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Rare Earth Elements are in growing demand globally. This paper presents a case study of applied mathematical modeling and multi objective optimization to optimize the separation of heavy Rare Earth Elements, Terbium-Lutetium, by means of preparative solid phase extraction chromatography, which means that an extraction ligand, HDEHP, is immobilized on a C18 silica phase, and nitric acid is used as an eluent. An ICP-MS was used for online detection of the Rare Earths. A methodology for calibration and optimization is presented, and applied to an industrially relevant mixture. Results show that Thulium is produced at 99% purity, with a productivity of 0.2 - 0.5 kg Tm per m
^{3} stationary phase and second, with Yields from 74% to 99%.

The Rare Earth Elements (REEs) are Scandium and Yttrium and the members of the Lanthanide group in the periodic table. Many of these metals are used in batteries, lasers, capacitors, superconductors [

The REEs are of growing economic importance, with China limiting the export quotas [

This work is a continuation of the work presented in [

To design and optimize the chromatographic system, a performance function is required. This in turn depends on a mathematical formulation of the physical chromatography system. This mathematical formulation requires retention parameters, mass transfer parameters and column capacity to be calibrated. To reduce the complexity of the calibration, only the retention parameters are calibrated with a less computationally expensive model, i.e. the retention volume model. These retention parameters are then used as a starting guess in the more advanced model of the chromatography column, which takes dispersion and mass transfer effects into account as well as the retention data. For the optimization, Productivity and Yield were selected as performance functions, subject to a Purity constraint.

The retention volume model is an ideal model, which can be seen as a simplified version of the film mass transfer model, presented in 2.2, that includes the isocratic modifier concentration effect on the retention volume:

where V_{R} is the retention volume, V_{R}_{0} is the retention volume under non-binding conditions, ε_{C} is the column porosity, ε_{P} is the particle porosity, K_{D} is the exclusion factor and H is the Henry constant [

where H_{0} is the Henry constant when the modifier concentration is unity, s_{modifier} is the modifier concentration (mol/m^{3}), ν is the stochiometric coefficient. This expression is valid only when the loaded amount is much smaller than the total adsorption capacity.

The model including mass transfer and non-linear effects is a Dispersive-Equilibrium model using Langmuir type binding with mobile phase modifier, consisting of a dispersive-equilibrium description of the mobile phase with all mass transfer effects lumped in the dispersion, and Langmuir equilibrium adsorption for the solid phase [

where c_{i} is the concentration of component i in the mobile phase (mol/m^{3}), D_{AX} is the apparent dispersion coefficient (m^{2}/s) see Equations (6)-(8). x is the axial coordinate along the column (m), v_{app} is the apparent velocity, which is the superficial velocity divided by the total void fraction (m/s), ε_{T} is the void fraction of the packed bed (m^{3} mobile phase/m^{3} column), q_{i} is the concentration of adsorbed species i (mol/m^{3} column) and t is the time (s).

The column equation is subject to two boundary conditions, one at the inlet and one at the outlet. A Robin boundary condition describes the concentration at the inlet:

where c_{inlet}_{,i} is the concentration of component i at the inlet. The concentration of species i at x = 0 may be lower than the inlet concentration due to dispersion. The other boundary condition is at the outlet of the column, at x = L, and is a von Neumann condition:

The concentration of species i adsorbed on the stationary phase is described by the following equation, which is a mobile phase modified version of the Langmuir adsorption description [

where H_{0} is the Henry constant, s is the modifier concentration, ν is the stochiometric constant, q_{max,i} is the maximum adsorption capacity for component i. This equation must at equilibrium satisfy the adsorption equilibrium isotherm, which means that the parenthesis must be 0 at equilibrium.

The flow rate dependence of the apparent dispersion, including mass transfer resistance and kinetics, is approximated with a second degree polynomial. This comes from the Knox equation [

where h is the reduced plate height, v_{app} is the apparent velocity, d_{p} is the particle diameter. A, B and C are constants coming from flow structure and non equilibrium effects. The apparent axial dispersion can then be described as:

where D_{AX} is the apparent dispersion coefficient. Combining equations 7 and 8 give:

Estimating the apparent dispersion with a second degree polynomial gives a small error in the second term, however this is acceptable with current model precision.

The target component for the optimization is Thulium, as producing pure Thulium would produce pure Ytterbium also, and then both of the major components have been purified, see

where t_{cut}_{,1} and t_{cut}_{,2} are the times between which the target component can be collected at required purity (s), F is the flow rate (m^{3}/s), c_{outlet} is the outlet concentration of the component (mol/m^{3}), t_{cycle} is the total cycle time (s) and V_{col} is the column volume (m^{3}). The yield can be described as:

In this case t_{load} is the time at the end of the loading(s), c_{feed} is the concentration in the feed (mol/m^{3}).

When two competing objectives, in this case productivity and yield are competing, the result will be a Pareto front, which means that for a given optimal point on the front, increasing one objective must decrease the other [

The experiments were performed on an Agilent 1260 Bio-Inert HPLC system, consisting of two Agilent 1260 Bio-Inert Quaternary pumps, one Agilent Bio-Inert 1260 Autosampler and one Agilent 1260 Bio-Inert UV/Vis detector connected to an Agilent 7700 ICP-MS for detection of the rare earths. The column preparation was performed using a GE Healthcare ÄKTA Purifier 100. The individual rare earths were acquired from Merck at 1 g/L in nitric acid. The concentrated nitric acid of HPLC grade was acquired from Merck. The columns used were Kromasil C18, 100 Å, 16µm columns with immobilized HDEHP acting as ligand. HDEHP was acquired from Merck. Methanol was acquired from Merck. To be able to do higher flow rate experiments, an extra isocratic pump was added to the system after the column, and used to feed the ICP-MS with a constant flow of 1 ml/min, the rest going to waste or collection. The heavy mixture of rare earths used in the Thulium optimization was mixed to mimic an industrially relevant REE-stream, with a composition as described in

The column was modified by passing a mixture of 22 mM HDEHP in 65% (v/v) Methanol-water, using the conductivity meter of the ÄKTA to determine the breakthrough volume, and thereby the ligand concentration of 345 mol/m^{3} column volume. As the ligand forms dimers, the concentration of sites was 172.5 mol/m^{3} column volume.

% (wt) | |
---|---|

Y | 0.67 |

La | 1.2 |

Ce | 0.1 |

Pr | 0.09 |

Nd | 0.13 |

Sm | 0.42 |

Eu | 0.06 |

Gd | 1.59 |

Tb | 0.13 |

Dy | 0.74 |

Ho | 4.23 |

Er | 6.54 |

Tm | 11.99 |

Yb | 71.89 |

Lu | 0.24 |

Isocratic experiments at 16 compositions of the mobile phase were performed to get retention volumes for all components, gradient experiments at different flow rates were performed to get peak broadening from mass transfer and kinetics.

16 isocratic experiments, 0.35 to 4.9 M nitric acid were performed, with 3 replicates on all experimental points. The experiments were performed at 0.35, 0.53, 0.56, 0.60, 0.63, 0.67, 0.70, 1.05, 1.40, 1.75, 2.10, 2.8, 3.50, 4.20 and 4.90 M nitric acid concentration, using the quaternary pump to create the isocratic eluent.

No overloading experiments were performed, as the column capacity had been decided during the preparation of the column.

The chromatographic experiments performed, consisted of four consecutive steps; loading, isocratic elution, Cleaning in Place (CIP), and reequilibration.

The injected amount during loading was 10 µl of a mixture containing 0.0357 g/L of each of the lanthanides, plus yttrium. This was well within the linear range of the isotherm, so any competitive effects could be neg- lected. After loading, a prepared eluent of a predetermined concentration was passed through the column for 120 minutes, this was due to a limitation of the ICP-MS software, after elution a CIP at 7M nitric acid was performed. After the CIP the system was reequilibrated with water. All experiments were performed with three replicates.

For the validation experiment, the experimental setup was modified so that it no longer had the post column pump. This decreased the post column dispersion. The validation experiment was adjusted somewhat from the very aggressive points on the Pareto front, to a longer gradient and with a lower flow rate. The validation experiment consisted of five consecutive steps: loading, wash, gradient elution, CIP and reequilibration. The injected amount during loading was 200 µl heavy mixture, diluted to 33.6 g/L with deionized water, then a 4.5 ml wash with deionized water, followed by a 17 ml gradient from 0 to 7 M nitric acid. Then 4 ml CIP at 7 M nitric acid, followed by 10 ml reequilibration with deionized water.

The chromatography model was discretized in the axial direction, to turn the partial differential equation into a system of Ordinary Differential Equations (ODE) and Algebraic equations, Differential Algebraic Equation (DAE). The discretization used a 2 point backward finite difference scheme for the flow, and a 3 point central finite difference scheme for the dispersion. The assembled DAE system was solved with MATLAB ODE/DAE solver ode 15 s, which solves stiff DAEs. Discretization and Simulation is handled by the Preparative Chromatography Simulator, PCS, developed at the Department of Chemical Engineering, Lund University [

First the Henry constant and the stochiometric coefficient were calculated from the slope and intercept of the logarithm of the modified retention volume, versus the logarithm of the modifier concentration. These were then used for the inverse method, i.e. minimizing the least squares error for the normalized detector response and the normalized simulated chromatograms, where the mass transfer effects were fitted, using a Nelder-Mead simplex algorithm. The maximum binding capacity of the columns was decided during preparation of the columns, and based on the HDEHP concentration in the backbone.

The calibrated model was used with the performance functions presented in 2.3, and used with a modified Differential Evolution (DE) algorithm and parallel computations in a cluster environment [

The modified columns had some known characteristics, such as the porosities, ε_{c} = 0.6, ε_{p}= 0.4, and the maximum binding capacities, which were decided during the impregnation of the column, and calculated from equation 12:

where λ is 172.5 (mol/m^{3} column), and ν_{i} is the stochiometric constant for component i.

The Henry constants and characteristic charges, presented in

Component | ν | H | q_{max} |
---|---|---|---|

Yttrium | 2.18 | 22.96 | 79 |

Terbium | 2.20 | 21.07 | 78 |

Dysprosium | 1.96 | 20.24 | 88 |

Holmium | 2.12 | 21.89 | 81 |

Erbium | 2.34 | 24.49 | 73 |

Thulium | 2.19 | 24.29 | 78 |

Ytterbium | 2.25 | 25.60 | 77 |

Lutetium | 2.64 | 29.47 | 65 |

To calibrate the mass transfer effects, flow rate experiments were performed at 1, 1.25, 1.5, 1.75, 2, 2.5 and 3 ml/min, the normalized detector response was used against a normalized simulated response from the PCS, and parameters fitted by a least squares approach. The experimental and model results are presented in ^{−8}, B = 1.00 ´ 10^{−8} and C = 0.9936 ´ 10^{−8}.

The multi-objective optimization was performed using Productivity and Yield, see Equations (10) and (11), as objective functions. The optimization output is a the Pareto front, presented in

performance for different operating points, comparing in this case Productivity and Yield, under a 99% Purity constraint for the Thulium.

All of the operating points are under overloaded conditions, as can be seen in

As described in 3.2.2, the validation experiment was performed at a slightly less aggressive operating point than max (Prod * Yield), with a longer gradient and a flow rate of 1 ml/min. The chromatogram from the validation experiment, presented in

Prod (kg/m^{3}・s) | Yield (%wt) | Load (ml) | Gradient length (ml) | Flow rate (ml/min) | |
---|---|---|---|---|---|

Chromatogram 1 | 0.181 | 98.5 | 0.0684 | 7.70 | 1.26 |

Chromatogram 2 | 0.263 | 95.1 | 0.0672 | 5.17 | 1.55 |

Chromatogram 3 | 0.396 | 84.5 | 0.110 | 5.62 | 1.68 |

Chromatogram 4 | 0.447 | 73.3 | 0.140 | 5.66 | 1.73 |

chromatogram has a calculated Productivity of 0.20 kg Thulium per cubic meter of column volume and second, with a Yield above 95%. The reason that the peaks are a lot sharper in the validation chromatogram than the simulated peaks, is that the post column pump was removed, which leads to much less post column dispersion.

Using the inverse method to fit retention data and kinetics at the same time for a system with 15 components is very hard, for this reason, performing isocratic experiments to determine the isotherm parameters, and using the inverse method for the mass transfer effects proved to be an easier path to follow. The model predicts preparative loads under gradient elution well, and therefore is applicable in optimizations of the separation system, for a wide range of mixtures and performance criterion. The performance of the system showed a Productivity range for Thulium from 0.18 kg/(m^{3}∙s) to 0.45 kg/(m^{3}∙s) at Yields of 98.5% to 73.3%.