domain: chim spi

text:
Serena Delgado

Alain Gaunand
email: alain.gaunand@minesparis.psl.eu

Christophe Coquelet
email: christophe.coquelet@mines-albi.fr.

Renaud Cadours

Céline Volpi

A New Reactive Absorption Model Using Extents of Reaction and Activities. I. Application to Alkaline-salts-CO 2 Systems Submitted to Chemical Engineering Science

Keywords: Gas-liquid absorption, Kinetic model, Non ideality, Hydroxide solutions, CO 2 absorption

CO 2 absorption into basic aqueous solutions is widely used for CO 2 separation from gas streams (e.g., for natural gas purification). CO 2 loading and ionic strength increase significantly along industrial columns. In absorption modelling, deviation from ideality should then be considered.

This study implements a general steady-state model for reactive gas-liquid absorption. Firstly, equilibrium relations, Nernst-Planck diffusion fluxes and reaction rates are written based on activities. Secondly, local fluxes are related by stochiometric constraints through extents of reaction.

 to alkaline salts-water-CO 2 systems. The following Arrhenius expression was found for the direct kinetic constant of reaction CO 2 + HO -↔ HCO 3 -: lnk (m 3 .mol -1 .s -1 ) = 19.84 -5248.8/T (K) -12% overall AAD. This kinetic law can be used in any system involving this reaction (e.g., aqueous amine solutions). This part I paves the way to the further study of CO 2 absorption into aqueous amine solutions.

Nomenclature

A

Logarithm of Arrhenius pre-exponential factor a i Activity of species i (mol.m -3 ) a 1,0 Activity of absorbed gas at gas-liquid interface (mol.m -3 ) c i Molar concentration of species i (mol.m -3 ) c 1,0 Concentration of absorbed gas at gas-liquid interface (mol.m -3 ) c abs,tot Total absorptive species concentration (mol.m -3 ) C i Normalised concentration of species i (-) c b,i Liquid bulk concentration of species i (mol.m -3 ) c N,i Normalisation concentration of species i (mol.m -3 ) D i Diffusivity of species i in solution (mol.m -3 ) E

Enhancement factor (-) E A Activation energy (J.mol -1 ) f obj Optimisation objective function (-) G Gibbs free energy (J) H 1

Henry's constant of absorbed gas at infinite dilution (Pa) Ha j

Hatta number of reaction j (-) j i Flux of species i through liquid film (mol.m -2 .s -1 ) j 1T,0

Total absorption flux of absorbed gas (mol.m -2 .s -1 ) k j Reaction j kinetic constant K j Reaction j equilibrium constant K N,j Reaction j normalising equilibrium constant k L,1

Liquid-side mass transfer coefficient regarding the absorbed gas (m.s -1 ) M Matrix of the differential problem for concentration-related equations n C Number of solute constituents in liquid phase n i mole number of species i (mol) n R,kin Number of finite-rate reactions in the reaction mechanism n R,eq

Number of reactions at equilibrium in the reaction mechanism o D ij Direct kinetic order of species i in reaction j (-) o R ij Reverse kinetic order of species i in reaction j (-) P Pressure (Pa) p 1,0

Pressure of absorbed gas at gas-liquid interface (Pa) r j

Reaction rate of reaction j (mol.m -3 .s -1 ) T Temperature (K) x i Mole fraction of species I (-) X i

Vector of the differential problem for concentration-related equations z Distance from interface (m) Z Normalised distance from interface (z) z i Number of charges of species I (-) Z Di Diffusion-concentration ratio of species i (-)

Constants F Faraday constant (C.mol -1 ) R Ideal gas constant (J.mol -1 .K -1 )

Greek letters γ i Activity coefficient of species i (-) δ ij

Kronecker delta between species i and j δ L,1

Film thickness regarding the absorbed gas (m) µ i Chemical potential of species i (J.mol -1 ) ν ij Stochiometric coefficient of species i in reaction j (-) ψ

Local electric potential (V) ξ j

Extent of reaction j (-) [START_REF] Bottoms | Organic bases for gas purification[END_REF]. Solvent screening and design are at the heart of the research effort to increase both CO 2 absorption rate and capacity, and to decrease solvent regeneration duty (e.g., aqueous amine blends). Industrial absorption column design with a given solvent requires the identification of the dissolved gas reaction mechanism. To do so, reactions at equilibrium (e.g., acid-base reactions) and finite-rate reactions need to be studied at pressure, temperature and composition conditions encountered in the absorption column.

Subscripts
Kinetic parameters are typically obtained from absorption or desorption flux measurements in laboratory gas-liquid contactors of known mass transfer coefficient and interfacial area. Most kinetic studies deal with absorption experiments into unloaded basic solutions. Only one irreversible reaction is considered (e.g., [START_REF] Pohorecki | Kinetics of reaction between carbon dioxide and hydroxyl ions in aqueous electrolyte solutions[END_REF], [START_REF] Pani | Kinetics of absorption of CO2 in concentrated aqueous methyldiethanolamine solutions in the range 296 K to 343 K[END_REF], [START_REF] Derks | Kinetics of absorption of carbon dioxide in aqueous piperazine solutions[END_REF]). Such studies disregard the increase in ionic strength and non-ideality when CO 2 is absorbed, due to bicarbonate, carbonate, and carbamate ion production. However, absorption models based on activities rather than molarities should lead to more relevant process simulation.

CO 2 absorption modelling into aqueous amine solutions first requires the study of CO 2 reaction with HO -ion. Indeed, this reaction takes place in all aqueous systems with CO 2 . Furthermore, kinetic constants between CO 2 and tertiary amines can be expressed as a function of amine pKa [START_REF] Couchaux | Kinetics of carbon dioxide with amines. I. Stopped-flow studies in aqueous solutions. A review[END_REF]. Through molecular simulation, [START_REF] Rozanska | Quantitative Kinetic Model of CO2 Absorption in Aqueous Tertiary Amine Solvents[END_REF] show that CO 2 kinetics in aqueous tertiary amine solutions can be written as a function of CO 2 and HO -concentrations only.

For CO 2 absorption into aqueous alkaline salts solutions, researchers choose a one-irreversiblereaction mechanism. In this case, non-ideality is usually included. In early studies such as [START_REF] Pinsent | The kinetics of combination of carbon dioxide with hydroxide ions[END_REF], [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF], [START_REF] Augugliaro | Kinetics of carbon dioxide absorption into catalysed potassium carbonate solutions[END_REF], [START_REF] Pohorecki | Kinetics of reaction between carbon dioxide and hydroxyl ions in aqueous electrolyte solutions[END_REF], and [START_REF] Kucka | Kinetics of the gas-liquid reaction between carbon dioxide and hydroxide ions[END_REF], empirical ionic-strength, alkaline-ion-dependent kinetic constant and CO 2 solubility relations are developed. When studying CO 2 absorption into 30wt% K 2 CO 3 solutions, [START_REF] Thee | Carbon dioxide absorption into unpromoted and borate-catalyzed potassium carbonate solutions[END_REF] fit one Arrhenius-type expression to their kinetic data between 313 and 353 K. The expression is found coherent with [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] correlation when considering identical salt and concentration range. Under relatively diluted Na 2 CO 3 and K 2 CO 3 concentrations (300 mol/m 3 ), [START_REF] Penders-Van Elk | Combined effect of temperature and p K a on the kinetics of absorption of carbon dioxide in aqueous alkanolamine and carbonate solutions with carbonic anhydrase[END_REF] find good agreement of their CO 2 absorption data between 283 and 313 K with pKa-dependent Bronsted kinetic law for tertiary amines. Following [START_REF] Kucka | Kinetics of the gas-liquid reaction between carbon dioxide and hydroxide ions[END_REF], [START_REF] Haubrock | The applicability of activities in kinetic expressions: a more fundamental approach to represent the kinetics of the system CO2-OH-in terms of activities[END_REF], [START_REF] Haubrock | The applicability of activities in kinetic expressions: A more fundamental approach to represent the kinetics of the system CO2-OH--salt in terms of activities[END_REF], [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] and [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF] develop an activity-based kinetic rate expression, using Pitzer-Debye-Hückel model, empirical activity coefficients and e-NRTL model, respectively. Thus, a unique Arrhenius-type expression applies to all systems (with Li + , Na + or K + ). [START_REF] Sheng | Effective Mass Transfer Area Measurement Using a CO2-NaOH System: Impact of Different Sources of Kinetics Models and Physical Properties[END_REF] apply current knowledge on CO 2 absorption into aqueous NaOH solutions to measure pilot effective mass transfer area. The influence of five literature sources is evaluated for kinetic and physical properties estimation. According to this study, different Arrhenius parameters result in large variations for the effective mass transfer area estimation. Additionally, inconsistencies between kinetics and physical parameter sources should be avoided.

Absorption involves reversible finite-rate reactions and acid-base reactions at equilibrium. Species simultaneously diffuse to and from gas-liquid interface and react. The diffusion driving force is the molar concentration gradient in the widely used semi-empirical Fick diffusion law (e.g., [START_REF] Versteeg | Mass transfer with complex reversible chemical reactions-I. Single reversible chemical reaction[END_REF], [START_REF] Bishnoi | Absorption of carbon dioxide in aqueous piperazine/methyldiethanolamine[END_REF], [START_REF] Derks | Kinetics of absorption of carbon dioxide in aqueous piperazine solutions[END_REF]). However, Maxwell-Stefan theory [START_REF] Ahmadi | Rigorous Multicomponent Reactive Separations Modelling: Complete Consideration of Reaction-Diffusion Phenomena[END_REF], or Nernst-Planck equation in case of unknown binary diffusivities (e.g., [START_REF] Glasscock | Numerical simulation of theories for gas absorption with chemical reaction[END_REF] and [START_REF] Littel | Modelling of simultaneous absorption of H2S and CO2 in alkanolamine solutions: the influence of parallel and consecutive reversible reactions and the coupled diffusion of ionic species[END_REF]) are more suitable to highly non ideal multiconstituent electrolytic systems. The diffusion driving force is then the chemical potential gradient [START_REF] Taylor | Multicomponent mass transfer[END_REF]. [START_REF] Thomas | Convective dissolution of CO2 in reactive alkaline solutions: Active role of spectator ions[END_REF] model CO 2 diffusion-reaction into stagnant water and aqueous alkaline salt solutions with one irreversible reaction. Instead of individual ion diffusion, salt diffusion across the liquid medium is represented. Thus, diffusivities are decoupled without the need for an electric potential gradient to keep electroneutrality.

Mass transfer problem formulation with any transient or steady-state mass transfer theory results in a set of local balances of each species. The differential problem is a boundary value problem. At the gas-liquid interface, absorbed gas physical equilibrium is assumed and in the liquid bulk, chemical equilibrium is assumed. In some models, acid-base reactions are represented as reversible reactions with a very high reaction rate constant (e.g., [START_REF] Glasscock | Numerical simulation of theories for gas absorption with chemical reaction[END_REF], [START_REF] Versteeg | Mass transfer with complex reversible chemical reactions-I. Single reversible chemical reaction[END_REF], [START_REF] Ahmadi | Rigorous Multicomponent Reactive Separations Modelling: Complete Consideration of Reaction-Diffusion Phenomena[END_REF]). The problem then includes a system of n C second order ordinary or partial differential equations. In other models, diffusion-reaction flux coupling is explicitly considered. The number of differential equations decreases, but this formulation lacks generality (e.g., [START_REF] Rinker | Kinetics and modelling of carbon dioxide absorption into aqueous solutions of N-methyldiethanolamine[END_REF], [START_REF] Bishnoi | Absorption of carbon dioxide in aqueous piperazine/methyldiethanolamine[END_REF], [START_REF] Servia | Modeling of the CO2 absorption in a wetted Wall column by piperazine solutions[END_REF]). This second formulation ensures the n eq acidbase equilibria by the addition of n eq algebraic equations: a differential-algebraic problem needs to be solved.

The purpose of this article is to propose a general reactive absorption model entirely based on activities, unlike previous reactive absorption models. Indeed, previous models are either based on Nernst-Planck diffusion with a complete mass transfer model, but activity coefficients are assumed equal to unity or, like [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF], use a simplified mass transfer model with Fick's diffusion law and constant activity coefficients. Activity coefficients are then calculated at equilibrium with a thermodynamic model. For steady-state absorption, extent-of-reaction variables are introduced to take advantage of flux coupling (section 2). This has two advantages compared to usual model formulations:

(1) the number of equations is reduced while still maintaining a general formulation, and (2) solving a system of algebraic equations and substituting the results into the differential equations takes care of acid-base equilibria. Extents of reactions have yet to be implemented in reactive absorption problems.

This model is then applied to CO 2 absorption in aqueous alkaline salts solutions with a full reaction mechanism considering two reversible reactions. This third section aims at fitting a unique Arrheniustype expression for the direct kinetic constant of reaction CO 2 + HO -↔ HCO 3 -to literature CO 2 absorption flux data with all three alkaline salt counter ions. This part I paves the way to the further study of CO 2 absorption into aqueous amine solutions. Indeed, this reaction takes place in all aqueous solvents. It is necessary to characterise it before focusing on more complex systems.

Modelling framework

2.1. Activity-based diffusion, fluxes, rates of reaction and equilibria Equation 1 gives the Nernst-Planck expression for the diffusion flux of a given species i [START_REF] Taylor | Multicomponent mass transfer[END_REF].

Eq. ( 1)

In case of different diffusion coefficients of ionic species, the electric potential ψ is to ensure local electroneutrality through the liquid film ( ):

Eq. ( 2)

The chemical potential of a given species i, µ i , is defined as its partial molar Gibbs free energy [START_REF] Prausnitz | Molecular thermodynamics of fluidphase equilibria[END_REF]. A thermodynamic convention consists in defining a reference state, then the activity a i (eq. 3) and the activity coefficient (eq. SM16 in Supplementary Material) of the species i.

Eq. ( 3) µ i REF and a i are expressed in the same chosen thermodynamic convention. The dimension of the activity depends on the composition scale, such as molalities, molarities, mass fractions, etc. We express reaction rates as functions of activities as well (eq. 4).

Eq. ( 4) Kinetic constants k j and equilibrium constants K j also need to be expressed according to the chosen thermodynamic convention. For reactions assumed to be at equilibrium: Eq. ( 5) ν ij is the stoichiometric coefficient of species i in reaction j. Consistency between orders of reaction and stoichiometric coefficients (eq. 4 and 5) yields .

Absorption problems are typically one dimension. At gas-liquid interface, the partial pressure and the activity of the absorbed gas (subscript 1) are assumed at equilibrium. A Henry's relation is obtained [START_REF] Prausnitz | Molecular thermodynamics of fluidphase equilibria[END_REF], assuming asymmetric thermodynamic convention for the absorbed gas (eq. 6).

Eq. ( 6)

Extents of reaction and resulting system of equations

For a steady-state mass transfer model such as the film model, a new algebraic variable is introduced: the extent of each reaction. The extent ξ j of a given reaction j measures its progress in the liquid film from the interface to a distance z. The relation between fluxes and extents of reaction is dictated by stochiometric constraints. The flux of a given species i through the film, as a function of the extents of reaction, is given in eq. 7.

Eq. ( 7)

Where j 1T,0 is the total flux of gas absorbed. Due to the normalisation by j 1T,0 , ξ j is dimensionless. This expression sums up the contribution of extents both finite-rate reactions and reactions assumed at equilibrium where species i is either a reactant or a product. At the gas-liquid interface (z=0), only reactions at equilibrium may have a non-zero extent. For non-volatile species, the extents of all finite-rate reactions must be zero. Eq. 7 then becomes eq. 8 at the gas-liquid interface.

Eq. ( 8)

Where index k refers to a reaction assumed at equilibrium. As a result, the flux j 1 (z=0) is not only the flux of 1 crossing the interface, j 1T,0 . It is modified by all reactions at equilibrium with a non-zero extent of reaction at the gas-liquid interface (eq. 9).

Eq. ( 9)

The local variation of the extent of a finite-rate reaction, is by definition proportional to its local rate (eq. 10).

Eq. ( 10)

Combining equations 3 and 7 yields eq. 11.

Eq. ( 11)

Differentiation of equilibrium relation (eq. 5) of reaction k generates an algebraic relation between its extent of reaction and the extents of other reactions. Substituting the expression of from eq. 11:

Eq. ( 12)

Where index k refers to a given reaction at equilibrium. The resulting system of n R,eq algebraic equations can be solved separately. The resulting extents ξ k of reactions at equilibrium are substituted back into the differential equations.

Thermodynamic framework and mass transfer and diffusion assumptions

The mass transfer framework used in this work is the stagnant film theory [START_REF] Whitman | The two-film theory of gas absorption[END_REF]. In aqueous solution, the asymmetric convention is more relevant for solute species. The selected thermodynamic model is described in the Supplementary Material section 2. Ionic diffusivities are known only in water at infinite dilution at 25°C. As a first application, equal diffusivities are assumed for all ions in solution (as did e.g., [START_REF] Derks | Kinetics of absorption of carbon dioxide in aqueous piperazine solutions[END_REF], [START_REF] Ahmadi | Rigorous Multicomponent Reactive Separations Modelling: Complete Consideration of Reaction-Diffusion Phenomena[END_REF], [START_REF] Luo | Mass transfer and kinetics of carbon dioxide absorption into loaded aqueous monoethanolamine solutions[END_REF]). The electric potential term is thus neglected. Since the problem is spatially one-dimensional (in z), eq. 13 is obtained.

Eq. ( 13)

Where is the activity coefficient of species i in the molality-based asymmetric convention, and δ ik is the Kronecker delta (1 if i=k, 0 otherwise). Reactions rates can be expressed as:

Eq. ( 14)

Where

is the activity coefficient of species i in the molarity-based asymmetric convention.

is the forward kinetic constant of reaction j ( ) and is the equilibrium constant of reaction j ( ). Both are written in the same convention.

Reaction equilibrium equations are then written as follows.

Eq. ( 15)

Combining equations 3 and 14 yields eq. 16.

Eq. ( 16)

Without electric potential gradient, the system of equations defined by eq. 12 becomes linear regarding the extents of reactions at equilibrium:

Eq. ( 17)

In the chosen thermodynamic convention, Henry's relation for the absorbed gas at the interface (z=0) is expressed as follows:

Eq. ( 18)

As experimental pressure is lower than 1 bar in the data considered in section 3, gas-phase is assumed ideal.

Solution

Before writing the differential problem to solve, dimensionless variables (eq. 19) and parameters (eq. 20) are introduced.

Eq. ( 19)

Where c N,i is the normalising concentration, such as if i=1, c N,i = c 1,0 , and c N,i = c abs,tot otherwise. c abs,tot is the total absorptive species concentration (mol.m -3 ).

Eq. ( 20)

Therefore, the flux of species i is expressed as a matrix product (eq. 21).

Eq. ( 21)

Square matrix M inversion yields:

Eq. ( 22)

Where M -1 is the inverse of matrix M and vector X is defined as .

At each resolution step, an algebraic system of equations is solved independently to obtain the extents of acid-base reactions (eq. 23). These values are used in the differential equations.

Eq. ( 23)

Boundary conditions are presented in Table 1. [START_REF] Kierzenka | A BVP solver based on residual control and the Maltab PSE[END_REF].

Application to alkaline salts-water-CO 2 systems

Four species are considered: CO 2 , HCO 3 -, CO 3 2-and HO -(n C =4). The alkaline ion does not take part in the reaction scheme, so that its concentration is constant through the film because the electric potential is disregarded. However, this ion contributes to non-ideality. The mechanism consists in two reversible reactions (n R =2):

CO 2 + HO -↔ HCO 3 - (R-I) HCO 3 -+ HO -= CO 3 2-+ H 2 O (R-II)
Reaction R-I is reversible and finite-rate (n R,kin =1):

Eq. ( 24)

Where -is its forward kinetic constant -to determine -(m 3 .mol -1 .s -1 ) and -its equilibrium constant (m 3 .mol -1 ).

In addition, the acid-base equilibrium between HCO 3 -and CO 3 2-is considered (reaction R-II; n R,eq =1).

The equilibrium constant parameters are taken from [START_REF] Kamps | Solubility of single gases carbon dioxide and hydrogen sulfide in aqueous solutions of N-methyldiethanolamine at temperatures from 313 to 393 K and pressures up to 7.6 MPa: New experimental data and model extension[END_REF]. Solution density is taken from [START_REF] Laliberté | Model for calculating the density of aqueous electrolyte solutions[END_REF]. Liquid phase CO 2 diffusivity is calculated with the Stokes-Einstein relation [START_REF] Versteeg | Solubility and diffusivity of acid gases (carbon dioxide, nitrous oxide) in aqueous alkanolamine solutions[END_REF]. [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] use this relation as well in combination with CO 2 diffusivity in pure water from [START_REF] Danckwerts | Gas-Liquid Reactions[END_REF]. Viscosity is taken from [START_REF] Laliberté | Model for calculating the viscosity of aqueous solutions[END_REF]. Hydroxide ion diffusivity is set to 1.7×D CO2 based on the ratio proposed by [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF], and D (HCO3-) and D (CO32-) are supposed equal to D (HO-) (see Supplementary Material section 1 for detailed correlations).

As a base case, the mass transfer model was tested with no deviation from ideality (γ=1) and [START_REF] Weisenberger | Estimation of gas solubilities in salt solutions at temperatures from 273 K to 363 K[END_REF]) CO 2 apparent solubility ("concentration-based model"). Secondly, the activity-based model was implemented.

Few CO 2 absorption flux measurements are available in these systems. [START_REF] Kucka | Kinetics of the gas-liquid reaction between carbon dioxide and hydroxide ions[END_REF], [START_REF] Pohorecki | Kinetics of reaction between carbon dioxide and hydroxyl ions in aqueous electrolyte solutions[END_REF] and [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] do not publish raw experimental data, such as CO 2 partial pressure. [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF] use raw absorption flux data from [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] in alkaline hydroxide solutions and from [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] in alkaline carbonate solutions. Note that data from [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] is not available in open literature. The only other raw dataset on CO 2 absorption in sodium hydroxide and carbonate solutions was found in [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] (see Table 2). [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF], SDC Li + 1.8.10

-4 -0.035 1.8.10 -4 -0.035 0-3.5.10 -4

298-338 0.17-0.28 30 [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF], Jet & WWC Na + 2.3.10 -3 -0.025 0-9.0.10 -3

2.3.10 -7 -0.017 298-303 100 157 [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF], SDC Na + 1.8.10 -4 -0.036 1.8.10 -4 -0.036 0-3.6.10 -4

298-333 0.18-0.32 38 [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF], SDC K + 1.6.10 -4 -0.033 1.6.10 -4 -0.033 0-3.6.10 -4

298-337 0.16-0.29 35 SDC: "String of Discs Column", Jet: "Laminar Jet Absorber", WWC: "Wetted-Wall Column", Alk + : alkaline ion.

For kinetic constant regression, [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] data was used, as it encompasses all three alkaline ions between 298 and 338 K. To keep datasets with each counter ion approximately to the same size, [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] data for sodium hydroxide and carbonate solutions between 298 and 303 K was used as a validation dataset. [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] data therefore also serves as a model prediction test. Modelling each flux measurement requires the value of the liquid-side mass transfer coefficient, which is taken from each study.

Kinetic parameter optimisation of reaction R-I direct kinetic constant is achieved using the objective function specified in eq. 25.

Eq. ( 25) ε is a constant set to 1.10 -4 mol.m -2 .s -1 to mitigate the error value for very small absorption fluxes. The two terms in the objective function (eq. 25) give this function a symmetrical behaviour for flux underestimation and overestimation. Optimisation function fminsearch from Matlab R2013a was used to fit the Arrhenius parameters of the kinetic constant (eq. 26). In fminsearch, the simplex algorithm by [START_REF] Lagarias | Convergence properties of the Nelder--Mead simplex method in low dimensions[END_REF] is used.

Eq. ( 26)

The fitted Arrhenius parameters as well as model performance is presented in Table 3: the average absolute deviation (AAD, eq. 27) and the bias (eq. 28) of regression [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] and validation [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] datasets. Model performance is also compared to [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF].

Eq. ( 27)

Eq. ( 28)

Parameter standard errors are calculated by variance-covariance analysis, as described in the Supplementary Material section 3.

Table 3 Optimised Arrhenius parameters of reaction R-I direct kinetic constant over [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] (Li + , Na + , K + ) AAD (%) 14 10 15 Bias (%) 6.2 -2.1 - [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF]) (Na + ) AAD (%) 17 14 -Bias (%) 12 -0.23 -Remark: [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF] do not provide parameter standard deviations.

Parity and deviation plots are shown in Figure 1 Considering solution non-ideality improves model performance for all three types of salts. The only exception is the data of [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] with NaOH data at 2 mol.L -1 . All data from [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] for all three salts (Li, Na and K-based) and most of the data from [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] are represented within a +/-30% relative deviation by the activity-based model. Only 13 of all 157 validation data points have a relative deviation exceeding +/-30%. AAD over validation dataset is 13,7% and bias -0,2%, which is on par with the model performance of [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF]. Concentration-based model performance is on par with that of [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF] when considering the data of [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF]. Ideal solution is assumed in the concentration-based model for reaction rate, diffusion flux, and chemical equilibria, but uses a correlation for CO 2 physical solubility at gas-liquid interface. Salting-out effect on CO 2 solubility due to increasing ionic strength is represented in both models: in the concentration-based model, by ion-concentration-dependent [START_REF] Weisenberger | Estimation of gas solubilities in salt solutions at temperatures from 273 K to 363 K[END_REF] correlation, and in the activity-based model, by the developed thermodynamic model (see Supplementary Material section 2). Finally, only the activity-based model unifies the representation of all equilibria with the chosen thermodynamic model, whereas the concentration-based model is correlation-dependent.

The activity-based model is better at predicting [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] measurements in NaOH and Na 2 CO 3 aqueous solutions than the concentration-based model. This suggests the activity-based model can more easily be extrapolated outside regression area than the concentration-based model.

Arrhenius parameters are compared to previous studies in Figure 3, specifically [START_REF] Pohorecki | Kinetics of reaction between carbon dioxide and hydroxyl ions in aqueous electrolyte solutions[END_REF], [START_REF] Kucka | Kinetics of the gas-liquid reaction between carbon dioxide and hydroxide ions[END_REF], [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] and [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF]. All these studies consider the solution non ideality in the reaction rate expression. Both concentration-based and activity-based models lead to close kinetic constants compared to [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF]. The Arrhenius law obtained with the activity-based model displays a -14% to -9% difference in the regression temperature range. Fitted Arrhenius law extrapolation to higher temperatures leads to significant underestimation compared to the kinetic constant obtained by [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF] with their own experimental data and methodology -94 vs. 157 m 3 .mol -1 .s -1 at 343 K. The value remains close to that of [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF] -102 m 3 .mol -1 .s -1 . Temperatures of 343 K and higher occur in industrial absorption columns. Therefore, having access to raw absorption flux data at higher temperatures (e.g., found in [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF]) is of interest to extend kinetic law application range. 4. The four selected points cover the examined data experimental conditions (type and concentration of salt, temperature, CO 2 partial pressure). As a reminder, CO 2 concentration is normalised by c 1,0 , its concentration at the interface, while all other concentrations are normalised by c abs,tot = c (Alk+) . Concentration and extent of reaction profiles calculated with non-ideality consideration (solid lines and symbols in Figure 4 to Figure 7) or with ideal solution hypothesis (dashed lines in Figure 4 to Figure 7) display similar trends. CO 2 concentration profiles are especially close in both models. Consequently, resulting Arrhenius parameters are very close in both models as well. As expected, in these systems and at these conditions, calculated activity coefficient profiles are mostly constant but largely different from unity. Other more complex systems, such as acid gas absorption into aqueous amine solutions, could lead to higher non-ideality influence. [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] experiments are compatible with pseudo-first order absorption regime assumption (Figure 4 and Figure 5). Concentration profiles are compatible with this assumption: HO - concentration barely drops at the interface. The main reaction product is not HCO 3 -, but CO 3 2-, and extents of both reactions are close along the liquid film. They are even the equal in Figure 5. This is because CO 3 2-(pKb(25°C) = 3,7) is more basic than HCO 3 -(pKb(25°C) = 7,3). At these conditions, activity coefficients are constant, as the overall solution composition remains close to bulk composition everywhere. Still, activity coefficients differ significantly from unity and non ideality needs to be considered. Therefore, according to the model, [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] absorption flux measurements can be interpreted with constant activity coefficients (as done by [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] and [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF]). The same behaviour, but amplified, is displayed in Figure 5. HO - excess is much more important, therefore, reaction R-II equilibrium shifts toward CO 3 2-formation. Activity coefficient divergence from unity is significantly higher, due to very high ionic strength. In Figure 6, resulting CO 2 absorption profiles are shown in aqueous Na 2 CO 3 at conditions researched by [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF]. In this case, the main reaction product is HCO 3 -from CO 3 2-conversion. This is also coherent with a negative extent of reaction R-II. Here, reversibility needs to be considered, which the model does, unlike models from the literature (e.g., [START_REF] Knuutila | Kinetics of the reaction of carbon dioxide with aqueous sodium and potassium carbonate solutions[END_REF], [START_REF] Gondal | Activity based kinetics of CO2--OH-systems with Li+, Na+ and K+ counter ions[END_REF]).

Activity coefficients display a gradient along the liquid film in Figure 7, with a maximum of 14% decrease from liquid bulk to interface for γ (CO32-) . In [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF], experimental CO 2 partial pressure is of several orders of magnitude higher than in [START_REF] Gondal | Kinetics of the absorption of carbon dioxide into aqueous hydroxides of lithium, sodium and potassium and blends of hydroxides and carbonates[END_REF] -100 vs. 0.2 kPa. HO - concentration drops significantly along the liquid film, more so close to the interface, in a similar behaviour to the instantaneous absorption regime. CO 3 2-is consumed in the vicinity of the interface. At these experimental conditions, it is important to consider the reaction mechanism with both reactions and varying activity coefficients along the liquid film. The high absorption driving force results in ionic strength variation. These conditions are closer to those expected in an industrial setting. This shows how the model could accurately represent mass transfer phenomena at industrial conditions.

Figure 6 Simulated absorption of CO 2 in Na 2 CO 3 aqueous solution at 0.06 mol.L -1 , 298 K and P CO2 = 100 kPa measured by [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF] [START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF]

Conclusion

This study implements a steady-state reactive absorption model applied to the stagnant film theory and focusing on solution non-ideality representation in equilibrium relations, Nernst-Planck diffusion fluxes and reaction rates. This approach is more complete than literature models where activity coefficients are assumed constant, and most often equal to unity. Extents of reactions, a new feature for reactive absorption models, reduces the number of differential problem variables while maintaining a general formulation. This formulation with Nernst-Planck equation for molecular diffusion and activity-based rate expressions consists in a system of n C +n R,kin first-order ordinary differential equations, whereas writing local balances per species leads to a system of n C secondorder differential equations. The relation between fluxes and extents of reactions is dictated by stochiometric constraints. This model is first applied to alkaline salts-water-CO 2 systems and the CO 2 absorption flux data from the literature. Arrhenius parameters for CO 2 + HO -↔ HCO 3 -direct kinetic constant are fitted. This is a necessary step to study amine-based systems in the same modelling framework. This progressive approach is made possible by the kinetic data availability in alkaline salts-water-CO 2 systems. The obtained Arrhenius parameters are coherent to previous studies in the regression temperature interval. Overall AAD is 12%. Besides, the consideration of activity coefficient variation along the liquid film is shown to be relevant when experimental conditions diverge from the simple pseudofirst order absorption regime. These conditions are closer to the industrial setting.

Model improvement perspectives include extending mass transfer representation to steady-state eddy-diffusivity theory, and assessing uncoupled ion diffusivities influence.

With this model, reaction mechanism investigation in any reactive absorption system is possible. Our further goal is to apply the model to CO 2 absorption in aqueous amines solutions (namely MDEA), especially for increasing CO 2 partial pressure and loading. In these solutions, the more complex reaction scheme could lead to higher variations in ionic strength and activity coefficients through the liquid film.

  and Figure2, respectively.

Figure 1 Figure 2

 12 Figure 1 Reactive absorption model parity plot for alkaline salts-water-CO 2 systems

Figure 3

 3 Figure 3 Arrhenius plot and comparison to previous studies Activity-based model results at specific conditions are then examined. Theses conditions are listed in in Table4. The four selected points cover the examined data experimental conditions (type and concentration of salt, temperature, CO 2 partial pressure). As a reminder, CO 2 concentration is normalised by c 1,0 , its concentration at the interface, while all other concentrations are normalised by c abs,tot = c (Alk+) .

Figure 4 Figure 5

 45 Figure 4 Simulated absorption of CO 2 in LiOH aqueous solution at 0.01 mol.L -1 , 336 K and P CO2 = 0.2 kPa measured by (Gondal, et al., 2015). C: normalised concentration, ξ: extent of reaction, γ: activity coefficient. Solid line & symbol: activity-based model, dashed line: concentration-based model (C and ξ). γ Li+ = 0.88

Figure 7

 7 Figure 6Simulated absorption of CO 2 in Na 2 CO 3 aqueous solution at 0.06 mol.L -1 , 298 K and P CO2 = 100 kPa measured by[START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF]. C: normalised concentration, ξ: extent of reaction, γ: activity coefficient. Solid line & symbol: activity-based model, dashed line: concentration-based model (C and ξ). γ Na+ = 0.68

  Figure 7Simulated absorption of CO 2 in NaOH aqueous solution at 0.13 mol.L -1 , 303 K and P CO2 = 100 kPa measured by[START_REF] Hikita | Absorption of carbon dioxide into aqueous sodium hydroxide and sodium carbonate-bicarbonate solutions[END_REF]. C: normalised concentration, ξ: extent of reaction, γ: activity coefficient. Solid line & symbol: activity-based model, dashed line: concentration-based model (C and ξ). γ Na+ = 0.76

  Chemical absorption into basic solutions has been in use since the 1930s for CO 2 separation from raw natural gas or from flue gases

	1. Introduction
		and superscripts
	app	Apparent
	calc	Calculated
	exp	Experimental
	REF	Reference state
	1	Absorbed gas
	¨	Molality-based asymmetric thermodynamic convention (see Supplementary Material section
	2)
	~c	Molar concentration-based symmetric thermodynamic convention

Table 1

 1 Film model boundary conditions with dimensionless variables -This formulation with Nernst-Planck equation for molecular diffusion and activity-based rate expressions consists in a system of n C +n R,kin first-order ordinary differential equations, thanks to steady-state extents of reaction, whereas writing local balances per species leads to a system of n C second-order differential equations. Extents of reactions at equilibrium are calculated from a set of n R,eq algebraic equations (eq. 23). Equilibrium relations, Nernst-Planck diffusion fluxes and reaction rates are activity-based. Therefore, this model reconciles kinetics and thermodynamics especially close to equilibrium. Simulations are performed in Matlab R2013a. The bvp4c function available in the software is used to solve the differential problem

Table 2

 2 Available CO 2 absorption flux data in alkaline hydroxide and carbonate aqueous solutions

	Ref, apparatus	Alk	+	x(Alk	+ ,app)	x(HO	-,app)	x(CO 2 ,app)	T (K)	P CO2 (kPa)	n data

  CO 2 absorption flux data in alkaline hydroxide solutions and performance

	Model		Concentration-based	Activity-based	(Gondal, et al., 2016)
		A	19.387 ± 0.004 19.841 ± 0.003	19.53
	Reference	E A /R (K)	-5020.99 ± 0.07 -5248.77 ± 0.05	-5111.2

Table 4

 4 Selected points for concentration, extent of reaction and activity coefficient profile simulation: system and conditions

	Figure	Reference	Salt	c salt (mol.L	-1 )	T (K)	P CO2 (kPa)	k L,1 (m.s -1 )	c 1,0 (mol.L	-1 )*
	Figure 4 Figure 5 Figure 6 Figure 7	(Gondal, et al., 2015) (Gondal, et al., 2015) (Hikita, et al., 1976) (Hikita, et al., 1976)	LiOH KOH Na 2 CO 3 0.06 0.01 1.76 NaOH 0.13		336 0.2 317 0.2 298 100 303 100	1.7.10 -4 1.1.10 -4 5.0.10 -5 4.5.10 -5	-5 3.1.10 -5 2.6.10 -3 32.0.10 28.2.10 -3
	*Calculated variable (eq. 18), in the activity-based model

Acknowledgements

Pr. Alain Gaunand and Pr. Christophe Coquelet would like to acknowledge TotalEnergies for their research partnership and financial support, and the authors would like to acknowledge the Agence Nationale de la Recherche et de la Technologie (ANRT) for their financial support (Cifre 2018/1069).


Dest Mscc

Harvest Sowing MSCC Destruction MSCC Sowing Development period Harvest Sowing MSCC Destruction MSCC Sowing

Depending on: cropping system, main goals, soil and climate conditions, … some examples of insertion of MSCC in cropping systems

Case 1: on clay soils in conventionnaly-tilled cropping systems ex. destruction of bristle oat in November Case 2: on loamy soils in cropping systems with reduce/no tillage ex. direct sowing of maize within fababean p.

Depending on: cropping system, main goals, soil and climate conditions, … some examples of insertion of MSCC in cropping systems

Case 3: sowing of the cover crop at the end of the summer to avoid the lack of water (but not taking advantage of the most intense period for the photosynthetic activity...) ex. white mustard sown at the end of August Case 4: sowing after a spring crop harvested in autumn: the choice of adapted species and varieties is quite reduced... Depending on: cropping system, main goals, soil and climate conditions, …

ex. fababean between two maize

Case 7: relay cropping of cover crops ex. forage sorghum sown after harvesting a cereal and then sowing a faba bean + pea in the forage sorghum fully restored to the plot

on: cropping system, main goals, soil and climate conditions, … some examples of insertion of MSCC in cropping systems Case 5: implantation

	Development period July August September October November December January February March April Harvest Sowing MSCC June Destruction MSCC Sowing June July August September October November December January February March April Depending Development period March April May Case 1 March April May Harvest Sowing MSCC Destruction MSCC Sowing Case 1
	Case 2 Case 2	Harvest Sowing MSCC Harvest Sowing MSCC	Development period Development period	Destruction MSCC Sowing Destruction MSCC Sowing
	Case 3 Case 3	Harvest Harvest	Sowing Sowing	Dev period Dev period	Destruc Destruc			Sowing Sowing
	Case 4 Case 4			Harvest Sowing MSCC Harvest Sowing MSCC	Development period Development period	Dest. MSCC Sowing Dest. MSCC Sowing
	Case 5 Case 5	Sowing MSCC Harvest Sowing MSCC Harvest	Development period Development period		Destruction MSCC Destruction MSCC	Sowing Sowing
	Case 6	Sowing Sowing MSCC	Development period 1	Harvest	Development period 2	Dest. MSCC	Sowing
								p. 5

Titre de la présentation Date / information / nom de l'auteur A wide range of sowing and destruction periods of MSCC What is the potential of cover crops and what do farmers say? 22/10/2021 of the cover inside the cash crop at the end of its cycle ex. clover sown within a soft wheat p. 6 Titre de la présentation Date / information / nom de l'auteur A wide range of sowing and destruction periods of MSCC What is the potential of cover crops and what do farmers say? 22/10/2021

Depending on: cropping system, main goals, soil and climate conditions, … some examples of insertion of MSCC in cropping systems Case 6: early

  undersown cover crop in a cash crop ex. Crimson clover undersown in maize at the 6-8 leaf stage

	Case 3	Harvest	Sowing	Dev period
	Case 7	Harvest	MSCC 1	MSCC 2	Sowing

Destruc Sowing p. 7 Titre de la présentation Date / information / nom de l'auteur A wide range of sowing and destruction periods of MSCC What is the potential of cover crops and what do farmers say? 22/10/2021

 There is a to effectively introduce cover crops at different times in cropping systems  A multi-criteria approach is necessary to reason their insertion  The amount of carbon captured by cover crops depends on (i) the of the plant, which varies according to species and phenological stage, and (ii) the produced, which depends on the species, varieties and practices used  A better knowledge of the is necessary to better choose them and thus optimize their performance  Plant breeding efforts on these plants now offer a wide range of possibilities that should be mobilized to p. 17 Thank you for your attention.