Gibbs energy minimization model for solvent extraction with

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Gibbs energy minimization model for solvent extraction with application to rare earths recovery Chukwunwike Iloeje, Carlos Jove-Colon, Joe Cresko, and Diane J. Graziano Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.9b01718 • Publication Date (Web): 03 Jun 2019 Downloaded from http://pubs.acs.org on June 4, 2019

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Environmental Science & Technology

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Gibbs energy minimization model for solvent

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extraction with application to rare earths recovery

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Chukwunwike O. Iloeje,*a Carlos. F. Jové Colón,b Joe Cresko,c Diane J. Grazianod

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aEnergy

Systems Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont IL

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60439-6903. bSandia

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cAdvanced

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National Laboratory, 1515 Eubank SE, Albuquerque, NM 87123.

Manufacturing Office, US Department of Energy, 1000 Independence Ave SW, Washington DC 20585

dDecision

and Infrastructure Sciences Division, Argonne National Laboratory, 9700 South Cass

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Avenue, Lemont, IL 60439-6903, US.

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KEYWORDS

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Rare earth recovery, critical materials recycling, solvent extraction, Gibbs energy minimization,

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process optimization, thermodynamic parameter fitting, chemical equilibrium

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ACS Paragon Plus Environment

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ABSTRACT

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The emergence of technologies in which rare earth elements (REEs) provide critical functionality

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has increased the demand for these materials, with important implications for supply security.

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Recycling provides an option for mitigating supply risk and creating economic value from the

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resale of recovered materials. While solvent extraction is a proven technology for rare earth

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recovery and separation, its application often requires extensive trial-and-error experimentation to

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estimate parameter values and determine experimental design configurations. We describe a

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modelling strategy based on Gibbs energy minimization that incorporates parameter estimation for

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required thermodynamic properties as well as process design for solvent extraction, and illustrate

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its applicability to REEs separation. Visualization analysis during parameter estimation revealed

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a linear relationship between the standard enthalpies of the extractant and respective organo-metal

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complexes, analogous to the additivity principle for predicting molar volumes of organic

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compounds. Establishing this relationship reduced the size of the parameter estimation problem


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and yielded good agreement between model predictions and reported equilibrium extraction data,

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validating the property estimates for the organic phase species. Design exploration and

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optimization results map out the space of feasible solvent extraction column configurations, and

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identify the set of optimal design parameter values that meet recovery and purity targets.

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1. Introduction

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Rare earth elements (REEs) have unique magnetic, catalytic, and phosphorescent properties that

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significantly improve performance of a wide range of technologies.1 These technologies span

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aerospace, clean energy, telecommunications, electronics, transportation, defence, and other

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diverse applications. For these applications, certain REEs are considered to be critical materials2–

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7

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REEs provides an option for closing the material flow loop, diversifying supply sources, and

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creating economic value from the resale of recovered material. Closing the material loop will

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conserve the primary resource;8–11 diversifying sources will insulate the supply chain from

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disruptions;12,13 and value recovery will benefit the local manufacturing industry.12,14

with important implications for supply security, economics and disposal management. Recycling

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Solvent-based extraction technologies are widely considered suitable for recovering high-purity

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fractions of REEs.15 These liquid-liquid extraction systems take advantage of the relative ability

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of solutes to distribute between immiscible aqueous and organic phases in equilibrium. The

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aqueous solvent promotes dissociation of ionic species while the organic phase contains a

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complex-forming extractant that combines with the target ionic species to form organo-metal

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complexes, with high solubility in the organic phase – and, conversely, negligible solubility in the

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aqueous phase.15,16 Depending on the type of extractant, this complex formation may involve

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solvation, anion exchange, or cation exchange. In solvation, the extractant displaces water

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molecules from the coordination sphere of a metal complex; in anion exchange, an anionic metal


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complex from the aqueous phase displaces an anion from the organic extractant; in cation

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exchange, the metal cation displaces a proton from the organic extractant.16 The 2-ethylhexyl

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hydrogen 2-ethylhexyl phosphonate (HEH-EHP, commonly known as PC88A) extractant

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considered in this study belongs to the cationic group, and is considered effective for REE

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recovery.15 Eqn. 1 illustrates the overall cationic exchange reaction for hydrometallurgical

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extraction where 𝑀𝑛 + represents the cation, (𝐻𝐴)2, the 𝑃𝐶88𝐴 extractant, and 𝑀(𝐻𝐴2)𝑛 , the

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organo-metal complex. 𝑒𝑞

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𝑀𝑛 + + 𝑛(𝐻𝐴)2,

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In practice, the physical and chemical similarities of REE mixtures make extraction difficult,

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needing multiple separation stages to meet design targets.12,17 The actual design of the separation

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process often requires extensive trial-and-error experimentation to characterize the extraction

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chemistry and to determine process configurations that meet separation and recovery targets. These

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factors motivate theoretical modelling strategies to predict extraction equilibria and identify

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optimal process configurations. Approaches for modelling the thermodynamics of liquid–liquid

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equilibrium in solvent extraction systems build on the theoretical approach proposed by Gibbs,18

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but differs on the choice of physical quantity and solution algorithms.19,20 Gibbs energy

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minimization (GEM) computes liquid–liquid equilibrium by determining the system’s minimum

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Gibbs free energy with respect to the amount of each species in each phase. Unlike equilibrium

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constant-based approaches, it does not require knowledge of the actual solubility and speciation

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reactions, but instead uses thermodynamic information about the chemical species composing in

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each phase at equilibrium.21 Yet it has seen limited use in REE solvent extraction literature,

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because, in many cases, essential standard thermodynamic properties of key organic phase species

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are not available in the open scientific literature.

𝑜𝑟𝑔

𝑀(𝐻𝐴2)𝑛,𝑜𝑟𝑔 + 𝑛𝐻 +


(1)

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In this study, we implement an approach – based on GEM – for estimating missing standard

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thermodynamic properties of organic phase species and predicting extraction equilibria in

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hydrometallurgical systems. This GEM approach – building on earlier work by Jové Colón et al.21

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– reduces the requirement for extensive regression of experimental data to retrieve parameters. We

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illustrate its applicability to the design, analysis, and optimization of REE separations, with

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emphasis on mixture compositions relevant to recycling Nickel Metal Hydride (NiMH) batteries.

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The first part of this paper describes the Gibbs minimization and standard thermodynamic property

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estimation method while the second part illustrates an application to rare earths separation by

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solvent extraction, covering multi-parameter sweep studies for process design exploration, and

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optimization analysis.

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2. Methods

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2.1. Gibbs Energy Minimization

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The minimum of Gibbs free energy with respect to the molar quantities of each species in each

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phase represents the equilibrium criterion for a multiphase, multicomponent system under the

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constraints of fixed temperature and pressure.19,22 For such a system, the Gibbs minimization

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problem takes the following form:

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min 𝐺 = ∑𝛼∑𝑖𝑛𝑝𝑖𝐺𝑝𝑖

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𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜.

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𝐺𝑖 = 𝐺0𝑖 + 𝑅𝑇𝑙𝑛(𝑎𝑖)

(3)

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𝑎𝑖 = 𝛾𝑖𝑚𝑖

(4)

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∑𝑖𝑛𝑝𝑖 = 𝑛𝑖

(5)

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∑𝑖𝑐𝑗,𝑖 = 𝑑𝑗

(6)

(2)


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𝑛𝑝𝑖 ≥ 0

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∑𝑖𝑛𝑖𝑧𝑖 = 0

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𝑖 ∈ {1, 2, …, 𝑁}; 𝑗 ∈ {1,2, …, 𝑀}

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𝑛𝑝𝑖 and 𝐺𝑝𝑖 represent the number of moles and partial molar Gibbs energy of species 𝑖 in phase 𝑝,

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𝐺0𝑖 is the molar Gibbs free energy of species 𝑖 in its standard state, ai is the activity, a measure of

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the effective concentration of species 𝑖 that accounts for any deviations from ideality, 𝛾𝑖 is the

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activity coefficient, 𝑚𝑖 represents molality, the number of moles of solute per kg of solvent, and

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𝑧𝑖 is the charge of species 𝑖 in the aqueous phase. Eqn. 4 defines activity as the product of the

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activity coefficient and molality. Eqn. 5 satisfies species balance, where 𝑛𝑖 is the total number of

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moles of species 𝑖 in all phases. Eqn. 6 ensures elemental balance, where 𝑑𝑗 is the total number of

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moles of element 𝑗 across all species and 𝑐𝑗,𝑖 represents the number of atoms of element 𝑗 per

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molecule of species 𝑖. Eqn. 7 satisfies the non-negativity criterion while equation 8 represents

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charge balance. Solving Eqn. 2–8 predicts the species composition and phase distribution at

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chemical equilibrium. To solve, we first define the appropriate thermodynamic models for the

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electrolyte and organic phases.

(7) (8)

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2.2. Electrolyte phase chemistry

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The activity coefficient model for the electrolyte phase is based on the Pitzer formulation for the

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excess Gibbs free energy in ionic solutions.23 The Pitzer formulation represents the non-ideal

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behaviour by accounting for short- and long-range interactions between ions in solution. In this

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study, the Pitzer model is used to represent the HCl electrolyte interactions, with electrolyte

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parameters refitted from Holmes et al.24 The ionic standard state for the aqueous solute is a

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hypothetical one molal solution referenced to infinite dilution25. Standard partial molal

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thermodynamic properties of the aqueous species in the electrolyte are computed using the


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Helgeson-Kirkham-Flowers equation of state (EoS)26–29. This EoS provides the standard partial

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molal property values as a function of pressure and temperature, accounting for solvation effects

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that arise from ion–solvent interactions. Its expressions for the electrostatic contribution derive

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from Born equations, while the structure effects are determined from semi-empirical expressions

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of the form in Eqn. 9, regressed from experimental data.30

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∆𝐸0𝑖 = 𝑓(𝛼𝑖, 𝑃,𝑇)

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Here, 𝐸 represents some partial molar quantity (volume, compressibility or specific heat); P and

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T are pressure and temperature, respectively, and 𝛼𝑖 represents the regressed virial coefficients.

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These coefficients have been documented for several electrolyte and ionic species by Shock et

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al.30,31 We provided the coefficient values for the ionic species as inputs to the electrolyte phase

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model defined in Cantera, an open source suite of object-oriented, constitutive modelling tools and

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property databases, used for equilibrium calculations.25 (See supplemental material section S0-A

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for the input data file specifications. For further information on defining the phase model in

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Cantera, see Moffat and Jové Colón 25). We validate the electrolyte model by predicting water

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activity and osmotic pressure for 𝐻𝐶𝑙 ― 𝐻2𝑂 systems and comparing these predictions with data

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from Hamer and Wu.32 Fig. 1 shows good agreement for the molality range of interest.

(9)

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Fig. 1 The electrolyte model validation. Predictions for activity and osmotic coefficients match

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data from Hamer and Wu. Deviations become significant as molality increases beyond 8 m, but

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the region of interest for this study is below 4 m, where the fit is very good.

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2.3. Organic phase chemistry

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The organic phase comprises the organic extractant (HA)2, an inert diluent, and the organo-metal

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complexes with the stoichiometry M(HA2)3. The corresponding solute and solvent standard states

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are the pure solute and pure organic solvent, respectively. We model this phase as an ideal mixture,

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which removes the complexity of predicting activities for the organo-metal complexes (𝛾𝑖 = 1 in

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Eqn. 4). This assumes that by representing the organic complex association equilibria, the mixing

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model indirectly captures any non-ideal behaviour.21,33–35 The ideal mixing model requires the

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standard Gibbs energy values (𝐺0𝑖 ) as input. However, the standard Gibbs energy for the extractant

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and organo-metal species are not readily available in open literature, so we estimate them by

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regression to reported equilibrium data as described in the following section.

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2.4. Standard Gibbs parameter estimation

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The standard Gibbs parameter estimation for the organic-phase species involves finding the

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minimum on a multidimensional residual function surface defined by the least squares difference

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of predicted and experimental isotherms. Eqn. 10 expresses the functional form of the least squares

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objective. 𝑁

𝑁

2

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min ∑𝑖 𝑑𝑎𝑡𝑎∑𝑗 𝑅𝐸(𝑓(∆𝐺0𝑘, ∆𝐺0𝑗 , 𝜑𝑖,𝑗,𝑘) ― 𝑌𝑒𝑥𝑝 𝑖,𝑗 )

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Here, 𝑖 refers to experimental data points, 𝑗 to the set of organo-metal species, and 𝑘 to the

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organic extractant. ∆𝐺0 = ∆𝐻0 ―𝑇∆𝑆0 is the vector of species Gibbs energy, expressed as a

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function of enthalpy (𝐻) and entropy (𝑆). 𝜑𝑖,𝑗,𝑘 includes all other input parameters like molar

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volumes, temperature, pressure, as well as the vector of experimental feed composition data. 𝑌𝑒𝑥𝑝

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is any set of experimentally determined equilibrium isotherms and 𝑓 represents the corresponding

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model predictions determined by solving the GEM problem (Eqn. 2-8). A typical equilibrium

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isotherm is the distribution ratio, 𝐷 - a ratio of the amount of solute in the organic phase to the

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amount left in the aqueous phase at equilibrium. For this problem, we formulate the residual

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function with 𝑙𝑜𝑔(𝐷) because the log-function improved gradient scaling, with 𝐷 defined as: 𝑛𝑗,𝑜𝑟𝑔 + 𝜀

(10)

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𝐷 = 𝑓(𝐷,𝜀) =

(11)

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𝑛𝑗,𝑜𝑟𝑔 and 𝑛𝑗,𝑎𝑞 are the organic and aqueous phase molar amounts of species j at equilibrium and

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𝜀 is a very small value used for numerical smoothening to avoid division-by-zero exceptions. Fig.

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2 illustrates the parameter-estimation process. It starts with an initial guess for the standard molar

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Gibbs energy for the organic species, which we provide indirectly as the standard entropy (𝑆0) and

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the standard enthalpy of formation (∆𝐻0𝑓). We assume a fixed dummy value for the absolute

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entropy data for the organic species (approximated from the correlation by Glasser et al.36 ). In this

𝑛𝑗,𝑎𝑞 + 𝜀


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way, ∆𝐻0 replaces ∆𝐺0 as the explicit model variable. The extractant standard molar enthalpy (∆𝐻𝑘

181

) was initialized with a default value (for our case, we used that of tri-butyl phosphate,21) while

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those for the organo-metal complexes were initialized as 𝑛∆𝐻𝑘, where 𝑛 is the ratio of the

183

extractant-to-metal mole proportion in the complex from Eqn. 1. The extractant molar volume

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was obtained from vendor data37. Those for the organo-metal complex were initially set at three-

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times this value, but later refined by refitting. The refitting results showed that equilibrium

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predictions were largely insensitive to the specified organo-complex molar volume.

187

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Fig. 2 Thermodynamic parameter estimation logic.

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The prediction step follows, where, for each experimental data point, the model minimizes the

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Gibbs energy of the system to predict multiphase equilibrium composition. Then, it uses the result

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to minimize the least-squares function of Eqn. 11. If the problem converges, the process exits with

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the current standard molar property values; otherwise, it proceeds to an update step and repeats the


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cycle. The modelling framework depicted in Fig. 2 was programmed in python and used the

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Cantera implementation of the Villars-Cruise-Smith (VCS) algorithm20,38.

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3. Results

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3.1. Standard property estimation

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We applied the GEM approach to the separation of REE mixtures relevant to recycling (NiMH)

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batteries (Neodymium, Nd; Praseodymium, Pr; Cerium, Ce; and Lanthanum, La). For this study,

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we collected a set of single-component and multi-component REE extraction equilibrium isotherm

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data from the literature,

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values, extractant fractions, and rare-earth concentrations (see supplemental material S0-B for data

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sources). Having set up the problem, we first use visualization experiments to gain insight on the

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least squares solution space, given the chemistry implicit in the experimental datasets. To this end,

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we generate contour plots of the least squares residual function for individual 𝑀𝑅𝐸𝐸

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―𝐻𝐶𝑙 ― 𝑃𝐶88𝐴 systems (with dodecane as organic phase diluent) using single component data

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from Li et al.39 To create each plot, we generate a two dimensional grid of standard molar Gibbs

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energy values for the extractant and organo-metal complex, each dimension representing a range

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(with log-scaled intervals) bounded by ± 10 × ∆𝐻0𝑖𝑛𝑖𝑡, where 𝑖𝑛𝑖𝑡 represents the respective initial

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values. Then, for all points on that grid, we evaluate the residual function in Eqn. 10, and use the

212

result to generate a contour plot like the example in Fig. 3a.

15,39–43

reported for 𝑀𝑅𝐸𝐸 ―𝐻𝐶𝑙 ― 𝑃𝐶88𝐴 systems, over a range of pH

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Fig. 3 Plots of least-squares residuals for the simple case of an Nd-HCl-PC88A system with two

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fit parameters: the standard molar enthalpy for the extractant (∆𝐻𝑃𝐶88𝐴), and that for the complex

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species (∆𝐻𝑁𝑑). (a) Contour plot illustrating degree of fit between predicted and experimental39

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values. (b) The locus of points with high degree of fit (function value where relative error < 4%).

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Axes units are in 𝐽𝑚𝑜𝑙 ―1.

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A simple visual trace of the low function value (darker blue) regions in Fig. 3a suggests that

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∆𝐻𝑁𝑑 varies linearly with ∆𝐻𝑃𝐶88𝐴. Zooming in further – by selecting points with residual function

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value below the specified threshold – confirms the linear relationship between these two

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parameters, as illustrated in Fig. 3b. We observed a similar trend for all the other REEs, with the

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general linear relationship described in Eqn. 12.

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(12)

∆𝐻𝑗 = 𝑀𝑗∆𝐻𝑘 + 𝐶𝑗


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M, C refers to the slope and intercept, respectively, for each line; 𝑘 stands for the extractant

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while 𝑗 identifies the corresponding REE complex. Table 1 summarizes the slope and intercept for

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each rare earth case. Note that each slope value in Table 1 is approximately equal to the ratio of

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the extractant-to-metal mole proportions in the organo-metal complex (𝑛 in Eqn. 1). The form of

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this relationship is analogous to that underlying the molar volume group-contribution additivity

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principle of Screttas et al.44 This suggests that the terms on the right in Eqn. 12 approximate

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respective contributions from the organic and metallic components of the organo-metal complex.

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Eqn. 12 also reduces the dimension of the parameter fit problem to just one unknown (Eqn. 13)

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which we solve using the logic in Fig. 2 for ∆𝐻0𝑘, and back-substitute into Eqn. 12 for each ∆𝐻0𝑗 .

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Table 1 summarizes these results.

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𝑁

𝑁

2

min ∑𝑖 𝑑𝑎𝑡𝑎∑𝑗 𝑅𝐸(𝑓(∆𝐻0𝑘, 𝜑𝑖,𝑗,𝑘) ― 𝑌𝑒𝑥𝑝 𝑖,𝑗 )

(13)

238

Table 1 Regression parameters describing the linear relationship between extractant (PC88A) and

239

organo-metal complex standard molar enthalpies, and the resulting partial molar value predictions. M

C (𝑱𝒎𝒐𝒍 ―𝟏)

∆𝑯𝟎 (𝑱𝒎𝒐𝒍 ―𝟏)*

(𝐻𝐴)2**

2.9984

-804498

-2,687,575

𝑁𝑑(𝐻𝐴2)3

2.9990

-812287

-8,862,925

𝑃𝑟(𝐻𝐴2)3

2.9983

-804162

-8,872,327

𝐶𝑒(𝐻𝐴2)3

2.9985

-807196

-8,862,321

𝐿𝑎(𝐻𝐴2)3

2.9984

-804498

-8,865,892

Organic species

240 241 242

*These results represent apparent thermodynamic values consistent with the assumed ideal solution mixing behavior in the organic phase **PC88A organic extractant

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Fig. 4 Parameter estimation validation results comparing predicted and reported equilibrium

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molar compositions in both the aqueous and organic phases. (a) compares predictions for single-

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component data from Li et al.39; (b) compares predictions for three-component data from Banda


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et al.45, and (c) compares steady-state predictions for aqueous and organic-phase rare earth

249

compositions with bench-scale laboratory data from Lyon et al. 42

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Fig. 4 shows selected validation plots comparing model predictions to extraction equilibrium

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data for single component systems in Fig. 4a, three-component systems in Fig. 4b, and a

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multistage, bench-scale solvent extraction unit in Fig. 4c (see the supplemental material, section

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S1-A, for additional validation plots). The multistage solvent extraction process model used for

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the results in Fig. 4c was implemented as described in the supplemental material section S2-A,

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where GEM model predicts equilibrium compositions for each separation stage. The plots show

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good agreement between our model and the reported equilibrium extraction data. However, the

258

prediction accuracy is not uniform across all data points. The assumptions – ideal mixing in the

259

organic phase and complete dissociation in the aqueous phase – , while consistent with other

260

studies15,33,35,39,40,42, might explain some of these deviations. Limitations in the least squares

261

algorithm may have also contributed, because in minimizing an averaged difference over the entire

262

dataset, it loses some local information at each data point that might have improved the fit. In

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addition, most of the data sources used in this analysis did not report any measurement error

264

estimates, which makes it impractical to apply appropriate weights to data segments when

265

resolving inconsistencies or conflicting measurements. However, the overall good agreement

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between model and data show the power of this GEM approach in estimating the missing

267

thermodynamic quantities and justify its use for predicting extraction equilibrium

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3.2. Solvent extraction process design exploration

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The physical similarities between REEs often necessitate combinations of multiple separation

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stages to achieve recovery targets.12 This complexity is hierarchical, and arises from

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combinations of inherently modular process units: as stages combine to form columns, columns

275

combine to form separation modules, and separation modules combine to meet the recovery and

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purity targets of the plant (Fig. 5).

277 278

Fig. 5 Solvent extraction system showing hierarchical modular combination of process units for

279

the separation of an Nd, Pr, Ce, La mixture. Module 1 governs the primary separation into two

280

rare earth groups: Nd, Pr and Ce, La. Module 2 further separates Nd from Pr, while module 3

281

separates Ce from La; n, m, and p are the number of extraction stages, number of extraction+scrub

282

stages, and total number of stages, respectively.

283 284

One challenge with designing these systems is that the proportion of configurations with low

285

recovery and high purity dominate the space of feasible options (Fig. 6) as the set of parameters

286

values that maximize recovery often trade-off on purity. Therefore, we use the GEM-based


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solvent extraction model to explore and map the space of design and operating parameter value

288

combinations, and incorporate resulting insights in optimizing the process design. (See

289

supplemental material sections S2- S4 for details on how we set up the process simulation,

290

design exploration and optimization models).

291 292

Fig. 6 Normalized distribution of (a) recovery predictions and (b) purity predictions from a

293

parameter sweep of configurations separating Nd, Pr from Ce, La. More than 90% of cases have

294

less than 20% recovery. The trend reverses for purity, highlighting the trade-off between recovery

295

and purity targets.

296 297

3.2.1. Design exploration and optimization

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The heat maps in Fig. 7 summarize the relationship between operating parameter combinations

299

and target recovery/purity specifications for the three separation modules (illustrated in Fig. 5).

300

Each box in the heat map contains a probability value that expresses the likelihood that selecting

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a given parameter value will meet the corresponding recovery and purity targets. The trends are

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similar for modules 1 and 2 (Figs. 7a and 7b), but contrasts emerge in some parameter categories

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for module 3 (Fig. 7c). For instance, whereas the former do not tolerate high H+ concentrations


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in the extraction and scrub columns, the latter is more forgiving and can find combinations of

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other parameters that overcome the solubility barrier that would otherwise prevent the metal ions

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from leaving the aqueous solution.. In general, these visualizations show high-level trends for

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operational parameter combinations useful for initial specification of the solvent extraction

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system.

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311 312

Fig. 7 Heatmaps showing the probability that a parameter value range (low, medium, high) is

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compatible with desired recovery and purity targets for the different REE separation modules.


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Values are rounded to the nearest single decimal place and zero means less than 5% chance of

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occurrence.

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The results in Fig. 8 (for module 1) show that the likelihood of meeting recovery specifications

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with the same numbers of stages plummets as recovery target increases. This is an important

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consideration in the economics of developing a REE separation plant: the marginal value derived

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from designing for higher recovery must be worth the additional capital outlay required. Note that

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these results only show trends, because the sweep analysis does not represent an exhaustive

321

exploration of the space of parameter combinations. Regardless, with this insight into the

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underlying space of solutions, we can make quick design predictions and provide better initial

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values and bounds for optimization runs.

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325 326

Fig. 8 Populations of total number of stages that meet increasingly stringent recovery targets for

327

separating Nd, Pr at 99% purity from Ce,La.

328 329

For optimization, we add directionality to the design exploration to maximize a design objective.

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Since the scope of this study excludes economics, we keep things simple and set the objective

331

function as the total number of equilibrium stages, a surrogate for the required equipment capital


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332

outlay. Fig. 9 shows the optimized recovery/purity results for the first module in the extraction

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process (separating Nd, Pr from Ce, La). Table 4 in section S4-A lists the corresponding result

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for all three modules.

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336 337

Fig. 9 Solvent extraction recovery and purity results for the first module separating Nd, Pr from

338

Ce, La. This design recovers 96.5% of the Nd and 96.7% of Pr in the feed at over 99% purity. The

339

extraction and scrub column raffinates contain relatively uncontaminated mixtures of the

340

remaining Ce and La.

341 342

These results reflect the simplified optimization logic and constraint specifications used in this

343

study. Practical considerations may necessitate different constraints, objective function

344

specifications and separation targets. For instance, Fig. 10 shows the relative equipment size

345

(number of equilibrium stages) for different separation target levels. There is more than twofold

346

increase in the total number of stages as the separation target transitions from clustered groups

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to individual components. This increase translates to higher capital cost, which must be justified

348

by the additional value derived from further recovery. Given current market prices (Nd, Pr,


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349

Nd+Pr ~ $60-$70; Ce, La, Ce+La ~ $5-$7)46–48, the existence of applications for Nd+Pr

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mixtures, may obviate further separation, except perhaps to adjust the relative composition.

351

Since Ce is a low-value product, separating it out is harder to justify.

352 353

Fig. 10 Cumulative number of stages as a function of separation objective: Additional separation

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requirements translate to a significant increase in total number of stages. Target 1: (Nd, Pr from

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Ce, La); Target 2: (Nd, Pr from Ce, La)+(Nd from Pr); Target 3: (Nd, Pr from Ce, La)+(Nd from

356

Pr)+(Ce from La).

357 358

4. Discussion

359

The parameter estimation analysis revealed a linear relationship between standard molar enthalpy

360

values for the extractant and organo-metal complexes, which suggests a strategy for estimating

361

organo-metal standard enthalpies analogous to the molar volume additivity principle of Screttas et

362

al.44 The solvent extraction exploratory design revealed a topology skewed severely towards

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suboptimal designs due to the inherent competition between recovery and purity targets. This

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analysis also mapped ideal parameter ranges for high recovery and purity targets for the different

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REE mixtures. By revealing optimal design combinations and operating parameters for REE


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separation, this study also highlights design considerations relevant to the technical feasibility of

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rare earth recycling as a pathway to closing the REE material supply loop. The GEM approach for

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both stnadard parameter estimation and process design can be extended to a wide range of

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hydrometallurgical applications involving mixtures of base, rare, and precious metals.

370

The findings from this analysis should be taken in context, particularly when generalizing to other

371

systems. The results from parameter estimation analysis represent our best model prediction given

372

the referenced datasets, and underlying model assumptions. If an experimentally determined

373

standard property measurement becomes available for the extractant, the corresponding values for

374

the organo-metal complexes can be updated using the strategy described in this study. Also, though

375

we set the optimization objective to minimize total number of stages, the feasibility of REE

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recycling requires broader considerations. Therefore, a natural extension of this study will

377

reformulate the optimization objective and constraints to incorporate economic indicators (capital

378

costs operating costs and revenues from product sales), the environmental footprint of toxic

379

process streams, and end-of-life feedstock supply logistics.

380 381

ASSOCIATED CONTENT

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A supplemental material file (PDF) accompanies this manuscript. The Supplemental material

383

contains the following sections: (S0-A) the complete Cantera xml input file with phase

384

definitions and property value specifications; (S0-B) Table of extraction equilibrium data

385

sources; (S1-A) Additional validation plots for the standard property estimates; (S2-A)

386

Description of the solvent-extraction process simulation implementation; (S3-A) Description of

387

the method for design exploration parameter sweep studies; (S4-A) Description of the

388

implementation strategy for process optimization, as well as a table of sample optimization


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results. This information is available free of charge via the internet at http://pubs.acs.org.

390 391

AUTHOR INFORMATION

392

Corresponding Author

393

Chukwunwike O. Iloeje

394

[email protected]

395

Author Contributions

396

The manuscript was written through contributions of all authors. All authors have given approval

397

to the final version of the manuscript.

398

ACKNOWLEDGMENT

399

We wish to acknowledge Artem Gelis and Candido Pereira (Argonne National Lab), Kevin Lyon,

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Mitchell Greenhalgh, Devin Imholte and Tedd Lister (Idaho National Lab) for providing valuable

401

discussions, data and references. The submitted manuscript has been created by UChicago

402

Argonne LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S.

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Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-

404

06CH11357. Sandia National Laboratories is a multi-mission laboratory managed and operated by

405

National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of

406

Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security

407

Administration under contract DE-NA0003525. The U.S. Government retains for itself, and others

408

acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to

409

reproduce, prepare derivative works, distribute copies to the public, and perform publicly and

410

display publicly, by or on behalf of the Government. The Department of Energy will provide public


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411

access to these results of federally sponsored research in accordance with the DOE Public Access

412

Plan. The design exploration studies were performed on Argonne’s Leadership Computing

413

Facilities.

414 415

REFERENCES

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Gibbs energy minimization model for solvent extraction with

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