Modelling and simulation of the ion exchange process for Zn 2+ (aq) removal using zeolite NaY

The treatment of water contaminated by toxic metals using ion exchange with zeolites is becoming attractive due to its low capital costs and high potential for removal capacity. Mathematical modelling of this process allows for operational control and estimation of the ability to remove these metals. In this work, the kinetic modelling was performed based on finite bath experimental data, with Intraparticle Diffusion (IPD) and External Liquid Film Mass Transfer (MTEF) models. The models Thomas (TH), Yoon-Nelson (YN) and Solid Film Mass Transfer (MTSF) were used to estimate the saturation time, ion exchange capacity and sizing variables of a fixed bed column. For the finite bath system, the results showed that the mass transfer was better represented by the IPD phenomenon. The breakthrough curve obtained by the Aspen Adsorption (MTSF) model presented the best fit, compared with experimental data, with R 2 ≥0.9923. The average ion exchange capacities calculated for MTSF, TH and YN were respectively 2.22, 2.12 and 2.07 meq Zn 2+ (aq)/ g of zeolite. The model simulated with Aspen Adsorption was also used to analyze the continuous system behaviour, by varying the height of the bed. It was observed that increasing the height, the saturation time and ion exchange capacity also increase, while reducing the height makes axial dispersion the predominant mass transfer phenomenon, which reduces the diffusion of Zn 2+ (aq) ions. The MTEF fit, a differential model that represents the transport of the ion within the solution to the liquid film layer around the zeolite particles, was performed using the MatLab software, which enabled to simulate the behaviour of MTEF by decreasing the concentration of Zn 2+(aq) during ion exchange with NaY zeolite in the finite bath system.


Introduction
Heavy metals are still causing environmental problems in the 21st century, such as the pollution of rivers connected to public water sources, due to unstructured urbanization and industrial processes (Oliveira et al., 2018).
According to the 2019 Statistical Yearbook of the Metallurgical Sector, the Brazilian production of zinc was in 2nd place among non-ferrous metals. This intense production process results in high waste disposal. The National Council for the Environment, through resolution CONAMA 430/2011 of Brazil, establishes that the maximum concentration for disposal of zinc in the receiving body is 5 mg/L. The EPA (United States Environmental Protection Agency) recommends the National Secondary Drinking Water Regulations (NSDWRs) 5 mg/L of Zinc. Based on Cupertino et al. (2020) zinc has nutritional importance, however in high doses it presents toxicity and health risks. Zinc oxide nanoparticles, for example, present in battery, biomedical and food production industries, are related to the decrease in cellular antioxidant levels (Lazzaretti & Hupffer, 2018).
Most effluent and water purification processes containing excess of toxic metals, which employ ion exchange technology, use fixed bed columns. To model this dynamic process, it is interesting to know the chemical balance, kinetics and mass balances of the system (Nakajima, 2013). For fixed bed modelling, the Thomas (TH) and Yoon -Nelson (YN) models are considered useful by some authors (Zheng et al., 2008;Trgo et al., 2011) however, according to Abdi and Abedini (2020), Aspen Adsorption software models have also been efficient.
The Aspen Adsorption (Adsim) software is a tool used to simulate dynamic adsorption and ion exchange processes, which provides models for the generation of breakthrough curves and estimation of performance parameters for continuous fixed bed. The Adsim model was used in recent researches, such as simulation of the ion exchange of Cu2+ in aqueous solution with the biomass Cucumis melo VAR. cantalupensis, (Nieva et al., 2018), Cu(II) and Cr(III) on charcoal (Zhang et al., 2019) and Cd (II) and Cu (II) in simple and binary metallic systems with the biomass Eichhornia crassipes (Adornado et al., 2016).

Objectives
In this work, an investigation was carried out regarding possible kinetic modelling using finite bath experimental data from literature (Intraparticle Diffusion and External Film Mass Transfer models). Fixed bed column models (MTSF, TH and YN) were obtained to represent the removal of Zn2+(aq) in aqueous medium, by ion exchange, with zeolite NaY. The model obtained with Aspen Adsorption (MTSF) was used to evaluate the influence of the bed height on the Zn2+(aq) ion removal capacity by the zeolite NaY, in a continuous fixed bed column. Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 3

Methodology
This research was carried out in four distinct steps: (1) bibliographic research of significant experimental data on the removal of Zn 2+ (aq) by NaY zeolite in finite bath and fixed bed column; (2) kinetic modelling of finite bath data using the Intraparticle Diffusion (IPD) and External Film Mass Transfer (MTEF) models to determine the limiting step for the mass transfer (TM); (3) kinetic modelling of fixed bed column data using the models: Solid Film Mass Transfer (MTSF) from Aspen Adsorption, Thomas (TH) and Yoon-Nelson (YN), to compare the significance of each and identify mass transfer and axial dispersion phenomena involved in the process; (4) simulation of the ion exchange process between Zn 2+ (aq) and NaY zeolite. Calculations of some other parameters were performed to satisfy specifications of the Aspen Adsorption model (MTSF). To this end, data referring to the physical characteristics of zeolite NaY were collected from Silva and Souza (2004) and are presented in Table 2. These data were used to calculate interparticle and intraparticle voids (εi and εp). Column porosity (εi), referring to the volume of voids in the fixed bed column, was determined by Equation (1), based on Silva and Souza (2004). Source: Silva and Souza (2004) Applying data from Table 2 to Equation (1), it was possible to calculate εi = 0.54 m 3 /m 3 , a value close to εi = 0.5, found by Ostroski et al. (2008), presented in Table 1. Due to the convergence of εi values, the characterization data in Table 2 Research Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 4 were also considered to calculate the particle porosity (εp), referring to the total pore volume of a material (Vpore), using Equation (2), based on Silva and Miranda (2003).

Sources of Experimental Data
The calculated εp value was 5.99x10 -4 m 3 /m 3 . Another variable calculated to satisfy the software model was the selectivity constant (KAB), using Equation (3) based on Abrão (2014) where the distribution coefficient is the total number of meq of the cation in the exchanger per gram of resin, and is the total ion concentration, in meq per mL of solution.
The selectivity constant (KAB) belongs to the ion exchange equilibrium governed by the Law of Mass Action, which was used, in Aspen Adsorption, to estimate the molar fraction of exchanged ions equivalent to the aqueous phase and the ion exchange bed, through Equation (4).
In Equation (4), the parameter IP1 is equal to KAB and the parameter IP2 is the stoichiometric coefficient of the ion exchange reaction; x represents the equivalent molar fraction, in the ion exchange resin, of the ion (component A) and counter ion (component B); y represents the equivalent molar fraction in the aqueous phase of the ion (component A) and counter ion (component B); C0 is the total ionic concentration (eq.m -3 ) and Q represents the maximum volumetric ion exchange capacity of the resin (eq.m -3 ).
The parameter Kd was calculated by fitting the models (Equations (5) to (8)) to the equilibrium data of the exchange between Zn 2+ (aq) and Zeolite NaY obtained from Ostroski et al. (2008) The linear isotherm model is described by Equation (5) (Calábria et al., 2017). Lambert (1967) suggested a polynomial function of the equilibrium concentration presented in Equation (6). The linearized Langmuir isotherm is given by Equation (7) and the linearized Freundlich model is given by Equation (8) (Calvet, 1989): Where qe (or y) is the amount of solute adsorbed by an amount of adsorbent, in equilibrium with a solution of concentration Ce (or x); Kd (or B) is the distribution coefficient of the solute between the solution and the solid surface; K1 and K2 (or B and C) are constant, and K1 is the estimate of Kd of the product; qm and b are constant and qmb (or 1/B) is the Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 5 estimated Kd of the product; Kf (or exp [A]) is the Freundlich constant, 1/n is an index of the adsorption intensity and since 1/n = 1, Kf = Kd is considered (Souza et al., 2001).
To determine the most suitable model for predicting the constant Kd, the following statistical criteria were evaluated: coefficient of determination (R 2 ) obtained by Equation (9) where SSres is the residual sum of squares and SStot is the total sum of squares, the Mean Squared Error (MSE) obtained by Equation (10) where n is the data points on all variables, Y is the vector of observed values of the variable being predicted with being the predicted values, and the Standardized Residual Scatter Plot in which it is desirable that the residuals are in the range between -2 and +2, as well as well distributed along the zero mean, without presenting clusters of points (Montgomery & Peck, 1982). (10)

Kinetic Modelling of Finite Bath
Adsorption kinetics studies demonstrate beneficial information about control mechanisms of adsorption processes, such as surface adsorption, chemical reaction and diffusion mechanisms (Montgomery & Peck, 1982). In this work, two kinetic models of mass transfer (IPD and MTEF) were fitted to the experimental data of the finite bath kinetics of ion exchange between Zn 2+ (aq) and NaY zeolite, from Ostroski et al. (2008) to define which phenomenon represented the limiting step of the process.

Kinetic equation of Intraparticle Diffusion (IPD)
First and second order kinetic models are not able to determine the diffusion mechanism. For this purpose, it is preferable to evaluate the kinetic results with the intraparticle diffusion model, which is represented by Equation (11) in its linearized form (Abdi et al., 2017): Where qt is the amount of solute in the solid phase at time t (meq/g); Kp is the intraparticle diffusion rate constant (meq/g/min -0.5 ); I is the intercept.

Kinetic model of External Film Mass Transfer (MTEF)
The External Film Mass Transfer model assumes that the solute is removed from the solution until equilibrium is reached in the liquid film formed on the solid surface, and that the equilibrium concentration varies with time (Puranik et al., 1999). The total mass balance of the finite bath system and the convective model that described the mass transfer between the solution and the adjacent film was described according to Equation (12), adapted from Puranik et al. (1999). Research, Society and Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 Where C is the concentration of adsorbate in solution (meq.L-1); Ktm is the liquid phase mass transfer coefficient (min -1 ); t is time (min); qm is the maximum equilibrium adsorption capacity; b is the Langmuir equilibrium isotherm constant; q(t) is the amount of solute in the film adjacent to the surface at time t (meq/g); C0 is the initial solution concentration (meq.L -1 ); V is solution volume (L); m is the mass of adsorbent (g). The differential Equation (12) was solved numerically, using the 'ode45' routine from the MATLAB software and the experimental values of each parameter, obtained from Ostroski et al. (2008).

Modelling of Breakthrough Curves
For modelling the fixed bed column, the TH, YN and the MTSF Adsim models were fitted to the Breakthrough Curves presented by Ostroski et al. (2008) in order to verify the significance of the MTSF model in relation to other dynamic models also adjustable to this process.

Thomas Model (TH)
One of the most used models for predicting the breakthrough curve and evaluating the performance of a fixed bed column was proposed by Thomas (1944), expressed in Equation (13).
Where q is the maximum solute concentration in the solid phase (meq/g); Q is the flow rate (mL/min); m is the exchanger mass (zeolite NaY) (g), C0 is the feed concentration (meq/L); C is the output concentration; t is the time and KTH is the Thomas model constant (mL/min.meq).

Yoon-Nelson Model (YN)
The Yoon-Nelson theoretical model has the characteristic of not requiring detailed information on the exchange process in the fixed bed column and still providing the process variable τ, which is the time required to reach the effluent concentration at 50% of the feed concentration (min), as well as calculate KYN, the Yoon-Nelson constant (min −1 ) (Jung et al., 2017). The fractional removal, C/C0, is represented by Equation (14).

Aspen Adsorption (MTSF) Solid Film Ion Exchange Mass Transfer Model
In Aspen Adsorption software, the process modelling was governed by a set of differential equations that configure the ion exchange between the ion to be removed from the effluent and the exchangeable cation of the zeolite NaY. The ionic species Zn 2+ (aq) in the liquid phase, fed into the ion exchange column, was governed by the MTSF model, represented by Equation (15), of mass balance and axially dispersed 'plug flow' moment (Tantet et al., 1994). The convection with estimated dispersion (Ez) was included in the mass balance through Equations (15) and (16) (Slater, 1991;Ruthven, 1984): Research, Society and Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 Where ck is the ion concentration in the liquid phase (eq/m 3 ); Ez is the Axial dispersion coefficient (m 2 /s); t is time (s); Z is the Axial Coordinate Axis; εi is bed void (porosity); MTCsk is the solid film mass transfer coefficient (1/s); wk is the resin ion loading (eq/m3); Wk* is the ion charge in equilibrium with the ion concentration in the liquid phase (eq/m3); vl is the liquid velocity (m/s); dp is the diameter of particles (m); Re is Reynolds number.

Process Simulation
The Aspen Adsorption software and MTSF model were also used to design new continuous systems by varying the column height, to assess what interference would occur under the saturation time and the ion exchange capacity.
Bed heights were based on the mass (mb) of NaY zeolite used in the bed ion exchange study, as shown in Equation (17) Where D is the column diameter (m); Hb is the bed height (m) and the bulk density of the exchanger (g). The choice of useful heights was established according to those that generated well-structured breakthrough curves, with a column saturation time different from zero (t[min]) and greater capacities for removing zinc from contaminated aqueous medium (q[meq/ g]). The flowchart shown in Figure 1, represents the entire simulation process carried out in this research.
The simulation was performed in transient regime with a time span of 1 second. The bed length was divided into 20 knots. The UDS1 was used as a discretization method and the system of discretized equations was integrated by the implicit Euler method with a variable step from 0.01 to 0.05. Furthermore, the thermodynamic model ELECNRTL (Electrolyte Non-Random Two Liquid) was used for the interaction of ions and water molecules and the parameters adjusted by the mass action law were used to estimate the equilibrium concentrations.
Assuming that the system has no interference with any obstruction, but only dissociation in pure water, the properties of Zn 2+ (aq) were determined using Aspen Properties Electrolytes Wizard. To add the counterion from the ion exchange process, the component list was saved and converted to a component set. Based on the characterization of zeolite NaY performed by Ostroski et al. (2008) its elemental composition has 12.6% Na2O, which was used as a counterion in the simulation. (aq) by NaY zeolite in a fixed bed column, using Aspen Adsorption.

Simulation of Breakthrough curves
What is the limiting step of the finite bath TM process?
It was assumed that the solution was very dilute and that metal ions present in the solution would not significantly affect the density and viscosity of the solution. The density and viscosity of water at 25°C were used, which are respectively 55.41 kmol/m 3 and 8.94×10 -4 Ns/m 2 , as well as molar mass 18.05 kg/kmol.

Column ion exchange capacity and saturation time
The calculation of ion exchange capacity in fixed bed columns was obtained by mass balance using Equation (18 Where qeq is the equilibrium concentration of metal ions in the zeolite (meq/g); ms is the dry mass of zeolite (g); Q is the volumetric flow rate of the solution (mL/min); t is time in minutes; Cout is the cation concentration at the column output (meq/L); C0 is the cation concentration at the column output (meq/L). Column saturation time was collected when the column reached at least 90% C0 at the column outlet. Research, Society and Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 9 Table 3 shows the estimated coefficients from the isotherms for ion exchange between Zn 2+ (aq) and NaY zeolite, as well as the statistical evaluation parameters. Observing only the R 2 parameter, it is biased to select the Freundlich and Lambert models as the best fits, to be used in the estimation of Kd.  (Figure 2), the Freundlich model showed fit failures characterized by a tendency to overestimate for low equilibrium concentrations, however it presented a residual value within the range of -1.5 to +2.0.

Determination of distribution (Kd) and selectivity ( )
The Lambert model, which resulted in the second highest R 2 and low MSE, showed better random dispersion of the standardized residuals along the X axis, as shown in Figure 2. It was observed that in the Lambert model, the standardized residuals ranged from -1.1541 to 2.0178 while in Freundlich's, they range from -1.1868 to 1.5558.
Regarding the Freundlich model, it is worth remembering that, in this case, the unit of estimated Kd is equal to that of Kf given in meq.g -1 /(meq.L -1 ) (Nkedi-Kizza & Brown, 1998). In this model, when the values of the index (1/n) of the equation are equal to or close to one, the value of Kf is equivalent to a partition coefficient of the solute between the solution and the solid surface, that is, Kf = Kd. However, from the result obtained by the adjustment of 1/n = 0.3153 shown in  (1980) this behaviour is expected, as the selectivity coefficient increases with the concentration of the external solution and with the content of heavy metal cations bound to the clays.
In tests 4 and 5, the values of of 0.76 and 0.93, respectively, with C0 = 0.844 meq/L, presented an increase due to a flow difference of 4 ml/min and 12 ml/min between them, which interfere in q0 (zeolite ion exchange capacity). The highest value of , referring to test 3 (C0 = 1.381 meq/L and Q=8 mL/min), theoretically characterized the condition of greater resin preference for Zn 2+ (aq) ion of the external solution. According to Abrão (2014) the higher the selectivity coefficient, the greater the preference of the exchanger material for this ion. The selectivity coefficient had a direct influence = 1.17 Research, Society and Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 10 on the modelling of the breakthrough curves; it was one of the data required for the adjustment of the MTSF model in Aspen Adsorption.

Adjustments of kinetic models from finite bath system
First and second order kinetic models for finite bath data are not able to determine the diffusion mechanisms (Abdi et al., 2017). According to Fagnani et al. (2017) the IPD model identifies the diffusion mechanisms in ion exchange processes with NaY zeolite and represents the ion transport through the zeolite pores. To adjust the IPD and MTEF models, it was necessary to first calculate q(t), with Equation (19) (Ruthven, 1984): Where q(t) is the amount of Zn 2+ (aq) in the solid phase at time t (meq Zn 2+ (aq)/ g of zeolite NaY); Ci is the initial concentration of Zn 2+ (aq); C(t) is the concentration of Zn 2+ (aq) in the liquid phase at time t (meq Zn 2+ (aq)/ L of sol.); V is the volume of 1.5 L of solution and m is the mass of 1g NaY zeolite, obtained from Ostroski et al. (2008). Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 11 Thus, when plotting the data of q(t) versus t 1/2 (minutes 0.5 ), it was possible to adjust the IPD model to the kinetic experimental data from Ostroski et al. (2008) and obtain a straight line with slope (I), as highlighted in Figure 3, with a multilinearity indicating more than one limiting step in the process. When this occurs, it can be observed three distinct steps. In the first one, an instantaneous removal of the Zn 2+ (aq) ions in solution referring to the diffusion in the pores of the external surface of the zeolite; a second stage of gradual removal referring to diffusion into the intraparticle pores; and, finally, a final stage when the ion exchange reaches equilibrium (Chen et al., 2003).  stage of the IPD model, with R 2 = 0.89, indicating that intraparticle diffusion also interferes with the speed of the process.
Furthermore, it was possible to observe that the value of I was not null, indicating that the boundary layer is representative in the exchange process, as the value of this deviation indicates an approximation of the boundary layer in meq.g -1 and thus, the higher this value, the greater its importance to the process. The first stage that obtained a value of R 2 = 0.9099, confirmed that the exchange process is also governed by ion diffusion on the outer surface of the solid (Silva et al., 2015).
On the adjustment of the external film mass transfer model (MTEF), when plotting qt versus t, it was possible to adjust it to the experimental kinetic data of Ostroski et al. (2008) and obtain different curves according to the variation of the initial concentration of Zn 2+ (aq) in the aqueous medium (Ci), as highlighted in Figure 4.  Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 12 The MTEF fit, a differential model that represents the transport of the ion within the solution to the liquid film layer around the zeolite particles, was performed using the MatLab software, which enabled to simulate the behaviour of MTEF by decreasing the concentration of Zn 2+ (aq) during ion exchange with NaY zeolite in the finite bath system. As can be seen in Figure 4, the MTEF model presented good adjustment when the Ci(t) reached the range of 1.85-2.0 meq of Zn 2+ (aq)/L of solution, with the highest R 2 = 0.8477, when Ci reaches 1.9 meq of Zn 2+ /L of solution, as can be seen in table 5. Thus, the mass transfer in the external liquid film is not the limiting step in the zinc removal process by zeolite NaY, since the phenomenon occurred only in a specific range of concentration of the contaminating solution. Table 5 shows the parameter estimated by the MTEF and the rate of mass transfer rate in the external liquid film (kTM). It was possible to observe that as Ci reduced from 2.0 meq/L to 1.85 meq/L, there was an increase in the kTM value from 0.05 min-1 to 0.0559 min-1, characterizing a decrease in the resistance to mass transfer through the external film to the zeolite NaY (Stephen et al., 2005). Source: Authors.
In general, both MTEF and IPD mass transfer processes are present in the ion exchange process, however, the limiting step was the intraparticle diffusion (IPD), with better fits, which was expected since, according to Ruthven (1984), pelletized materials, such as NaY zeolite from Ostroski et al. (2008), show resistance to diffusion in their micro and macropores.
From this pre-definition, it was possible to later choose the most representative model for the modelling and simulation of a fixed bed column in Aspen Adsorption, since the software has two options for ion exchange processes, fluid film model (limited by the MTEF phenomenon) and solid film model (limited by the IPD phenomenon). Thus, the previous study of finite bath kinetics allowed to define that the solid film mass transfer model (MTSF) is the most representative for the process. Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v10i12.20362 13

Breakthrough curves and calculated parameters of MTSF, TH and YN models
At this stage of the research, the breakthrough curves obtained by Ostroski et al. (2008) were modelled for Zn 2+ (aq) removal in a column filled with NaY zeolite, using solid film mass transfer models (MTSF), Thomas (TH) and of Yoon-Nelson (YN), in order to define the most representative model for the process. The parameters estimated by adjusting the models to the breakthrough curves are listed in Table 6 and each one of them allowed different considerations.  On the Thomas model, it was possible to observe that the velocity constant (KTH) decreased as the feed concentration (C0) increased and increased as the flow (Q) increased. Zheng et al. (2008) and Trgo et al. (2011) obtained similar behaviours when evaluating the ion exchange in zeolites. The highest zinc removal capacity by the NaY zeolite bed, estimated by the Thomas model, was q0 = 2.33 meq/g under lower C0 and reduced as the flow increased. The TH model, which disregards axial dispersions in the bed, presented R 2 = 0.9683 to 0.9985, while the MTSF model that considers them stood out as being able to predict the curve advance with R 2 = 0.9923 to 0.9977, as C0 and Q varied.
In the MTSF model of Aspen Adsorption, which considers the presence of axial dispersion in the hydrodynamic conditions of the column, as can be seen in Table 6, it resulted in Ez values sensitive to flow variation. As Q increased, Ez increased since their estimates are linked to the velocity of the fluid flow, which also increases as flow increases. This same behaviour for Ez was found by Abdi and Abedini (2020) when estimating it in a fixed bed column using Aspen Adsorption.
The Reynolds numbers of the process, regardless the variation of Q, resulted in Re < 1.3 and according to Slater (1991) for fixed bed columns, a Re < 2 characterizes a non-turbulent liquid flow. Vermeulen (1958) suggests that the axial mixture can be neglected if Re > 0.1 under VL = 0.00014 m/s, however the liquid velocity reached values of 0.001048, 0.002091 and 0.003144 m/s for Q = 4, 8 and 12 mL/min, respectively, that is, the axial dispersion was significant for the column that is not operating under a plug flow condition. Figure 5 shows the breakthrough curves for each model. (aq), KYN (min -1 ), consequently increased when increasing C0 and Q. This behaviour is similar to that found in the literature for application of this model in adsorption with zeolites (Trgo et al., 2011). However, the Yoon-Nelson model resulted in R 2 between 0.9683 and 0.9985, varying in relation to changes in C0 and Q.
Therefore, given the above discussions, the greater significance of using the MTSF model of Aspen Adsorption to simulate the ion exchange process in the removal of Zn 2+ (aq) by zeolite NaY was validated, as it could predict the experimental data with good accuracy and, regardless of the variation of the C0 and Q. Figure 6 shows the flowchart implemented in the Aspen Adsorption environment to simulate the breakthrough curves.
The results of the breakthrough curves show that as the bed height increased, the breakthrough moment increased.
Thus, there was a greater efficiency in removing Zn 2+ (aq), since the zinc ions had greater contact with the NaY zeolite, as expected. A higher column has greater available surface area, which provides more active site for ion exchange, resulting in an increase in time to reach breakthrough point. Thus, a greater removal of metal is observed.
In quantitative terms, Table 7 compares the experimental ion exchange capacity from Ostroski et al. (2008) with q0 = 2.36 meq/g, and calculated with Equation (18) under the curves estimated with the MTSF Adsim model with q0 = 2.38 meq/g, demonstrating the good fit of the model. The proximity of estimated q0 with the experimental q0, both for Hb= 0.03 m, as well as presenting q0 calculated for the simulated curves with variations of Hbs = 0.012, 0.015, 0.06 and 0.08 meters (m). Research, Society and Development, v. 10, n. 12, e310101220362, 2021 (CC BY 4  Analyzing data from Table 7, it was noticed that q0 gradually increased with the increase in height and mass of NaY zeolite, as expected. The authors Nieva et al. (2019) and Adornado et al. (2016) found similar behaviours with the variation of Hb, for the estimated breakthrough curves for ion exchange of metallic ions with biomass, using the MTSF model of Adsim. A column with lower height implies a lower capacity of the bed to remove metal ions from the solution and, therefore, resulting in a faster rupture and saturation time, as shown in Table 7, where the shortest saturation time (t = 282 min) was detected for the lowest height (Hb = 0.012 m). Also, at lower bed heights, axial dispersion is considered as the predominant mass transfer phenomenon that reduces the diffusion of metallic ions (Singha et al., 2012).

Conclusion
The kinetic modelling with finite bath data allowed to stablish that the limiting step of the exchange process, between the IPD and MTEF models, is the intraparticle diffusion, with R 2 = 0.9099 for the 1st stage of instantaneous ion exchange between Zn 2+ (aq) and the surface of zeolite NaY, and R 2 = 0.8908 for the 2nd stage of gradual ion exchange within the pores of the zeolite. The kinetic modelling of fixed bed column using the MTSF, TH and YN models demonstrated that the Adsim MTSF model, which also considers intraparticle diffusion as one of the limiting phenomena, presented the best fit (R 2 ≥ 0.9923). TH and YN resulted in R 2 ≥ 0.9683. The mass transfer (MTC) and axial dispersion (Ez) phenomena identified by the Adsim MTSF model demonstrated that N0 and t, from the NaY zeolite bed, increased as the values of MTC and Ez decreased, as well as N0 and t decreased as MTC and Ez increased. The increase of C0 and Q resulted in high MTC and Ez. Due to the bed's sensitivity to these phenomena, the importance of including these terms in the mass balances became evident, but only for values of C0 and Q studied in this work. Based on the results of the simulation, it was noticed that the increase in Hb increases q0 and t. For lower Hb values, the axial dispersion was considered as the predominant mass transfer phenomenon, thus reducing the diffusion of Zn 2+ (aq) metallic ions. It can be concluded that a multi-scale treatment system design for the removal of Zn 2+ (aq) ions from wastewater streams using a fixed bed column filled with NaY zeolite can be achieved using the validated model obtained in Aspen Adsorption.