Factors influencing kinetic and equilibrium behaviour of sodium ion exchange with strong acid cation resin

Separation and Purification Technology 163 (2016) 79–91

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Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur

Factors influencing kinetic and equilibrium behaviour of sodium ion exchange with strong acid cation resin Graeme J. Millar a,⇑, Gavin L. Miller a, Sara J. Couperthwaite a, Shannon Papworth b a b

Institute for Future Environments, Science and Engineering Faculty, Queensland University of Technology (QUT), Brisbane, Queensland, Australia School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology (QUT), Brisbane, Queensland, Australia

a r t i c l e

i n f o

Article history: Received 26 September 2015 Received in revised form 30 January 2016 Accepted 24 February 2016 Available online 26 February 2016 Keywords: Sodium Strong acid cation Resin Equilibrium Non-linear least squares Kinetics Isotherm

a b s t r a c t This study reports an investigation of the ion exchange treatment of sodium chloride solutions in relation to use of resin technology for applications such as desalination of brackish water. In particular, a strong acid cation (SAC) resin (DOW Marathon C) was studied to determine its capacity for sodium uptake and to evaluate the fundamentals of the ion exchange process involved. Key questions to answer included: impact of resin identity; best models to simulate the kinetics and equilibrium exchange behaviour of sodium ions; difference between using linear least squares (LLS) and non-linear least squares (NLLS) methods for data interpretation; and, effect of changing the type of anion in solution which accompanied the sodium species. Kinetic studies suggested that the exchange process was best described by a pseudo first order rate expression based upon non-linear least squares analysis of the test data. Application of the Langmuir Vageler isotherm model was recommended as it allowed confirmation that experimental conditions were sufficient for maximum loading of sodium ions to occur. The Freundlich expression best fitted the equilibrium data when analysing the information by a NLLS approach. In contrast, LLS methods suggested that the Langmuir model was optimal for describing the equilibrium process. The Competitive Langmuir model which considered the stoichiometric nature of ion exchange process, estimated the maximum loading of sodium ions to be 64.7 g Na/kg resin. This latter value was comparable to sodium ion capacities for SAC resin published previously. Inherent discrepancies involved when using linearized versions of kinetic and isotherm equations were illustrated, and despite their widespread use, the value of this latter approach was questionable. The equilibrium behaviour of sodium ions form sodium fluoride solution revealed that the sodium ions were now more preferred by the resin compared to the situation with sodium chloride. The solution chemistry of hydrofluoric acid was suggested as promoting the affinity of the sodium ions to the resin. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Ion exchange remains a key technology, especially in the area of water and wastewater treatment. Of particular interest at present is the rapid growth of the coal seam gas (CSG) industry worldwide [1] especially in countries such as USA, India, China and Australia [2]. Extraction of the coal seam gas is achieved by reduction in water pressure which holds the gas in the coal cleats. As such, inevitably significant volumes of associated water are collected which contains salt concentrations that may require to be reduced to allow the water to be allocated for beneficial reuse options [3]. The primary contaminant is sodium ions which mainly arise from ⇑ Corresponding author at: Science and Engineering Faculty, Queensland University of Technology, P Block, 7th Floor, Room 706, Gardens Point Campus, Brisbane, Queensland 4000, Australia. E-mail address: [email protected] (G.J. Millar). http://dx.doi.org/10.1016/j.seppur.2016.02.045 1383-5866/Ó 2016 Elsevier B.V. All rights reserved.

dissolution of sodium chloride and sodium bicarbonate species. Reverse osmosis has been predominantly installed in Australia for the desalination of coal seam water [4] whereas in the USA ion exchange processes have been successfully used [5]. The composition of coal seam water in the USA is typically different from that in Australia in terms of the relative quantities of sodium chloride and sodium bicarbonate present [5–7]. Ion exchange is attractive for coal seam water treatment because it may produce relatively small volumes of waste; is able to remove dissolved ions to very low levels (<1 mg/L); requires low energy input; and, is comparatively simple to operate. However, at present insufficient information exists regarding the performance of synthetic resins for coal seam water demineralization. Recently our research group has reported a study regarding the equilibrium and column behaviour of sodium ions with a strong acid cation resin supplied by Lanxess (S108H) [8]. Often, a strong acid cation

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is placed in a fixed bed with either a weak or strong base anion resin located in a second column downstream to ensure major removal of both cations and anions from solution [9]. Millar et al. [8] found that the exchange of sodium ions from sodium chloride solutions and actual coal seam water was dependent upon several variables including solution normality and the presence of competing cations in solution. Drake has studied the kinetics of sodium ion exchange from contact of sodium bicarbonate solution with a strong acid cation resin [10]. It was reported that increasing sodium ion concentration or resin present in the kinetic tests accelerated the rate of sodium removal from solution with a pseudo first order relationship determined. As alluded to above, the core criteria evaluated for ion exchange processes are typically kinetics [11,12], equilibria [13,14] and column behaviour [15,16]. However, concern has been expressed by several authors regarding the robustness of the approaches used. For example, Tien [17] lamented the significant number of manuscripts concerning sorption studies which are rejected for publication due to a range of outlined deficiencies. A lack of understanding of the subject matter, limited data interpretation, and formulaic approach to experimental design were all cited as problems. Chu [18] has similarly emphasised issues associated with a lack of rigour concerning the often quoted Thomas model which is used to analyse sorption column data. The realization of the challenges faced in sorption studies has also been elegantly explained by Lehto and Harjula [19] who illustrated factors which may interfere with the collection of valid equilibrium isotherm data. Literature evaluation reveals that Langmuir [20,21] and Freundlich [22,23] equations are routinely applied to equilibrium isotherm interpretation along with a random assortment of other models including Temkin [24], Redlich Peterson [25], Toth [26], Sips [27], Dubinin Astakhov [28], Dubinin Radushkevich [29], Khan [30] and Brouers Sotolongo [31]. Indeed, the papers by Hadi et al. [32], Rangabhashiyam et al. [33] and Hamdaoui and Naffrechoux [13,14] describe numerous two, three and four parameter isotherm expressions which can be applied to interpret sorption equilibria. Recent publications by our research group have focussed on the impact of factors such as the bottle-point method used for equilibrium isotherm generation [34], influence of the initial ratio of sorbate ions in solution to the resin mass and equilibrium loading of sorbate species [35] and impact of solution pH upon sorbate uptake [36]. Overall, it is apparent that collection of equilibrium isotherm data is not a trivial exercise. Ion exchange kinetic studies typically accompany investigations of equilibrium performance [37]. Similar to the case with equilibrium isotherm investigations, the methodology employed for analysis of exchange kinetics in solution appears not to have a standard approach [38–40]. Nevertheless, the pseudo first order and pseudo second order models invariably are found in many ion exchange publications [37,41,42]. Notably, both adsorption and ion exchange methods tend to use common approaches for data interpretation despite the fact that ion exchange is inherently limited by stoichiometric considerations. Consequently, ion exchange isotherms for example are often represented in stoichiometric plots, albeit care must be taken to ensure that interpretations of the exchange process take into account the possibility of super-equivalent ion exchange [43,44], non-structural ion exchange [35], and accompanying adsorption processes [45]. The use of solution concentrations instead of activities is also a factor which can make the analysis of sorption behaviour challenging, as the use of molarities is only applicable when the solutions involved are highly dilute. Hence, it is perhaps understandable why confusion may exist as to the merits of each model and validity of information which it provides. Compounding this

latter situation, is the fact that many studies have focussed on use of linearized equations when fitting experimental data and also limited assessment of the statistical validity of this approach [46– 48]. Several authors have reported the inherent flaws in linearizing equations such as El-Khaiary and Malash [49], Bolster and Hornberger [50] and Bolster [51], yet this practice is still commonplace [52,53]. The object of this paper was to answer the following problems: (1) what is the impact upon ion exchange behaviour by using different strong acid cation resins; this issue relates to reported studies that resins of the same class can exhibit different exchange behaviour [54]: (2) does the identity of the anion the uptake of sodium ions from solution; this problem relates to the fact that coal seam water samples contain other ions apart from chloride species: (3) what is the impact upon data interpretation of both kinetic and equilibrium sorption information when using linearized and non-linearized equations; this target is based upon the previous discussion of problems with data management in the literature. 2. Background theory 2.1. Kinetics 2.1.1. Pseudo first order Lagergren [55] first described the application of a pseudo first order kinetic expression pertinent to sorption studies wherein the rate limiting step is usually a diffusion controlled process (Eq. (1)).

dqt =dt ¼ k1 ðqe  qt Þ

ð1Þ

Integration of Eq. (1) using appropriate boundary conditions results in Eq. (2).

qt ¼ qe ð1  expðk1 tÞÞ

ð2Þ

Many authors have used a linearized version of Eq. (2) which can be represented by Eq. (3).

logðqe  qt Þ ¼ logðqe Þ 

k1 t 2:303

ð3Þ

2.1.2. Pseudo second order The pseudo second order equation has been described in detail by Ho [56] and is represented as shown in Eq. (4).

qt ¼

k2 q2e t 1 þ k2 qe t

ð4Þ

Or in a linear form more convenient for analysis of kinetic data as illustrated in Eq. (5).

t 1 1 ¼ þ t qt ðk2 q2e Þ qe

ð5Þ

Eq. (5) is actually one of four expressions which have been reported as linearized versions of the pseudo second order kinetic model. In the notation of Ho [57], Eq. (5) is termed Type 1, with the remaining three forms displayed in Eqs. (6)–(8):

!

Type 2 :

1 ¼ qt

1

Type 3 :

qt ¼ qe 

Type 4 :

qt 2 ¼ kqe  kqe qt t

2

kqe



1 1 þ t qe

ð6Þ

 1 qt kqe t

ð7Þ ð8Þ

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2.2. Equilibria

qe;Na ¼

Background information will be restricted to analysis of the Langmuir and Freundlich models in an effort to maintain clarity to the reader. The arguments presented in this study can also be readily applied to other isotherm equations with two parameters [13] and those with more than two parameters [14]. Consideration of more complex models involving multi-component exchange processes and two-site models will also not be explored at this time for the sake of brevity. In addition, molar concentrations are used throughout this study and not the more correct solution activities, an assumption which is only valid at low solution concentrations where the activity coefficient is approximately 1. 2.2.1. Langmuir adsorption isotherms If we consider the ion exchange process in this study as being represented as if it were an adsorption process, we can write the expression illustrated in Eq. (9).

Naþ þ R $ Naþ R

ð9Þ

Consequently, we can derive Eq. (10).

qe ¼

K L qmax C e 1 þ K LCe

ð10Þ

The Langmuir model as shown in Eq. (10) can be linearized in at least five different ways to allow easy plotting of data in terms of the traditional ‘‘y = mx + c” expression [13] as represented in Eqs. (11)–(15).

  1 1 1 1 ¼ þ qe qmax K L C e qmax Ce ¼ qe





1 qmax K L

 qe ¼ qmax 

þ

Ce qmax

 1 qe K L Ce

  1 1  KL ¼ K L qmax Ce qe qe ¼ K L qe þ K L qmax Ce

  1 1 plot vs: qe Ce   Ce plot vs: C e qe

 plot qe  plot

v s: 1 Ce

 q plot e Ce

qe Ce

ð11Þ

ð12Þ



v s:

v s: qe

ð13Þ 1 qe

 ð14Þ

 ð15Þ

2.2.2. Competitive Langmuir equation For ion exchange processes, the simple Langmuir model is problematic in that it does not include the stoichiometry of ion exchange. The Competitive Langmuir equation as described by Petrus and Warchol [58] appears more relevant as it takes into account the presence of the second ion involved. For example, with respect to the exchange of sodium ions from sodium chloride solution with a cation resin material in aqueous solution we can describe the exchange as in Eq. (16). At equilibrium, the solution contains both sodium ions and protons (hydronium ions, more correctly). þ

þ

Na ðaqÞ þ HR $ RNa þ H ðaqÞ

ð16Þ

By the law of mass action we can derive Eq. (17).

kNaþ =Hþ ¼

qe;Na C e;H C e;Na qe;H

ð17Þ

Application of the mass balance, qmax = qeNa + qeH and C = CeH + CeNa, gives rise to Eq. (18).

kNaþ =Hþ qmax C e;Na C o þ ðkNaþ =Hþ  1ÞC e;Na

ð18Þ

2.2.3. Freundlich equation The Freundlich expression (Eq. (19)) was originally proposed as an empirical model, although in later years it was demonstrated to be based upon physical principles [59–61].

qe ¼ K F C 1=n e

ð19Þ

Often, Eq. (19) is linearized to the form shown in Eq. (20).

log qe ¼ log K F þ

1 log C e n

ð20Þ

2.2.4. Langmuir Vageler model Vageler and Woltersdorf [62,63] first introduced their isotherm model in 1930 in relation to sorption studies of soils (Eq. (21)).

qe ¼

ðVC o =mÞqmax ððVC o =mÞ þ K LV Þ

ð21Þ

Notably, in contrast to other isotherm equations the Langmuir Vageler expression relates the quantity of ions loaded on the resin to the initial concentration of those ions in solution. The usefulness of the Langmuir Vageler model in relation to ion exchange isotherms has been recently demonstrated [34,35]. Millar et al. [34] showed that determination of appropriate experimental parameters for creation of ion exchange isotherms was aided by the application of the Langmuir Vageler expression. 2.3. Error functions Outlined above are various linearized forms of the fundamental Langmuir and Freundlich models. Application of non-linear least squares (NLLS) fitting procedures requires the use of error functions as discussed by Foo and Hameed [64]. Several software tools are available which facilitate NLLS analysis of sorption data with the Solver function in Microsoft Excel probably the most common package used. Based upon the work of Ho et al. [65] five error functions were employed as summarised in Table 1. The ‘‘sum of normalized errors” (SNE) procedure was used to determine which error function represented the optimal isotherm fit [65]. 3. Materials and methods 3.1. Resins A strong acid cation (SAC) resin was supplied by DOW and termed ‘‘Marathon C”, which was a gel type resin in the hydrogen

Table 1 Error functions used for NLLS analysis of ion exchange data. Error function Sum of the Squares of the Errors (ERRSQ or SSE) Hybrid Fractional Error Function (HYBRID) Marquardt’s Percent Standard Deviation (MPSD) Average Relative Error (ARE) Sum of the Absolute Errors (EABS)

Equation Pp ðq  qe;meas Þ2i i¼1 e;calc 100 pn



Pp

100

i¼1

ðqe;meas qe;calc Þ2i qe;meas



i ffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp hqe;meas qe;calc i2 1 pn

i¼1

qe;meas

Pp ðqe;calc qe;meas Þ 100  qe;meas p i¼1  i Pp jq  q e;meas Þji i¼1 e;calc

i

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form based upon styrene–divinylbenzene co-polymer with a stated ion exchange capacity of 1.8 eq/L. The mean bead size was 0.60 mm, the bulk density 800 g/L and quoted water retention of 50–56 wt%. Upon replacement of H+ species with sodium ions the resin contracted by 8 vol%. 3.2. Chemicals

3.7. Abbreviations

calc C Ce CeH

Aqueous solutions were prepared using triply distilled water to which appropriate amounts of salt were added. Analytical reagent grade sodium chloride was supplied by Rowe Scientific. 3.3. ICP analysis Samples were analysed using an Agilent ICP-MS 7500CE instrument. Samples were diluted to a concentration between 10 and 50 mg/L using a Hamilton auto-dilutor with 10 and 1 mL syringes. A certified standard from Australian Chemical Reagents (ACR) containing 1000 ppm of sodium was diluted to form a multi-level calibration curve. An external reference was used to monitor instrument drift and accuracy of the results obtained. Results were obtained using an integration time of 0.15 s with 10 replications. 3.4. Kinetic tests 200 mL of the stock sodium chloride solution to be tested was placed in a plastic sample container and agitated at 400 rpm by means of a magnetic stirrer. Resin was rapidly dosed from a large volume syringe and continuous solution measurements made by a conductivity meter (TPS Aqua-C). Calibration curves were created by means of analysis of a series of sodium chloride/hydrochloric acid solutions. Subsequently, the recorded conductivity value could be translated to an equivalent sodium ion concentration. In order to check the validity of the latter approach, the final equilibrated solution was also analysed by means of ICP-MS.

CeNa Co ICP k1 k2 KF KL KLV k+Na/H m meas n nF NLLS p qe qeNa qeH qmax qt ODR R SNE t V

calculated value total liquid concentration of ions equilibrium concentration of sodium ions in solution (mg/L) equivalent concentration of protons in solution at equilibrium (meq/L) equivalent concentration of sodium ions in solution at equilibrium (meq/L) initial concentration of sodium ions in solution (mg/L) inductively coupled plasma pseudo first order rate constant (min1) pseudo second order rate constant (g/mg min) Freundlich equilibrium coefficient (mg11/n L1/n g1) Langmuir equilibrium coefficient (L/mg) ‘‘half value” Competitive Langmuir equilibrium coefficient mass of resin (g) actual measured value degrees of freedom in the isotherm equation (e.g. 2 for Langmuir isotherm and 3 for Toth isotherm) Freundlich exponent (dimensionless) non-linear least squares number of data points in the isotherm equilibrium loading of sodium ions on the resin (mg/g) equivalent concentration of the sodium ions on the resin at equilibrium (meq/g) equivalent concentration of the protons on the resin at equilibrium (meq/g) maximum capacity of ions on the resin (mg/g) loading on the resin at time t (mg/g or mmol/g) orthogonal distance regression resin sum of normalized errors time solution volume (L)

3.5. Equilibrium studies Resins in the ‘‘wet form” were used in order to avoid structural degradation of the resins which can be induced by oven drying. A defined mass of resin was introduced into 250 mL Nalgene flasks and the concentration of sodium ions in solution kept at 1405 mg/L. The solution pH was not adjusted. Based upon previous kinetic studies equilibrium was found to be acquired within 1 h in a temperature controlled incubator (Innova 42R, New Brunswick Scientific), hence exchange was conducted for 24 h to ensure equilibrium was obtained. The temperature was maintained at 30 ± 0.1 °C and the solutions were shaken at 200 rpm. Concentrations of sodium ions remaining in solution at equilibrium (Ce (mg/L)) were then measured and the equilibrium concentration of sodium ions on the resin phase (qe (mg/g)) inferred from Eq. (22).

qe ¼

V ðC o  C e Þ m

ð22Þ

To ensure experimental accuracy experiments were repeated under identical conditions.

4. Results and discussion 4.1. Kinetic studies A standard methodology by which to examine ion exchange kinetics does not appear to exist [66], and indeed several models have been suggested to violate physical reality [67]. Fundamentally, even the experimental protocol for kinetics evaluation has not been clarified. Typically a batch process is performed wherein a solution of ions is placed in contact with a suitable sorbent material. In this situation, it is apparent that two scenarios can exist: (1) constant solution concentration and variable masses of sorbent present; or (2) constant sorbent mass and variable solution concentrations. With ion exchange processes it is well known that solution normality is important in determining the ion exchange behaviour [19], indeed ion exchange should be studied under conditions of constant solution normality and constant temperature. Interestingly, this latter statement indicates that scenario (1) may be more relevant to ion exchange studies. Hence, we have shown both scenarios in this investigation in order to reveal the inherent differences.

3.6. Data management The Solver function in Microsoft Excel was used for all linear and non-linear least squares fitting of the equilibrium exchange data.

4.1.1. Pseudo first order kinetics – constant sodium concentration and variable resin mass Figs. 1 and 2 illustrates kinetic data for the exchange of sodium ions for a solution of sodium chloride with various masses of

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diminished as the mass of resin present increased (Table 2), which was expected based upon the reduction in the ‘‘driving force” of the exchange process (i.e. the ratio of sodium ions in the initial solution relative to the resin mass decreased as the resin mass was increased) [45]. Notably, estimates for the equilibrium loading of sodium ions on the resin were different using the linear least squares and non-linear least squares methods (Tables 2 and 3). It was apparent that the linearized pseudo first order equation over-estimated both the equilibrium loading of the sodium ions on the resin and the pseudo first order rate constant.

Fig. 1. Linear Least Squares (LLS) Lagergren pseudo first order fit of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K: mass of resin from 1 to 6 g as indicated.

Fig. 2. Non-linear Least Squares (NLLS) Lagergren pseudo first order fit of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K: mass of resin from 1 to 6 g as indicated.

proton exchanged strong acid cation resin. With Fig. 1, the data was analysed using the linearized version of the pseudo first order equation (Eq. (3)). Generally, the data fits appeared reasonable, albeit it was evident that the linear correlations did not simulate the equilibrium information precisely. In contrast, Fig. 2 employed non-linear least squares analysis of Eq. (2). Visually, the pseudo first order kinetics expression appeared to simulate the experimental data to a high degree of accuracy (Fig. 2) and this was confirmed by the relatively low error values reported in Table 2. According to Ho and McKay [68] the pseudo first order expression best fitted sorption systems wherein the sorbate loading attained equilibrium within 20–30 min, thus avoiding problems with estimating values of the equilibrium sorption capacity. The observation that less than 5 min was typically required for sodium ions to equilibrate with the strong acid cation resin exchange sites in this study was in harmony with the latter statement. As a general observation, the equilibrium loading of sodium ions on the resin

Table 2 Summary of non-linear least squares (NLLS) analysis of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K. Model

Resin mass (g)

qe (g Na/g resin)

k

SSE

r2

1st Order 2nd Order 1st Order 2nd Order 1st Order 2nd Order 1st Order 2nd Order

1 1 2 2 4 4 6 6

54.2 71.5 44.9 56.3 35.1 43.3 28.8 35.0

1.0009 0.0126 0.9988 0.0179 1.0600 0.0254 1.1329 0.0355

15.50 41.50 13.36 47.00 8.22 28.95 2.85 16.74

0.999 0.998 0.999 0.997 0.999 0.997 0.999 0.998

4.1.2. Pseudo second order kinetics – constant sodium concentration and variable resin mass Fig. 3 the displays application of the four types of linearized pseudo second order kinetic expressions. It was observed that Type 1 and Type 2 versions of the model were substantially better representations of the experimental data compared to Type 3 and 4 (Fig. 3 and Table 3). The values of r2 (Table 3) for resin masses of 1, 2 and 4 g were slightly less than those for the pseudo first order equation, and for 6 g of resin there was no discernible difference in the goodness of fit. The range in predicted equilibrium values for sodium ions on the resin was broad, for example from 73.5 to 102 g Na/kg resin when 1 g of resin was used in the kinetic studies. The latter data was therefore not viable for calculating equilibrium loading capacity of the resin for sodium ions. In each instance of resin mass evaluated, the Type 2 model estimated that the equilibrium loading of sodium ions was greater than values calculated with the remaining three linearized types of the pseudo second order equation. The best fit of the kinetic data was dependent upon the experimental parameters used, with Type 2 being optimal for the lowest resin mass of 1 g, however Type 1 favoured a resin mass of 4 or 6 g. Non-linear least squares analysis of the same kinetic data using the pseudo second order equation (Fig. 4) indicated that this model did not fit the profile as well as the pseudo first order model. The magnitude of the error values shown in Table 2 was substantially higher for all resin masses studied compared to when a 1st order fit was applied. In contrast, with linearized equations, when 6 g of resin was studied the goodness of fit was actually the same for the 1st and 2nd order expressions, thus no conclusion as to the most appropriate kinetics description could be made on that basis.

Table 3 Summary of linear least squares (LLS) analysis of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K. Model 1st Order 2nd Order 2nd Order 2nd Order 2nd Order 1st Order 2nd Order 2nd Order 2nd Order 2nd Order 1st Order 2nd Order 2nd Order 2nd Order 2nd Order 1st Order 2nd Order 2nd Order 2nd Order 2nd Order

Type Type Type Type

1 2 3 4

Type Type Type Type

1 2 3 4

Type Type Type Type

1 2 3 4

Type Type Type Type

1 2 3 4

Resin mass (g)

qe (g Na/g resin)

k

r2

1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 6 6 6 6 6

70.4 73.5 102.0 76.6 86.0 58.8 56.5 76.9 59.7 66.4 44.9 43.5 60.2 45.6 38.9 30.8 34.1 42.4 36.4 38.9

1.6137 0.0110 0.0043 0.0096 0.0068 1.4594 0.0169 0.0062 0.0137 0.0096 1.479 0.0242 0.0083 0.0199 0.0133 1.2897 0.0383 0.0167 0.0287 0.0223

0.986 0.964 0.980 0.791 0.791 0.991 0.975 0.975 0.778 0.778 0.993 0.976 0.965 0.750 0.750 0.995 0.988 0.973 0.830 0.830

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Type 1

Type 3

Type 2

Type 4

Fig. 3. Linear Least Squares (LLS) pseudo second order fit of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K: mass of resin from 1 to 6 g as indicated.

with DOW Marathon C strong acid cation resin. Non-linear least squares methods were demonstrated to be superior to linearized approaches at interpreting the kinetics data. Usually, first order kinetics are associated with an reversible process between liquid and solid phase [70], whereas second order processes are taken to infer that chemical interactions at the sorption site play a role in the rate limiting step.

Fig. 4. Non-linear Least Squares (NLLS) pseudo second order fit of exchange kinetic data for 1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K: mass of resin from 1 to 6 g as indicated.

It was observed that the non-linear least squares analysis of the pseudo second order kinetic expression was superior to linearized fits (c.f. Tables 2 and 3) based upon the magnitude of the goodness of fit (r2). Detailed statistical evaluation of the application of linearized versions of the pseudo second order kinetic model compared to the corresponding non-linear equation has been completed by El-Khaiary et al. [69]. These authors concluded that due to modifications in the error structure of the transformed pseudo second order equations when linearized, that they were not recommended for comparing goodness of fit values. In each case studied, non-linear regression analysis of the pseudo second order equation provided optimal estimates of the model parameters, in harmony with this work. The kinetics studies revealed that the pseudo first order expression best simulated the experimental data for sodium ion exchange

4.1.3. Pseudo first and second order kinetics – variable sodium concentration and constant resin mass In this instance, the resin mass was kept constant at 6 g and the concentration of sodium ions in solution varied from 638 to 1288 mg/L. Non-linear least squares analysis of the kinetic data was completed for both the pseudo first and second order models (Fig. 5). The pseudo first order model simulated the experimental data to a significantly higher degree compared to the pseudo second order expression. For example, residual errors when 1288 mg/L sodium ions present were 2.8 and 15.4 for pseudo first and second order fits of the kinetic data. Closer inspection of the pseudo first order fitting data revealed that the calculated rate constants were 0.57, 0.90 and 1.13 min1 for sodium concentrations of 638, 953 and 1288 mg/L, respectively. The general trend of higher equilibrium sodium ion loading as a function of increasing sodium ion content in the initial solution was consistent with ion exchange studies of other species in solution such as nickel ions [71]. Similarly, the observed increase in the rate constant as the initial concentration of sodium ions was elevated was in harmony with the results of Dizge et al. [71]. These authors noted that when the concentration of nickel ions in solution was raised from 50 to 200 mg/L in the presence of a strong acid cation resin the pseudo first order rate constant increased from 0.04 to 0.12. Sen Gupta and Bhattacharyya [70] noted in their review of kinetics of metal ions on

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(a) Pseudo First Order Fit

(b) Pseudo Second Order Fit

Fig. 5. Non-linear Least Squares (NLLS) (a) Lagergren pseudo first order fit and (b) pseudo second order fit of exchange kinetic data for 638–1288 mg/L sodium (from sodium chloride) with DOW Marathon C strong acid cation resin at 297 K: mass of resin 6 g.

inorganic materials that the rate constant k1 normally varies with initial solution concentration of the ion of interest. 4.2. Equilibrium isotherm tests with sodium chloride 4.2.1. Langmuir Vageler equation The Langmuir Vageler expression is unlike other models used to describe sorption equilibria in that it relates conditions in the initial solution to equilibrium loading of the sorbate on the resin. In this study, the ratio of the number of moles of sodium ions in solution to the mass of resin present was termed the ‘‘driving force” of exchange in accord with the definition by Helfferich [45]. It can be observed that as the driving force increased above a value of approximately 15 mmol Na ions per g resin, the sodium loading gradually plateaued, suggesting that monolayer coverage of the resin surface had indeed occurred (Fig. 6). The Langmuir Vageler equation predicted a value for maximum sodium ion loading of 3.03 mol/kg (69.7 g Na/kg resin) (Table 4). 4.2.2. Langmuir isotherm equation Fig. 7 illustrates the Langmuir fits of the equilibrium data obtained by using linearized Langmuir Eqs. (11)–(15). Notably the goodness of fit ranged from high (r2 = 0.9941) to relatively low values (r2 = 0.8585) and the value of the maximum loading of sodium ions (qmax) on DOW Marathon C estimated to be from 126.4 to 145.3 mg/g (Table 5). Similarly, the Langmuir equilibrium coefficient ranged from 0.0006 to 0.0073 L/mg. Immediately, there existed confusion as to which of the five values for each of the unknown parameters was correct. Simply using the r2 term as an indication of goodness of fit did not resolve the latter situation as

Fig. 6. NLLS Langmuir Vageler fit of the equilibrium data when 1405 mg/L sodium ions (from sodium chloride solution) exchanges with DOW Marathon C strong acid cation resin.

Table 4 Non-linear least squares (NLLS) fit of Langmuir Vageler isotherm for 1405 mg/L sodium exchange from sodium chloride solution with DOW Marathon C resin. Numbers in bold indicate minimum error values and minimum sum of normalized error values.

qmax (mol/kg) KLV SSE HYBRID MPSD ARE EABS SNE

SSE

HYBRID

MPSD

ARE

EABS

2.97 2.42 0.21 0.45 4.45 3.41 1.64 4.77

2.97 2.44 0.22 0.44 4.39 3.43 1.67 4.78

2.97 2.47 0.22 0.45 4.37 3.42 1.70 4.80

3.03 2.55 0.23 0.48 4.53 3.07 1.50 4.75

3.02 2.48 0.23 0.48 4.63 3.14 1.49 4.79

Eqs. (11) and (14), although of the same r2 value, predicted different numbers for the maximum sodium ion loading on the resin and the Langmuir equilibrium coefficient. One must also consider the physical reality of the estimated parameters. For example, the resin supplier indicated based upon titration measurements that the DOW Marathon C material should uptake a minimum of 2.0 eq/L sodium ions or 56.1 g Na per kg resin (based upon a packing density of 0.82 kg/L). Consequently, it can be seen in this instance that the Langmuir fit of the equilibrium data substantially over estimated the resin capacity. Application of non-linear least square (NLLS) fitting methods (Fig. 8) again revealed a range of values for the variable parameters in the Langmuir isotherm equation with qmax from 130.0 to 142.4 mg/g and KL from 0.0006 to 0.00069 L/mg (Table 6). Inspection of the Sum of Normalized Errors (SNE) suggested that the HYBRID error function provided the most accurate fit of the experimental data and as such the resultant estimated values of qmax and KL were 136.2 mg/g and 0.00064 L/mg. Regardless of whether, linear or non-linear least squares fitting methods were employed, the estimated loading capacity was excessive.

4.2.3. Competitive Langmuir isotherm equation The NLLS Competitive Langmuir fit of the equilibrium points was of similar quality to the NLLS Langmuir fit of the same data (Fig. 8). The most notable difference was the estimate of the maximum loading capacity of the resin for sodium ions (Table 7), which was calculated as 64.7 g Na per kg resin. This latter value was of the magnitude expected based upon the data from the resin supplier. Hekmatzadeh et al. [72] examined the exchange of nitrate ions with a strong base anion resin and compared the conventional Langmuir fit of the equilibrium data with the Competitive Langmuir model (which they termed the ‘‘mass action law” isotherm). In harmony with our study, they reported that the Competitive

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Linearized Langmuir Equation 10

Linearized Langmuir Equation 11

Linearized Langmuir Equation 12

Linearized Langmuir Equation 13

Linearized Langmuir Fits of Experimental Isotherm Data

Linearized Langmuir Equation 14

Fig. 7. Illustration of the various isotherm data fits resulting from application of the 5 linearized versions of the Langmuir equation to the equilibrium data when 1405 mg/L sodium ions (from sodium chloride solution) exchanges with DOW Marathon C strong acid cation resin.

Table 5 Summary of parameters derived from the 5 linearized forms of the Langmuir sorption isotherm for the equilibrium data when 1405 mg/L sodium ions (from sodium chloride solution) exchanges with DOW Marathon C strong acid cation resin. Linearized Langmuir equation  1 1 1 1 qe ¼ qmax K C e þ qmax  Ce Ce 1 qe ¼ qmax K þ qmax

 qe ¼ qmax  K1 Cqee  1 1 K C e ¼ Kqmax q

qmax (mg/g)

KL (L/mg)

r2

126.6

0.00072

0.9941

135.1

0.00065

0.9058

126.4

0.00073

0.8585

129.6

0.0007

0.9941

145.3

0.0006

0.8585

e

qe Ce

¼ Kqe þ Kqmax

Langmuir approach calculated a value for the maximum loading of ions on the resin which was less than that which resulted from the Langmuir model.

Fernández-Olmo et al. [54] noted that the identity of the strong acid cation used to remove cationic impurities from hydrofluoric acid solutions. Amberlyst 15 resin was discovered to exhibit greater capacity than either Dowex HCR-W2 or Lewatit SP112 H strong acid resin. Millar et al. [8] found that the sodium ion capacity from sodium chloride solutions with Lanxess S108H SAC resin was ca. 61.7–67.5 g Na/kg resin, depending upon solution normality. The value of 64.7 g Na per kg of Dow Marathon C resin in this investigation suggested that there was no significant difference in resin capacity between the two resins studied. 4.2.4. Freundlich model Fitting the equilibrium data with the linearized version of the Freundlich equation (Fig. 9) resulted in a reasonable approximation of the isotherm. Notably, the goodness of fit (r2) term of

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87

Fig. 8. NLLS Langmuir & Competitive Langmuir fits of the equilibrium data when 1405 mg/L sodium ions (from sodium chloride solution) exchanges with DOW Marathon C strong acid cation resin.

Table 6 Non-linear least squares (NLLS) fit of Langmuir isotherm for 1405 mg/L sodium exchange from sodium chloride solution with DOW Marathon C resin. Numbers in bold indicate minimum error values and minimum sum of normalized error values.

qmax KL SSE HYBRID MPSD ARE EABS SNE

SSE

HYBRID

MPSD

ARE

EABS

142.4 0.00060 146.8 13.3 5.1 3.9 43.5 4.94

136.2 0.00064 148.6 13.0 4.9 3.7 43.1 4.89

131.8 0.00068 152.8 13.2 4.8 3.7 43.7 4.93

130.0 0.00069 154.3 13.3 4.8 3.7 43.9 4.95

140.1 0.00062 148.1 13.3 5.0 3.8 42.7 4.93

Table 7 Non-linear least squares (NLLS) fit of Competitive Langmuir isotherm for 1405 mg/L sodium exchange from sodium chloride solution with DOW Marathon C resin. Numbers in bold indicate minimum error values and minimum sum of normalized error values. SSE qmax kNaþ =Hþ SSE HYBRID MPSD ARE EABS SNE

65.2 1.842 146.7 13.3 5.1 3.9 43.1 4.77

HYBRID 64.7 1.909 148.6 13.0 4.9 3.7 43.1 4.69

MPSD 64.2 1.960 153.7 13.2 4.8 3.7 43.8 4.74

ARE 64.2 1.983 155.1 13.3 4.9 3.7 43.9 4.76

EABS 66.3 1.798 156.8 14.6 5.4 3.9 42.3 4.96

0.9877 was less than the optimal value of 0.9941 for the linearized Langmuir model. Consequently, the standard conclusion adopted by many previous authors would be that the Langmuir expression

was the most accurate representation of the sorption of sodium ions on strong acid cation resin [73]. Non-linear least squares (NLLS) analysis was also conducted and the graphical outcome shown in Fig. 9. Inspection of the magnitude of the errors in Table 8 indicated that not only was the HYBRID error function the optimal fit of the equilibrium data but also that the residual error values were lower compared to those calculated for the NLLS fit of the isotherm using the Competitive Langmuir expression. In harmony with the warning by El-Khaiary and Malash [49] which noted that direct comparison of error values should only be conducted when the equations used have the same number of unknown parameters, it can in fact be deduced that the Freundlich model was the optimal fit.

4.2.5. General interpretation of data There have been literally thousands of papers published in relation to the application of the Langmuir isotherm equation since the initial development in 1918 [74]. However, it can be said that researchers may not have fully understood the limitations of the equation and the need to carefully consider precisely what the results derived from the Langmuir equation actually represent. Kinniburgh [75] probably explained the situation best when he said ‘‘. . .ease of fitting may have led to the Langmuir. . ..isotherms enjoying rather more success than they deserve. . .”. In relation to this latter statement, Kinniburgh [75] also highlighted the need to use non-linear fitting of isotherm data even when employing the Langmuir equation which has only two unknown parameters. El-Khaiary and Malash [49] succinctly detailed the issues associated with linearization of the Langmuir isotherm equation, including alterations to the error structure, changes to the weighting of the errors for each data point, and spurious correlations due

Fig. 9. LLS and NLLS Freundlich fits of the equilibrium data when 1405 mg/L sodium ions (from sodium chloride solution) exchanges with DOW Marathon C strong acid cation resin.

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Table 8 Non-linear least squares (NLLS) fit of Freundlich isotherm for 1405 mg/L sodium exchange from sodium chloride solution with DOW Marathon C resin. Numbers in bold indicate minimum error values and minimum sum of normalized error values.

KF nF SSE HYBRID MPSD ARE EABS SNE

SSE

HYBRID

MPSD

ARE

EABS

0.457 1.4569 136.093 12.177 4.797 3.788 42.625 4.67

0.432 1.4392 135.160 12.097 4.748 3.692 41.993 4.63

0.424 1.4342 135.837 12.124 4.739 3.680 42.187 4.64

0.360 1.3822 154.309 14.184 5.159 3.363 38.629 4.82

0.364 1.3846 153.955 14.141 5.148 3.364 38.575 4.81

to errors introduced to the independent variable. Consequently, there were fundamental reasons why linearization of non-linear equations should not be attempted despite the advantage of ease of use in software such as Excel. In terms of least-squares regression analysis one of the central tenets is that the independent variable x is fixed [76], in this case the measured equilibrium concentration of the sodium ions. Therefore, there should be no error in the measurement of the sodium concentration in the solution following equilibrium exchange with the resin material. However, the process of analysing the concentration of sodium ions by the ICP method involved use of an auto-diluter to prepare the samples in the correct concentration range and also operation of the ICP equipment itself which may have had an error in analysis (despite careful calibration and repetition of sample analysis). Thus we must address the question of whether the variance in sodium ion equilibrium concentration in solution must be considered. If the answer is ‘‘yes”, then application of orthogonal distance regression (ODR) may be appropriate [76,77]. The ODR error function can is shown in Eq. (23).

!2 2 n  X qei;meas  qei;calc C ei;meas  C ei;calc þ C ei;meas qei;meas i¼1

ð23Þ

In contrast to the error functions shown in Table 1, the ODR error function evaluated the impact of errors in both Ce and qe. Reliance upon r2 as a measure of the goodness of fit has also been shown to be misleading in this study, assuming we ignore the inherent flaws in linearizing equations. Based solely upon the magnitude of r2 values, the Langmuir model appeared to fit the experimental data better than the Freundlich equation. Yet, nonlinear least squares fitting of the data suggested that the Freundlich expression was optimal. This latter conclusion had significant impact on the interpretation of the exchange process as the Freundlich model assumed a heterogeneous resin surface with the heat of sorption exponentially decreasing as the extent of exchange increased. In contrast, the Langmuir model was based upon a homogeneous resin surface with a constant heat of sorption during the ion exchange. Interestingly, El-Khaiary [77] also reported similar issues in the study of the sorption of methylene blue on water hyacinth, where determination of the correct sorption model was dependent upon the fitting procedure employed. Tellinghuisen and Bolster [78] illustrated causes of the inability of the correlation coefficient r2 to compare fitting models such as improper weighting of data. Bowser and Chen [79] reported that for equations which were in the form of rectangular hyperbolae such as the Langmuir model, that high values of r2 can be very misleading. These latter authors also noted the value for the binding coefficient influenced the accuracy of the fitting data as did the range of additive concentrations used. A subsequent paper by Bowser and Chen [80] further confirmed that the test conditions had to include a range of experimental conditions which were substantially greater than the error in the individual points.

Despite being proposed many years ago [62], the Langmuir Vageler model has barely been reported in subsequent sorption studies. Significantly, in line with the discussion of errors above, Tellinghuisen and Bolster [81] and Lehto and Harjula [19] both noted the importance of choosing appropriate experimental parameters, otherwise no matter how careful the data analysis was, spurious results would be obtained. The driving force principle inherent to the Langmuir Vageler approach is intuitive and allows standardization of data from a range of equilibrium experiments. For example, if sorption equilibrium tests were conducted with different concentrations of sodium ions or alternate range of resin masses, the data could still potentially be reduced to one series of driving force values which could be related to loading of sodium ions on the resin surface. Our conclusion that the pseudo first order kinetic expression and Freundlich isotherm model best represented the sodium ion exchange with strong acid cation was worth contemplating. The basis of the Freundlich model was the assumption that the surface of the sorbent was heterogeneous with the heat of sorption decreasing exponentially as the surface coverage increased. Rudzinski and Plazinski [82] argued based upon statistical rate theory that the pseudo second order equation was essentially equivalent to their generalized Elovich equation. The significance of this latter statement was that the underlying derivation of the Elovich equation was based upon the premise of a sorbent with a heterogeneous surface. It may therefore be surprising to discover that the pseudo first order expression was a better representation of the exchange kinetics. However, Azizian [83] has examined the fundamental theory underpinning sorption kinetic models and noted that at higher initial sorbate concentrations (hundreds of mg/L) the pseudo first order model fits the data best. Conversely, at low initial sorbate concentrations (tens of mg/L) the pseudo second order model normally simulates the exchange data optimally. 4.3. Equilibrium isotherm tests with sodium fluoride Based upon the methodology described above, the influence of the anion associated with the sodium ions in solution was investigated. Fluoride ions are often found in coal seam water [3,6,7] thus equilibrium tests with SAC resin were conducted using solutions of sodium fluoride (Fig. 10). The Langmuir Vageler plot revealed a distinct plateau in the sodium ion uptake on the resin which was predicted to correspond to a value of 3.39 mol Na/kg resin (77.9 g Na/ kg resin). This latter value appeared to be an overestimate of the exchange capacity of the resin as the Langmuir Vageler model was not able to accurately simulate the plateau region. Nevertheless, as previously highlighted, the Langmuir Vageler model indicated that the experimental conditions were sufficient to ensure the resin was optimally loaded with sodium ions. The corresponding Langmuir representation of the sodium exchange data in Fig. 9 was shown to be an excellent fit of the equilibrium data. The maximum loading of sodium ions was calculated to be 69.0 g Na/kg resin and the equilibrium coefficient was 16.51 L/mg. Comparison of the equilibrium isotherm for sodium exchange from sodium fluoride solution with the corresponding case with sodium chloride (Fig. 8), revealed that the affinity of the resin exchange sites for sodium ions was remarkably promoted in the presence of fluoride ions. In particular the equilibrium coefficient was only 1.909 L/mg when chloride was the counter-ion and 16.51 L/mg when fluoride was the counter-ion in solution. The uptake of sodium ions was also slightly higher when fluoride ions were present in solution (69.0 compared to 64.7 g Na/kg resin). Fernandez-Olmo et al. [54] examined the use of Dowex HCR-W2 strong acid cation to remove a range of impurities including sodium ions from hydrochloric acid solutions. The affinity of the resin for sodium ions in the presence of hydrofluoric acid was

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Fig. 10. NLLS Langmuir Vageler and Competitive Langmuir fits of the equilibrium data when 1144 mg/L sodium ions (from sodium fluoride solution) exchanged with DOW Marathon C strong acid cation resin.

demonstrated but no equilibrium studies were reported. As indicated by these latter authors, hydrofluoric acid is a weak acid which means that the concentration of fluoride ions in solution would be expected to be low. The question arises as to reasons why sodium ions were preferred by the resin when fluoride ions were present in solution. One possibility was that when the sodium ions exchanged with the protons on the resin surface, the subsequently formed protons (more correctly hydronium ions) and fluoride ions would recombine to create hydrofluoric acid. Thus equilibrium would be perturbed in manner which would promote release of protons from the resin surface and thus enhance the affinity of the resin for sodium ions. Viswanathan and Meenakshi [84] contacted a strong acid cation resin with sodium fluoride solutions comprising of 3 mg/L fluoride ions. Notably, these authors noted a small quantity of fluoride ions were removed from solution to give fluoride loadings on the resin in the range 0.06 to 0.25 g F/kg resin. Inspection of the fluoride ion concentration in the equilibrium solutions produced in this study showed that up to 11% of the fluoride ions were removed from solution which was in agreement with Viswanathan and Meenakshi [84]. It was proposed that hydrogen bonding of fluoride ions with the sulphonic acid groups on the resin was responsible for fluoride uptake. Changing the exchange site to include sodium ions actually promoted fluoride ion loading on the resin, and the same phenomenon may have occurred in the current investigation. Based upon the aforementioned discussion as to the reason for the enhanced affinity of sodium ions on SAC resin being due to the secondary transformation of dissolved sodium and fluoride ions to molecular hydrofluoric acid, a second equilibrium experiment was conducted wherein the sodium fluoride concentration was reduced by an order of magnitude. Fig. 11 shows the resultant Langmuir Vageler and Competitive Langmuir fits of the exchange data. Since the ratio of sodium ions to resin mass was maintained in

approximately the same range as the higher sodium ion concentration test, the Langmuir Vageler model again exhibited a distinct plateau in sodium ion uptake. The Competitive Langmuir fit of the isotherm profile suggested that the maximum loading capacity of sodium ions was 62.6 g Na/kg resin and the equilibrium coefficient was 6.66 L/mg. Notably, the affinity of the resin for sodium ions was reduced when the sodium fluoride concentration was diminished. This latter result was consistent with the reduced driving force of the exchange process. In accord with Viswanathan and Meenakshi [84] the extent of fluoride ion removal from solution was also lowered when the concentration of sodium fluoride was reduced, namely 4.6–5.0% reduction in this situation. 5. Conclusions Analysis of simple ion exchange processes such as sodium ion exchange from sodium chloride solutions with a strong acid cation resin, has been demonstrated to be surprisingly complex. For example, this paper highlighted the care that must be undertaken when modelling ion exchange isotherm data using the Langmuir equation. Discrepancies in the estimation of the important maximum loading capacity of a strong acid cation resin have been exemplified, with ten different values noted in this study from the one equilibrium data set. Random selection of a linearized Langmuir isotherm equation was arguably inaccurate and even a non-linear least squares approach to simulating the experimental data although an improvement, was influenced by the range of error functions selected. A critical aspect of this study was the recommendation that the Langmuir Vageler model should be applied in conjunction with standard equilibrium isotherm expressions such as the Langmuir and Freundlich models. When equilibrium isotherms do not display a clear plateau in the sorbate loading on the resin, then a means is required to ensure the test

Fig. 11. NLLS Langmuir Vageler and Competitive Langmuir fits of the equilibrium data when 122 mg/L sodium ions (from sodium fluoride solution) exchanged with DOW Marathon C strong acid cation resin.

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parameters were sufficient to facilitate full loading of the functional sites on the resin surface. The Langmuir Vageler method appeared to be useful in respect of this latter requirement. Isotherm analysis using linearized versions of the Langmuir and Freundlich isotherm models suggested that the Langmuir equation best represented the equilibrium data, based upon the magnitude of the r2 goodness of fit. Notably, application of non-linear least squares data fitting procedures indicated that the Freundlich expression was actually the optimal representation of the isotherm profile. Hence, researchers must take care in determining what isotherm models actually reveal about the sorption process. The maximum loading of sodium ions on the strong acid cation resin was concluded to be 64.7 g/kg form equilibrium studies by application of the Competitive Langmuir model, which was in accord with the capacity specifications provided by the resin manufacturer. Similarly, analysis of ion exchange kinetics using linearized versions of the pseudo first and second order equations was highlighted to be undesirable. Moreover, the batch testing conditions were shown to be important, with the ‘‘constant solution normality, variable resin mass” approach possibly more appropriate for ion exchange studies. In either testing scenario, the pseudo first order kinetic expression was shown to simulate the exchange process optimally. This study, in harmony with existing literature, suggested that the practice of linearizing equations used in sorption studies should not be pursued further. Acknowledgements We thank DOW Water and Process Solutions for supply of the Marathon C resin. The Science and Engineering Faculty at Queensland University of Technology is recognised for its financial support for equipment used in this study. References [1] T.A. Moore, Coalbed methane: a review, Int. J. Coal Geol. 101 (2012) 36–81. [2] I. Hamawand, T. Yusaf, S.G. Hamawand, Coal seam gas and associated water: a review paper, Renew. Sustain. Energy Rev. 22 (2013) 550–560. [3] G.J. Millar, S.J. Couperthwaite, C. Moodliar, Strategies for the management and treatment of coal seam gas associated water, Renew. Sustain. Energy Rev. 57 (2016) 669–691. [4] S. Chalmers, A. Kowse, P. Stark, L. Facer, N. Smith, Treatment of coal seam gas water, Water 37 (2010) 71–76. [5] R. Dennis, Ion exchange helps CBM producers handle water, Oil Gas J. 105 (2007) 41–43. [6] L.D. Nghiem, T. Ren, N. Aziz, I. Porter, G. Regmi, Treatment of coal seam gas produced water for beneficial use in Australia: a review of best practices, Desalination Water Treat. 32 (2011) 316–323. [7] G.J. Millar, J. Lin, A. Arshad, S.J. Couperthwaite, Evaluation of electrocoagulation for the pre-treatment of coal seam water, J. Water Process Eng. 4 (2014) 166– 178. [8] G.J. Millar, S.J. Couperthwaite, M. de Bruyn, C.W. Leung, Ion exchange treatment of saline solutions using Lanxess S108H strong acid cation resin, Chem. Eng. J. 280 (2015) 525–535. [9] I.R. Radosavlevici, D.N. Robescu, Demineralization of water with mixed-layer ion-exchangers, Environ. Eng. Manage. J. 12 (2013) 137–145. [10] R.N. Drake, Ion exchange: novel IX process to remove salts from wastewaters, Ultrapure Water 28 (2011) 13–20. [11] L.C. Lin, R.S. Juang, Ion-exchange kinetics of Cu(II) and Zn(II) from aqueous solutions with two chelating resins, Chem. Eng. J. 132 (2007) 205–213. [12] A.W. Marczewski, Kinetics and equilibrium of adsorption of organic solutes on mesoporous carbons, Appl. Surf. Sci. 253 (2007) 5818–5826. [13] O. Hamdaoui, E. Naffrechoux, Modeling of adsorption isotherms of phenol and chlorophenols onto granular activated carbon: Part I. Two-parameter models and equations allowing determination of thermodynamic parameters, J. Hazard. Mater. 147 (2007) 381–394. [14] O. Hamdaoui, E. Naffrechoux, Modeling of adsorption isotherms of phenol and chlorophenols onto granular activated carbon: Part II. Models with more than two parameters, J. Hazard. Mater. 147 (2007) 401–411. [15] C.E. Borba, E.A. Silva, S. Spohr, G.H.F. Santos, R. Guirardello, Application of the mass action law to describe ion exchange equilibrium in a fixed-bed column, Chem. Eng. J. (Amsterdam, Neth.) 172 (2011) 312–320. [16] P. Gomathi, C. Priya, Ahmed Basha, V. Ramamurthi, Removal of Ni(II) using cation exchange resins in packed bed column: prediction of breakthrough curves, Clean – Soil, Air, Water 39 (2011) 88–94.

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