Model-Based Comparison of Batch and Continuous Preparative

Article pubs.acs.org/IECR

Model-Based Comparison of Batch and Continuous Preparative Chromatography in the Separation of Rare Earth Elements Niklas Andersson, Hans-Kristian Knutson, Mark Max-Hansen, Niklas Borg, and Bernt Nilsson* Department of Chemical Engineering, Lund University, SE-221 00 Lund, Sweden S Supporting Information *

ABSTRACT: The demand for rare earth elements (REEs) is growing, while the future supply is uncertain. Their unique electronic characteristics make them irreplaceable, and the commercial value of pure fractions is high. A model-based simulation study is presented that compares batch chromatography with the twin-column MCSGP (multicolumn countercurrent solvent gradient purification) process for ion-exchange chromatography of the four-component system neodymium, samarium, europium, and gadolinium. The last three components are considered products with individual purity requirements of 99%. The twin-column process has been shown to be a good alternative to the batch process regarding modifier consumption and productivity since it enables internal recycling to achieve high purities. A new cut strategy for MCSGP is applied where subfractions are taken from each outlet. Two multiobjective optimizations with yield, solvent productivity, and productivity objectives show that the MCSGP process is a better alternative than batch chromatography.

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INTRODUCTION The rare earth elements (REEs) are found in many products such as batteries, permanent magnets, superconductors, lasers, and capacitors.1 The sources of REEs are minerals that contain a mixture of several rare earth elements that need to be separated. Their specific properties make them irreplaceable, and the commercial demand is increasing. The use of rare earth elements often requires high purity, and the price increases with purity.2 Attaining a high purity is challenging because of similar chemical and physical properties.3 Current separation methods include liquid−liquid extraction, selective oxidation or reduction, and ion-exchange chromatography (IEC). In the past decade, global demand has increased and so will the demand for new sustainable separation methods.4 A number of chromatographic processes can be used for multifraction collection, such as batch chromatography, simulated moving bed,5 intermittent simulated moving bed,6 steady-state recycling,7 gradient with steady-state recycle,8 multicolumn countercurrent solvent gradient purification (MCSGP),9 and twin-column MCSGP.10 The MCSGP method allows three fractions to be collected and also allows solvent gradients to be implemented. The twin-column MCSGP is a variant of the conventional six-column MCSGP setup. The columns are run in four different steps, (I1, B1, I2, and B2), where I is the interconnected step and B is the batchwise step as shown in Figure 1. A column is cycled through eight different sections (s1−s8) during a full cycle. Two sections are run simultaneously, s1 and s5 during I1, s2 and s6 during B1, s3 and s7 during I2, and finally s4 and s8 during B2. The sections s1−s8 form a full cycle, and the combined outflow from the sections produce a chromatogram that is similar to a batch chromatogram. A column switch is carried out when step B2 is finished, i.e., just before the very end of the first half-cycle. The columns are switched so that column 1 starts the second half-cycle in section s5 and column 2 in section s1. The weakly adsorbed compound (W) is collected © 2014 American Chemical Society

Figure 1. Schematic figure of the twin-column MCSGP. The columns are run in four different steps (I1, B1, I2, and B2). There are eight different sections s1−s8 that together produce a chromatogram. The flows are denoted q1−q8, each corresponding to its section. The loading (L) is done in s2, the weakly adsorbed compound (W) is collected in s1−s4, the product (P) is collected in s6, and the strongly adsorbed compound (S) is collected in s8. The chromatogram illustrates in which section W (light gray), P (gray), and S (dark gray) elute.

from s1 to s4, the product (P) is collected from s6, and the strongly adsorbed compound (S) is collected from s8. The outflow from section s5, containing a mixture of W and P, is Received: Revised: Accepted: Published: 16485

June 11, 2014 September 5, 2014 September 24, 2014 September 24, 2014 dx.doi.org/10.1021/ie5023223 | Ind. Eng. Chem. Res. 2014, 53, 16485−16493

Industrial & Engineering Chemistry Research

Article

⎛ ⎛ = k kin, i⎜ciHi⎜⎜1 − ⎜ ∂t ⎝ ⎝

recycled to s1. The outflow from section s7, which contains a mixture of P and S, is recycled to s3. The subject of this study was the separation of the elements neodymium (Nd), samarium (Sm), europium (Eu), and gadolinium (Gd) by preparative ion-exchange chromatography. The separation of europium by ion exchange using an acid as a modifier has previously been investigated.11,12 Optimization of the modeled system has recently been performed in batch chromatography with promising results.13−15 This work is a model-based simulation study that compares the performance of the twin-column MCSGP with conventional batch chromatography. Both processes have been optimized using a genetic algorithm, and the Pareto solutions of yield/ productivity and yield/solvent productivity rate are compared. The simultaneous collection of the three products (Sm, Eu, and Gd) was optimized with respect to product value and element abundance. The first contribution is the comparison between the twincolumn MCSGP process and the conventional batch process, where the three objective functions of productivity, specific productivity, and yield are optimized. Two different multiobjective optimizations were performed, where the first was productivity−yield and the second was specific productivity− yield. Second, a model-based approach to simulate and optimize the MCSGP process was performed. Third, pooling was applied to reach a higher performance compared to taking the whole outflow as a product in the MCSGP process. In the batch process a multicomponent pooling strategy has been used where the products have been ranked in priority order with respect to product value.

∂qi

(1)

(2)

Yi =

(3)

where Pe is the Péclet number and dp is the mean particle radius. The rate of adsorption was described with the following relation: rads, i = −

∂qi 1 − ϵc ϵc + (1 − ϵc)ϵpKd, i ∂t

(7)

where L is the column length, μ is the viscosity of the mobile phase, ϵc is the column void fraction, vsup is the superficial velocity, dp is the particle diameter, and ρ is the density of the mobile phase. In this work, the pressure drop limit was set to 10 bar. Optimization. Multiobjective optimization aims to simultaneously maximize multiple objective functions Q(u) = [Q1(u), ..., Qni(u)] by tuning the decision variables (DVs), u. The following conditions were given for the optimizations: • The objective functions Q(u) are maximized. • The model equations (eqs 1−7) are solved. • The DVs are subject to lower and upper bounds that act as inequality constraints. • All inequality and equality constraints need to be fulfilled. When the objectives are incommensurate, there will be no single solution that is optimal with respect to all objectives simultaneously. The solution of the optimization problem will instead be a Pareto solution, with multiple equally optimal points, where no objective can improve without worsening another objective. The decision maker can then choose the most suitable point from the Pareto solution. Objective Functions. Three objective functions were studied in this work. The yield of a component is described as

d pvi 2Pe

(6)

150μ(1 − ϵc)2 vsup 1.75(1 − ϵc)ρvsup2 ΔP = + L ϵc 3d p2 ϵc 3d p

where F is the flow rate, R is the column radius, ϵc is the void in the column, ϵp is the particle porosity, and Kd is the exclusion factor. The axial dispersion was calculated with Dax, i =

(5)

where Hi° is the Henry constant coefficient. The partial differential equations were discretized with the method of lines to create a set of ordinary differential equations (ODEs) that was solved with ode15s in MATLAB. A flux limiter implementation17 was utilized to reduce the number of grid points in the axial direction while maintaining a low numerical dispersion. The maximum flow in a chromatography column can be calculated from the pressure drop limitation, ΔP, as given in Ergun’s equation for spherical particles:

where z is the axial coordinate along the column and Dax describes the axial dispersion for i ∈ {Nd, Sm, Eu, Gd, modifier}. The linear velocity, vi, is given by F 1 2 R π ϵc + (1 − ϵc)ϵpKd, i

j=1

Hi = Hi°s−βi

THEORY The Model. The column was simulated using a convective dispersive model where the concentration, c, in the column was described with

vi =

∑

⎞ qj ⎞ ⎟ − q⎟ i⎟ qmax, j ⎟⎠ ⎠

where nc is the number of REEs modeled, qmax is the column capacity, s is the concentration of the modifier, kkin governs the adsorption kinetics, β describes the ion-exchange characteristics, and Hi is the Henry constant. The Henry constant is dependent on the modifier concentration

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∂ci ∂ 2c ∂c = Dax 2i − vi i − rads, i ∂t ∂z ∂z

nc

1 VLc L, i

∫captured ci dt

for i ∈ {Sm, Eu, Gd} (8)

where VL is the load volume and cL,i is the load concentration. To get a combined yield for all products, a weighted yield was defined as Y=

∑ i = {Sm,Eu,Gd}

(4)

WY i i (9)

where Wi is a weight factor to compensate for the different prices and Y = 1 means 100% yield for all three products. The weighted yield can be seen as an economic value rather than as a yield because the weight factors reflect the commercial value

for i ∈ {Nd, Sm, Eu, Gd}, where the adsorbed concentration, q, is described by the Langmuir mobile phase modulator (MPM) isotherm:16 16486

dx.doi.org/10.1021/ie5023223 | Ind. Eng. Chem. Res. 2014, 53, 16485−16493


Article

of the products. The second objective was the productivity, and the last was the solvent productivity.18,19 Since the elements are valued differently from a commercial perspective, the products are weighted accordingly. The productivity was defined as

∑

Pr =

Wi

i = {Sm,Eu,Gd}

Puri =

captured

(10)

where Vcol is the total volumes of all columns and τ is the time for a complete cycle. The specific productivity was defined as

∑

Sp =

Wi

i = {Sm,Eu,Gd}

VLc L, iYi Vsolv

(11)

where Vsolv is the total concentration of solvents used. The Pareto solution can roughly be seen as a weighting of objectives that, for two objectives, can be written as a normalized earning objective:20−22 Q 12 = w

Q1 Q 1,max

+ (1 − w)

Q2 Q 2,max

(12)

where the Pareto solution is the set of solutions obtained from optimizations with the weighting factor, w, ranging from 0 to 1. The normalized earning will henceforth be used with Y as Q2 and Pr or Sp as Q1 depending on context. Setting w = 0 means maximum yield; w = 1 means maximum productivity. Decision Variables. The elution gradient, loading factor, flow rate, and cycle time were chosen as decision variables since they influence the same behaviors in both processes. A total of four DVs was used for the batch optimization, and nine were chosen for MCSGP since it is a more advanced process. The DVs are listed in Table 1 for the batch case and in Table 2 for the MCSGP case.

notation (unit)

lower limit

upper limit

loading factor gradient length low buffer strength elution buffer strength

θ (CV) Vgrad (CV) BA (mol m−3) BB (mol m−3)

0.01 10 1.0 1000

10 200 1000 20000

decision variable

notation (unit) sL (mol m−3) sg1

lower limit 1 0

45 0.3

gradient end buffer ratio

sg2

0.1

1.0

cycle time flow rate in s4 flow rate in s5 flow rate in s6 flow rate in s7 loading time ratio

τ (h) q4 q5 q6 q7 θL

1.6 0.7 0.3 0.7 0.3 0.1

11.2 1.0 0.99 1.0 0.99 1.0

(13)

MATERIALS AND METHODS Materials. An Agilent 1200 series HPLC system (Agilent Technologies, Waldbronn, Germany) was used together with a Kromasil H4 (Eka, Bohus, Sweden) column that is 25 cm long and has a diameter of 4.6 mm. The columns were delivered as is with a stationary phase of spherical silica particles coated with C18, a diameter of 16 μm, and a pore size of 100 Å. Bis(2ethylhexyl) phosphoric acid 38 (HDEHP) was used as a ligand due to its versatile ability to separate REE,23,24 and each column was filled with HDEHP (Sigma-Aldrich, St. Louis, USA) to a concentration of 342 mM. Nitric acid was used as modifier (eluent), and the modifier concentration gradient was varied between 6 and 13 vol % of 7 M acid. The length of the modifier gradient was set to 5 column volumes (CV) in order to avoid diluted product pool concentrations while still enabling sufficient separation. Each elution was followed by a regeneration step of 2.5 CV of 7 M nitric acid and an equilibration step of 2.5 CV of water. An inductively coupled plasma mass spectrometry (ICP-MS) system (Agilent Technologies, Tokyo, Japan) was used for in-line postcolumn REE detection due to its documented capability for this purpose.24,25 As can be seen in Figure 2, the simulated model used in this work was in good agreement with the batch experiments. The experiment shown in Figure 2 was performed with a REE composition similar to that of monazite ore26 with 4.6, 58.2, 12.0, 24.3, and 0.9 wt % neodymium, samarium, europium, gadolinium, and terbium, respectively. Simulation. The simulations of this work used the Preparative Chromatography Simulator, a tool developed in MATLAB at Lund University, for simulations of chromatographic separation.27 The parameters of the model, p = [qmax, β, kkin, H0]T, were based on the calibrations by Ojala et al.14 and are presented in the Supporting Information, Table S1. The kinetic constant, kkin, was modified to be inversely proportional to the liquid velocity.28 The parameters ϵc and ϵp were assumed to be 0.4 and 0.6, while Kd was set to 1. A constant Péclet number of 0.6 was used.

Table 2. Decision Variables for the MCSGP Case acid concn in load gradient start buffer ratio

captured

cj dt )−1

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Table 1. Decision Variables for the Batch Chromatography Case decision variable

j=1

∫V

where Vcaptured is the volume between the optimized pool cut points. Each optimization was performed on two levels: the simulation level where the chromatogram is simulated and the postprocessing level where the cut points are optimized to reach the purity requirement. Pooling. The optimization of the cut points is referred to as pooling and maximizes the objective while meeting the purity requirement. The best choice of objective corresponds to the weighting of the objectives in the multiobjective optimization (eq 12). In order to maximize the yield, a constant value of w = 0 was chosen for the pooling optimization. Genetic Algorithms. The optimizations were performed with the multiobjective genetic algorithm pSADE. The algorithm finds Pareto solutions, where one objective cannot be reduced without making another objective worse. The genetic algorithm creates an initial population where each individual has its unique set of the decision variable values. After the population has been evaluated, a new population is created by combining the best individuals of the population.

VLc L, iYi Vcolτ

∫V

nc

ci dt (∑

upper limit

Constraints. An optimization usually has some constraints that need to be fulfilled. The product purity requirement was used as a constraint since it is a central property in separation. For all components there was a product purity requirement of 99%. The purity requirement was calculated as 16487



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are found with a Nelder−Mead algorithm. The interpolation gives continuous and more accurate solutions. A cut priority order based on the element value was given for the components. The component with the highest priority got the first cut. Then the cut for the second component was made with what was left of the chromatogram after the first cut. The cut strategy continued until all components had received their cuts. A cut strategy where subfractions are taken from each outlet was applied for the MCSGP process. The strategy was favored over taking the whole outflow from the section due to an increased process purity flexibility. With the pooling, the desired purity can be chosen and this improves the process performance. The components were only allowed to elute at the section in which they were designed to elute. The pooling tool was called once for every component: sections s1−s4 for samarium, s6 for europium, and s8 for gadolinium. Neodymium was considered an impurity and was removed in steps s1−s4. Decision Variables. The optimization was performed for a number of preselected decision variables (DVs), with lower and upper bounds, given in Tables 1 and 2. Four DVs were chosen for the batch case. The first one was the loading factor, θ. The modifier gradient was affected by the other three DVs: the gradient length, Vgrad, the low buffer strength, BA, and the elution buffer strength, BB. The batch chromatography optimizations were run in a cycle, starting with the load followed by elution and cleaning in place. The simulation was performed and a pooling of the system to decide optimal cut points was done. The last cut point, when all target components had eluted, was considered as the stop time for the elution. Finally, a cleaning in place of 2 CV was added to calculate the total cycle time. There are many inflows of modifiers in the MCSGP process that affect the modifier concentration profile. To limit the number of DVs, only two variables for the modifier profile were chosen. The first was the initial concentration in the section s4 called sg1, and the second was the final concentration in the section s7 called sg2. Between these two points, the gradient was linearly interpolated. The variables sg1 and sg2 are ratios of the low and high concentration modifier (1 and 500 mM) buffers to obtain the desired concentration. In s8, the modifier concentration was set to the high modifier concentration buffer to be able to elute gadolinium, which is the strongest adsorbed element. In s1 and s3, the modifier concentration was assumed to have the lowest concentration. The modifier concentration in the load section, sL, was also chosen as a decision variable. Eight flow rates, one for each section, needed to be determined in MCSGP. For simplicity, most flows were chosen to be at maximum, Qmax. An assumption was that the flow rate of the last interconnected bed was the maximum allowed, so that

Figure 2. An overloaded experiment (solid line) together with simulations (dashed line). Some lines are shown in gray for an easier overview.

The MCSGP process was simulated until cyclic steady state was reached, meaning that the simulated concentration profiles were the same in two succeeding cycles. Parallel Computing Methodology. A computer cluster was constructed to provide an environment for distributing parallel simulations. The platforms supported by the cluster are python, MATLAB, and COMSOL. It consists of a server, clients, and working computers, communicating with files written at a shared RAM memory. The server is implemented as a script that handles the queue of jobs by distributing them to the next available computer. The client is a script that is utilized to distribute the working files. The working computers have 60 cores, composed of five 64-bit computer nodes (Intel Core2 Quad core with 2 GB RAM and running at 2.33 GHz), five 64-bit computer nodes (Intel Core i5 750 4 cores with 4.00 GB RAM and running at 2.67 GHz), and five 64-bit computer nodes (Intel Core i5-3450, 4 cores with 8 GB RAM and running at 3.10 GHz). Optimization. Four optimizations were performed with two different objective combinations on both the batch process and the MCSGP process. The objective combinations were production rate and yield (Y−Pr) and specific productivity and yield (Y−Sp). The purity requirement of 99% was implemented as an inequality constraint in the pooling of the chromatograms. No active equality constraints were set for the chosen decision variables. The same columns and maximum flow rate were used in the batch and twin-column MCSGP cases to secure a fair comparison. The weight factors were set to 0.26 for samarium, 0.65 for europium, and 0.09 for gadolinium, in order to resemble the components’ internal value distribution. Neodynium was not considered a product, and the weight was thus 0. The concentration of the load was 1.5 g L−1 for neodymium, europium, and gadolinium and 15 g L−1 for samarium. Pooling. A pooling algorithm was developed to find the optimal cut points of the chromatograms. The pooling is a suboptimization of each simulated chromatogram, and optimizes the target component with a purity requirement. The discrete points from the simulation are interpolated (monotone piecewise cubic interpolation), and the cut times

q1Q max + q5Q max = Q max

(14)

q3Q max + q7Q max = Q max

(15)

where qiQmax is the flow rate in section si. The flow rates q5 and q7 were chosen as decision variables, determining how much was recycled in the process. The higher the value, the more was recycled back into the process. When the values of q5 and q7 became close to Qmax, the values q1 and q3 became small. Small values of q1 and q3 mean that the recycled flows q5 and q7 cannot be diluted and the modifier concentrations will be the 16488



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Figure 3. Optimal Pareto solutions for the batch (−●−) and MCSGP (−□−) cases for Y−Pr and Y−Sp optimizations.

Figure 4. Simulations of the batch for the (a) best productivity in Y−Pr, (b) best yield in Y−Pr, (c) best solvent productivity in Y−Sp, and (d) best yield in Y−Sp for (−●−) Nd, (−□−) Sm, (−△−) Eu, (−○−) Gd, and the modifier (). Cut points for the pooling are seen as shaded peaks with dotted borders. Sm is scaled with 0.1.

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same as in the corresponding sections s1 and s3. This may be unwanted because the modifier concentration usually increases with the sections to elute harder adsorbing components. This shows the difference in complexity between the DVs in the MCSGP and the batch optimization. The intermediate flow ratios q4 and q6 were used as DVs as they control the section volume where Sm and Eu are eluted and thereby assure that the sections are cut optimally. The cycle time, τ, was used as a DV because it has a great impact on all of the studied objective functions. The last DV was the ratio between the load time and cycle time, θL.

RESULTS AND DISCUSSION

The optimal Pareto solutions for yield−solvent productivity (Y−Sp) and yield−productivity (Y−Pr) are shown in Figure 3, and the objective values for some operating points are summarized in Tables S2 and S3 in the Supporting Information. For the Y−Sp optimization, the Pareto solution for MCSGP has much higher values than the batch case in the whole range. This means that the solvent consumption is lower in the MCSGP case than the batch case, and this has a positive impact on downstream processing costs. The result was expected because MCSGP utilizes the solvent more efficiently through internal recycling. For the Y−Pr Pareto fronts it can be 16489



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Figure 5. Simulations of the two-column MCSGP for the (a) best productivity in Y−Pr, (b) best yield in Y−Pr, (c) best solvent productivity in Y− Sp, and (d) best yield in Y−Sp for (−●−) Nd, (−□−) Sm, (−△−) Eu, (−○−) Gd, and the modifier (). The chromatogram is built up by joining s1 to s8. Cut points for the pooling are seen as shaded peaks with dotted borders. Sm is scaled with 0.1.

Figure 6. Simulation of the two-column MCSGP for the Y90% Pareto point for the Y−Pr optimization with (−●−) Nd, (−□−) Sm, (−△−) Eu, (−○−) Gd, and the modifier (). The chromatogram is formed by joining s1 to s8. The x-axis is volume eluted. Cut points for the pooling are seen as shaded peaks with dotted borders. The modifier gradient consists of five sections denoted as g1−g5. Sm is scaled with 0.1.

seen that the Pareto solution for MCSGP also has higher values than the batch case in the whole range. The best yield should be approximately the same for both the Y−Sp and Y−Pr optimizations because pure yield is optimized for the Pareto point corresponding to w = 0. The best yields for the batch case are 98.8 and 99.0%, and those for MCSGP are 99.5 and 100.0%. At high yields, Pr and Sp drop very quickly for the batch process compared to the MCSGP process. For w = 0, Sp and Pr are 5.4 and 3.7 times higher for MCSGP than for batch. In the other end of the Pareto front, w = 1, Sp is 1.38 times higher for MCSGP compared to batch. Sp and Pr are roughly

the same for both batch and MCSGP, but MCSGP provides a much higher yield of 61.7% as opposed to 50.5% for batch. In Figure 4 the simulated extreme point chromatograms of the batch process Pareto fronts are shown. The best yields (in Figure 4b,d) show similar chromatogram profiles. The elution step is long to allow for good separation with fulfilled purity requirements. The loading factors are low with smaller peaks as a result. The highest values of solvent productivity in Figure 4 a and productivity in Figure 4c show similar chromatogram profiles with a shorter cycle time. The loading factors are higher in these simulations, which results in higher peaks. However, the peaks are not baseline separated, which leads to a loss of yield. 16490



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of it. The pooling also results in higher product concentration, which reduces the cost for downstream processing steps. The DVs in the MCSGP case were chosen to correspond to the batch DVs. However, the twin-column MCSGP needed to have some extra DVs because it is a more complex process. In batch chromatography, it is optimal to run at maximum flow rate in the column, whereas in MCSGP, there are eight different sections with different flow rates that need to be optimized. The modifier gradient is described with three DVs in the batch case: Vgrad, BA, and BB. In the MCSGP case, it is very important to control the modifier gradient so that every component flows out from the correct section. In the MCSGP case, three modifier DVs directly affects the gradient, sL, sg1 and sg2, and the other DVs affect it indirectly. For example, the recycle parameters q5 and q7 affect the penetration of the high concentration sections s5 and s7 into the low modifier concentration sections s1 and s3. The θL was chosen to correspond to the load volume, VL, in the batch case.

The extreme points of the Pareto fronts for MCSGP are shown in Figure 5. The chromatograms of the MCSGPs are similar to the corresponding batch chromatograms. At a first glance it looks like a lot of yield is lost because nothing is pooled from sections s5 and s7. However, these are the outflows in the interconnected beds and are recycled to sections s1 and s3. The highest yields in Y−Pr and Y−Sp are shown in parts b and d, respectively, of Figure 5. The cycle time has high values, and the recycle parameters q5 and q7 are high. This means that the flow rates q1 and q3 have low values and cannot dilute the high modifier concentration from sections s5 and s7. This is visible as a plateau of the modifier between sections s3 and s4 in Figure 5b,d. The load factors are low to avoid overloading, and this results in lower peak heights. In Figure 5a,c the load factors are high and the cycle times are kept short to increase productivity. Section s6 is pooled, contrary to the maximum yield chromatograms where the whole section is used. The pooling is required to reach the desired purity and corresponds to most of the yield loss. The extreme points of the Pareto solutions may not be optimal as operating points because the competing objective is completely discriminated. A reasonable choice of operating point could be at 90% yield (denoted Y90%), which results in a solvent productivity of 0.16 kg m−3 for batch and 0.30 kg m−3 for MCSGP. The productivity at 90% yield was 0.27 kg m−3 h−1 for batch and 0.39 kg m−3 h−1 for MCSGP. Both these cases rule heavily in favor of MCSGP. The simulation of the Y90% point for the Y−Pr optimization can be seen in Figure 6. The shaded peak areas in s4, s6, and s8 are the collected fractions. Note that sections s5 and s7 cannot be collected, but are not lost because they are recycled to s1 and s3. The values of the scaled flows q4, q5, q6, and q7 are 0.87, 0.89, 0.74, 0.66 and correspond to the section widths in Figure 6. The cycle time was 35 min, and the load fraction of the cycle, θL, was 0.39. The modifier profile consists of five parts, marked with g1, g2, g3, g4, and g5 in Figure 6. Gradient g1 has the highest concentration due to the high inlet concentration for section s8. The volume added during one step shift is just slightly higher than the column volume and causes the profile to be shifted and eluted in the next section instead. The g2 part stems from the mixing of the inflow of s1 and the connected s5 section. The g3 part is the concentration in the load and is determined by sL that was 29 mol m−3 for this simulation. The g4 part is high because of the high concentration in s7 that is connected to s3. Finally, the g5 part is decided by setting the left and right concentrations with sg1 and sg2 in the optimizations; these were 0.30 and 0.17 in the simulation. All objective functions are weighted sums of the three products. For the Y90% Pareto point in the Y−Pr optimizations for batch and MCSGP, yields and productivities for the individual products are shown in the Supporting Information, Table S4. Sm has the highest yield and Eu has the lowest yield for both MCSGP and batch. This is in line with the expectation of that Eu should be difficult to separate since it is the middle eluting component. Samarium has the highest productivity for both batch and MCSGP, which is expected since it is 10 times more abundant in the load than the other products. The pooling of the outlets is apparent in all shown MCSGP chromatograms for all pools of samarium and gadolinium. This is also true for all but two chromatograms for europium, where the whole sections are used. The pooling is most important for samarium, because the impurity neodynium is eluting in front

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CONCLUSIONS For the separation of neodymium, samarium, europium, and gadolinium, the MCSGP process shows better performance than the batch process for all Pareto solutions, regardless of whether the Y−Sp or Y−Pr objectives are optimized. The efficient solvent utilization for MCSGP makes it preferable since it will improve the production cost. The pooling of each section in the MCSGP process clearly shows that the optimal cuts are not equal to the whole outlet and adds an extra degree of freedom that improves the optimal solution. The twin-column setup is faster to start up compared to the previous six-column setup and is also easier to get synchronized. These results show that MCSGP could be a good alternative compared to batch, when high purity is desired and solvent consumption is expensive.

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ASSOCIATED CONTENT

S Supporting Information *

Table S1, calibrated model parameters values used in the simulations; Table S2, objective values [Y, Sp] for Y−Sp optimizations; Table S3, objective values [Y, Pr] for Y−Pr optimizations; Table S4, individual yields and productivities for Y90% in Y−Pr optimizations for batch and MCSGP. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support of the Swedish Foundation for Strategic Research is gratefully acknowledged. NOMENCLATURE

Latin Symbols

BA = low buffer strength (mol m−3) BB = elution buffer strength (mol m−3) c = mobile phase concentration (mol m−3)

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cL = load concentration (mol m−3) Dax = axial dispersion (m2 s−1) dp = particle diameter (m) F = flow rate (m3 s−1) H° = Henry constant coefficient ((mol m−3)β) H = Henry constant kkin = kinetic constant (s−1) Kd = exclusion factor L = column length (m) nc = number of components np = number of points Pe = Péclet number Pr = productivity (kg m−3 h−1) q = concentration on the surface of the stationary phase (mol m−3 gel) q = flow rate fraction of Qmax in section si Q = objective qmax = column capacity (mol m−3 gel) R = column radius (m) sg1 = gradient start buffer ratio sg2 = gradient end buffer ratio sL = modifier concentration in the load (mol m−3) Sp = solvent productivity (kg m−3) t = time (s) u = decision variables v = linear velocity (m s−1) Vgrad = gradient length (CV) VL = load volume (m3) w = weighting factor between two objectives W = weighting factor for a component in the objective functions Y = yield (%) z = axial coordinate of the column (m)

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Greek Symbols

β = parameter describing the ion-exchange characteristics ϵc = void in the column ϵp = porosity of the particles ρ = mobile phase density (kg m−3) θ = loading factor (CV) τ = cycle time (h) θL = loading time ratio of a cycle Index

i = component

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REFERENCES

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length and the linear velocity of the mobile phase. J. Chromatogr. A 1999, 831 (1), 17.

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