Majid Mehrasa
email: majid.mehrasa@g2elab.grenoble-inp.fr

Khaled Hajar
email: khaled.hajar@grenoble-inp.fr

Mehrdad Gholami
email: mehrdad.gholami@grenoble-inp.fr

Antoine Labonne
email: antoine.labonne@g2elab.grenoble-inp.fr

Ahmad Hably
email: ahmad.hably@grenoble-inp.fr

Seddik Bacha
email: seddik.bacha@g2elab.grenoble-inp.fr.

Reza Razi
email: reza.razi@grenoble-inp.fr

A Dynamic Real-Time Optimization Algorithm for the Revenue Assessment of a Vehicle-To-Grid System in Presence of Wear Cost Model

come

Introduction

Nowadays, Vehicle-to-Grid system has been paid attention due to the technological needs of various structures of power systems including microgrids [START_REF] Razi | A Novel Graph-Based Routing Algorithm in Residential Multimicrogrid Systems[END_REF][START_REF] Holari | Power Management Using Robust Control Strategy in Hybrid Microgrid for both Grid-Connected and Islanding Modes[END_REF] and smart grids [START_REF] Das | Charging Scheduling of Electric Vehicle Incorporating Grid-to-Vehicle and Vehicle-to-Grid Technology Considering in Smart Grid[END_REF]. The smart charging/discharging strategies designed for Electric Vehicles (EVs) as a vital part of V2G systems have faced different kinds of technical and control design challenges [START_REF] Ovalle | Grid Optimal Integration of Electric Vehicles: Examples with Matlab Implementation[END_REF][START_REF] Turker | Optimal Minimization of Plug-In Electric Vehicle Charging Cost With Vehicle-to-Home and Vehicle-to-Grid Concepts[END_REF]. In order to attain reliable and cost-effective smart grids in presence of smart charged EVs, these challenges must be taken into account to the extent possible. One of these challenges is associated with the battery degradation [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF][START_REF] Lehtola | Electric Vehicle Battery Cell Cycle Aging in Vehicle to Grid Operations: A Review[END_REF]. A battery aging management system was proposed by [START_REF] Corno | Active Adaptive Battery Aging Management for Electric Vehicles[END_REF] to regulate the battery capacity degradation for EVs by noticing the driving range, recharge time, drivability, maximum battery current, and the depth of discharge. Ref [START_REF] Kim | Battery-Wear-Model-Based Energy Trading in Electric Vehicles: A Naive Auction Model and a Market Analysis[END_REF] concentrated on an auction-based energy commerce for reaching a specified optimized energy sharing to minimize the final payment of the present EVs by considering their practical battery condition. A model-free deep reinforcement learning was proposed in [START_REF] Cao | Deep Reinforcement Learning-Based Energy Storage Arbitrage with Accurate Lithium-Ion Battery Degradation Model[END_REF] to accurately estimate the battery degradation cost while the control challenge was shaped using Markov Decision Process (MDP) as well as the charging/discharging technique was designed based on a noisy network based deep reinforcement learning method. A vehicle-to-grid (V2G) system was optimized in [START_REF] Ginigeme | Distributed Optimal Vehicle-To-Grid Approaches with Consideration of Battery Degradation Cost Under Real-Time Pricing[END_REF] through a multi-objective function with variables and specifications including the peak demand, load variations, the battery degradation cost, EV charging/discharging cost, the driving distance of EV, the driving periods of EV, and the charging/discharging levels and also locations. The disparate vehicle-to-building (V2B) applications were discussed in [START_REF] Nazari | Electric Vehicles for Smart Buildings: A Survey on Applications, Energy Management Methods, and Battery Degradation[END_REF] wherein the battery capacity faded due to the bidirectional power flow and also the extra discharging cycles with V2B were taken into account leading to the battery degradation integration into energy management development. Apart from the importance of battery lifespan consideration for EVs [START_REF] Connor | Impact of Connectivity on Energy Consumption and Battery Life for Electric Vehicles[END_REF], selecting an appropriate optimization method has also a key role at providing a suitable charging/discharging strategy for a V2G system [START_REF] Ovalle | An Electric Vehicle Load Management Application of the Mixed Strategist Dynamics and the Maximum Entropy Principle[END_REF][START_REF] Razi | A User-Friendly Smart Charging Algorithm Based on Energy-Awareness for Different PEV Parking Scenarios[END_REF].

The linear programming algorithm that is taken into account as a solver for optimization problem and global optimal solutions [START_REF] Kwon | Optimal V2G and Route Scheduling of Mobile Energy Storage Devices Using a Linear Transit Model to Reduce Electricity and Transportation Energy Losses[END_REF][START_REF] Razi | Limiting Discharge Cycles Numbers for Plug-in Electric Vehicles in Bidirectional Smart Charging Algorithm[END_REF], have been applied to many V2G systems in disparate conditions [START_REF] Wang | Coordinated Planning Strategy for Electric Vehicle Charging Stations and Coupled Traffic-Electric Networks[END_REF]. In an on-demand EV fleet, a binary linear programming has combined with deep reinforcement learning to provide an optimal solution for a Markov decision process aiming to the EV fleet operator welfare maximization [START_REF] Liang | Mobility-Aware Scheduling for Shared On-Demand Electric Vehicle Fleet Using Deep Reinforcement Learning[END_REF]. In Ref [START_REF] Zeng | Bilevel Programming Approach for Optimal Planning Design of EV Charging Station[END_REF], a bi-level optimization algorithm was redrafted as an equivalent single-level linear program for an optimized planning process of a EV charging station to reach optimal configuration of the renewable energy resources (RCS), the maximized total profits of RCS owner, and the planned charging intentions of EV users. A single mixed integer linear programming algorithm acted as a model predictive energy management system in [START_REF] Chandra Mouli | Integrated PV Charging of EV Fleet Based on Energy Prices, V2G, and Offer of Reserves[END_REF] for the EV fleet charging process from PV panels to provide the grid stability, employ a single EV charger for charging numerous EVs, and prevent the grid overload. It was also aimed in [START_REF] Yao | Joint Routing and Charging Problem of Multiple Electric Vehicles: A Fast Optimization Algorithm[END_REF] to divide a mixed integer programming algorithm into two linear programming problems reaching an effective charging method for multiple EVs. In spite of the related discussions regarding the EVs-based and battery-based systems [START_REF] Mehrasa | Power Management of a Smart Vehicle-to-Grid (V2G) System Using Fuzzy Logic Approach[END_REF][START_REF] Gholami | An Efficient Control Strategy for the Hybrid Wind-Battery System to Improve Battery Performance and Lifetime[END_REF][START_REF] Hajar | Smart Charging Impact on Electric Vehicles in Presence of Photovoltaics[END_REF], in this paper, a linear programming optimization algorithm with changeable weighting factors is presented for a smart Vehicle-to-Grid (V2G) system to reach maximum revenue in the peak-value duration of the PV power and electricity price. The paper is organized as follows. The introduction is given in Section I. Section II focuses on the wear model of EV battery. In section III, the proposed dynamic optimization algorithm is discussed. Simulation results and conclusions are written in Sections IV and V, respectively.

Wear Model of EV Battery

The estimation process of the lithium-ion battery actual lifespan can be executed using two major factors including: 1) Calendar life that determines how many years the lithium-ion battery can be expected to be retained. 2) Cycle life that indicates the number of charge-discharge cycles wherein the lithium-ion battery is able to possess a persistent function before its exploitable capacity reaches 80% of its initial rated capacity. However, the calendar life can somewhat confine the actual lifetime, but this life is not taken into account in the battery degradation formulation. It is worth mentioning that if the cycle life is higher than the calendar life, the V2G/G2V cycles can be increased to provide more revenue for the EV owner. The following components will affect the battery wear for both V2G and G2V applications [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF]:

 Temperature: When the resistance of the electrode film in the lithium-ion batteries is raised, the charging capacity is mitigated with an accelerated ramp. In order to hamper the resistance increment, the ambient temperature, as well as the charge and discharge currents, must be placed within acceptable ranges.  Charge and Discharge Currents: High charge and discharge currents will speed up the lithium-ion battery degradation.  Depth of Discharge (DOD): The DOD is equalized to (1-SOC) meaning that as DOD increases, the SOC decreases.  Cycle Numbers: High cycle number along with high charge-discharge current can make maximum degree of battery degradation. Since it is assumed that the temperature is attempted to remain at standard level and the charge and discharge currents are kept below the moderate value, only DOD effect is employed to attain the Daily Wear Cost (DWC) [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF]. The relation [START_REF] Razi | A Novel Graph-Based Routing Algorithm in Residential Multimicrogrid Systems[END_REF]

            (2)
In (2), the coefficient k is deliberated for when the higher capacity loss occurs in the driving state. The Kes is named as the discharge coefficient that can be 0.00015 (kW•h/kW•h) with an acceptable approximation. Fig. 1 depicts the estimated cycle life based on the factors EDRV and EDIS while coefficient k is diminished. The three-dimension curves in Fig. 1 verify that the γECL can reach its maximum value when both factors EDRV and EDIS approach to their minimum values. Noticing the inequalities k1>k2>k3>k4>k5 as the variation trend of k in Fig. 1, it can be found that the estimated cycle life increment can be intensified through the decrement of the coefficient k. It can be also investigated how the wear cost can impact on the cycle life. To this end, the relation ( 1) is rewritten to obtain the ECL as a function of DWC.

 

1 EDD EDD ECL EDD DWC BSV Ln DWC BCC Ln               (3)
Where Ln(.) is the natural logarithm (Ln(.)=loge(.), e=2.718281828459). In order to reach authentic relation for (3), the equivalent daily discount must comply with the conditions (4).

Assuming β= BCC/BSV, the estimated cycle life is plotted in Fig. 2 based on the daily wear cost while β is increased or decreased accordingly. In this figure, while β1(β2)<<<<1 (meaning that BCC<<<<BSV), the estimated cycle life will be equalized to negative which is an unacceptable outcome. But, when β>>>>1 that happens for {β3<β4<β5< β6}, the estimated cycle life can be significantly increased as well.

Proposed Dynamic Real-Time Optimization Algorithm

The Vehicle-to-Grid (V2G) system under study is shown in Fig. 3. There are three types of electricity sources (ESs) in this system including 1) The PV panels as the ESs that only generate power, 2) The variable load as an ES that only consumes power, and 3) The power grid and EVs as other It is realized from the categorized ESs that the flexible power exchange can be provided by both power grid and EVs. The relation between the flexible power generation units and other ESs is first written as follows,

3 1 i Grid EV Load PV i P P P P      (5)
The relation ( 5) is regarded as one of the optimization constraints. The forecasted profiles shown in Fig. 4(a), and (b) are employed respectively for load (PLoad), and PV (PPV). The grid power is demonstrated by PGrid and the i th EV has the power P i EV. Since there is a flexible power feature for the grid and EVs, the objective function can be fortified by assigning dynamic coefficients to the related optimization variables. For this reason, a dynamic price profile (Cgrid(t)) for the grid power is aimed in the proposed objective function [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF] as depicted in Fig. 5(a

                          (6)
The set of {λ 1 EV, λ 2 EV, λ 3 EV} are the dynamic coefficients of EV battery power. The V2G system in Fig. 3 is persistently encouraged to max out the utilization of PV power during the smart charging process. Noticing this aim, the dynamic coefficients [START_REF] Lehtola | Electric Vehicle Battery Cell Cycle Aging in Vehicle to Grid Operations: A Review[END_REF] are proposed for the EV battery power.

                          (7) (a) (b) (c)
Fig. 5 The dynamic coefficients for the optimization variables including (a) The grid power, (b) The EVs power for cases 1, 3 and 4, and (c) The EVs power for case 2. Supposing i=1,2,3, four cases are defined to obtain the dynamic coefficients (λ i EV). These cases that are given as follows, are exerted to the relation of the dynamic coefficeint in [START_REF] Lehtola | Electric Vehicle Battery Cell Cycle Aging in Vehicle to Grid Operations: A Review[END_REF] leading to the results depicted in Figs. 5(b) and(c).

 Case1: α 1 EVi=1, α 2 EVi=10000, and α 3 EVi=1.

 Case2: α 1 EVi=1, α 2 EVi=↑(increasing), and α 3 EVi=1.

 Case3: α 1 EVi=↑(increasing), α 2 EVi=10000, α 3

EVi=1.

 Case4: α 1 EVi=1, α 2 EVi=10000, α 3 EVi=↑(increasing).

In the case 1, all the parameters α 1 EVi, α 2 EVi, and α 3 EVi are assumed to be constant. The importance of the parameter α 2 EVi is specified when the PV power reaches zero. Consequently, in this condition, the dynamic coefficients λ i EV are equalized to 1/α 2 EVi. The dynamic coefficient in case 1 during the simulation interval is seen in Fig. 5(b). By noticing the PV panel power profile in Fig. 4(b), it can be realized that the coefficient in case 1 is completely inverted in comparison with the PV power profile. Based on Fig. 5(b), the EV battery power dynamic coefficients for the case 3 is more fluctuated compared to other coefficients. It may lead somewhat to the flucutaed optimal responses for the EV power as the solutions of the proposed linear optimization problem. However, as can be undrestood from Fig. 5(b), the case 3 provides smaller coefficients during the optimization run time compared to the coefficients achieved from the first case. When the PV power is placed within its peak value, the case 4 will enforce the dynamic coefficients λ i EV to reach its smallest values to the extent possible in comparison with other cases as depicted in Fig. 5(b). According to this figure, irregular inverse curve compared to the PV power profile is indeed obtained for the dynamic coefficients of EV battery power in the fourth case. On the other hand, the dynamic coefficients λ i EV for the case 2 is approximately kept at its smaller values that is not suitable for the variable profile of the PV power.

Simulation Results

The V2G system shown in Fig. 3 is simulated in MATLAB/Simulink environment. The EVs parameters are given in Table I. The first part concentrates on comparing the revenue achieved from the constant weighting factors with the revenue due to the proposed dynamic weighting factors. In next part, the effects of the EV battery wear cost model is taken into consideration as well for assessing the revenue obtained from the dynamic weighting factors-based optimization algorithm. Fig. 6 illustrates the revenue obtained by using various values of the constant weighting factors in the proposed objective function. As it can be understood from this figure, when the constant weighting factors λ i EV are chosen as small as possible, more revenue in the peak-value duration of the PV power can be obtained. However, this decreasing trend does not impact the revenue in both the peak-value duration of electricity price and the end of the simulation. It is worth mentioning that the extent of this decrement is restricted because of the charging/discharging cycle numbers as will be discussed in the final version of the paper. In Fig. 7, the revenue achieved from various values of the coefficients involved with the dynamic weighting factors are exhibited. The inequalities and equalities that are exerted for attaining these results are

(α 2 EVi)2 > (α 2 EVi)1 > (α 2 EVi)3 and (α 3 EVi)2=(α 3 EVi)3 > (α 3 EVi)1.
Noticing the logical relations, the small and big values for respectively α 2 EVi and α 3 EVi, can lead to higher revenue within the peak-value duration of the PV power as depicted in Fig. 7. By paying attention to both Fig. 6 and Fig. 7, it can be realized that using the dynamic weighting factors provides at least around two times more revenue in comparison with when the constant ones are employed. The accurate comparison can be carried out when the wear cost model of the EV battery is also considered.

Revenue Assessment in Presence of Wear Cost Model

In this sub-section, the revenue achieved from the proposed smart charging with dynamic weighting factors in presence of wear cost model is assessed. The battery and EV specifications are extracted from references [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF] and [START_REF] Zhou | Modeling of the Cost of EV Battery Wear Due to V2G Application in Power Systems[END_REF]. The cost per kW. h due to wear model is calculated through the relation (KesDWC)/ϕ. Fig. 8 presents the optimal power of EVs while the dynamic weighting factors in [START_REF] Lehtola | Electric Vehicle Battery Cell Cycle Aging in Vehicle to Grid Operations: A Review[END_REF] are used for the objective function [START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF]. The revenues of these results for the EVs power when the wear model is/is not considered are illustrated in Fig. 9. As it can be understood from Fig. 8, the aims of proposed smart charging strategy are obtained. However, when the wear cost model is taken into account, the revenue is decreased as depicted in Fig. 9.

Conclusions

A dynamic optimization algorithm with changeable weighting factors has been proposed in this paper to attain its maximum revenue in three times i.e., the peak-value duration of the PV power, the peak-value duration of electricity price and the end of the simulation. The EV battery wear model has also exerted to the revenue achieved from optimization algorithm using the parameters including the equivalent daily discount, estimated cycle life, the battery capital cost and battery salvation value. The proposed dynamic optimization algorithm has consisted of a linear objective function with a changing pre-defined electricity price and the dynamic weighting factors for the EV power variables formed based on the forecasted PV power profile. Simulation results without considering the wear cost model have shown the high ability of proposed algorithm at reaching the specified aims. The results along with considering the EV battery wear model will be presented in the final version of the paper.

Fig. 1

 1 Fig. 1 The estimated cycle life based on the factors EDRV and EDIS while

Fig. 2

 2 Fig.2The estimated cycle life based on the daily wear cost while β is increased or decreased.

Fig. 3

 3 Fig.3The V2G system including EV station, power grid, load, and PV panels.

Fig. 4

 4 Fig.[START_REF] Ovalle | Grid Optimal Integration of Electric Vehicles: Examples with Matlab Implementation[END_REF] The power profiles for (a) Variable load, and (b) PV panels.

Fig. 6

 6 Fig.[START_REF] Farzin | A Practical Scheme to Involve Degradation Cost of Lithium-Ion Batteries in Vehicle-to-Grid Applications[END_REF] The revenue achieved from various values of the constant weighting factors.

Fig. 7

 7 Fig.[START_REF] Lehtola | Electric Vehicle Battery Cell Cycle Aging in Vehicle to Grid Operations: A Review[END_REF] The revenue achieved from various values of the coefficients involved with the dynamic weighting factors.

Fig. 8

 8 Fig. 8 Optimal power of EVs while the dynamic weighting factors are used.

Fig. 9

 9 Fig.9Revenue in the states of with/without wear model.

  is used to calculate DWC wherein the λEDD and γECL are respectively the Equivalent Daily Discount (EDD) and Estimated Cycle Life (ECL).

	DWC		 1 11    ECL ECL BCC    EDD  EDD EDD   	BSV		$/ US day
	ECL			0.2	0 E days ()
	K	es		  1 k E	DRV	2	E	DIS		( . / kW h day	)

(1) In addition, the BCC and BSV are respectively the Battery Capital Cost and Battery Salvation Value while the cycle life is ended. The ECL is achieved through

[START_REF] Holari | Power Management Using Robust Control Strategy in Hybrid Microgrid for both Grid-Connected and Islanding Modes[END_REF]

. It is assumed that the EVs with initial energy (E0) possess the average daily amounts of the consumed energy in the driving and discharging modes respectively as EDRV and EDIS.

  ).

	Min	t	t	2 EV ( ) grid C t P t 2 0 ( EV P t ( ) grid	t	11 3 EV EV EV P t 0 ( EV 3 ( P t  )	0	t	)	t	)

Table 1

 1 The V2G system parameters

	The EVs	EV1	EV2	EV3
	Max and Min SOC	20%, 90%	20%, 90%	20%, 90%
	Nominal EV Energy 55 kWh	48.2 kWh	55kWh
	Max and Min P i EV	-8.5/8.5 kW	-25/25 kW	-15/15 kW
	Arrival Time	09:15	11:00	03:30
	Departure Time	21:15	23:30	23:00

Acknowledgements This work has been supported by the aVEnir project of the PIA operated by ADEME.


Pedro Arana-Agudelo

Marwen Moussa
email: marwen.moussa@agroparistech.fr

Ioan-Cristian Trelea

Kevin Lachin

Violaine Athès

Reactive liquid-liquid extraction of bio-based 3-hydroxypropionic acid using a biocompatible organic phase containing a tertiary amine: a model-based approach to elucidate dominant mechanisms

Keywords: organic acids, N, N-didodecylmethylamine (DDMA), apparent amine basicity, equilibrium modeling, infrared spectroscopy Nomenclature

This work studies reactive liquid-liquid extraction of 3-hydroxypropionic acid (3-HP) from aqueous solutions using a biocompatible organic phase consisting in N,Ndidodecylmethylamine (DDMA) (20% v/v) diluted in 1-dodecanol (40% v/v) and dodecane (40% v/v). The objective was to propose an equilibrium model based on the law of mass action accounting for the main phenomena occurring in the system. A set of equilibrium extraction experiments was performed in the initial acid concentration range 0.0028 -1 mol L -1 in order to determine the extraction yield, equilibrium pH and to collect infrared spectra of the loaded organic phases. FT-IR spectra allowed to identify and elucidate the main mechanisms of acid-amine interaction, which were taken into account in the model. The formation of 1:1 acid-amine stoichiometry complex through ion-pairing was determined as the dominant mechanism in all the concentration range. For initial acid concentrations above 0.11 mol L -1 , the formation of 2:1 acid-amine complexes through H-bonding of acid molecules with 1:1 complexes was also observed and determined to be almost as important as ion-pair formation at 1 mol L -1 . The presence of impurities in the organic phase was taken into account by introducing a side reaction essential to represent low extraction yields and high equilibrium pH values observed at low initial acid concentrations ( 0.02 mol L -1 ). A global sensitivity analysis of the model's parameters pointed out three concentration regions based on the relative contribution of each phenomenon: (i) for initial acid concentrations below 0.02 mol L -1 , the main phenomena are 1:1 ion-pair formation and the side reaction; (ii) between 0.02 mol L -1 and 0.11 mol L -1 , mainly 1:1 ion-pair formation; and (iii), for concentrations above 0.11 mol L -1 both considered complexation reactions i.e., 1:1 ion-pair formation and 2:1 Hbonding. The proposed model allowed thus to evaluate the relative importance of the phenomena occurring in the system in all the studied acid concentration range.

Introduction

Due to growing concerns about climate change and the dependence on fossil resources [START_REF] Mcglade | The geographical distribution of fossil fuels unused when limiting global warming to 2 °C[END_REF], there is an increasing interest in the substitution of crude oil derivatives by bio-based products obtained from renewable feedstocks [START_REF] Stegmann | The circular bioeconomy : Its elements and role in European bioeconomy clusters[END_REF]. Historically, organic acids have been widely used in industry. The bioeconomy context has reinforced the interest in their production through biotechnological routes from renewable resources, for the development of bio-based materials [START_REF] Panda | Research and production of organic acids and industrial potential[END_REF]. Carboxylic acid biological production is inhibited by acid accumulation during the process, limiting both its productivity and titer. The inhibition can be alleviated by the implementation, among other techniques, of In Situ or In Stream Product Recovery (ISPR) processes. On the subject of extractive production of carboxylic acids, reactive liquid-liquid extraction has been extensively studied due to its high selectivity even at low acid titers, mild operation conditions, the possibility of continuous extractant regeneration and its low energy consumption [START_REF] Kumar | Reactive extraction as an intensifying approach for the recovery of organic acids from aqueous solution: a comprehensive review on experimental and theoretical studies[END_REF][START_REF] Wasewar | Reactive extraction: an intensifying approach for carboxylic acid separation[END_REF].

Several extractants have been studied for the recovery of organic acids in literature.

Kertes and King [START_REF] Kertes | Extraction chemistry of fermentation product carboxylic acids[END_REF] classified the different extractants into (i) carbon-bonded oxygendonor, (ii) phosphorous-bonded oxygen-donor and (iii) high molecular weight aliphatic amines. Among the diversity of extractants suitable for this application, tertiary aliphatic amines solubilized in organic solvents are the most common. Tertiary amines are highly hydrophobic molecules that interact with the acid present in aqueous phase through successive reversible complexation reactions at the liquidliquid interface. The distribution ratios that can be achieved using tertiary amines are usually higher than those obtained with other extractants when the organic phase is properly formulated [START_REF] Kurzrock | New reactive extraction systems for separation of bio-succinic acid[END_REF]. As amines show poor solvating properties, the formulation of organic phases generally includes an active diluent such as long-chain alcohols, capable of strongly solvating the acid-amine complexes through polar interactions (e.g. formation of H-bonds) [START_REF] Inyang | Kinetic studies on propionic and malic acid reactive extraction using trioctylamine in 1-decanol[END_REF].

3-hydroxypropionic acid (3-HP) emerged as one of the most promising chemical building blocks for the production of acrylic acid, biopolymers, and other chemicals with a wide range of applications in industry [START_REF] Werpy | Top value added chemicals from biomass Volume I -Results of screening for potential candidates from sugars and synthesis gas top value added chemicals from biomass[END_REF][START_REF] Bozell | Cutting-edge research for a greener sustainable future Technology development for the production of biobased products from biorefinery carbohydratesthe US Department of Energy ' s " Top 10 " revisited[END_REF][START_REF] Jers | Production of 3-hydroxypropanoic acid from glycerol by metabolically engineered bacteria[END_REF]. In this context, the interest in microbial production of 3-HP has increased in the last years [START_REF] De Fouchécour | Process engineering for microbial production of 3-hydroxypropionic acid[END_REF][START_REF] Bhagwat | Sustainable production of acrylic Acid via 3-hydroxypropionic acid from lignocellulosic biomass[END_REF]. 3-HP recovery from microorganisms culture broths is still challenging because of the hydrophilic nature of this molecule 0.89) and the low titers reached in the process( 10% w/w) due to low pH and acid inhibition [START_REF] Chemarin | Recovery of 3hydroxypropionic acid from organic phases after reactive extraction with amines in an alcohol-type solvent[END_REF]. This is why the implementation of reactive liquid-liquid extraction shows encouraging prospects.

The main drawback of reactive liquid-liquid extraction as ISPR technique remains the toxicity of the extraction phases in close contact with culture media. Recently, several studies have been published on the screening and formulation of organic phases accounting for the biocompatibility with some bacteria of the genus Lactobacillus [START_REF] Sánchez-Castañeda | Organic phase screening for in-stream reactive extraction of bio-based 3hydroxypropionic acid: biocompatibility and extraction performances[END_REF][START_REF] Pérez | Liquid -liquid equilibrium and molecular toxicity of active and inert diluents of the organic mixture tri-iso-octylamine/dodecanol / dodecane as a potential liquid membrane for lactic acid removal[END_REF]. In spite of the identification of more biocompatible organic phases, most experimental and equilibrium modeling studies so far focused on trioctylamine (TOA)

or the commercially available Alamine 336 (a blend of C 8 and C 10 trialkylamines) in different mixtures of active and inert diluents, which are mostly deleterious for microorganisms.

The objective of this work is thus to propose an equilibrium model of the extraction of 3-hydroxypropionic acid using a new formulation of organic phase that has been identified from literature to be more biocompatible than other organic phases employed for 3-HP extraction [START_REF] Sánchez-Castañeda | Organic phase screening for in-stream reactive extraction of bio-based 3hydroxypropionic acid: biocompatibility and extraction performances[END_REF][START_REF] Burgé | Towards an extractive bioconversion of 3-hydroxypropionic acid: study of inhibition phenomena[END_REF]. Among promising biocompatible organic phases, we chose to focus on N,N-didodecylmethylamine (DDMA) diluted in a mixture of 1-dodecanol and dodecane as active and inactive diluent, respectively, according to the recent study of Sánchez-Castañeda et al. [START_REF] Sánchez-Castañeda | Organic phase screening for in-stream reactive extraction of bio-based 3hydroxypropionic acid: biocompatibility and extraction performances[END_REF]. The authors found that DDMA showed better performances than TOA in terms of distribution coefficient as well as biocompatibility towards the natural 3-HP producer microorganism Lactobacillus reuteri. They attributed better extraction performances to the reduction of steric hindrance for reacting with acid molecules, while biocompatibility was linked to more hydrophobic nature of DDMA. The inclusion of long chain aliphatic alcohols as active diluents improved extraction performance. Nevertheless, depending on the chain length, primary alcohols could increase toxicity. As was the case for extractants, diluents' biocompatibility was found to be related to hydrophobicity, longer carbon chains being less toxic. The inclusion of alkanes as inert diluents was found to decrease extraction yield but was useful to further reduce toxicity, improving mass transfer. Thus, the authors concluded that the organic phase containing DDMA

Materials and Methods

Amine purification

The commercial DDMA was purified by several steps of acid-base washing. Equal volumes of commercial amine and aqueous H 2 SO 4 0.1 mol L -1 were vigorously mixed for 5 minutes and centrifuged during 1 h at 11 000 g and 25°C. The organic phase was then recovered and washed with equal volume of NaOH 0.3 mol L -1 before being centrifuged at 11 000 g for 20 minutes. These steps were done twice. The resulting organic phase was washed twice with Milli-Q ultrapure water and centrifuged for 1 h at 11 000 g. The purified DDMA was diluted to 20% v/v (0.45 mol L -1 ) in 1-dodecanol (40% v/v) and dodecane (40% v/v) and used as organic extraction phase.

Extraction equilibrium experiments

Physical extraction experiments were performed by putting in contact 25 mL of aqueous phases containing 3-HP in concentrations ranging from 0.0056 to 0.28 mol L -1 with equal volume of an organic phase consisting in 50% v/v 1-dodecanol and 50% v/v dodecane. The sample flask containing both phases was slightly mixed by gentle shaking for 5 minutes and then settled to reach the equilibrium for 72 h at 25°C. After equilibration, acid concentration in the aqueous phase was analyzed by HPLC and acid concentrations in organic phase were estimated through a mass balance. The partition coefficient was determined as the slope of the regression line of the graph of acid concentration in the organic phase versus acid concentration in the aqueous phase (Figure A1). Reactive extraction was evaluated by performing the same phase contacting protocol as for physical extraction. A volume of 2 mL of aqueous 3-HP phases solutions ranging from 0.0028 to 1 mol L -1 was put in contact with equal volume of an organic phase containing 20% v/v purified DDMA, 40% v/v 1dodecanol and 40% v/v dodecane. Under these mild contacting conditions, the formation of a stable emulsion difficult to separate was avoided. The time required to reach equilibrium inside the flask was determined through a preliminary experiment (not shown) and was found to be around 24 h. After reaching the equilibrium, the aqueous phase was analyzed by HPLC and its equilibrium pH was measured using a Mettler Toledo SevenCompact pH meter (Greisensee, Switzerland), equipped with a Mettler Toledo InLab Viscous Pro-ISM probe (Greisensee, Switzerland). The loaded extraction phase was recovered for FT-IR analysis. Each experiment was performed in triplicate and the reported concentrations and FT-IR spectra are averages. The extraction yield based on aqueous phase analysis was defined as Eq. ( 1):

(1)

Analytical methods

The concentration of 3-hydroxypropionic acid in aqueous phases was measured by High-Performance Liquid Chromatography (HPLC) using a Biorad Aminex HPX-87H column (CA, USA). The mobile phase used was 0.5 mmol L -1 H 2 SO 4 at a flowrate of 0.4 mL min -1 . The temperature was set at 65 °C. The samples were detected by a 210 nm UV detector.

FT-IR spectra acquisition

A drop (~20 µL) of the liquid sample to analyze was enclosed between two CaF 2 windows (ISP Optics, Riga, Latvia). IR spectra of the samples were recorded using a Nicolet Magna-IR 750 spectrometer (MA, USA) in the mid-IR region (wavenumbers between 800 and 4000 cm -1 ) at ambient temperature. The resolution of the spectra was set to 2 cm -1 .

Gaussian peak deconvolution was performed using the Multiple peak fit tool in Origin 2021b software (OriginLab Corporation, MA, USA).

Amine basicity quantification

The amine basicity was estimated by determining the amine half-neutralization point as described by Canari and Eyal [START_REF] Canari | Extraction of carboxylic acids by amine-based extractants: Apparent extractant basicity according to the pH of halfneutralization[END_REF]. The cited method for determining the strength of highly hydrophobic amines was first introduced by Grinstead and Davis [START_REF] Grinstead | Base strengths of amine-amine hydrochloride systems in toluene[END_REF] as an analogy to acid-base equilibria in aqueous media. This method is based on the complete reaction of tertiary amines diluted in an organic phase after equilibration with aqueous hydrochloric acid, as expressed by Eq. ( 2).

(

) 2
From Eq. (2) it is possible to define as Eq. ( 3):

(

The half-neutralization point corresponds to the point where half of the amine in the organic phase is neutralized by hydrochloric acid. Hence, at the half-neutralization point , and is given by Eq. ( 4): [START_REF] Kumar | Reactive extraction as an intensifying approach for the recovery of organic acids from aqueous solution: a comprehensive review on experimental and theoretical studies[END_REF] Aiming to measure the pH at the half-neutralization point, a volume of 1 mL of organic phase was put in contact with HCl 1 mol L -1 in an acid/amine molar ratio of 1:1. After 72 h of contact, NaOH 0.5 mol L -1 was added to the aqueous phase in an acid/base ratio of 2:1. The resulting pH of the aqueous phase recovered corresponds to .

Theory

At the standard state (pure liquid at 25 °C), reactive liquid-liquid extraction of 3-HP using DDMA involves several physicochemical phenomena, namely physical partitioning, self-ionization of water, acid dissociation, reactive extraction by tertiary amine, electroneutrality and organic impurities impact. All these phenomena are considered below for modeling purposes.

3.1. Physical partitioning When aqueous 3-HP phases are in contact with an organic phase containing an extractant, almost all the acid transferred to the organic phase is extracted as acidextractant complexes (reactive extraction). Nevertheless, a small part of the acid is present in the organic phase as a result of physical partitioning with the organic phase diluents, leading to the definition of the partition coefficient introduced in Eq. ( 5).

(5) where represents the concentration of total free acid in the organic phase, which is extracted by neat diluents, and the total acid concentration in aqueous phase. As the organic phase's total volume contains also the extractant, it is necessary to take into account the diluent's fraction expressed in Eq. ( 6).

(6)

Self-ionization of water

The equilibrium between hydronium and hydroxide ions in solution is determined by the auto-ionization constant of water. This equilibrium is presented in Eq. ( 7):

(7)

Acid dissociation

For a weak acid in an aqueous medium, the distribution of the undissociated and dissociated species is given by its dissociation constant , presented in Eq. ( 8):

Based on the hydronium ion concentration (Eqs. (7-8)), pH in aqueous phases is defined as:

(9)

Complexation reactions by a tertiary amine

Reactive extraction of 3-HP is achieved through a set of reversible complexation reactions taking place between the acid in the aqueous phase and the tertiary amine in the organic phase. As reported by different authors ( [START_REF] Puttemans | Extraction of organic acids by ion-pair formation with tri-n-octylamine. Part 3. Influence of counter-ion and analyte concentration[END_REF][START_REF] Tamada | Extraction of carboxylic acids with amine extractants. 2. chemical interactions and interpretation of data[END_REF]), when the amine basicity is high enough to bind a proton, the formed cation interacts with the dissociated acid in aqueous phase to form a complex with 1:1 stoichiometry soluble in the organic phase. This mechanism is known as ion pair formation. The global reaction mechanism is presented in Eq. ( 10): [START_REF] Bozell | Cutting-edge research for a greener sustainable future Technology development for the production of biobased products from biorefinery carbohydratesthe US Department of Energy ' s " Top 10 " revisited[END_REF] For simplification purposes, represents three generic alkyl groups, possibly different from each other. In the case of DDMA, , where represents a dodecyl group and represents a methyl group.

At high initial acid concentrations, an undissociated acid molecule can interact with the 1:1 complex by hydrogen bonding thus forming a 2:1 complex. [START_REF] Jers | Production of 3-hydroxypropanoic acid from glycerol by metabolically engineered bacteria[END_REF] In the same manner as 2:1 complex formation, higher stoichiometry complexes could be formed through successive aggregation of acid molecules. This phenomenon depends on the acid nature, the extractant and the interaction between both of them.

Considering that 1:1 complex is formed by ion-pairing, the global reaction of higher stoichiometry complexes can be described as Eq. ( 12):

where

Complexes containing more than one amine molecule can also be formed in some cases but are not considered in this work as this mechanism is likely to occur only for secondary amines [START_REF] Tamada | Extraction of carboxylic acids with amine extractants. 2. chemical interactions and interpretation of data[END_REF].

The global extraction yield presented in Eq. ( 13) is defined as the ratio of all acid species in the organic phase and the initial acid concentration in the aqueous phase.

(13)

Mass balances 12

The mass balances of the acid and the tertiary amine are given by Eqs ( 14) and ( 15)

(14) (15)

Amine loading ratio

The reactive extraction of the acid by a tertiary amine can be characterized using the loading ratio . The loading ratio of the extractant is defined as the ratio between the total concentration of acid in the organic phase (extracted by the amine) and the initial concentration of amine in the organic phase. The loading ratio is a function of the strength of the acid-amine interaction, the stoichiometry of the overall equilibrium reactions and the initial acid and extractant concentrations [START_REF] Datta | Status of the reactive extraction as a method of separation[END_REF].

The general expression of the loading ratio is given by Eq. ( 16):

When 1, i.e., ion-pair 1:1 complex formation, is defined as Eq. ( 17):

Combining Eq. ( 10), ( 15) and ( 17): [START_REF] Canari | Extraction of carboxylic acids by amine-based extractants: Apparent extractant basicity according to the pH of halfneutralization[END_REF] If reactive extraction follows ion-pair 1:1 complex formation, the plot of versus yields a straight line with slope equal to .

For the general case of the formation of higher stoichiometry complexes, if

, can be simplified as given in Eq. ( 19): [START_REF] Grinstead | Base strengths of amine-amine hydrochloride systems in toluene[END_REF] Combining Eq. ( 12), ( 15) and ( 19): [START_REF] Puttemans | Extraction of organic acids by ion-pair formation with tri-n-octylamine. Part 3. Influence of counter-ion and analyte concentration[END_REF] Consequently, the plot of versus is a straight line with slope , if the main assumption of predominance of is valid.

Electroneutrality principle

Considering that there is no net charge in the aqueous phase, the sum of the concentration of positively charged species is equal to the sum of the concentrations of negatively charged species:

(21)

Modeling Reactive Liquid-Liquid Extraction

Organic impurities impact

As a result of their synthesis process (alkylation of primary amines), commercial tertiary amines contain unreacted primary and secondary amines [START_REF] Katritzky | Reaction of aliphatic amines with 49% formic acid . l ) N,N-dimethyl-1-dodecylamine[END_REF]. These lower amines, considered as impurities, are more soluble in water than tertiary amines and have an effect on the pH and the extraction yield. In order to model the effect of organic impurities on the extraction, a neutralization reaction was considered to occur between a pseudo-species , representing organic impurities (primary amines), and the undissociated acid. Impurities involved in the considered side reaction are solubilized into the aqueous phase. Eq [START_REF] Datta | Status of the reactive extraction as a method of separation[END_REF] presents the side reaction: [START_REF] Datta | Status of the reactive extraction as a method of separation[END_REF] 14

The mass balance for impurities can be defined as Eq. ( 23): [START_REF] Katritzky | Reaction of aliphatic amines with 49% formic acid . l ) N,N-dimethyl-1-dodecylamine[END_REF] Given that a new positively charged species is introduced in the aqueous phase, the electroneutrality principle is modified as stated in Eq. ( 24):

(24)

Numerical solution of the mathematical models

Three different models of increasing complexity were compared in this study: firstly, a model considering 1:1 complex formation and neglecting the presence of impurities, secondly, the formation of a 2:1 complex was added to the first model, and finally, a model including 1:1 and 2:1 complexes and the side reaction. Table 1. presents the model equations, their parameters and their calibration datasets. 5), ( 7)-( 8), ( 10), ( 14)-( 15), [START_REF] Tamada | Extraction of carboxylic acids with amine extractants. 2. chemical interactions and interpretation of data[END_REF] Extraction yield 21:1 stoichiometry model ( 1) + 2:1 stoichiometry model ( ) without impurities (5), ( 7)-( 8), ( 10)-( 11), ( 14)-( 15), 5), ( 7)-( 8), ( 10)-( 11), ( 14)-( 15), ( 22)- [START_REF] Lide | CRC Handbook of chemistry and physics[END_REF] and Extraction yield

Equilibrium pH

The parameters of models 1 and 2 were calibrated using the extraction yield measured in the experimental section, by minimizing the difference between measured and calculated values in the least-squares sense. For model 3, measured equilibrium pH was also included as a calibration data set because one of the main consequences of the side reaction was pH increase, compared to the theoretical values without impurities. The values of the two experimental datasets were divided by their respective ranges of variation in order to obtain values of similar order of magnitude suitable for simultaneous use in parameter calibration. Given that the correlation between and was greater than 98%, it was not possible to estimate the two parameters simultaneously. For this reason, an external loop was implemented to make vary between 0.5 and 5 mmol L -1 . For each fixed , parameters , and were determined. The combination of parameters giving the best fit was selected among the solutions significantly different from zero (i.e. zero was not included in the 95% confidence interval of either parameter).

The goodness of fit of the proposed models has been evaluated using the mean absolute error (MAE), defined as Eq. ( 25) [START_REF] Jansen | Analysis of variance designs for model output[END_REF] where is the number of experimental points, are the observed values and the predicted values. -( 5)-( 6), ( 13)-( 14)

* (8) (7)
** -

(15)-( 17), ( 19) ( 14), (23) ( 14), [START_REF] Katritzky | Reaction of aliphatic amines with 49% formic acid . l ) N,N-dimethyl-1-dodecylamine[END_REF] * Taken from [START_REF] Lide | CRC Handbook of chemistry and physics[END_REF] ** Described in sec. 2.3 (Figure A1 in appendices).

Sensitivity analysis

Global Sensitivity Analysis was performed through the estimation of Sobol indices as described by Jansen [START_REF] Jansen | Analysis of variance designs for model output[END_REF] and Saltelli et al. [START_REF] Jansen | Analysis of variance designs for model output[END_REF]. For each initial acid concentration, the model was solved 5 10 4 times with random parameter sets drawn from a normal distribution with standard deviation equal to 10 % of the mean estimated value of each parameter. Confidence intervals for Sobol indices were estimated using the bootstrap method.

Results and discussion

Experimental results: extraction yield and equilibrium pH

Table 3 and Figure 1 present the experimental yield, the observed equilibrium pH and loading ratios obtained from the equilibrium experiments described in section 2.3.

Table 3. Experimental equilibrium results

Initial acid concentration (mol L -1 )

Extraction yield (%)

Equilibrium pH (-)

Loading ratio (mol mol The extraction yield measured experimentally varied between 40% to 60% in all the studied concentration range except for concentrations lower than 0.008 mol L -1 , for which the observed yield is lower. This behavior has already been observed by some authors [START_REF] Kaur | Development of reactive extraction systems for itaconic acid: A step towards in situ product recovery for itaconic acid fermentation[END_REF][START_REF] Chemarin | New insights in reactive extraction mechanisms of organic acids: An experimental approach for 3-hydroxypropionic acid extraction with tri-noctylamine[END_REF] and is generally related to the presence of impurities originated in commercial tertiary amines synthesis process. Impurities consist mainly in traces of primary amines as mentioned previously. Primary amines are less hydrophobic than tertiary amines (e.g. 3.06 for octylamine and 10.94 for TOA)

and have a negative impact on the extraction performance by reacting with the acid in the aqueous phase. The availability of acid molecules to reactive extraction into the organic phase is thus decreased and its effect is especially noticeable at acid concentrations low enough to be comparable to the concentration of primary amines in the organic phase. effective extractants were the most hydrophobic ones. Inversely, the less hydrophobic extractants showed poor extraction yields. As an example of this, those authors found that the distribution ratio obtained using TOA in 1-decanol was almost 8 times higher than that using octylamine in the same diluent. These results were attributed to the partial solubility of primary amines in the aqueous phase that decreased their availability for acid extraction into the organic phase. Chemarin et al.

[28] obtained similar results for 3-HP extraction using TOA diluted in 1-decanol. They found that in presence of octylamine in the organic phase at concentrations near 2 mmol L -1 , the extraction yield of 3-HP decreased from 70% to 10% for initial acid concentrations lower than 0.1 mol L -1 . They proposed the acid-base purification protocol presented in 2.2 and obtained a purified TOA containing less than 0.06 mmol L -1 octylamine. The organic phase prepared using purified TOA showed no decrease in extraction yield at lower acid concentrations. In the present study, although DDMA was purified using the same protocol, the decrease in the extraction yield was still observed at lower acid concentrations, suggesting that DDMA purification was not optimal. Therefore, further studies need to be conducted to precisely quantify the impurities in commercial DDMA aiming to design a more adapted purification protocol.

The transfer of amine impurities to the aqueous phase also had an effect on the equilibrium pH [START_REF] Tamada | Extraction of carboxylic acids with amine extractants. 1. equilibria and law of mass action modeling[END_REF]. It can be observed that experimental pH measured at equilibrium is indeed systematically higher than the theoretical pH estimated considering the 3-HP concentration at equilibrium measured by HPLC (Figure 1b). The discrepancy is especially high for acid concentrations below 0.008 mol L -1 .

In the range 0.08 mol L -1 -0.11 mol L -1 the extraction yield seemed to reach a plateau and increased again for 0.11 mol L -1 with a maximum reached at 0.45 mol L -1 (Figure 1a). This behavior could indicate that the reactive extraction in all the studied range is the combination of different acid-amine interactions. When initial acid concentrations are high enough ( 0.5), which corresponds to higher or equal to 0.45 mol L -1 (Table 3), it is considered that more than one acid molecule is complexed by the amine [START_REF] Datta | Reactive extraction of pyridine carboxylic acids with N,Ndioctyloctan-1-amine: experimental and theoretical studies[END_REF].

Nature of 3-HP-DDMA complex: an FT-IR spectroscopy study

In order to gain insights into the interactions between 3-HP molecules and amine in the organic phase, several IR spectra were recorded. This section presents the analysis of the FT-IR spectra of loaded extraction phases and results are further used to support the mechanistic model of reactive liquid-liquid extraction of 3-HP using DDMA in 1-dodecanol/dodecane. FT-IR spectra of different compounds used in this study are presented in the appendices section.

Organic acids characteristic IR bands

The spectral analysis of samples containing organic acids is particularly interesting in the region 1100 -1800 cm -1 in which several carbonyl/carboxyl characteristic bands of organic acids appear. As an example, Figure 2 presents FT-IR spectra of commercial 3-HP aqueous acid and the same product neutralized with NaOH. pH of the commercial 3-HP solution (3.2 mol L -1 ) is about 2.0, consequently more than 99% of the acid is in its undissociated form (estimated using Equation ( 8)). This product showed a strong absorption peak at about 1720 cm -1 . The region comprising 1700 -1780 cm -1 corresponds to carbonyl stretch ( in a carboxyl functional group and has been extensively linked to the presence of undissociated forms of organic acids [START_REF] Max | Infrared spectroscopy of aqueous carboxylic acids: comparison between different acids and their salts[END_REF]. When this region is present at lower wavenumbers (1700 -1720 cm -1 ), carbonyl groups are considered to be involved in hydrogen bonds. In certain cases, they are associated to acid molecules forming cyclic dimers although cyclization has not been yet proved by means of FT-IR data [START_REF] Ziegenfuß | Distribution of acetic acid between water and organic solutions of tri-n-octylamine[END_REF]. Commercial product (solid line), Commercial product neutralized with NaOH (dashed line)

Regarding neutralized 3-HP (pH 7, more than 99% dissociated), spectra lack a peak in the 1700 -1720 cm -1 region. Instead, a peak in the region 1550 -1620 cm -1 appeared. In this case, the peak at about 1565 cm -1 is assigned to the asymmetric stretch of the carboxylate group ( ) and hence attributed to the presence of dissociated acid molecules [START_REF] Max | Infrared spectroscopy of aqueous carboxylic acids: comparison between different acids and their salts[END_REF]. A common peak between the two solutions could be found at about 1650 cm -1 and is related to the bending of water molecules in the sample ( ).

Loaded organic phases

In the light of the previously described absorption bands, a similar analysis was done

for the loaded extraction phases aiming to elucidate the nature of the acid-amine interaction. FT-IR spectra of some loaded organic phases are presented in Figure 3.

As it can be observed, all the spectra of loaded extraction phases present a peak at 1571 cm -1 . This indicates the presence of the acid in its dissociated form and thus that the acid forms an ion-pair with the amine in the organic phase. This mechanism has been reported to be mostly dependent on the basicity (strength) of the extractant in the diluents, the dissociation constant of the acid and its hydrophobicity (for 3-HP, 4.51, 0.89) [START_REF] Qin | Extraction of monocarboxylic acids with trioctylamine: equilibria and correlation of apparent reactive equilibrium constant[END_REF]. Due to the difficulty of estimating acid-base properties of water-insoluble tertiary amines diluted in organic phases, it is convenient to define an apparent basicity with respect to the equilibrium with protons in the aqueous phase. The apparent basicity of a given amine in given diluents depends on amine properties (substituents, steric hindrance), diluents' properties (polarity, proton donor or acceptor) and the extracted acid [START_REF] Canari | Extraction of carboxylic acids by amine-based extractants: Apparent extractant basicity according to the pH of halfneutralization[END_REF]. Canari and Eyal [START_REF] Eyal | pH dependence of carboxylic and mineral acid Extraction by Amine-Based Extractants: Effects of pKa, Amine Basicity, and Diluent Properties[END_REF] proposed an equilibrium constant of the reaction of the tertiary amine with HCl (section 2.6). Those authors stated that for amines with higher or similar basicity than the conjugated base of the acid to extract ( ), ion-pairing is the predominant extraction mechanism.

The value of measured for the organic phase used in this study was found to be equal to 6.42 0.08. According to these results, DDMA (20% v/v) in 1-dodecanol (40% v/v) and dodecane (40% v/v) is a stronger base than the conjugate base of 3-HP in the aqueous phase. The amine is strong enough to bind a proton and the dissociated acid form is therefore extracted in order to keep electroneutrality in the organic phase. The presence of a polar diluent, especially a protic solvent such as 1dodecanol, plays an essential role in this mechanism by facilitating the solvation of ion-pairs in the organic phase by polar interactions [START_REF] Qin | Extraction of monocarboxylic acids with trioctylamine: equilibria and correlation of apparent reactive equilibrium constant[END_REF]. Recently, Chemarin et al [START_REF] Chemarin | Mechanistic modeling and equilibrium prediction of the reactive extraction of organic acids with amines: A comparative study of two complexation-solvation models using 3hydroxypropionic acid[END_REF] proposed this mechanism for 3-HP extraction based on FT-IR data obtained using TOA in 1-decanol. Ion-pair formation has been widely studied for several organic acids using different tertiary amines [START_REF] Inyang | Reactive extraction of malic acid using trioctylamine in 1decanol: equilibrium studies by response surface methodology using Box Behnken optimization technique[END_REF][START_REF] Dhongde | Experimental study on reactive extraction of malonic acid with validation by Fourier Transform Infrared Spectroscopy[END_REF][START_REF] Thakre | Modeling and optimization of reactive extraction of citric acid[END_REF].

For initial acid concentrations higher than 0.11 mol L -1 , spectra of loaded organic phases show a shoulder at about 1716 cm -1 . The apparition of this peak indicates the presence of the undissociated acid form in the organic phase, more particularly, carboxyl groups participating in H-bonds. Different explanations of this phenomenon can be formulated: (i) the presence of free acid extracted by physical partitioning, (ii) the existence of a tautomeric equilibrium between the ionic complex and a hydrogenbonded form of this complex, and (iii) the formation of high stoichiometry complexes.

Concerning hypothesis (i), the presence of free acid molecules in the organic phase can be neglected due to the low partition coefficient of 3-HP in the organic phase diluents ( 0.004). In the studied concentration range (0.0028 -1 mol L -1 ), the acid extracted by physical partitioning is so low that it is undetectable by FT-IR.

Regarding hypothesis (ii), the existence of an equilibrium between a complex formed by proton transfer and a molecular complex formed by hydrogen-bonding (see Figure 4) have been reported to happen in several acid-amine couples in different organic phases [START_REF] Tamada | Extraction of carboxylic acids with amine extractants. 2. chemical interactions and interpretation of data[END_REF][START_REF] Gusakova | A spectroscopic study of the interaction of isobutyric acid with pyridine and dioxan[END_REF][START_REF] Barrow | The nature of hydrogen bonded ion-pairs: the reaction of pyridine and carboxylic acids in chloroform[END_REF]. Gusakova et al. [START_REF] Gusakova | Spectroscopic investigation of the reaction of acetic and isobutyric acids with tertiary amines[END_REF] recorded spectra from an equimolar mixture of isobutyric acid and triethylamine diluted in C 2 Cl 4 . They observed a decrease of the band and an increase of proportional to the addition of methanol (a polar and protic solvent). Hence, they concluded that tautomeric equilibrium is highly dependent on the diluents, and it is mostly shifted to the ion-pair form in presence of a polar solvent. These observations were also validated by the same authors using isobutyric acid in dimethyl sulfoxide and acetonitrile, both polar solvents. In this study, the presence of 1-dodecanol as a polar and protic diluent (40% v/v) would shift the equilibrium showed in Figure 4 to the ion-pair form of the complex.

Ion-pair complex Hydrogen-bonded complex As discussed previously, hypothesis (i) and (ii) are unlikely to occur, consequently, the apparition of the shoulder at ~1716 cm -1 can be attributed to hypothesis (iii), i.e., to the aggregation of undissociated acid molecules to the ion-pair complex through hydrogen bonding [START_REF] Smith | Infrared spectroscopic investigations of acid-base interactions in aprotic solvents. Part I. The interaction of tri-n-propylamine and some carboxylic acids[END_REF]. This seems to be in agreement with another phenomenon observed in spectra of loaded organic phases (Figure 3) for 0.11 mol L -1 : the apparition of a bond stretch band ( ) coupled with the scissoring of a hydrogen-bonded hydroxyl group ( ⋯ ) at 1189 cm -1 [START_REF] Wierzejewska-Hnat | Infrared studies of complexes between carboxylic acids and tertiary amines in argon matrices[END_REF]. The ratio between the surface of the band at 1716 cm -1 and that at 1570 cm -1 decreases while increasing initial concentration of 3-HP. For this reason, the formation of high stoichiometry complexes is considered to gain importance with respect to 1:1 ion-pair formation at higher acid concentrations. The structure of the different acid-amine complexes considered is presented in Figure 5.

Water co-extraction

All the loaded organic phases showed the characteristic peak of bending of water molecules at 1650 cm -1 . Tamada and King [START_REF] Tamada | Extraction of carboxylic acids with amine Extractants[END_REF] observed water co-extraction in a set of reactive extraction experiments of several mono and dicarboxylic acids using Alamine 336 in different active diluents. They reported that this phenomenon was highly dependent on the solubility of free water molecules in active diluents. They also observed that monocarboxylic acids such as lactic and acetic acid carried less water than dicarboxylic acids, inferring that water molecules were tightly bound to the carboxylate part of organic acids. In this work, when the organic phase was priorly equilibrated with pure water, the observed water band is negligible (Figure 6) and it increases while increasing the initial acid concentration. Therefore, the presence of the water band in loaded organic phases is due to the co-extraction of water molecules bound to acid-amine complexes. Figure 6. Deconvoluted water bands in loaded organic phases Pure water (dashdotted line), organic phases after being equilibrated with 3-HP solutions: 0.11 mol L -1 (dotted line), 0.55 mol L -1 (dashed line), 1 mol L -1 (solid line)

In Figure 6, a deconvolution of the peaks showed in Figure 3 was done for the sake of better data visualization than in the case of the overlapping peaks. Figure 7 shows the predicted extraction yield and equilibrium pH for the three models, calibrated based on experimental data. Table 4 presents the estimated parameters of each model. It appears that model 1 predicts a slight increase of the extraction yield while increasing initial acid concentrations until 0.02 mol L -1 . For higher concentrations, amine becomes the limiting reactant, and consequently, a strong decline is observed. Recall that Model 1 includes only the 1:1 acid-amine complex formation, and accordingly is unable to predict the behavior observed for acid concentrations exceeding 0.1 mol L -1 (increase in yield) and below 0.08 mol L -1 (strong yield decrease and high equilibrium pH). Model 1 shows thus poor accuracy for predictions of both extraction yield and equilibrium pH. In average, the absolute difference between predicted values and observed values is 0.11 in the case of the [AH] (mol L -1 ) magnitude as for initial acid concentrations higher than 0.13 mol L -1 and becomes almost equal to at the highest acid concentration of the studied range. Experimentally, was detected from the stretching of the carbonyl group for acid concentrations higher than 0.11 mol L -1 , thus being in accordance with the behavior depicted by the model. Experimental data (symbols), regression line (solid line).

Modeling results

Despite the high uncertainty, the relative difference of the graphically estimated values (Table 5) remains less than 10% compared to the values in Table 4 obtained by calibration of model 3.

Table 5. Graphically estimated complexation constants and

Parameter

Estimated value Coefficient of variation [%]

[L 2 mol -2 ]

[L mol -1 ]

Sensitivity analysis of the model

The goal of global sensitivity analysis is to investigate the changes of model outputs induced by modifications of model parameters. In this work, global sensitivity was evaluated using Sobol indices. Sobol indices, estimated using Monte-Carlo simulations, represent the fraction of the variance of each output that can be explained by variations of each parameter. While decomposing the variance, it is possible to estimate an effect associated to the parameter itself (main effect ) as well as the interaction between all the parameters. Regarding the interactions, as a matter of simplicity, a total index is defined for each parameter as the sum of its main effect and all the interactions of this parameter with the others ( ). In region 1, impurities have a moderate effect on extraction yield ( 20%) but act strongly on the equilibrium pH. Indeed, explains up to 85% of the variance of the equilibrium pH. In this region, the ratio of estimated impurities to the initial acid is equal to 0.71 mol mol -1 for the lowest acid concentration and is equal to 0.02 mol mol - 1 at 0.08 mol L -1 . At higher acid concentrations, impurities become insignificant compared to acid molecules. The considered side reaction thus significantly contributes to the behavior observed at initial acid concentrations lower than 0.02 mol

L -1 .
In region 2, the whole variance is explained by . This was expected because is the predominant form of the acid in the organic phase. As has no influence on the model outputs in regions 1 and 2, the model could be simplified to 1:1 complex formation plus the side reaction for initial acid concentrations below 0.11 mol L -1 .

When initial acid concentration exceeds 0.11 mol L -1 , the effect of increases as the initial 3-HP increases (region 3). This behavior is explained referring back to Figure 3 and Figure 9, where it was observed that the 2:1 complex becomes significant at high acid concentrations. In regions 2 and 3 the presence of primary amines can be neglected, and the model could be simplified to 1:1 and 2:1 complex formation without the side reaction (model 2).

The sensitivity analysis also explains the observations concerning the uncertainty in parameter estimation (Table 4). The equilibrium constant related to the formation of the 1:1 complex ( ) has a high sensitivity for both model outputs in all concentration regions; its estimated value is thus based on the full set of measurements and has the lowest uncertainty (3.2%). The equilibrium constant related to the formation of the 2:1 complex ( ) has a good sensitivity for both model outputs but in region 3 only, explaining its higher uncertainty (8.7%). As for the equilibrium constant of the side reaction ( ), its sensitivity is only significant for the pH output in region 1; the number of relevant measurements is thus small and its uncertainty the largest (23.9%).

Overall, the sensitivity analysis aligns well with the observations of sections 5.1 and 5.2 about the contribution of each complexation mechanism and the uncertainty on the parameter estimation (Table 4).

Conclusion

In this work, we studied the reactive extraction of 3-hydroxypropionic acid with a newly designed biocompatible organic phase consisting in 20% v/v DDMA, 40% v/v 1-dodecanol and 40% v/v dodecane. The loaded organic phase, analyzed by FT-IR spectroscopy, allowed to elucidate the reactive extraction mechanism of 3-HP in initial acid concentration range 0.0028 -1 mol L -1 . DDMA in these diluents was found to be a stronger base than the dissociated 3-HP ( ) being capable to bind a proton from the aqueous phase. A 1:1 stoichiometry complex is then formed from the interaction of the protonated amine and the dissociated acid. Besides, the 1:1 complex is able to extract undissociated acid molecules by the formation of hydrogen bonds leading to the formation of a complex of stoichiometry 2:1. The mathematical model proposed showed that the 1:1 complex predominates in all the studied concentration range. The 2:1 complex becomes significant for initial acid concentrations higher than 0.11 mol L -1 .

The extraction yield and equilibrium pH prediction were improved by taking into

p

  Number

diluted in 1 -

 1 dodecanol and dodecane presented the best trade-off between extraction yield and biocompatibility. In order to elucidate the underlying 3-HP extraction mechanisms, a set of extraction equilibrium experiments was performed at different initial acid concentrations. Fourier Transform Infrared spectroscopy (FT-IR) of loaded organic phases was employed as a means to understand the nature of acid-amine interactions. In order to validate and quantify the extraction mechanisms, a model built on the law of mass action, considering different acid-amine interactions was developed and its parameters were calibrated based on experimental data. The distribution of acid species in the organic phase was thus studied. A global sensitivity analysis of the model was ultimately performed through the estimation of main and total Sobol indices. The sensitivity analysis quantified the importance of each parameter in different acid concentration ranges and supported the discussion of the relative importance of the considered mechanisms.

Figure 1 .

 1 Figure 1. Experimental results as a function of initial 3-HP concentration a. Extraction yield, b. Equilibrium pH: experimental (symbols), theoretical (solid line).Error bars represent standard deviation

Figure 2 .

 2 Figure 2. FT-IR spectra of 3-HP in solution (3.2 mol L -1 )

Figure 3 .

 3 Figure3. FT-IR spectra of organic extraction phases after prior equilibration with aqueous 3-HP solutions at different initial concentrations. : 0.01 mol L -1 (dashdotted line), 0.11 mol L -1 (dotted line), 0.55 mol L -1 (dashed line), 1 mol L -1 (solid line)

Figure 4 .

 4 Figure 4. Tautomeric equilibrium of ionic and molecular 1:1 acid-amine complexes.

Figure 5 .

 5 Figure 5. Proposed structures for acid-amine complexes and their characteristic IR bands a.1:1 complex, b. 2:1 complex

  yield ( 11%) and 0.34 pH units for equilibrium pH ( 0.34). Model 2 additionally includes the 2:1 acid-amine complex and captures the general trend of the extraction yield increase in the 0.08 -0.3 mol L -1 range and the subsequent decline from 0.3 mol L -1 . Nevertheless, model 2 is unable to describe the behavior of the extraction yield and equilibrium pH towards lower concentrations. The inclusion of the 2:1 complex formation improved the accuracy on the estimation of the extraction yield ( 4.5%) but no significant improvement was observed in the prediction of the pH ( 0.33). Model 3 includes the side reaction with water soluble amines and captures correctly all the observed variations of the extraction yield and equilibrium pH in all the studied concentration range. This model was more accurate in the prediction of both the extraction yield ( 2.5%) and the equilibrium pH ( 0.15). This model overestimates pH for initial acid concentrations between 0.008 and 0.02 mol L-1. As pointed out by sensitivity analysis (section 5.4), pH estimations are highly related to the considered parasite reaction. The parasite reaction presented in this work is a simplification of a series of acid-base reactions that could occur between acid molecules and different primary amines transferred from the organic phase to the aqueous phase. Even if the inclusion of the parasite reaction improved pH prediction, real phenomena are expected to be more complex than those considered in the model and cause a deviation from a behavior predicted using a single global pseudo-species. Despite this simplification, model 3 accounted for the observed phenomena with satisfactory accuracy, in agreement with the extraction mechanism proposed in 5.2.2.

Figure 7 .Table 4 .

 74 Figure 7. a. Extraction yield and b. equilibrium pH as a function of initial 3-HP concentration. Experimental data (symbols), model 1 (dotted line), model 2 (dashed line), model 3 (solid line). Error bars represent standard deviation

Figure 8 .

 8 Figure 8. Loading curve of 3-HP extraction with DDMA (0.45 mol L -1 ) in 1-dodecanol (40% v/v) and dodecane (40% v/v). Experimental (symbols), Predicted by model 3 (solid line)

Figure 9 .

 9 Figure 9. Distribution of acid species in the organic phase calculated with model 3, as a function of initial 3-HP concentration Left y-axis:(solid line), (dashed line). Right y-axis:(dotted line)

Figure 10 .

 10 Figure 10. Graphical determination of the complexation constants and a. 1:1 ion-pair complex formation, b. 2:1 hydrogen bonded complex formation.Experimental data (symbols), regression line (solid line).

  The main and total Sobol indices of the parameters were estimated for two model outputs: extraction yield and equilibrium pH. Only main indices are presented in this section as no significant interactions between the parameters were observed. Three main regions can be identified in Figure11. Region 1 covers initial acid concentrations lower than 0.02 mol L -1 and the variance of both outputs is mainly explained by and . Region 2 covers concentrations in the range 0.02 mol L -1 -0.11 mol L -1 , where the variance is completely explained by . Finally, region 3 covers acid concentrations above 0.11 mol L -1 , where and are the key parameters explaining the variance.

Figure 11 .

 11 Figure 11. Sensitivity analysis of the model outputs. a. Extraction yield, b. Equilibrium pH (squares), (circles), (triangles)

1 )

 1 Figure A1. Estimation of physical partitioning constant . Experimental data (symbols), Regression line (solid line). Error bars correspond to standard deviation ( )

Table 1 .

 1 Definition of the three considered models.

		Model		Equations	Parameters	Calibration datasets
		1:1 stoichiometry	(
	1	model (	1)
		without impurities

Table 2 .

 2 Fixed parameters for calculations

	Parameter	Value	Unit	Present in Eq.

MATLAB R2020b (The MathWorks Inc., Natick, MA), equipped with Statistics and Optimization toolboxes was used for numerical calculations. For each model, the systems of nonlinear equations listed in Table

1

were solved with the 'trust region' algorithm implemented in the fsolve function. The nlinfit function implementing the Levenberg-Marquardt algorithm was used to solve the nonlinear least-squares problem for parameter estimation. The values of the fixed parameters for calculations are listed in Table

2

.

Acknowledgements

This work was funded by a doctoral grant awarded from the ABIES doctoral school of Université Paris-Saclay.


Guiquan Wang

Micheline Abbas

Éric Climent

Numerical Simulation of Neutrally Buoyant Particles in Turbulent Couette Flow

Open Archive TOULOUSE Archive Ouverte (OATAO)

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse. Fluids 2013, Yu et al. Phys. Fluids 2013, Lashgari et al. PRL 2015). However this effect depends on the particle concentration and on the particle size in a non-monotonous way while the nature of the interaction between the particles and the flow rotational structures is not yet fully understood.

  fr/ Eprints ID : 14286 To cite this version : Wang, Guiquan and Abbas, Micheline and Climent, Eric Numerical Simulation of Neutrally Buoyant Particles in Turbulent Couette Flow. (2015) In: Dispersed two-phase flows 2015, 22 September 2015 -24 September 2015 (Toulouse, France). (Unpublished) Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr Numerical Investigation of Particle Laden Plane Couette Flow Authors : Guiquan Wang 1 , Micheline Abbas 2 &EricCliment 1 1 Institut de Mécanique des Fluides de Toulouse. CNRS-INPT-UPS -France 2 Laboratoire de Génie Chimique, Toulouse. CNRS-INPT-UPS -France Proposed topic: Hydrodynamics of dispersed flows Key words: Transition to turbulence, particle suspensions, two-way coupling, plane Couette flow. Speaker: Guiquan Wang/ gwang@imft.fr Nowadays, the mechanisms of turbulence modulation by particles and their dependence on different physical properties (particle size, concentration and inertia) are still poorly understood and are widely open for fundamental investigation (Balachandar& Eaton, Ann. Rev. Fluid Mech. 2010). After the experiments of Matas et al. PRL 2003, fully resolved numerical simulations of pressure-driven particle-laden flows clearly revealed that particles of finite-size (with respect to the hydraulic diameter)at moderate concentration have a significant impact on the unsteady nature of the flow reducing the threshold of the transition from laminar to turbulent regimes. Particles enhance the transverse turbulent stress components and modify the flow rotational structures (Loisel et al. Phys.

  This work aimsto understand the modulation of turbulence by particles under controlled macroscopic shear stress and bulk Reynolds number (plane Couette flow).We perform resolved two phase flow simulations with the aid of the Force-Coupling Method. The minimum flow unit is used (Hamilton et al.JFM 1995), with different ratios of Couette gap width to particle diameter, Reynolds numbers and concentrations. Results on the particle distribution (fig. 1 left), their interaction with the large scale structures (fig. 1 right), and the induced modification of the flow turbulence properties will be analyzed and discussed.

Figure 1 :

 1 Figure 1: Simulations of plane Couette flow seeded with finite size particles. (From left to right: Particle positions, Particle trajectories in laminar flow, Flow structures)