Recovery of acetate by anion exchange with consecutive CO2-expanded methanol desorption A model-based approach

Production of bio-based acetate is commonly hindered by the high costs of the downstream processing. In this paper, a model is developed to describe a new method that recovers acetate salts using anion exchange resins, and subsequently desorbs and upgrades them using CO 2 -expanded alcohol. The model consists of equilibrium parameters for both the adsorption and desorption step. The calculated parameters are: for the adsorption and 0.674 mol/kgresinmol/kgsolution , and for the desorption 3.71. The maximum experimental concentration of acetic acid obtained in CO 2 -expanded methanol is 0.427mol/kg (20g/ L MeOH ) at an operating pressure of 31bar. The model represents the expected trends for all species, and can be used to design a multicolumn system for the recovery and upgrading of carboxylates.


Introduction
Bio-based production of carboxylic acids via fermentation is a route to a wide variety of chemicals [1]. Examples of carboxylic acids that can be fermented from renewable materials, and for which large scale production exists, are acetic acid, citric acid, lactic acid, and itaconic acid [2]. Commercial production of carboxylic acids by fermentation is only possible if the recovery from the aqueous solution is efficient.
Some fermentation methods to produce carboxylic acids (concentrations 2-100 g/L) require titration with a base to maintain neutral pH, and as a result produce a carboxylate salt [2]. Traditional recovery of carboxylic acids from these carboxylate salts involves high energy consumption and waste co-production. One method to capture carboxylates from a dilute solution is to use strong anion exchange resins. Anion exchange resins are used to recover carboxylates because of the high affinity of the positively charged functional group. To avoid the use of strong mineral acids during desorption, a novel process for the recovery of carboxylic acid using the strong anion exchange resins and desorption with CO 2 -expanded alcohols was developed [3]. At the end, the resin is regenerated to the bicarbonate form and the carboxylic acid dissolves in the CO 2 -expanded alcohol solution for further processing (e.g. ester formation).
The main advantages of the method are: high solubility of CO 2 in the alcohol, no stoichiometric waste salt production (if the liberated bicarbonate is reused to control the pH of the fermentation), and integration with further downstream steps such as esterification, distillation or crystallization. The method was tested to work with different alcohols and carboxylates recovered from aqueous solutions and paper mill wastewater. However, the main limitation of the method is the high dilution of the produced carboxylates and esters (0.1-0.3 wt. %) after desorption/esterification [3,4]. A higher concentration of products in the CO 2 -expanded alcohol solution would facilitate the further purification. However, no theoretical or empirical data are available to predict the maximum concentration achievable in desorption at a given amount of methanol and CO 2 . For these reasons, a model that can predict the equilibrium concentration of carboxylate desorption using CO 2 -expanded methanol is needed. In this paper, acetic acid is used as example for the determination of the model parameters. Acetic acid is one of the carboxylic acids that can be produced via fermentation. It is industrially used in the synthesis of vinyl acetate, cellulose acetate, and other acetate esters [5].
To predict the maximum achievable concentration during desorption, the current research aims to develop an equilibrium model for both the adsorption of aqueous acetate and chloride to a strong anion exchange resin in the bicarbonate form and subsequent desorption of acetic acid with CO 2 -expanded methanol. After desorption with CO 2 -expanded methanol, the resin is regenerated to the bicarbonate form. In this chapter, chloride is used as model anion to study the effect of impurities in the system. The equilibrium model is used to check the maximum concentration of acetate achievable in the CO 2 -expanded methanol solution. Furthermore, an equilibrium dispersive model is used to model the transport through the column during adsorption and desorption. This mathematical model is needed for the design of a multicolumn system for the continuous recovery of carboxylates from diluted streams.

Ion exchange and adsorption equilibrium
In this study, recovery of acetate from a dilute aqueous solution at pH above the pK a is performed using a strong anion exchange resin with a quaternary ammonium functional group (Type I). In addition to exchange of the acetate ion with the counter-ion of the resin's functional group, adsorption of the un-dissociated acetic acid to the resin backbone occurs. For this reason, the adsorption mechanism of acetic acid at a pH above the pK a is described by ion exchange (Eq. (6)) and adsorption (Eq. (8)). The counter-ion of the functional group is chloride, and the interaction with acetate and bicarbonate (regenerated version of the resin) is studied. Additionally, the dissociations of all the species involved in the recovery of acetate are represented, and shown by the equilibrium reactions in Table 1, in which γ and m are the activity coefficient and molality in the aqueous phase, and q is the molality in the wet resin phase.
Ion exchange equilibria are described by a homogeneous mass action model, Eqs. (6) and (7). This originates from the common approach to treat the process like a reversible chemical reaction [8]. The exchanger is assumed to be a homogeneous phase, and non-idealities for the liquid phase are taken into account by introducing activity coefficients [9].
In this study, the activity coefficients for neutral species in solution are considered ideal (γ HAc and = γ 1 ), since the deviation from ideality in aqueous systems is small and the error in the calculation is not significant [10]. For electrolyte solutions, deviations from ideality, even at low concentrations, can be important because of strong ion interactions with other ions, solvent and exchange resin. To quantify solution non-idealities, Debye-Hückel or Pitzer like models are generally used [11]. In this study, the activity coefficients for acetate, bicarbonate, carbonate, chloride, proton, hydroxide and sodium acetate in the water phase are calculated by the modified Debye-Hückel model of Truesdell and Jones (Eq. (9)) [7,12].
The ionic strength, I, is calculated as Eq. (10), where z i is the charge of ionic species.
The fermentation broths that we consider have an acetate concentration < 0.5 mol/kg, in which the Truesdell-Jones model is valid [12]. The parameters used for the activity coefficient model are reported in Appendix A.1 (Table A.1 and Table A.2).
The activity coefficients in the resin phase are resin dependent, and the simultaneous calculation of the activity coefficient and equilibrium constant exhibits a disturbing interdependence [9]. For this reason, it was assumed that the activity coefficients of the acetate, bicarbonate and chloride ions in the resin phase are approximately the same, leading to Eqs. (6) and (7) without activity coefficients in the resin phase.
For the non-dissociated acetic acid species, the adsorption is described through the linear region of the isotherm by a constant (K OV HA , ), Eq. (8). All the parameters are calculated at 25°C and 1 atm CO 2 and for this reason the fugacity coefficient (for the adsorption step) is assumed to be 1. The results of the batch adsorption experiments are used to calculate the selectivity and distribution coefficient for acetate, bicarbonate and acetic acid, respectively. The calculated parameters are further used to describe the effect of pH on the acetate and bicarbonate recovery. This information is used to model the recovery of the ions with the anion exchange resin at different pH and concentrations.
To solve the system, additional assumptions and mass balances are implemented: • Electroneutrality in the liquid phase is assumed during all the experiments.
• The difference between inlet molalities and aqueous equilibrium molalities is assumed to be in the resin phase for each species i. In which W resin is the mass of resin added in the liquid phase, and W solution is the mass of the solution, which are assumed to be constant. q in i , is the initial value of each species on the resin, which is

Desorption equilibrium
The anion exchange resin is regenerated using CO 2 -expanded methanol as described in our previous publications [3,4]. However, in the current study no catalyst is used for the formation of esters, because the aim is to study the desorption equilibrium (without esterification). As a result, the desorption with CO 2 -expanded methanol is described by five equilibrium reactions: the transfer of CO 2 from the gas to the methanol phase, the formation and deprotonation of methyl carbonic acid, the ion exchange of methyl carbonate with acetate and the protonation to acetic acid. Table 2 shows the equilibrium reactions considered in our desorption model.
For simplicity, the activity coefficients of neutral species are again taken as unity (γ γ γ , , ) HMeCO HAc MeOH

3
. The mean ionic activity coefficient, ± γ , is estimated using Eq. (17), which is the first term of Pitzer's model, as proposed by others [13], and used in Eqs. (14) and (16). This term of the Pitzer model includes the electrostatic far field interactions between ions, but does not represent the specific binary, near-field interactions between pairs or ternary interactions (Ac − , MeCO 3 − and H + ), which are not reported elsewhere.
A is the Debye-Hückel parameter; which is estimated as proposed by Gmehling et al. [15].
where ρ solv is the density of methanol (0.786 kg/L), T is the temperature in K (298 K) and ∊ r is the relative dielectric constant of methanol (33.05).
As previously, ion exchange equilibria for the batch experiments are described by a homogeneous mass action model. Since the mean activity coefficients of ions in the methanol phase are calculated by Eq. (17), the activity coefficients of acetate and methyl carbonate in methanol are equal. Furthermore, the activity coefficients of acetate and methyl carbonate on the resin phase are assumed to be the same. These assumptions lead to = K 1 γ in Eq. (15). In this study, the mass action law model is derived to express the equilibrium concentration of acetic acid in methanol as a function of CO 2 pressure and the methanol/resin ratio. The concentration of anions in both phases are expressed in terms of acetic acid concentration, CO 2 pressure and methanol/resin ratio. Finally, the parameter is determined from experimental data to model the ion exchange between acetate and methyl carbonate. Electroneutrality in the liquid phase is assumed in the batch calculations (Eq. (19), and the amount of each component bound to the resin is calculated as stated in Eq. (12).

Dispersive model
The performance of a chromatographic column depends on factors that belong to two broad categories: equilibrium and dispersive factors. The equilibrium factors for our system have been described in Section 2.1 and 2.2. In this section, a term describing axial dispersion is included in the mass balance of the mobile phase, and the bed porosity is included [16,17]. In the model, the effect of several parameters are lumped into the dispersion coefficient D app . The lumped parameter D app includes peak broadening effects caused by the fluid dynamics of the packing (axial dispersion) and all other mass transfer effects. Using these assumptions, the differential mass balances for the liquid phase of all components are given by equation (20).
Here, the required parameters are the dispersion coefficient (D app ), the interstitial velocity (u int ) and the bed porosity (ε). The bed porosity is determined by residence time distribution experiments with potassium chloride and dextran as discussed in Section 3.5.1. It is further calculated using three characteristic times: the time of a tracer that enters the particles (t ) KCl , the time of a tracer that should not enter the particles (t ) Dextran and the time of a tracer without column (dead volume until detector) (t ). detector In a typical experiment, = t 130s KCl , = t 38s detector , and = t Dextran 117 s.
were V̇is the flow rate and V c is the packed volume of the column, and calculated from the packed length and the internal diameter of the column. The interstitial velocity (u int ) is calculated from the operating flow rate (V̇), the column internal cross-sectional area (A c ) and the extraparticle void fraction (ε) as: The dispersion coefficient (D app ) is calculated by moment analysis of an injection of potassium chloride and their relation with the van Deemter plot, and assumed to be the same for all components. The first two moments were calculated directly by numerical integration as: And the height equivalent to a theoretical plate (HETP) is calculated by the two moments and the column length (L c ) as: Exploiting the connection between the apparent dispersion coefficient and the second moment, the dispersion coefficient can be calculated as proposed by others [17]. This apparent dispersion coefficient is used directly for the adsorption experiments. In the case of the desorption experiments, the apparent dispersion coefficient is calculated by fitting it to the experimental data. (29) For the solid phase, the differential mass balance for each component is given by Eq. (30).
From which the m i eq is given by the specific equilibrium equation (Section 2.1, Eq. (12) and k for ion exchange is considered to be an instantaneous reaction (assumed to be 1000 s −1 ). An additional required parameter is the total capacity of the resin (q max ), which is used to close the resin mass balance. The resin q max is obtained experimentally as explained in Section 3.5.2.
The assumptions for all the experiments are: • The feed is homogeneous so that the concentration of each species at the inlet is equal to m i feed , at all times.
• The apparent dispersion coefficient is equal for all species.
• The concentration of carbonate and carbonic acid are negligible at these conditions.
The initial and boundary conditions are: • At time zero, the concentration of the binding component (acetate and chloride for adsorption, methyl carbonate for desorption) in the liquid phase throughout the column is zero.
• The concentration of bicarbonate (for adsorption) and acetate (for adsorption and desorption) was given a small initial concentration (0.01 mmol/kg) to avoid division by zero.
• The concentrations at the inlet were increased from zero to the feed value by using a step function.
• At the outlet boundary, a zero gradient of the liquid concentration was assumed.
• For adsorption, the resin phase was initially assumed to be fully loaded with bicarbonate and this is equal toq max .
• For desorption, the methyl carbonic acid inlet concentration is assumed to be constant at the operating pressure of 31 bar CO 2 . This was calculated with Eq. (13) and the result is shown in Table A.3.
• For desorption, the resin phase was initially assumed to be fully loaded with acetate (acetate loading) or with acetate, chloride and bicarbonate in which acetate is loaded with a ratio of 0.14⋅q max .
The additional required equations necessary to solve the system are the thermodynamic equilibrium that is used as calculated in Section 2.1 and 2.2. The simulation is done in COMSOL Multiphysics as explained in Section 3.7, in which the molalities are converted to mol/L to be solved in the software using the density of water (1 kg/L) and methanol (0.786 kg/L), respectively.

Materials
Sodium bicarbonate was purchased from J.T. Baker. Anhydrous potassium acetate (99%), methanol (≥99.9%), dextran blue and anhydrous methanol (≥99.8%) were obtained from Sigma-Aldrich. Potassium chloride (> 99.5%) was purchased from Merck. Carbon dioxide (≥99.8%) was supplied by Linde as compressed gas. The strong anion exchange resin (Dowex Marathon MSA, macroporous) in the chloride form was purchased from Sigma Aldrich. The nominal total exchange capacity is at least 1.1 eq/L (wet basis). All aqueous solutions were prepared with deionized water from a Milli-Q water purification system (Millipore). Deionized water was used from a Milli-Q purification system.

Adsorption batch experiments
Adsorption experiments were performed in 50 mL flasks with 1 g of wet resin (chloride form) added to 10 mL aqueous solutions of a carboxylate salt/carboxylic acid at different concentrations. Sodium acetate solution concentrations were between 0.006 and 0.72 mol/kg (0.5-60 mg/g solution ), acetic acid between 0.012 and 3.66 mol/kg (0.7-180 mg/g solution ) and sodium bicarbonate between 0.006 and 0.34 mol/kg (0.5-28 mg/g solution ). The flasks were then shaken at 200 rpm and 25°C for about 18 h in which equilibrium was reached (as checked with preliminary experiments). Each experiment was performed in duplicate. The pH of the samples before and after the reaction was measured. After the reaction each system was filtered using a Millex-GV Syringe Filter Unit and acetic acid, bicarbonate and chloride concentration in the liquid phase were quantified (Section 3.6).

Resin preparation for batch desorption
Column elution was used to convert the resin to the bicarbonate form. Fresh resin in the chloride form was hydrated in a beaker with deionized water for 30 min. The resin was then filtered under vacuum for 2 min in a glass filter. The hydrated resin was weighed and tightly packed in an Omnifit glass column (1 cm internal diameter, 15 cm height) and the column was placed in a Dionex Ultimate 3000 HPLC (Thermo Scientific). The resin was sequentially washed with 2 mL/min deionized water for 30 min, 4 mL/min sodium bicarbonate (20 g/L) for 240 min, then with 4 mL/min of deionized water for 240 min, and finally is converted to the acetate form with a solution of 0.10 mol/kg (10 g/L) potassium acetate at 2 mL/min. The absorbance of the outflow was continuously measured with the internal VWD-3400RS UV-Visible detector, and 2 mL samples were taken at 40 min intervals with the AFC-3000 automatic fraction collector. The last sample was analyzed for acetate concentration (Section 3.6). The breakthrough curve was constructed using the online measured absorbance and final sample acetate concentration, and the resin capacity for acetate was calculated by integration. The resin was removed from the column and washed 3 times with 50 mL deionized water and filtered at 20 mbar using Millipore Steriflip 60 μm nylon net filtration unit. The washing-filtering procedure was repeated two times with 30 mL of methanol and one time with 30 mL of anhydrous methanol. The resin was dried in an oven at 60°C for 4 h to remove residual water and allow to cool in a desiccator.

Desorption batch experiments
Desorption experiments were performed at varying CO 2 pressures and methanol to resin ratios. CO 2 pressures of 2.1, 5, 10 and 20 bar and methanol to dry resin ratios of 5, 10, 15, 20, and 25 g/g were used. The desorption of acetate from the resin was performed by adding 0.5-2 g of resin (dry) and the required mass of anhydrous methanol in a 50 mL Büchi glass stirred autoclave. The (dry) resin has a water content of 8 w/w% as reported previously [3]. The vessel was equipped with a magnetically driven four blade impellers, an overhead motor, a pressure sensor, a pressure relief valve, a carbon dioxide inlet, and a sampling port. The reactor was flushed 4 times with CO 2 . Agitation was set to 250 rpm and then CO 2 was added until the pressure stabilized at the desired value. The experiments were performed for 4 h in duplicate. Final pressure and temperatures (20-22°C) were recorded. Liquid samples were obtained at the set pressure and analyzed for acetic acid concentration (Section 3.6).

Dispersion and porosity determination
Hydrated resin was packed in an Omnifit glass column (15 cm height × 1 cm internal diameter). A 20 µL tracer pulse of aqueous potassium chloride (3 mol/L) was added at a constant flow (0.15-6 mL/ min) to the system with (resin in chloride form) and without column at 20-22°C. The mean time of passage (τ ) and variance (σ 2 ) of the system with the column were calculated by numerical integration of the conductivity response, and used for the determination of total porosity as explained in Section 2.3. The particle porosity was measured in the system by repeating the experiment with 6.6 g/L of dextran blue as tracer.

Total resin capacity
The total anion exchange capacity of the resin was determined based on the ASTM D2187-94 Standard method-Test H. The method consists of the conversion of a sample to the chloride form using a concentrated solution of hydrochloric acid. The sample was washed with water and isopropanol. Elution of chloride from non-salt-splitting group was done using ammonium hydroxide. The sample was changed to the chloride form again washing it with sodium chloride (50 g/L) and the subsequent elution of chloride from salt splitting group was performed using sodium nitrate. Determination of chloride in the separate eluents was done by titration with silver nitrate (0.1 mol/L).

Adsorption dynamic experiments
An Omnifit glass column (1 cm internal diameter × 15 cm height) was used, with approximately 5 g of wet resin in the chloride form (∼60 wt%/wt water). The dispersion of the column was measured as mentioned in Section 3.5.1. The column was converted to the bicarbonate form with a solution of 0.24 mol/kg (20 g/L) sodium bicarbonate (Section 3.3). Dynamic experiments were performed in a Thermo Scientific Dionex Ultimate 3000 system. Carboxylate solutions were pumped through the column at 0.3 and 2 mL/min and 25°C. The carboxylate inlet solutions contained acetate 14.2 mmol/kg (0.84 g/L) and chloride 9 mmol/kg (0.32 g/L) at a pH of 5 or 7.6 Fractions of 2 mL were collected. Feed samples and collected fractions were analyzed for acetate and chloride, and pH, conductivity and absorbance (210 nm) were measured online.

Desorption dynamic experiments
After an adsorption experiment, the column was washed with Milli-Q water (4 mL/min) until the conductivity was below 0.007 mS/cm. Then, the column was washed with methanol (2 mL/min) for 30 min. The desorption dynamic experiments were performed in a modified Thermo Scientific Dionex Ultimate 3000 system. The system was modified with a high-pressure pre-mixing vessel to equilibrate the methanol with 10 bar of carbon dioxide during ∼30 min. A backpressure was installed after all sensors to the modified system to assure that the pressure remained above the set-point of the pre-mixing vessel (10 bar). The methanol/carbon dioxide solution was pumped through the column at 1.5 mL/min at 20-22°C. Fractions of 2 mL were collected. Collected fractions were analyzed for acetic acid as explained in Section 3.6, and absorbance (210 nm) was measured online.

Analytical methods
The concentration of total acetate in the systems was evaluated using a Waters HPLC system equipped with a UV/Visible Detector (Waters 2489) and a Refractive Index Detector (Waters 2414) for both water and methanol samples. The column used was the Bio-Rad Aminex HPX-87H column (7.8 × 300 mm). A mobile phase of phosphoric acid (1.5 mmol/L) was used in isocratic mode at 0.6 mL⋅min −1 . The injection volume was 10 μL and the duration of the run was 30 min, the column temperature was 60°C and the detection was at 210 nm. The experimental aqueous acetate and acetic acid concentration were determined using the total acetate measurement, the experimental pH and Eq. (1). The methanol samples were diluted 10 times to avoid deterioration and interference with the column material. Chloride concentrations (Sigma-Aldrich MAK023) were determined spectrophotometrically using the Sigma Aldrich MAK023 test kit in 96 well plates. Total carbonate was measured as carbon dioxide with a Hach-Lange LCK388 kit, which includes all carbonate species (carbonate, bicarbonate and carbon dioxide).

Regression and model calculations
The determination of the parameters from each set of equations was done by minimizing the sum of mean squared error of the total acetate (chloride concentration for the bicarbonate experiment) with MATLAB® optimization algorithm fminsearch. The objective functions were the set of equilibrium equations that were solved in a separate MATLAB® function using the built-in nonlinear solver fsolve.
The dynamic model was implemented on the COMSOL Multiphysics platform (v5.2a, Comsol Inc., Burlington, MA). Equations for a onedimensional dispersion and ion exchange were solved with a variable time step with a mesh size of 2 ⋅ 10 −4 m. Fig. 1 shows the isotherms for acetate, acetic acid and bicarbonate with the corresponding model. The maximum loading obtained was 169 mg/g (wet resin in chloride form) for acetate and 120 mg/g wet resin for acetic acid. Thus, the resin shows a lower capacity for acetic acid than for acetate. The bicarbonate isotherm does not reach a plateau because of low solubility of sodium bicarbonate (∼1.3 mol/kg at 20°C). The loading obtained for bicarbonate is 49 mg/g wet resin . The total exchange capacity (q max ) was 1.9 mEq/g wet resin , and measured as explained in Section 3.5.2. Some of the reported loadings for the adsorption of acid species with a quaternary group resin are: 24 mg/g resin for acetic acid with a Sepra SAX resin (silica matrix, quaternary amine group, 0% water), 66 mg/g resin for the Sepra ZT SAX resin (polymeric matrix with quaternary amine group, 0% water), and 41.2 mg/g wet resin on the A26OH resin (polystyrene matrix, quaternary ammonium group) [18][19][20]. In these reported values, the type of the functional group of the resin leads to different interactions of the functional group with the acid at the low pH experiments [21]. For the acetate recovery (pH above the pK a ), the obtained capacities are comparable with reported values, 28 mg/g wet resin (IRA-910 chloride form) and 112 mg/g wet resin (IRA-910 hydroxide form) [22]. Experiments reported in dry resin basis were converted to wet basis assuming 60 wt.% water content. Fig. 1 shows that the model follows the experimental values for chloride, bicarbonate, acetate and pH. The pH in all the experiments is lower than the pH of the feed solutions (data not shown), because chloride is a weaker base than bicarbonate and acetate. For acetic acid, which binds to the resin by hydrogen bonding [2], this was an indication that ion exchange also occurs, confirmed by the chloride release (Fig. 1c). The model could describe this behavior and the change of pH with an average error of 10.6% for total carbonate, 5.22% chloride and 4.92% for pH in the bicarbonate model (Fig. 1a); 5.6% acetate and 4.0% for the pH in the acetate model (Fig. 1b); 4.4% acetic acid, 9.6% for pH and a high error of 60% for the chloride in the acetic acid model (Fig. 1c). The high chloride error in the acetic acid model is assumed to be due to measurement error at these low concentrations (1-9 mmol/L). A reason for the discrepancy might be that the isotherms were not developed at isonormal conditions. Further studies should considered developing the isotherms at isonormal conditions to check if the correlation can be improved as reported by others [23].

Parameters for the ion exchange adsorption equilibrium model
The selectivity for acetate and bicarbonate over chloride and the overall distribution coefficient of acetic acid were estimated as: These parameters are comparable with reported values of 0.10 for the selectivity of acetate over chloride (calculated from their reported acetate/hydroxide and chloride/hydroxide selectivity) [24], 0.11-0.16 for propionate over chloride and 0.28 to 0.33 bicarbonate over chloride [25]. The selectivity of bicarbonate over chloride is lower than the one reported by others (0.21 compared to 0.23-0.33) [25]. The reason is that the equilibrium equation of carbon dioxide to carbonic acid (Eq. (3)) was added independently, and in other reported work [25] both equilibrium equations (Eqs. (3) and (4)) are included in one apparent equilibrium constant (K app = 6.3). The resin supplier reports selectivity of 0.145 for acetate over chloride and 0.272 for bicarbonate over chloride [26]. , for the acid species is higher than reported values with other strong anion exchange resins [24]. The obtained parameters values are used in Section 4.3 to build the dispersive-equilibrium model for the adsorption step.

Parameters for the desorption equilibrium model
In this section, the effect of excess methanol and CO 2 pressure in the desorption equilibrium of acetate from the strong anion exchange resin is studied. The aim is to estimate the parameter value was used to predict the equilibrium desorption concentration at different pressures. The error of the predictions ranges from 0.9% to 10.8% with an average of 6.5% and 6.7% for 5 bar and 10 bar of CO 2 , respectively. Additional experiments at 20 bar (5 and 30 g methanol per g of resin (dry)) were performed to check the validity of the model at higher pressures. The main deviations occur at high acetic acid concentration, and the reason may be that near-field binary and ternary interactions were not considered for the calculation of the activity coefficient (Section 2.2), which means that the model for activity coefficient calculations has the same accuracy as a Debye-Hückel model [15]. As well, high acetic acid concentrations imply high concentrations of dissolved CO 2 such that the activity coefficient might deviate and CO 2 might adsorb to the resin like acetic acid does.
In Fig. 2, the experimental and predicted equilibrium concentrations and recoveries are presented. In Fig. 2a, the acetic acid equilibrium concentration is plotted as a function of the methanol/resin ratio for the four experimental CO 2 pressures. In Fig. 2b, the recovery (mol HAc in liquid/mol total acetate originally in resin) is presented as a function of the same variables. As expected, increasing the amount of methanol decreases the concentration of acetic acid but increases its recovery. As m MeOH is constant, the concentration of methyl carbonate increases with CO 2 pressure. The ion exchange equilibrium is then displaced towards the acetate side, hence more acetic acid appears in solution. At 2.1 bar CO 2 , reducing the methanol/resin ratio from 25 to 5 increases the dissolved HAc concentration from 0.063 to 0.147 mol/kg, but decreases the recovery (from 49.1 to 22.6%). That represents a 2.3-fold increase in concentration at the expense of a 2.2-fold decrease in recovery. At 5 bar CO 2 , for the same decrease in the methanol/resin ratio, the concentration increases 2.47-fold at the expense of a 2.05 fold decrease in recovery. At 10 bar CO 2 the trade-off improves slightly; a 2.72fold increase in concentration leads to a 2.02-fold decrease in recovery.  The effect of CO 2 pressure on concentration is more pronounced for lower methanol/resin ratios. However, it can be seen that as CO 2 pressure increases, its effect on both concentration and recovery levels off. Higher recoveries deplete the acetate bound to the resin, which in turn requires a larger concentration of methyl carbonate ions to achieve further desorption. The highest recovery achieved in this study is 81% when using 29 g MeOH /g resin and 20 bar CO 2 . At these conditions, a concentration of acetic acid 0.098 mol/kg (4.6 g of acetic acid/L methanol) is obtained. This result is in line with Rebecchi et al., where 1 g of acid was desorbed with 0.2 L of basified ethanol (5 g of acid per L of ethanol) [18].
In Fig. 3, the model is used to extrapolate the results to 61 bar CO 2 , where it becomes supercritical. The goal is to check the theoretical maximum concentration and recovery that can be achieved in one equilibrium stage. The recovery of acetate is plotted as a function of the equilibrium concentration for different methanol/resin ratios. Isobaric curves are also drawn to allow for a quick estimation of the recoveries and concentrations at different experimental conditions. The values are a useful approximation for design and operation of the column system.
The highest concentration is obtained at 61 bar, and it is about 0.48 mol/kg (22 g of acetic acid/L of methanol) with a recovery of about 70%. This indicates the possibility to increase the concentration in comparison with previous data [3,4]. However, lower pressures are desired for an industrial application. For this reason, a column operation (eventually a multicolumn system) is analyzed to improve the recovery further, which is studied in Section 4.3.

Equilibrium dispersive model for adsorption and desorption
Dynamic adsorption experiments were performed at different flows (0.3 and 2 mL/min) and different feed pH (5 and 7.6) with the resin initially in the bicarbonate form. The results are shown in Fig. 4. Fig. 4 confirms that chloride is more selective to the ion exchange resin than acetate. This causes an overshooting of acetate after the breakthrough of chloride (Fig. 4a), in line with data for recovery of carboxylates from paper mill wastewater [4]. In general, overshooting of an ion occurs when the feed solution contains ions with higher selectivity for the resin group than the overshot ion, resulting in a portion of eluted solution with a concentration of the latter above its feed concentration [27].
At pH 5, the loading of total acetate to the resin is higher, caused by the additional adsorption of acetic acid to the backbone of the resin. Additionally, a lower elution of bicarbonate at pH 5 than at pH 7 supports the hypothesis that part of the total acetate is removed by hydrophobic adsorption and not by ion exchange. An equilibrium dispersive model was used to study the behavior of the adsorption to the column in the bicarbonate form. Fig. 4 shows that the model represents the expected trends for all species. The equilibrium constant (selectivity) of acetate over bicarbonate was calculated from the values obtained of acetate over chloride and bicarbonate over chloride (Section 4.1). The model presents deviations caused by the assumptions used to simplify the system, namely: equal dispersion coefficient for all compounds, all mass transfer effects lumped into the dispersion coefficient, the bed void fraction calculated using dextran blue, and negligible concentrations of carbonate and carbonic acid. The apparent dispersion coefficient was calculated, as mentioned in Section 2.3, using Eq. (29). To improve the model, one might use correlations for the lumped dispersion coefficient that take into account diffusion rates into the particles. Bicarbonate was the species that has the highest deviation from the model. Main reasons can be experimental variations caused by measurement errors (inaccuracy of the carbonate measurement method), and the fact that divalent carbonate was not considered in the model. The dynamic desorption experiments are shown in Fig. 5. For these experiments, the resin is pre-loaded with an acetate and chloride solution (as in the dynamic adsorption experiments), or fully pre-loaded with acetate. For the resin pre-loaded with acetate and chloride, the resin had all three anions (acetate, chloride and bicarbonate) prior to desorption. The ratio of acetate on the resin compared with the total resin capacity was 0.14. The desorption was performed using CO 2 -expanded methanol pre-equilibrated at 10 bar at an operational pressure of 31 bar in the column controlled by the back-pressure. The maximum experimental concentration of acetic acid achieved was 0.427 mol/kg (20 g/L MeOH ), which indicates that the CO 2 concentration is dictated by the operation pressure (31 bar), since the maximum expected methyl carbonic acid concentrations are 0.183 mol/kg (at 10 bar of CO 2 ) and 0.515 mol/kg (at 31 bar of CO 2 ), calculated from the methyl carbonic acid concentration of the equilibrium reactions (Section 4.2). All the equilibrium constants (K ACA , − − K MeCO Ac 3 , K m ) were combined to represent the equilibrium as one reaction with a constant of 1.97 (Appendix A.5). In this way, dissolved ions were not taken into account, which can be justified because their proportion was relatively low.
The CO 2 -expanded methanol cannot desorb bound chloride. This is confirmed by the loss of acetate capacity after several adsorption and desorption cycles (Appendix A.6). The reason that chloride cannot be desorbed is because of its low pK m (pK a of HCl in water is −5.9) [28]. To remove the chloride, the resin was additionally regenerated using an aqueous bicarbonate solution.
For the resin loaded with acetate and chloride, the inlet concentration of methyl carbonic acid was assumed to be the ratio of acetate bound to the resin and the concentration of methyl carbonic acid ( * m 0.14 HMeCO3 ) at the operating pressure of 31 bar. The reason was that methyl carbonate seemed to exchange not only with acetate, but also with the bicarbonate bound to the resin. This caused a higher consumption of methyl carbonic acid than the expected if only acetate was on the resin. To prove this, the resin was fully pre-loaded with acetate instead of an acetate-chloride solution. Fig. 5 shows that the model presents a trend closer to these experimental data. This states the importance of having a fully pre-loaded resin during desorption, which is comparable to other desorption-reaction techniques [29].
The dispersion coefficient of the desorption step was fitted to the fully loaded acetate experiment and was not calculated using empirical correlations. A more detailed study of the dispersion coefficient of desorption with CO 2 -expanded methanol is needed.
The equilibrium and the dynamic experiments show that the resulting concentration of acetic acid in CO 2 -expanded methanol is limited by the methyl carbonic acid concentration. The actual column system presents an improvement from our previous reported data, from 0.8 to 2.4 g/L to 20 g/L HAc, and the model can be used to design a simulated moving bed multicolumn system. A multicolumn system can improve the total resin usage and the breakthrough capacity can be increased [30], but the concentration of HAc remains limited by the methyl carbonic acid concentration (pressure of CO 2 ) and ratio between acetate and other anions in the resin. It would be useful to extend the proposed model to the desorption of mixtures of organic acids, as they are commonly encountered in fermentation and acidogenic anaerobic digestion. [18]

Conclusions
This chapter presented an equilibrium-dispersive model to describe and validate the recovery of acetate by anion exchange with a consecutive desorption with CO 2 -expanded methanol. The equilibrium parameters were estimated as: for batch desorption. Using these parameters, the maximum equilibrium concentration of acetic acid (after batch desorption) was predicted to be 0.48 mol/kg at 61 bar. The dynamic behavior for all species were described using the equilibrium-dispersive model. For column adsorption, it was concluded that acetic acid also binds to the resin, since there was a higher loading of total acetate at pH 5 in comparison with pH 7, complemented with a lower elution of bicarbonate. Additionally, the higher selectivity of chloride in comparison to acetate caused an overshooting of acetate. For column desorption, it was concluded that the acetic acid concentration in the CO 2 -expanded methanol was limited by the methyl carbonic acid concentration and the ratio of acetate to chloride loaded to the resin. The maximum achieved acetic acid concentration was 0.427 mol/kg (20 g/L MeOH ) at an operating pressure of 31 bar. The concentrations reported in this chapter are an improvement from previous reported data. The model presented the equilibrium and dispersive parameters required to design the specific columns within a multicolumn operation scheme.