Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger Green Process Synth 2019; 8: 480–487 Self-dual Leonard pairs Laura Turri*, Karine Gérardin, Hervé Muhr, François Lapicque, Alvaro Saravia, Stéphanie Szenknect, Adel Mesbah, Régishttps://doi.org/10.1515/spma-2019-0001 Mastretta, Nicolas Dacheux, Daniel Meyer, Anaëlle Cloteaux, Antoine Gérard and Salvatore BertucciReceived May 8, 2018; accepted September 22, 2018 Abstract: Let F denote a eld and let V denote a vector space over F with nite positive dimension. Consider CO2 sequestrationa pair byA, A ∗carbonationof diagonalizable F-linear of maps olivine: on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which a new process forthere optimal exists an automorphism separation of the endomorphism of the algebra of V that swaps A and A∗. Such an automorphism is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this solids produced ↔ duality. In particular, we display an invertible F-linear map T on V such that the map X TXT− is the duality → A A∗. We express T as a polynomial in A and A∗. We describe how T acts on ags, decompositions, https://doi.org/10.1515/gps-2019-0016 ↔ Keywords: CO sequestration; olivine; products separa- and 24 bases for V. 2 Received September 25, 2018; accepted November 28, 2018. tion; carbonation; direct/indirect processes Keywords: Leonard pair, tridiagonal matrix, self-dual Abstract: CO2 sequestration by reaction with abundant, reactive minerals such as olivine hasClassication: often been consi17B37,15A21- dered. The most straightforward, direct process consists 1 Introduction in performing the reaction at high temperature and CO2 pressure, in view to producing silica, magnesium and iron The increasing concentration of CO2 in the atmosphere carbonates and recovering the traces of nickelIntroduction and chro- is attributed to the rising consumption of fossil fuels for mite contained in the feedstock mineral. Most of direct energy generation or use around the world. In order to processes were found to have an overallLet Fcostdenote far larger a eld develop and let steelV denote industry a with vector low space carbon over dioxideF with emissions, nite positive dimension. We consider a than the CO removal tax, because of incomplete carbona- chemical processes by reaction of CO with minerals derived 2 pair A, A∗ of diagonalizable F-linear maps on V, each2 of which acts on an eigenbasis for the other one in an tion and insufficient properties of theirreducible reaction products. tridiagonal from fashion. silicate Suchminerals, a pair e.g. is olivine called can a Leonard be considered pair (see [1,2]. [13, Denition 1.1]). The Leonard pair Similar conclusions could be drawn in a previous inves- Because of their alkaline nature, silica-based minerals A, A∗ is said to be self-dual whenever there exists an automorphism of the endomorphism algebra of V that tigation with a tubular autoclave. An indirect process can react with acidic carbon dioxide at high pressure and swaps A and A∗. In this case such an automorphism is unique, and called the duality A A∗. has been designed for high conversion of olivine and the temperature to generate carbonates and silica [3]. Various ↔ production of separate, profitable productsThe e.g. literature silica, containsprocesses many have examples been investigated of self-dual as for Leonard instance pairs. in [4,5]. For instance (i) the Leonard pair associ- carbonates, nickel salts, so that the overallated withprocess an irreduciblecould However, module the for reaction the Terwilliger has a significant algebra ofcost, the hypercubeestimated (see [4, Corollaries 6.8, 8.5]); (ii) a Leonard pair of Krawtchouk type (see [10, Denition 6.1]); (iii) the Leonard pair associated with an irreducible be economically viable: the various steps of the process in the range 80-200 €/tCO2 depending on the process are described in the paper. Olivine particlesmodule (120 for µm) the can Terwilliger considered algebra [3]. In of this a distance-regular study, the recovery graph of the products that has has a spin model in the Bose-Mesner alge- be converted at 81% with a low excessbra of acid (see within [1, Theorem], 3 h been [3, considered Theorems in 4.1, the 5.5]); global (iv) carbonation an appropriately process, normalizedfor an totally bipartite Leonard pair at 95°C. The silica quantitatively recovered(see [11,exhibits Lemma a BET 14.8]); economic (v) the profitability Leonard pair and consisting an environmental of any acceptability. two of a modular Leonard triple A, B, C (see [2, 2 -1 area over 400 m g , allowing valuableDenition applications 1.4]); to be (vi) theMore Leonard precisely, pair the consisting conventional of adirect pair route of opposite for olivine generators for the q-tetrahedron alge- considered. Besides, the low contentsbra, of actingnickel cations on an evaluation carbonation module has been (see [5,compared Proposition to an 9.2]). indirect The exampleprocess (i) is a special case of (ii), and the could be separated from the magnesium-rich solution by currently under investigation and relying upon olivine examples (iii), (iv) are special cases of (v). ion exchange with a very high selectivity. acidic leaching followed by carbonation and separation Let A, A∗ denote a Leonard pair on V. We can determine whether A, A∗ is self-dual in the following way. stages. Comparison of the two processes is made in terms d By [13, Lemma 1.3] each eigenspace of A, A∗ has dimension one. Let θ denote an ordering of the eigen- of reaction yields and quality of the products obtained.{ i}i= * Corresponding author: Laura Turri, Laboratoire Réactions et d values of A. For ≤ i ≤ d let vi denote a θi-eigenvector for A. The ordering θi i= is said to be standard Génie des Procédés, CNRS-Université de Lorraine, BP 20451, 1 rue d { } d whenever A∗ acts on the basis vi i= in an irreducible tridiagonal fashion. If the ordering θi i= is standard Grandville, 54000 Nancy, France, d { } { } e-mail: [email protected] then the ordering θ2d− i Directi= is also carbonation standard, and no of further olivine ordering is standard. Similar comments apply to d { } d Karine Gérardin, Hervé Muhr and François Lapicque,A∗. Let Laboratoireθ denote a standard ordering of the eigenvalues of A. Then A, A∗ is self-dual if and only if θ { i}i= { i}i= Réactions et Génie des Procédés, CNRS-Université de Lorraine, BP is a standard ordering of the eigenvalues of A∗ (see [7, Proposition 8.7]). 20451, 1 rue Grandville, 54000 Nancy, France 2.1 Feedstock Alvaro Saravia, Stéphanie Szenknect, Adel Mesbah, Régis Mastretta, Nicolas Dacheux and Daniel Meyer, Institut de Chimie The olivine used is produced in Norway and can be Séparative de Marcoule, CEA-CNRS-Université Montpellier, ENSCM, considered as a mixed Mg-Ni and Fe silicate. The 30207 Marcoule, France *Corresponding Author:main Kazumasa phase Nomura: correspondsTokyo Medicalto the and stoichiometry Dental University, of Ichikawa, 272-0827,Japan, Anaëlle Cloteaux and Antoine Gérard, ProgepiE-mail: – Process [email protected] Lab, Mg Fe Ni (SiO ), with chromite particles present BP 20451, 1 rue Grandville, 54000 Nancy, FrancePaul Terwilliger: Department1,838 of Mathematics,0,156 0,006 4 University of Wisconsin, Madison, WI53706, USA, E-mail: at approx. 0.38 wt% and other inert minerals at trace Salvatore Bertucci, ArcelorMittal Research Center,[email protected] Voie Romaine, 57500 Maizières-les-Metz, France levels. The particle size distribution of the GL30 ® grade Open Access. © 2019 Turri et al., published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. L. Turri et al.: CO2 sequestration by carbonation of olivine 481 used ranged from 70 to 250 µm with an average size of 2.3 Pretreatment of olivine (Step 1) 120 µm (d(0.5) = 121 µm) to limit the grinding costs. More details about olivine characterization and properties are Being both observed in olivine and in the carbonation given in [6,7]. products, chromite particles are not affected by high temperature carbonation. Various mechanical operations can be considered for the separation of chromite from 2.2 Introduction to the direct process olivine as magnetic or gravimetric separation [8,9]. Flotation of chromite, based on surface phenomena, Direct carbonation using olivine as feedstock has been has been extensively investigated for the recovery of fine considered as an alternative to convert CO2 emissions particles in the range 25-100 µm [10,11]. In a previous study into solid carbonates and silica. In this study, separation [7], flotation has been studied in a lab-scale column, of carbonation products has been investigated for an using a sodium carbonate solution at pH 11 with cetyl environmental aspect and is specifically decisive for the trimethyl ammonium bromide (CTMAB). The enrichment economic viability of the process. The overall process of ratio of chromite was found to be as large as 10 which direct carbonation consists of the following steps (Figure 1): demonstrated the potential of flotation for the recovery of chromite in a multistage flotation process [7]. • Olivine pretreatment has been considered before Further, leaching of nickel for the pretreatment of the carbonation reaction, due to the presence of olivine has been tested in view to purify the carbonation impurities (Step 1). Two separation processes have products. In various studies, dissolution of nickel been investigated successively: (i) flotation of contained in laterites (mixed iron and aluminium chromite particles, for which the results are presented hydroxides and silicates) has been carried out using in [7], (ii) selective leaching of nickel which replaces ammoniacal solutions [12,13], as for example in the Caron magnesium in the forsterite matrix (Mg SiO ) largely 2 4 process.