg/L), bed height (1.86, 3.72, and 5.58 cm), and feed injection rate (1 and 3 mL/min). A numerical model is developed by combining convective flow with hydrodynamic dispersion and sorption dynamics in a porous bed. By fixing the sorption equilibrium constant, the remaining sorption parameters are estimated with inverse modeling of the MB concentration breakthrough curves. The variability of BP bed sorption parameters is interpreted referring to interactions of reactive with mass-transfer processes at the grain-and bed-scale. Finally, the estimated MB sorption capacity of BP beds is compared with the corresponding capacities of other biosorbents from earlier studies. Table 1. Dye sorption capacity for banana peels and other biosorbents.

Materials
Methylene blue (molecular formula: C 16 H 18 N 3 ClS) was supplied by Merck and used without any other purification. MB was dissolved in distilled water to prepare stock solutions with a concentration of 1000 mg/L, which were diluted to obtain solutions over the concentration range of experiments The goal of the column sorption experiments was to quantify the effects of flow and masstransfer parameters (e.g., Peclet number) on the sorption capacity and kinetics. Such information could be exploited when designing a pilot adsorption system and for scaling-up and selecting the geometrical (e.g., diameter, column, and bed lengths) and flow (e.g., flow rate) parameters.

Sorption Isotherms
The Langmuir model assumes monolayer biosorption on identical and energetically equivalent active sites without any interaction between adsorbed molecules [43] and is written where Seq (mg/g) is the mass of dye adsorbed per unit mass of adsorbent at equilibrium, Ce (mg/L) is the equilibrium concentration of dye in solution, Smax (mg/g) is the maximum adsorption capacity of the adsorbent corresponding to monolayer coverage, and KL (L/mg) is the Langmuir adsorption constant. The Freundlich isotherm describes the multilayer biosorption with the nonuniform distribution of sorption sites over the heterogeneous surface along with interactions between adsorbed molecules [44] and is expressed by where F K ((mg/g)(L/mg) 1/n ) is the Freundlich adsorption constant related to the maximum adsorption capacity of the adsorbent, and the exponent  varies with the adsorbent heterogeneity. The goal of the column sorption experiments was to quantify the effects of flow and mass-transfer parameters (e.g., Peclet number) on the sorption capacity and kinetics. Such information could be exploited when designing a pilot adsorption system and for scaling-up and selecting the geometrical (e.g., diameter, column, and bed lengths) and flow (e.g., flow rate) parameters.

Sorption Isotherms
The Langmuir model assumes monolayer biosorption on identical and energetically equivalent active sites without any interaction between adsorbed molecules [43] and is written where S eq (mg/g) is the mass of dye adsorbed per unit mass of adsorbent at equilibrium, C e (mg/L) is the equilibrium concentration of dye in solution, S max (mg/g) is the maximum adsorption capacity of the adsorbent corresponding to monolayer coverage, and K L (L/mg) is the Langmuir adsorption constant. The Freundlich isotherm describes the multilayer biosorption with the nonuniform distribution of sorption sites over the heterogeneous surface along with interactions between adsorbed molecules [44] and is expressed by where K F ((mg/g)(L/mg) 1/n ) is the Freundlich adsorption constant related to the maximum adsorption capacity of the adsorbent, and the exponent β varies with the adsorbent heterogeneity. The Langmuir-Freundlich isotherm is a hybrid model describing the distribution of sorption energy onto the heterogeneous surface of the adsorbent [45]. This model transits to the Freundlich Processes 2020, 8, 1467 6 of 22 or Langmuir isotherm at low or high adsorbate concentrations, respectively. The isotherm can be expressed as follows where K LF is the equilibrium constant for heterogeneous solid, and β is a heterogeneous parameter ranging from 0 to 1. For a reversible adsorption/desorption process, the dependence of the equilibrium constant K I (I = L, F, LF) on the absolute temperature T (K) is governed by the van't Hoff equation [46] dln( where R represents the universal gas constant (8.314 J K −1 mol −1 ), and ∆H 0 represents the change of the standard enthalpy (heat) of adsorption (J mol −1 ). Over a limited temperature range, ∆H 0 can be assumed constant, the integration of Equation (4) yields and ∆H 0 can be estimated from the slope of the plot of ln(K I ) versus 1/T (Equation (5)).

Kinetic Models
The analysis of kinetic data of MB sorption onto BP with conventional (first-order/second-order/etc.) models has already been performed [27]. To maintain consistency in kinetic with equilibrium models, instead of using the conventional models, the dynamic form of the aforementioned isotherms was chosen [47,48]. Sorption is regarded as a reversible reaction with fundamental steps: the adsorption/desorption of dye molecules between the aqueous phase and the surface of adsorbent.
(1) In the Langmuir kinetic model, the adsorption rate is regarded proportional to the fraction of vacant sites, and the desorption rate is proportional to the fraction of sites covered by adsorbed molecules. The overall rate is written where k a , k d are the kinetic constants of adsorption and desorption steps, respectively. At equilibrium, dS/dt = 0, and Equation (6) coincides with the Langmuir isotherm with (2) In the Freundlich kinetic model, the adsorption rate changes nonlinearly with the concentration of dissolved molecules, and the overall sorption rate is given by [48] At equilibrium, Equation (8) coincides with the Freundlich isotherm with At equilibrium, Equation (10) coincides with the Langmuir-Freundlich isotherm with Starting from the integrated form of the van't Hoff relationship (Equation (5)), and after some manipulation [46], we get Arrhenius-type equations for the rate constants where E a , E d are the activation energies for the adsorption and desorption step, and A a , A d are the corresponding pre-exponential factors, interrelated by

Modeling the Performance of a Fixed-Bed Column
The one-dimensional (1D) transport of a solute (MB) through a nonhomogeneous porous medium, such as the bed of packed BP grains, is governed by convection, dispersion, and adsorption [49] and described by the following mass balance: where t is the time (s), C is the MB concentration (kg/m 3 ) at axial distance x (m) from the column inlet, D L is the longitudinal hydrodynamic dispersion coefficient (m 2 s −1 ), u 0 is the pore velocity (m s −1 ), defined as u 0 = Q/(φA), A is the cross-sectional area of the column, φ is the bed porosity, S is the concentration of MB adsorbed per unit mass of solid (kg/kg), and ρ b is the bulk density of the dried adsorbent. If normalized, the aforementioned kinetic models (Equations (6), (8), and (10)) can be merged into a general expression for the overall sorption rate as where a is a rate constant that is proportional to k d . The hydrodynamic dispersion coefficient is commonly described by the relationship [50]: where D m is the molecular diffusion coefficient of MB (m 2 s −1 ), F is the electrical formation factor of the porous bed of adsorbing grains, and a L is the longitudinal dispersion length (m). For homogeneous porous media, for the sake of simplicity, we can assume that a L ≈ d g [50]. If L is the total height of the column (m), and using the dimensionless variables τ = tu 0 /L, ξ = x/L, C* = C/C 0 , Equations (14) and (15) are finally transformed into the following dimensionless relationships where Pe d = u 0 d g /D m is the grain-scale Peclet number.
Equations (17) and (18) describe the mass transfer along the bed of adsorbing material extending over the length, ξ 0 ≤ ξ ≤ 1.0, where the distance ξ 0 specifies the upper surface of the adsorbent, ξ 0 = 1 − (L b /L), and L b is the bed height ( Figure 1c). On the other hand, the first segment of the column, 0 ≤ ξ < ξ 0 , is fully occupied by the aqueous phase. Over this region, no sorption occurs, namely whereas the MB mass transfer is described by the classical diffusion-convection equation, which in dimensionless form is written where Pe L = u 0 L/D m is the column-scale Peclet number, which can be replaced by Pe d through the relationship The foregoing Equations (17)- (20) are subject to the initial condition and the boundary conditions

Surface Chemistry and Sorption Mechanism
Potential changes on the BP surface due to the BP sorption are evident in the FE-SEM images (Figure 2a  The surface chemistry of BP and its interactions with MB were analyzed by examining the ATR spectra of BP grains before and after MB adsorption (Figure 2e), and an analytic interpretation was given elsewhere [27]. A high number of hydroxyl, oxygen, and carboxyl functional groups (OH, C-O, C=O) were identified on the surface of BP. The oxygen atoms of these groups might create hydrogen bonds with the nitrogen atoms of MB [51]. The characteristic bands of MB were overlapped by stretching vibrations of BP (inset of Figure 2e). Given that the adsorbed MB mass is much less than The surface chemistry of BP and its interactions with MB were analyzed by examining the ATR spectra of BP grains before and after MB adsorption (Figure 2e), and an analytic interpretation was given elsewhere [27]. A high number of hydroxyl, oxygen, and carboxyl functional groups (OH, C-O, C=O) were identified on the surface of BP. The oxygen atoms of these groups might create hydrogen bonds with the nitrogen atoms of MB [51]. The characteristic bands of MB were overlapped by stretching vibrations of BP (inset of Figure 2e). Given that the adsorbed MB mass is much less than the BP mass, the peaks associated with BP were prevalent in the spectrum. The band at~1600 cm −1 is associated with the stretching vibrations of C=O and C-O of carboxylate groups of BP and the stretching vibration of the C=N bond of MB (Figure 2e). In this manner, the amplitude and intensity of this band changed after the MB sorption due to the electrostatic interactions between BP and MB. Respectively, the oxygen atoms of the C=O carboxylate group of BP could participate in hydrogen bonding (a form of electrostatic interactions) with the nitrogen atoms of MB, leading to the disappearance of the band at 1374 cm −1 in the spectrum of BP-MB (Figure 2e). The electrostatic interactions of the negatively charged BP surface with the positively charged cations of MB is the main mechanism of adsorption onto BP.

Equilibrium and Kinetic Parameters from Batch Tests
Nonlinear regression analysis of sorption isotherms revealed that the Langmuir (Equation (1)) and Langmuir-Freundlich (Equation (3)) models fit the data better than the Freundlich model (Equation (2)) (Table 2, Figure 3). This is reflected in the lower value of the coefficient of determination, R 2 , and broad confidence intervals for the Freundlich parameters (Table 2). of this band changed after the MB sorption due to the electrostatic interactions between BP and MB. Respectively, the oxygen atoms of the C=O carboxylate group of BP could participate in hydrogen bonding (a form of electrostatic interactions) with the nitrogen atoms of MB, leading to the disappearance of the band at 1374 cm −1 in the spectrum of BP-MB (Figure 2e). The electrostatic interactions of the negatively charged BP surface with the positively charged cations of MB is the main mechanism of adsorption onto BP.

Equilibrium and Kinetic Parameters from Batch Tests
Nonlinear regression analysis of sorption isotherms revealed that the Langmuir (Equation (1)) and Langmuir-Freundlich (Equation (3)) models fit the data better than the Freundlich model (Equation (2)) (Table 2, Figure 3). This is reflected in the lower value of the coefficient of determination, R 2 , and broad confidence intervals for the Freundlich parameters (Table 2).  The foregoing estimated values of equilibrium constants, ln(K L ), ln(K F ), ln(K LF ) ( Table 2), were fitted to the van't Hoff equation (Equation (5)) to estimate the heat of adsorption (Table 3). It seems that the reversible MB adsorption/desorption in BP is an exothermic reaction (∆H 0 < 0), whereas the fitting to the Freundlich equilibrium constant is poor (Table 3). However, depending on the type of adsorbent, the potential pretreatment, and changes on pore space morphology, the dye sorption onto sorbents may be dominated by one or more mechanisms (e.g., electrostatic interactions, pore diffusion), and such differences are commonly reflected in the different values of standard heat of adsorption (Table 4). The equilibrium parameters estimated from sorption isotherms (Table 2) were fixed, and the kinetic constant k d was estimated (Table 5) with the nonlinear fitting of the transient MB sorption datasets to the numerical solution of Langmuir (Equation (6)) Freundlich (Equation (8)) and hybrid (Equation (10)) kinetic models (Figure 4a-c) by the ATHENA Visual Studio 14 software [52]. Due to the limited number of degrees of freedom (only one parameter to estimate), discrepancies were observed between the predicted curves and measured data (Figure 4a-c), while the confidence intervals were quite high (Table 5) for all kinetic models. observed between the predicted curves and measured data (Figure 4a-c), while the confidence intervals were quite high (Table 5) for all kinetic models.  Using linear regression analysis, the kinetic constant vs. temperature datasets ( for the adsorption step were obtained from Equation (13) along with the results of Table 3. All parameters of the sorption rate constants are summarized in Table 6. Based on 2 R values and the relative size of the standard deviation (Table 6), it seems that the fitting is very good for the Langmuir model, satisfactory for the hybrid model, and poor for the Freundlich model (Table 6). For all cases, the activation energy for the desorption step is quite high and exceeds the heat of adsorption, in agreement with the theory for exothermic reactions [46]. On the other hand, the activation energy for Using linear regression analysis, the kinetic constant vs. temperature datasets (Table 5) were fitted with Equation (12) to estimate the pre-exponential factor ln(A d ) and activation energy E d for the desorption step. Subsequently, the pre-exponential factor of ln(A a ) and activation energy E a for the adsorption step were obtained from Equation (13) along with the results of Table 3. All parameters of the sorption rate constants are summarized in Table 6. Based on R 2 values and the relative size of the standard deviation (Table 6), it seems that the fitting is very good for the Langmuir model, satisfactory for the hybrid model, and poor for the Freundlich model (Table 6). For all cases, the activation energy for the desorption step is quite high and exceeds the heat of adsorption, in agreement with the theory for exothermic reactions [46]. On the other hand, the activation energy for adsorption step is quite low, indicating that MB adsorption onto BP can be regarded as a weakly activated step (Table 6). From Tables 2, 3, 5 and 6, it is evident that the Langmuir isotherm and kinetic model agree with the mechanism of monolayer chemisorption of MB cations on the surface of BP grains by attractive electrostatic forces, providing the best fitting to (i) the experimental data of batch tests and (ii) generated thermodynamic parameters therefrom. It is worth mentioning that the relatively high values of heat of adsorption (Table 3) and desorption activation energy (Table 6) along with the low adsorption activation energy are consistent with the abovementioned mechanism of chemical sorption of MB on BP surface through electrostatic interactions.

Parameter Estimation from Column Tests
To simulate the MB fate in a fixed-bed column (Figure 1c), the Langmuir isotherm and kinetic model were selected to describe the sorption process. Under these conditions, Equations (17) and (18) are transformed to where the rate constant a is defined by and its value is expected to vary over the region. The full set of input parameters for the operation of the fixed-bed column are given in Table 7. For each set of experimental conditions (C 0 , Q, L b ), the Langmuir equilibrium constant K L was kept equal to that estimated from batch tests at 25 • C (Table 2), and ATHENA Visual Studio 14 software was used to estimate the sorption parameter values (S max , a) that minimize the distance of experimental from numerically predicted breakthrough curve. Specifically, the partial differential equations (Equations (20), (25), and (26)) were solved with finite differences, under the adequate initial (Equation (22)) and boundary (Equations (19), (23), and (24)) conditions. The numerically predicted transient response of the MB concentration at the column outlet was fitted to the measured breakthrough curve by a stochastic Bayesian estimator [52], and the results are shown in Table 8. There is an excellent agreement between the experimental and numerically predicted breakthrough curves (Figure 5a-c). Table 7. Geometrical parameters and physical properties of the experimental system.   The maximum MB sorption capacity of BP bed, S max , estimated from flow-through tests (Table 8) is comparable to that determined by batch experiments (Figure 6a). The rate kinetic constant estimated from column tests, a, is comparable to the lower limit of inequality (Equation (27)), k d , and much less than the upper limit k d (1 + K L C 0 ) (Figure 6b). At the microscopic scale of the grain surface (Figure 7a), the near-surface flow and solute concentration fields can be approximated by the boundary layer theory so that the dynamics of solute mass-transfer toward the solid surface is correlated with the fluid properties and flow conditions [53]. Statistical thermodynamics and molecular dynamics [54] approaches are required to quantify the interactions (e.g., electrostatic attraction of MB cations to negatively charged sites of the solid surface) at the submicroscopic scale (Figure 7b).  At the microscopic scale of the grain surface (Figure 7a), the near-surface flow and solute concentration fields can be approximated by the boundary layer theory so that the dynamics of solute mass-transfer toward the solid surface is correlated with the fluid properties and flow conditions [53]. Statistical thermodynamics and molecular dynamics [54] approaches are required to quantify the interactions (e.g., electrostatic attraction of MB cations to negatively charged sites of the solid surface) at the submicroscopic scale (Figure 7b).
It seems that S max increases with the flow velocity (Table 8, Figure 6a). As the flow velocity increases, the thickness of the boundary layers around the grain surfaces decreases (Figure 7a), the access of MB cations to the active sites of the solid surface is facilitated (Figure 7b), and the overall sorption capacity of the bed, S max , has the tendency to increase (Table 8, Figure 6a). As the bed length increases, nonuniformities of the flow field caused by nonrandom pore space heterogeneities, associated with the quality of bed packing, may prevent the MB access to a fraction of grain surface sites and lead to lower bed sorption capacity (Table 8, Figure 6a). On the other hand, the longer water retention times and the conditions of fully developed flow in long beds increase the probability of MB cations to access the active surface sites, increasing the overall MB sorption capacity (Table 8). Therefore, the estimated S max value may be an increasing or decreasing function of the bed length (Table 8, Figure 6a).
It seems that the kinetic constant, a, increases with the flow velocity and with initial concentration and bed length decreasing (Table 8, Figure 6b). It is a macroscopic parameter depending on the diffusive transport of dissolved molecules from the bulk phase to the external surface of the grains (Figure 7a) and the kinetics of the purely reactive process of adsorption/desorption of solute molecules surrounding the surface (Figure 7b). The diffusion rate is proportional to the concentration gradient across the boundary layer, the thickness, δ c , of which is inversely proportional to the square root of the local pore velocity [53]. Therefore, the higher the flow velocity, the thinner the concentration boundary layers around the grains and the higher the expected a values (Table 8, Figure 6b). On the other hand, the higher the feed concentration, C 0 , the higher the concentration gradient across the boundary layer (Figure 7a), the faster the diffusive rate of MB toward the solid surface, the higher the expected a values (Table 8, Figure 6b).
At the microscopic scale of the grain surface (Figure 7a), the near-surface flow and solute concentration fields can be approximated by the boundary layer theory so that the dynamics of solute mass-transfer toward the solid surface is correlated with the fluid properties and flow conditions [53]. Statistical thermodynamics and molecular dynamics [54] approaches are required to quantify the interactions (e.g., electrostatic attraction of MB cations to negatively charged sites of the solid surface) at the submicroscopic scale (Figure 7b). As the bed height increases, the spatial fluctuations of the flow velocity are enhanced due to either more frequent nonrandom heterogeneities of the pore space or higher spatial deviations of the pore velocity. Such an enhancement of the nonuniformity of the flow field may delay the MB access to grain surfaces leading to a lower sorption rate constant, a (Table 8, Figure 6b For the sake of homogenization, the MB maximum sorption capacity estimated from flow tests on BP beds is expressed as a function of Pe b and compared with the corresponding literature data obtained from flow tests on the beds of various biosorbents (Figure 8a). It is evident that the highest S max values were achieved with bed columns operating at a relatively low Pe b < 1000 (Figure 8a). On the other hand, for the majority of other biosorbents, the MB sorption capacity was weaker, while most studies were conducted over high values of Peclet number (Pe b >> 1000) (Figure 8a). However, due to the different origins of adsorbents and grain sizes, it is difficult to give a full interpretation of the observed MB sorption capacity in terms of the prevailing flow and mass-transfer regime. obtained from flow tests on the beds of various biosorbents (Figure 8a). It is evident that the highest max S values were achieved with bed columns operating at a relatively low 1000  b Pe (Figure 8a).
On the other hand, for the majority of other biosorbents, the MB sorption capacity was weaker, while most studies were conducted over high values of Peclet number ( 1000  b Pe ) (Figure 8a). However, due to the different origins of adsorbents and grain sizes, it is difficult to give a full interpretation of the observed MB sorption capacity in terms of the prevailing flow and mass-transfer regime.

0.30
This study / banana peels Rice husk [23] Peanut husk [24] Eukalyptus bark [25] Rice straw [34] Maize stem ground [35] Water yakinth [36] Coffee residues [37] Watermelon rind [38] Pinecone [39] S max (kg/kg) Regarding the sorption dynamics in BP beds, the smaller than 1.0 Da values confirm that sorption is always much slower than convective flow at the bed-scale, and sorption kinetics is a decreasing function of Pe b , tending to an asymptotic value at very high Pe b values (Figure 8b).

Conclusions
A methodology was proposed and applied to the case of methylene blue sorption onto banana peels, where sorption parameters needed for the design of pilot-scale biosorbents were obtained from the combination of batch and column studies with mutually consistent and true-to-the mechanism sorption models. Equilibrium and kinetic sorption tests of MB onto BP grains were conducted in batch reactors at temperatures of 15-45 • C to estimate thermodynamic/kinetic properties and assess the goodness-of-fit for three models used to describe the isotherms and sorption kinetics: Langmuir, Freundlich, and Hybrid (Langmuir-Freundlich).
MB sorption tests were conducted in fixed-bed columns packed with BP grains by varying the flow rate, the MB concentration in feed solution, and bed length, measuring the relevant MB concentration breakthrough curves. To estimate the maximum sorption capacity along with the sorption rate constant as functions of all pertinent parameters, the breakthrough curves were fitted to the numerical solution of a 1-D advection-dispersion-sorption model coupled with the Langmuir kinetic model.
The most important conclusions are outlined below • MB sorption onto BP is an exothermic and reversible process, with a strongly activated desorption step and weakly activated adsorption step.

•
Equilibrium and kinetic data are better fitted by the Langmuir model, which is consistent with the monolayer chemisorption of MB cations onto BP solid surface by electrostatic forces.

•
The experimental breakthrough curves and MB sorption capacity onto BP grains of the bed are predicted satisfactorily by the numerical model.

•
The sorption capacity of adsorbents under continuous flow conditions is comparable to the corresponding one estimated from batch experiments.

•
The sorption rate constant estimated from continuous flow tests on a column bed is fully compatible with corresponding parameters estimated from batch tests when the same sorption model is used.

•
The kinetics of sorption in column beds is affected by the thickness of the velocity and concentration boundary layers surrounding the grains of sorbent and nonuniformities of the flow field caused by local nonrandom heterogeneities. This parameter is an increasing function of flow velocity and feed concentration and a decreasing function of bed length. It is a decreasing function of Pe b , tending to an asymptotic value at high values of this parameter.
• Compared to earlier studies, the highest MB sorption capacity was achieved in the present study with fixed-bed columns of BP operating over relatively low Pe b values.
It is worth mentioning that, depending on the sorbent/sorbate system, a broad variety of equilibrium/kinetic models might be used in the aforementioned methodology. However, it is critical to use equilibrium and kinetic sorption models that are mutually consistent with each other so that, finally, a unique set of parameters is estimated. In this manner, when designing a pilot-scale system, the uncertainty associated with the choice of parameter values is minimized, and the cost-efficiency and performance of large-scale sorption systems are ensured.

Conflicts of Interest:
The authors declare that they have no conflict of interest.