X Hai

C Mayer

S Tencé

S Miraglia

Phase transformations in Fe-Cr-Mn alloys for magnetocaloric applications

Keywords:

The sequence of phase transformations in a Fe-Cr-Mn alloy (known as an austenitic stainless steel) are investigated in order to explore the potential of this rare earth-free material for magnetocaloric applications at high temperature. In the lack of high temperature X-ray diffraction the magnetic response and transition temperatures of alloys have been determined by using a Faraday's Balance apparatus and an extraction vector magnetometer. These characterizations have been complemented by DCS and DTA measurements as well as thermodynamic calculations using Thermocalc. The retained nominal composition is Fe0.59Cr0.16Mn0.25 (referred to as FeCrMn 15/25) shows the targeted reversible transformation.

Carbon addition is shown to shift the transition temperature to lower values. The effect of carbon addition on the phase transition and on the magnetocaloric response is discussed.

A comparison is made with the chromium-poor Fe-Cr-Ni system.

Introduction

The magnetocaloric effect (MCE) is the change in temperature of a material due to the adiabatic application or removal of an external magnetic field. This temperature change is related to the magnetic entropy change (ΔSM). Generally, MCE is large in the vicinity of the Curie temperature (TC), where the magnetic spins undergo an order -disorder phase transition.

Historically this effect found its application for cooling in low temperature physics by using the adiabatic demagnetization of paramagnetic salts in 1933 [1]. The study of room temperature MCEs associated with a magnetic phase transition was revived in 1997 by Pecharsky and Gschneidner [2] who observed a 'giant' entropy change in Gd5Si2Ge2. From then on, the MCE has been considered for room temperature refrigeration technology using materials with a transition near room temperature or for heat-to-power conversion, where an appropriate thermodynamic cycle is chosen to use the magnetocaloric material for power generation. Today, a large set of magnetic materials show significant magnetocaloric effects. Most of such materials contain rare earths. In order to be considered for application, the magnetocaloric material should be sustainable and environmentally-friendly. Considering the fact that the economic and strategic issues surrounding rare earths are very complex, there exists a motivation to find novel rare earth free magnetocaloric materials. In particular there is a considerable interest in rare earth free, cost effective and readily available Fe-based materials.

This paper intends to investigate the potential of some Fe-based alloys for magnetocaloric applications. A literature review, summarized in the next section, prompted us to investigate the Fe-Cr-Mn system which has not been reported so far.

State of the art

Among the Fe-containing alloys, FexPt100-x alloys exhibit a temperature and magnetic field induced phase transition from disordered γ phase to ordered α phase. This gives rise to a magnetic entropy change up to 39.8 J/kg. K for alloy Fe79Pt21 under a field change of 0 to 7 T [3]. Their magnetocaloric effect near room temperature makes FexPt100-x alloys interesting candidates for magnetic refrigeration applications. Nonetheless, the price of Pt remains a challenge for the upscaling of these alloys.

In 2012, P. Souvatzis and colleagues [4] reported a theoretical prediction to achieve a structural and magnetic phase transition in alloy systems Fe1-y-xCryNix and Fe1-y-xCryMnx where 0.1<y<0.3 and 0.1<x<0.3. They predicted that it was possible to tune the composition so that the bcc to fcc phase transition took place in a temperature range close to room temperature, accompanied with a large change in entropy of the spin-system. By alloying with Ni or Mn, the ferromagnetic bcc-phase of Fe1-yCry (y < 20) can degenerate with the paramagnetic fcc-phase [5,6]. As the ferrite bcc-phase is being heated, it becomes destabilized and results in transforming into the paramagnetic fcc-phase at the transition temperature [6]. This structural and magnetic transition is expected to be accompanied by a large entropy change. Therefore, it should be possible to tailor the bcc to fcc transformation with the composition and microstructure engineering. and co-workers have shown that γ-FeNi nanoparticles exhibit a near room-temperature secondorder magnetocaloric response [7]. The peak entropy change is in the order of 2.8 J/kg·K for a maximum applied field of 5 T.

Later in 2013, Ucar and colleagues studied mechanically alloyed Fe70Ni30 and Fe72Ni28 and characterized them in terms of their structural and magnetic properties [8]. The maximum magnetic entropy change observed for Fe70Ni30 and Fe72Ni28 are 0.65 and 0.5 J/kg·K, respectively, at a field of 5 T. The peak temperatures are around 90 and 60 °C, respectively.

The Curie temperature of these nanoparticles were tuned by controlling the oxidation kinetics of Fe during ball milling. The Fe-Ni phase diagram shows that the magnetic properties and Curie temperature of these alloys strongly depend on the composition. Chaudhary et al. reported the magnetocaloric effect and magnetic properties of (Fe70Ni30)95Mn5 nanoparticles prepared by ball milling [9]. The quenched sample shows a Curie temperature at 65 °C. They show that the maximum entropy change is 1.45 J/kg·K for a field change of 5T.

A multiphase Fe-Ni-B bulk alloy has been investigated as a solution for inadequate temperature span in developing a magnetic cooling system [10]. The bcc phase forms during slow cooling from the fcc phase region to room temperature in the furnace. The coexistence of bcc, fcc, and spinel phases results in large working temperature spans of 439 °C for magnetic field change of 5 T. The mass fraction of the phases can be tuned by controlling the synthesis parameters.

Likewise, MCE in cementite (Fe3C) has been explored in a recent publication by B. Kaeswurm et al. [11]. While the transition temperature (202 °C) is still high for room-temperature magnetic refrigeration, the material has a measured magnetocaloric effect of ∆𝑇 𝑎𝑑 = 1.76 K and ∆𝑆 = 3 J/kg·K for a field change from 0 to 2 T. The Curie temperature of cementite can be adjusted by substitution of various elements. For instant, small concentration of Molybdenum can reduce 𝑇 𝐶 from 202 °C down to 147 °C [12]. Other variations have been examined, including (Fe0.9Mn0.1)3C , with a reported ∆𝑆 of 3.4 J/kg·K at 32 °C for a field change of 5 T, and Fe2.85Cr0.05C [12], which crystallizes in the cementite type structure and shows a temperatureinduced second-order magnetic phase transition at 87 °C.

Experimental details

After preliminary studies, Fe-Cr-Mn alloys selected for this study have been prepared by highfrequency induction melting ultra-pure elemental compounds in appropriate nominal compositions. Carbon insertion was performed via a solid-solid reaction with an intermediate alloy. The as-cast alloys were then wrapped in Mo foil and sealed in quartz ampoule, filled with partial pressure of argon. Annealing was performed for 15 days at 1000 ˚C. The magnetic response and transition temperatures of alloys were determined by a Faraday's Balance apparatus and an extraction vector magnetometer. Calorimetric measurements were carried out using Differential Scanning Calorimeter (DSC) in the range from room temperature to 800 °C and Differential Thermal Analysis (DTA) from room temperature to 1300 °C. The sample mass for the measurement is about 100 mg and pure Al2O3 is used as a reference material. Qualitative information on the phase presence given by X-ray diffraction at room temperature using a PANalytical X'Pert MRD X-ray diffractometer operated at the Cu K wavelength.

Metallographic characterization was done by optical and electronic microscopes with appropriate chemical attack of the surface. Since the alloying elements are all neighboring elements with very similar atomic number, the accurate composition of the different phases was difficult to obtain in EDX analysis. The magnetization measurements were performed by using a high-temperature VSM. The isothermal magnetic entropy change was obtained from isothermal magnetization measurements from 50 K below the transition temperature to 50 K above the transition temperature with a temperature step size of 3K and a field variation from 0 to 20 kOe. Knowing that this indirect method leads to the estimation that depends on the measuring procedure, we used a protocol in which, after each temperature step, the sample is cooled from its paramagnetic state before starting a new isothermal measurement.

Results and discussion

Only α-ferrite and α'-martensite phases are known to possess ferromagnetic properties at and above room temperature. The respective transition temperatures depend on the composition of the phases. In order to evaluate a large range of compositions, several alloys with different Cr, Ni, and Mn content were synthesized and characterized in the Fe1-x-yCrxNiy (Cr wt%=5,15,25,35 and Ni wt%=10,15,20,25) and Fe1-x-yCrxMny (Cr wt%=5,15 and Mn wt%=10,20,25) systems respectively. The compositions were selected on the basis of Thermocalc predictions and phase diagrams in order to position near the two-phase (bcc and fcc) region. The TCFE6 database is specifically applicable for various types of steels/Fe-alloys with a minimum Fe-content of 50 wt%. The database has been based on complete reassessments of binary and many ternary systems; however, many intermediate compounds that do not occur in steels/Fe-alloys have been deleted from the database. Therefore, it is not suitable to calculate complete binary and ternary systems, but rather only in the iron-rich corner (in our case the iron content is 65wt%).

In this study, the pressure in the simulated system is set to standard atmospheric condition and the system size is 1 mole.

These thermodynamic simulations are used to predict stability phases within a certain temperature range as a reference to the experimental data and to select viable compositions to test experimentally. In addition, the ThermoCalc simulations cannot provide information on the kinetics of phase transformation. Experimentally, the system is not in equilibrium during heating and cooling so this needs to be taken into consideration. Therefore phase fractions from Thermocalc cannot be directly compared with DSC figures. In this paper, we focus on results from Fe0.59Cr0.16Mn0.25 (Fe65Cr15Mn25) due to the possible relevance in magnetocaloric application. The other compositions were found to exhibit disappointingly low MCEs and are therefore not reported, those results may be found in [START_REF] Hai | Magnetocaloric materials for room temperature refrigeration[END_REF]. A shorthand notation of 15/25 is used in the following section to refer to this Fe-based alloy with 15 wt%Cr and 25 wt%Mn.

Alloy 15/25

The phase formation is first simulated with Thermocalc © software with TCFE6 database. Based on the thermodynamic calculation (Figure 1) for alloy 15/25, austenite (γ-fcc) phase is the primary solidification phase, together with a small percentage of σ phase (20%). The ferrite phase starts to form below 500 °C. The σ phase is present both before and after the transformation between bcc and fcc phases. The XRD pattern confirms that the sample contains a majority phase in a bcc structure with a Fe7Ni3-type lattice. The lattice parameter is about 2.86 Å. The microstructure of the annealed sample was observed in optical microscope after HNO3 electrolytic etching on the surface. The microstructure shown in Figure 2 shows austenite phase with ferrite grains and small dark dots of sigma phase. The phase transition is examined by means of Faraday's balance and DSC. Figure 3 (a) shows a transition from a ferromagnetic ferrite phase to a paramagnetic austenite phase during heating for sample 15/25. The DSC measurements in figure 3b display a sluggish endothermic signal in the temperature range [500°C-600°C] upon heating whereas no significant thermal event can be detected in the same temperature range upon cooling. For a first order phase transformation between equilibrium crystallographic phases (the common type) an endothermic heat effect will occur upon heating and the opposite effect is expected to occur during cooling. Our respective observations, magnetic (fig 3a) and calorimetric (fig 3b), are consistent with the following assumptions :

a very sluggish structural transformation is likely to take place around 600°C since the ferromagnetic to paramagnetic transformation is a second order transformation no heat effect should accompany it, therefore the magnetic signal observed in fig. 3a is likely to be Curie point effect. In this α/γ material, we manage to find the reversible structural transformation inducing a magnetic transition, even though the transformation temperature is too high for roomtemperature application. Alloy 15/25 was treated in thermal cycles to observe its cyclic behavior. The transition is quite stable from both (a) cycling with cooling to RT at each cycle and (b) cycling without cooling down to RT at each cycle, as shown in Figure 4. As shown in the thermodynamic calculation (Figure 5), the sigma phase is stabilized at lower temperature with increasing carbon content. According to Thermocalc calculations (figure 5)

the effects of carbon addition in the FeCrMn alloy are the following :

-A shrinking of the stability range of sigma phase which is no longer stable at temperatures above 600°C -In the mean time the precipitation of M23C6 carbides is favoured -Finally the transition between alpha-ferrite and gamma-austenite is shifted from 400°C for pristine alloy to about 600°C for C-doped ones Our magnetic measurements (figure 6) show a lowering of magnetic transition from 600°C down to 400°C which does not further depend on the carbon content. These experimental observations are consistent with Thermocalc calculations for the pristine alloy. For this alloy the calculations predict the presence of austenite and sigma phase and the lack of ferrite above 600°C. At 400°C ferrite, sigma phase and austenite may coexist.

For the carbon-doped samples the calculations predict the presence of austenite, sigma phase, M23C6 carbide and the presence of ferrite at 600°C (boundary). At 400°C ferrite, sigma phase and M23C6 carbide may coexist. The weakness and/or lack of magnetic signal above 400°C is therefore inconsistent with the presence of ferrite above 400°C for the carbon-doped samples.

The common austenitization process is a ferrite to austenite transformation accompanied by a dissolution of the M23C6 particles. Recently [START_REF] Bettanini | [END_REF] the extent of interaction of Cr-rich carbides dissolution and the austenitic growth has been reassessed for Cr-containg steels (our case). It was found that small carbide paticle size promote faster austenite transformation leading to ferrite-free microstrucures at lower temperatures (because of kinetics effects). Therefore our current experimental observations (figure 6) are consistent with such a scenario.

Magnetocaloric characterization

Figure 8 shows the temperature dependence of the magnetic entropy change under a magnetic field ranging from 0 to 2O KOe. It can be seen that carbon addition resulted in a shift of the transition temperature to lower values but the measured entropy change is very low and out of consideration for any application. The non-doped alloy instead, exhibits an entropy change comparable to that of chromium-doped Fe70Ni30 nanoparticles [START_REF] Chaudhary | Magnetocaloric Properties of Fe-Ni-Cr Nanoparticles for Active Cooling[END_REF] at the corresponding magnetic field change. Such composition is therefore suitable for high temperature cooling. It is our contention that there exists a solicitation in regard to the development of compact, low weight magnetic refrigerators that can operate at high temperatures. Development of such systems would allow for a more local, and more efficient, cooling process. The primary initial market opportunity for such magnetocaloric refrigeration systems would be for systems where localized cooling at high temperatures is required.

Conclusion

The Fe-Cr-Mn alloy 15/25 shows the targeted reversible structural transformation between ferrite and austenite, but this transformation occurs around 550-600°C and the trials with C doping did not allow to decrease this transition temperature while maintaining a large magnetic change and a reversible transformation.

The carbon addition to Fe-Cr-Mn alloy 15/25 can effectively decrease the transition temperature from ferrite to austenite during heating. However, this is accompanied by a change of structural transitions in the sample. The carbon-containing alloys show bcc to fcc transition upon heating.

However, the Mn and C concentrations further stabilize austenite phase, which is not transformed into ferrite upon cooling. Therefore, carbon alloying has compromised the reversibility of the magneto-structural transition observed in 15/25 alloy.

Figure 1 .

 1 Figure 1. Phase fraction as a function of temperature for alloy 15/25 calculated using Thermocalc © .

Figure 2 .

 2 Figure 2. Microstructure after HNO3 electrolytic etching for alloy 15/25 annealed at 1000 °C for 15 days.

Figure 3 .

 3 Figure 3. (a) Magnetic response in arbitrary units for samples 15/25 and (b) DSC measurement for alloy 15/25.

Figure 4 .Figure 5 .

 45 Figure 4. The magnetic response in arbitrary units of annealed sample 15/10 is plotted as a function temperature during (a) thermal cycles cooling down to RT at each cycle and (b) thermal cycles without going back to room temperature at each cycle 4.2 Effect of carbon addition Carbon is one of the γ-promoting alloying elements, which means that the addition of carbon stabilizes the high temperature austenite phase, lowering the ferrite to austenite transition temperature. Seen from the previous section, sample 15/25 shows very little thermal hysteresis, which is an advantage to study the influence of carbon on the transition temperature. Different concentrations of carbon were added by induction melting graphite with the other starting elements in order to elaborate samples with 0.05, 0.1, and 0.2 wt% of C. We use the shorthand notations 15/25/0.05, 15/25/0.1, and 15/25/0.2 to distinguish the samples. The alloys were annealed at 1100 °C and then quenched

Figure 6 .Figure 7 .

 67 Figure 6. Magnetic response in arbitrary units for sample Fe0.59Cr0.16Mn0.25 and the carbon-doped counterparts. Since the magnetic signals from the Faraday's balance are qualitative only, the magnetization level of the carbon-containing alloys has been further verified using an extraction vector magnetometer. The magnetization level decreases with increasing carbon content. Alloy 15/25/0.05 has a magnetization value of 31.36 A·m 2 /kg and for alloy 15/25/0.1, this value is dropped to 20.35 A·m 2 /kg. The paramagnetic behavior upon cooling of some carbonated samples also suggests the presence of sigma or carbide phases. The martensite starting temperatures calculated for these samples are all below 0 °C, which explains the absence of the recovering of martensite phase upon cooling. The transition temperature for sample 15/25/0.1 is 362 °C. (see figure 7a) and displays a small hysteresis. The magnetization loop shown in Figure 7 (b) illustrates the soft magnetic behavior of the sample.

Figure 8 .

 8 Figure 8. Magnetic entropy change dependence on temperature for sample 15/25/0.1 and 15/25

Acknowledgements

This work was supported by the CIFRE Project No.2013/0827.

Highlights

Low cost austenitic steels

Rare earth-free magnetocaloric materials High temperature solid state cooling


Pablo Ugalde-Salas

Jérôme Harmand

Elie Desmond-Le Quéméner

Modelling Interactions of a Microbial Community through an Optimal Tracking Control Approach

Genetic sequencing measurements of the existing species in a reactor are incorporated into a classic chemostat model through a non linear optimal tracking approach. The framework is applied to data coming from a nitrification process, resulting in a dynamical system with a high dimensional state. The observed ecological dynamics from the bacterial species are able to explain the substrates dynamics.

I. INTRODUCTION

In the last decades genetic sequencing techniques have become an affordable measurement in bioprocesses. However using this measurements for the prediction and control of the bioreactors performance (in terms of proportions of chemicals consumed and produced) lacks a modelling framework [START_REF] Widder | Challenges in microbial ecology: Building predictive understanding of community function and dynamics[END_REF].

It has been theorised that interactions between microorganisms could help explain the difference in performance of reactors operating under very similar environmental conditions [START_REF] Wade | Perspectives in mathematical modelling for microbial ecology[END_REF]. Some types of interactions have already been accounted in chemostat theory, such as inhibition in the presence of other species, or density dependence. Nevertheless this represent just a fraction of the possible ecological interactions [START_REF] Zuñiga | Elucidation of complexity and prediction of interactions in microbial communities[END_REF] that could take place in a microbial community.

The central idea behind the method presented in this work is that if one can accurately follow the dynamic of the species present in an ecosystem for a certain time horizon then one can recover the underlying interactions that might be driving the system.

II. MODEL DEFINITION AND PROBLEM STATEMENT

The following conventions are used: [START_REF] Widder | Challenges in microbial ecology: Building predictive understanding of community function and dynamics[END_REF] Let m be an integer then [m] := {1, . . . , n}, (2) For integers m 1 and m 2 and a ∈ R, a m 1 ×m 2 represents a matrix of m 1 rows and m 2 columns with a in every entry, (3) let m be an integer then I m is the identity matrix of size m, (4) given a vector v = (v 1 , . . . , v n ) ∈ R n the function diag(v) returns a square matrix M with entries M ii = v i , ∀i ∈ [n], and M i j = 0, ∀i, j ∈ [n], i = j, and (5) Let a 1 , a 2 ∈ R then a 1 ∨ a 2 := max{a 1 , a 2 } and a 1 ∧ a 2 := min{a 1 , a 2 }.

A. Stoichiometric Equations

Consider the situation where n different Operative Taxonomic Units (OTU) are present in a system. A group of OTU (G 1 ⊂ [n]) consumes a substrate s 1 for growth and as a result s 2 is produced, while another group of OTU (G 2 ⊂ [n]) consumes s 2 for growth and produces s 3 . G 1 and G 2 are called functional groups. This situation is often referred to 1 LBE, INRAE, Univ. Montpellier, 11100, Narbonne, France. * Corresponding author : pablo.ugalde-salas@inrae.fr as a cascade bioreaction. The number of organisms in G 1 and G 2 will be denoted n 1 and n 2 , respectively. It will be assumed that sets G 1 and G 2 satisfy:

G 1 ∩ G 2 = / 0 and G 1 ∪ G 2 = [n].
The situation is schematically described in bioreactions (R G1) and (R G2). The terms y i are known as yields, they represent the number of moles of biomass produced per mole of substrate consumed. The term M x represents a molecule of biomass and several expressions can be found in the literature (e.g. CH 1.613 O 0.557 N 0.158 [START_REF] Battley | Heat capacity measurements from 10 to 300 K and derived thermodynamic functions of lyophilized cells of Saccharomyces cerevisiae including the absolute entropy and the entropy of formation at 298.15 K[END_REF]). s = (s 1 , s 2 , s 3 ) and x = (x 1 , . . . , x n ) denote the vectors of concentrations of substrates and of OTU present in the system, respectively. Each reaction has its own process rate (also known as growth function or kinetics) µ i (s, x). Note that it depends on x to show the influence of other OTU on the growth function. A classic treatment of general reactions and kinetics in biotechnology can be found in [START_REF] Savageau | Energetics and Kinetics in Biotechnology[END_REF].

s 1 µ i (s,x) -→ s 2 + y i M x ∀i ∈ G 1 (R G1) s 2 µ i (s,x) -→ s 3 + y i M x ∀i ∈ G 2 (R G2)

B. Mass Balance Equations

The system where reactions (R G1) and (R G2) take place is a container of fixed volume V containing a liquid medium with the same input and output flow Q = Q(t), which can vary in time. The dilution rate is then defined as D = D(t) = Q V . The input flow contains a time dependent concentration s in = s in (t) of the same chemical composition than s 1 , for readability the time dependence notation is dropped. For chemostat theory detailed modelling and mathematical analysis one can refer to [START_REF] Lobry | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF]. The dynamics of the described situation are seen in [START_REF] Widder | Challenges in microbial ecology: Building predictive understanding of community function and dynamics[END_REF].

ẋi = (µ i (s, x) -D) x i ∀i ∈ G 1 ẋi = (µ i (s, x) -D) x i ∀i ∈ G 2 ṡ1 = (s in -s 1 )D -∑ i∈G 1 1 y i µ i (s 1 , x)x i ṡ2 = -s 2 D + ∑ i∈G 1 1 y i µ i (s, x)x i -∑ i∈G 2 1 y i µ i (s, x)x i ṡ3 = -s 3 D + ∑ i∈G 2 1 y i µ i (s, x)x i (1)

C. Kinetics and Interactions

Given a vector (v 1 , . . . , v n ) the interaction function I is noted as:

I : R n → [0, 1] n v →      I 1 (v) I 2 (v) . . . I n (v)      (2)
The growth function of each OTU will be assumed to be a product of substrate dependent function and the interaction function. Let f i (s) be a bounded, positive, and continuous function of s (e.g. Monod, Haldane). The growth equation of OTU i becomes:

µ i (s, x) = f i (s)I i (x) (3)
The notation f (s) := ( f 1 (s), . . . , f n (s)) will be used.

Since the growth of a single strain in chemostat experiments is limited by the substrate concentration , when no interactions are present one should recover expression f i (s). Therefore if there are no interactions then I i (x) = 1. From this reasoning, note that lim x→0 I i (x) = 1 since if there is minimal presence of OTU, interactions can not exist. Furthermore it is assumed that I i (•) is a continuously differentiable function on x. For making explicit all of the former:

The interaction function I previously defined satisfies:

1) I (0, . . . , 0) = (1, . . . , 1) 2) Let Ω be an open subset of R n then I ∈ C 1 (Ω).
By forcing the positivity of the interaction function then the growth functions are still positive and the system remains well posed [START_REF] Lobry | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF], meaning that if system (1) is provided with non-negative initial conditions, the trajectories remain nonnegative.

The problem is then to reconstruct the interaction function from the observations of the microbial community composition in time.

D. Optimal Control Setting for Unravelling the Interaction Function

Suppose that the functions f i (s), and the yields y i are well-known. By using experimental measurements of x, represented by z(t), the objective is to reconstruct function I(x). For doing so, the terms I i (x) are replaced by a control u i (t) (u i (t) = I i (x(t))) and a state feedback control law is obtained by solving a nonlinear optimal tracking problem. Consider the observable system (4).

ẋi = ( f i (s)u i (t) -D) x i ∀i ∈ G 1 ẋi = ( f i (s)u i (t) -D) x i ∀i ∈ G 2 ṡ1 = (s in -s 1 )D -∑ i∈G 1 1 y i f i (s)u i (t)x i ṡ2 = -s 2 D + ∑ i∈G 1 1 y i f i (s)u i (t) -∑ i∈G 2 1 y i f i (s)u i (t)x i ṡ3 = -s 3 D + ∑ i∈G 2 1 y i f i (s)u i (t)x i y(t) = x(t) (4)
Let Q, and R be positive definite matrices, and let • Q and • R be the weighted norms by the corresponding matrices. The optimal tracking problem is defined as:

min T 0 y -z Q dt + T 0 (u -1 n×1 ) R dt s.t. (x, s 1 , s 2 , s 3 ) solution of (4) u i (t) ∈ [0, 1] (5)
The control u(t) is intended to drive the system to be near a desired output z(t), which in this context are the measurements of the concentrations of OTU. The term (u -1 n×1 ) R , was added for two reasons:

First because the interest is testing the idea that interactions could be driving the system. Therefore adding a penalization in the objective function for each control to remain near 1 can be seen as an attempt to explain data without any interaction.

Second, to force a regularized control. Otherwise note that u is linear in the state equations (4), therefore if the integral cost does not have a non-linear expression of u the optimal control will be of a bang-bang type with possibly singular arcs. Since the objective is to find a differentiable expression of I(x) the addition of the regularization term is deemed necessary.

The problem of approximating the solution of the system to a desired reference (z in this case) is called the optimal tracking problem. For solving such a problem the approach developed by Cimen et al. [START_REF] Banks | Nonlinear optimal tracking control with application to super-tankers for autopilot design[END_REF] was adapted to our problem, using [START_REF] ¸imen | State-Dependent Riccati Equation (SDRE) control: A survey[END_REF]. The method proposed involves the resolution of Approximating Sequences of Ricatti Equations.

The method consists on iteratively calculating trajectories of system (4) with a certain control and feeding a nonautonomous Ricatti differential equation with the resulting trajectory. Then a control law that uses the Ricatti equation is proposed and a new trajectory is calculated. The iteration is stopped when a convergence in the output or in the control is observed.

First the tracking problem does not consider a constrained control, so one should consider a reformulation as proposed by the same author in other works [START_REF] ¸imen | State-Dependent Riccati Equation (SDRE) control: A survey[END_REF]. Nevertheless, the methods in [START_REF] Banks | Nonlinear optimal tracking control with application to super-tankers for autopilot design[END_REF] were directly used, and an explicit constraint in the synthesis of the control was added for the control to remain within the desired bounds (see equation ( 21)).

Even though this is probably suboptimal it worked for our purposes.

Second in their problem they used an approximation of their original dynamics for calculating the trajectories and subsequently proved that such linearisation converged to the original dynamics. In the case here presented the linearisation of the dynamics was not necessary.

E. Tracking Problem Reformulation

For applying the methods developed in [START_REF] Banks | Nonlinear optimal tracking control with application to super-tankers for autopilot design[END_REF]. Define the system state X = (x, s), and the change of variables v i = u i -1 with v = (v 1 , . . . , v n ). System (4) may be rewritten then as:

ẋi = ( f i (s)(1 + v i (t)) -D) x i ∀i ∈ G 1 ẋi = ( f i (s)(1 + v i (t)) -D) x i ∀i ∈ G 2 ṡ1 = s 1 s in s 1 -1 D -∑ i∈G 1 1 y i f i (s)(1 + v i (t))x i ṡ2 = -s 2 D + ∑ i∈G 1 1 y i f i (s)(1 + v i (t)) -∑ i∈G 2 1 y i f i (s)(1 + v i (t))x i ṡ3 = -s 3 D + ∑ i∈G 2 1 y i f i (s)(1 + v i (t))x i y(t) = x(t) (6)
Define

k G 1 = 1 y 1 , . . . , 1 y nA ∈ R n 1 (7) k G 2 = 1 y n 1 +1 , . . . , 1 y n ∈ R n 2 (8) A (X) = A 11 (X) A 12 (X) A 21 (X) A 22 (X) (9)
B (X) = B 1 (X) B 2 (X) (10)
A 11 (X) = diag( f (s) -D n×1 ) (11)
A 21 (X) =    f (s) -diag(k G 1 ) 0 n 1 ×n 2 f (s) diag(k G 1 ) -diag(k G 2 ) f (s) 0 n 2 ×n 1 diag(k G 2 )    (12)
A 12 (X) = 0 n×3 (13)

A 22 (X) =    s in s 1 -1 D 0 0 0 -D 0 0 0 -D    (14) B 1 (X) = diag( f (s))diag(x) (15) B 2 (X) =    f (s) diag(x) -diag(k G 1 ) 0 n 1 ×n 2 f (s) diag(x) diag(k G 1 ) -diag(k G 2 ) f (s) diag(x) 0 n 2 ×n 1 diag(k G 2 )    (16) C(X) = I n 0 n×3 (17)
Then system (4) becomes:

Ẋ = A(X)X + B(X)v (18) y = C(X)X
Let z(t) ∈ R n be the vector containing the experimental measurements of the OTU concentrations in time. The cost functional is now rewritten as (19).

J(v) = t f t 0 (z(t) -X(t)) Q (z(t) -X(t)) dt + t f t 0 v(t) Rv(t)dt (19)
Define the dynamic sequences for i ∈ N, X [i] as :

Ẋ[i] = A(X [i] )X [i] + B(X [i] )v [i]
(20)

y [i] = X [i] X [i] (t 0 ) = X 0
And for i = 0 let X [0] (t) = X 0 . The control law is given by

v [i] (t) j = 0 ∧ -1 ∨ -R -1 B X [i-1] (t) P [i] (t)X [i] (t) -s [i] f (t) j ∀ j ∈ [n] (21)
Where P [i] (t) ∈ M n+3×n+3 (R) and s

[i] f (t) ∈ R n+3 are the solution to the systems of differential equations ( 22) and (23), respectively.

Ṗ[i] = -C T X [i-1] (t) QC X [i-1] (t) (22)
-P

[i] A X [i-1] (t) -A X [i-1] (t) P [i] + P [i] B X [i-1] (t) R -1 B X [i-1] (t) P [i] P [i] (t f ) = 0 ṡ [i] f = -C X [i-1] (t) Qz(t) -A X [i-1] (t) (23) -B X [i-1] (t) R -1 B X [i-1] (t) P [i] (t) s [i] f s [i] f (t f ) = 0
Note that system ( 22) and (23) consists of (n + 3) 2 + (n + 3) differential equations. In the following the inspection of system ( 22) and ( 23) allows us to reduce the number of differential equations to 2n. Replacing matrices ( 9), [START_REF] Ugalde-Salas | Asymptotic Observers and Integer Programming for Functional Classification of a Microbial Community in a Chemostat[END_REF], and (17) in ( 22) and ( 23) one obtains:

Ṗ[i] (t) = - Q 0 n×3 0 3×n 0 3×3 -P [i] A X [i-1] (t) (24) -A X [i-1] (t) P [i] + P [i] B X [i-1] (t) R -1 B X [i-1] (t) P [i] P [i] (t f ) = 0 ṡ f [i] (t) = - Qz(t) 0 3×1 -A X [i-1] (t) (25) -B X [i-1] (t) R -1 B X [i-1] (t) P [i] (t) s [i] f s [i] f (t f ) = 0
The fact that certain entries are autonomous in the dynamic and that the constantly zero function is a solution for them, implies by existence and uniqueness that they should be constantly zero. Then P [i] has n × n non zero entries and s [i] has n non zero entries, explicitly:

P [i] (t) = P[i] (t) 0 n×3 0 3×n 0 3×3 (26) s [i] f (t) = s[i] f (t) 0 1×3 (27)
Where P[i] (t) and s f [i] (t) satisfy differential equations (28) and (29), respectively. Note that matrices depend on the previous state iteration i.e. A 11 = A 11 X [i-1] (t) and

B 1 = B 1 X [i-1] (t) . Ṗ[i] (t) = -Q -P[i] A 11 -A 11 P[i] + P[i] B 1 R -1 B 1 P[i] (28) P[i] (t f ) = 0 ṡ[i] f (t) = -Qz(t) -A 11 -B 1 R -1 B 1 P[i] (t) s[i] f (29) s[i] f (t f ) = 0
Assume now that matrices Q and R are diagonal matrices. Inspecting the equation (28) one notices that if i = j, P i j (t) = 0 is a solution of all non diagonal entries. And therefore, once again, by existence and uniqueness they are constantly zero. Hence in system (28) only the diagonal entries should be calculated as seen in equation (30) .

Ṗ[i] j j (t) = -Q j j -2 f j s [i-1] -D P[i] j j (30) + R -1 P[i] j j 2 f j s [i-1] 2 x [i-1] j 2 P[i] j j (t f ) = 0
And the control law is given by

v [i] (t) j = (31) 0 ∧ -1 ∨ -R -1 B 1 X [i-1] (t) P[i] (t)x [i] (t) - s[i] f (t) j ∀ j ∈ [n]
Differential equations ( 30) and (29) were solved using standard backward numerical integration for each iteration. Note that from the former computations the method relies in calculating 2n differential equations instead of (n+3) 2 +(n+ 3) as the straightforward application of the method would have meant.

III. APPLICATION AND DISCUSSION

The choice of matrices Q and R are always of discussion in a quadratic regulator. Q represents the covariance matrix of measurements, while the interpretation of R is not clear. In the simulations here presented Q = I n and R = 0.0001I n , found by trial and error in order for the control to fit accurately the species abundance. The method stopped after 110 iterations where each iteration took approximately 600 seconds of computing time in a computer equipped with 8gb of RAM memory and Intel core i3-7100U CPU 2,40 GHz.

The tracking problem was applied to data coming from a nitrification process; the experimental conditions are described in [START_REF] Dumont | Observers for microbial ecology-how including molecular data into bioprocess modeling?[END_REF]. For exploring the hypothesis of interactions as drivers of bioreactors performance environmental conditions should be kept as constant as possible. Therefore only data from day 183 onwards was used because a change in the operating temperature happened at that point, which is known to have an effect on the kinetics. For choosing which species belong in which functional group, the procedure described in [START_REF] Ugalde-Salas | Asymptotic Observers and Integer Programming for Functional Classification of a Microbial Community in a Chemostat[END_REF] was used: from day 183 to day 230: 17 OTU were identified in the G 1 group (known as Ammonium oxidizing bacteria AOB) and 8 in the G 2 functional group (known as Nitrite oxidizing bacteria NOB). The system state dimension becomes 28.

The knowledge of functions f i (s) was based on a study of nitrification's kinetic parameters [START_REF] Wiesmann | Biological nitrogen removal from wastewater[END_REF]. Particularly given the system's ammonium and nitrate concentration a Monod function (eq (32)) was used for G 1 and G 2 .

f i (s) = μ2 s 1 K 1 + s 1 ∀i ∈ G 1 (32) f i (s) = μ2 s 2 K 2 + s 2 ∀i ∈ G 2
For the reader to gain understanding of the situation, a simulation of the system using the experiments operating parameters (D and s in ) is presented without control (i.e. v(t) = 0) in Figure 1. s 3 accumulates all along the trajectory, but when compared to data it is clear that s 3 stops accumulating after a while.

When applying the presented method one obtains the simulation that can be seen in figure 2. The method captures the tendencies in the measured substrates, but not their absolute values. The simulation of each functional G 1 (AOB), and G 2 (NOB) can be seen in figures 3 and 4, respectively. Note that the fit of G 1 seems better than the fit of G 2 , which can be explained by the 1 order of magnitude difference of abundance between the members of the group. One can observe that in figure 2 the predicted s 3 around day 240 is much higher than the measurement. If one looks at figure 4 it suggests attributing this extra production of s 3 around day 240 to the growth of OTU 24. What would have happened if OTU 24 had been classified in the other functional group? One would have seen a decrease in s 1 , an increase in s 2 , and a decrease in s 3 which could have better fitted the measurements. Thus suggesting that OTU 24 was misclassified from the method presented in [START_REF] Ugalde-Salas | Asymptotic Observers and Integer Programming for Functional Classification of a Microbial Community in a Chemostat[END_REF].

In figures 5 and 6 it can be seen the obtained controls for each functional group. Both controls saturate at certain points, but in general terms they vary throughout the dynamic.

Recalling that u(t) is supposed to represent I(x(t)) (the interactions terms), one could try to fit different forms of I(•) for example I(x) = 1 + Ax, where A is a real valued square matrix. In figure 7 the simulation of the system after the identification of matrix A is presented, suggesting that an affine interaction function is not able to capture the behaviour of the system.

IV. CONCLUSION AND PERSPECTIVES

Considering the dimension of the system analysed, the approach here presented seems satisfactory for the integration of genetic sequencing in the context of bioreactors, particularly the Ricatti equations grow linearly in the number of species, rendering the method scalable for a bigger number of species. However some issues should be taken care of: 1) test the method on synthetic data for a class of interaction functions 2) the optimality of the control should be improved, considering is just a truncated control from a state feedback 3) A clear rational on the choice of matrices Q and R. Finally the hypothesis of misclassification presented for OTU 24 proposes as well a retroactive way to reclassify an OTU if the substrates predictions go astray.

Fig. 1 :

 1 Fig. 1: Simulation when u = 1.

Fig. 2 :

 2 Fig. 2: Simulation applying the control strategy.

Fig. 3 :

 3 Fig. 3: Abundance of G 1 (AOB) simulation applying the control strategy.

Fig. 4 :

 4 Fig. 4: Abundance of G 2 (NOB) simulation applying the control strategy.

Fig. 5 :

 5 Fig. 5: Control obtained for OTU in G 1 (AOB).

Fig. 6 :

 6 Fig. 6: Control obtained for OTU in G 2 (NOB).

Fig. 7 :

 7 Fig. 7: Simulation obtained for the interaction function I(x) = 1 + Ax, where A was fitted with the obtained controls from figures 5 and 6.


Farouk Aichouche
email: aichouche.farouk@gmail.com

Boumediéne Benyahia
email: b.benyahia.ut1@gmail.com

Marc Héran
email: marc.heran@umontpellier.fr

Jérôme Harmand
email: jerome.harmand@inra.fr

Flexibility of Waste Resource Recovery Facilities for water Reuse

In this paper, a new flexible wastewater treatment scheme is proposed to be used within the framework of wastewater reuse (REUSE). From a configuration where the water is treated to guarantee normative constraints, we propose a flexible configuration able to adapt, in minimal time, the quality and the quantity of water to usage. In other terms, the proposed scheme is able to treat water for purpose. Using model analysis and simulations, we show how a simulation platform may be used to demonstrate the flexible characteristics of the scheme, in particular when dealing with the decoupling of water fluxes and the different forms of nitrogen contents in the output treated water.

I. INTRODUCTION

Actual wastewater treatment plants (WWTP) aim at treating wastewater in such a way they comply with normative constraints before discharged into environment (rivers lakes etc). However there is the need nowadays for a paradigm shift within the framework of circular economy: from a waste to be treated wastewater can be viewed as new resources to be processed for purpose. To do so, WWTP have to be operated with more 'flexibility'. Although intensive attention has been given to WWTP, especially to biological nitrogen and carbon removal processes [START_REF] Kraume | Nutrients removal in MBRs for municipal wastewater treatment[END_REF], a large number of alternative configurations have been proposed [START_REF] Hreiz | Optimal design and operation of activated sludge processes: State-of-the-art[END_REF] in order to improve the performance of the treatment either in terms of cost [START_REF] Zarrad | Comparison of advanced control strategies for improving the monitoring of activated sludge processes[END_REF] or in terms of quality [START_REF] Van Kempen | SHARON process evaluated for improved wastewater treatment plant nitrogen effluent quality[END_REF]. In any case, the configuration should contain an aerobic capacity to promote the nitrification while an anoxic zone must be used to promote denitrification. The two-steps nitrification-denitrification process revealed to be the most economical viable for the total elimination of nitrogen and carbon. Conventional WWT is accomplished by combining denitrification and nitrification steps requiring oxygen and organic carbon, respectively. The separation of conditions may be done in time or in space by imposing sequentially anoxic and aerobic conditions in a single reactor [START_REF] Du Toit | Design and performance of BNR activated sludge systems with flat sheet membranes for solid-liquid separation[END_REF], or, by having a minimum of 2 reactors of which one is maintained anoxic and the other aerobic. The pre-denitrification and the postdenitrification configurations are the most widespread [START_REF] Barker | COD and nitrogen mass balances in activated sludge systems[END_REF]. In the pre-denitrification design, the anoxic tank is placed upstream of the aerobic reactor in order to use the carbon present in the effluent for the denitrification. Therefore, nitrified liquor recycling to the anoxic reactor is needed to recirculate nitrate from the aerobic reactor. In the postdenitrification configuration, the anoxic tank is placed downstream of the aerobic reactor. This configuration allows a better nitrate removal efficiency than the pre-denitrification process. However, since carbon source is consumed in the aerated tank, it may become limiting in the anoxic reactor. Therefore, the addition of carbon source may be required in order to achieve an efficient denitrification. The flexibility concept in WWTP relates to its capacity to adjust -as quickly as possible -to changes in demand, corresponding for instance to plant needs within the context of treated wastewater reuse. On one side, the flexible configuration allows to operate the WWTP in a mode to eliminate the components from wastewater or, on the other side, to keep nutrients or to decouple the water content from the water quantity (play with nutrient contents in the output effluent). In classical configurations, the nitrogen in ammoniacal form (NH + 4 ) is oxidized into nitrates (NO - 3 ) and then reduced into nitrogen gas (N 2 ). In a flexible system, it is expected that the level of the different forms of nitrogen may be adjusted with respect to agriculture requirements. for instance, in some cases, the ammoniacal form of nitrogen (NH + 4 ) will be favored while in others the nitrate form will be preferred. Moreover, keeping (NH + 4 ) requires lower energy consumption than when water is treated to provide nitrates (NO - 3 ). Indeed, performing nitrification requires much more energy because nitrifying bacteria need more time to develop and thus induce a higher oxygen consumption. However,the ammoniacal nitrogen is not directly assimilable by plants: microorganisms in the soil are responsible for transforming (NH + 4 ) into (NO - 3 ) that are more bioavailable. In the present paper, a new flexible configuration is suggested in the aim to control the transition between the conventional treatment (full treatment mode) and the adaptive treatment (reuse mode) by considering different scenarios. Our first objective is to analyze the best combination between valves positions, recirculation flows and sludge age for each operation mode, and to use this different degrees of freedom to minimize the time response to go from one mode to another one. When designed with the concern of recovering material and/or energy from a given effluent, a WWTP is renamed WRRF for Waste Resource Recovery Facilities. From now and upward, we will thus use WRRF instead of WWTP.

II. A FLEXIBLE PLATFORM OF WRRF A. Platform description

The flexible platform configuration consists of an anoxic tank (V 4 = 56 L) followed by aerobic tank (V 5 = 56L) and aerobic tank with membrane (MBR) (V 6 = 40 L). The membrane is supposed to be ideal (perfect separation of soluble and particular matters) to promote sludge separation (fouling phenomena were not taken into account). The plant was designed to treat a flow Q in = 270 L/d. The platform (represented in Figure 1) can be operated as a one (only tank 6) -or two (tanks 5 and 6) -or three (Tank 4, 5 and 6) -stage process by adjusting influent feeding location (step feeding) and recirculation rates. Denitrification takes place in the anoxic tank whereas nitrification and carbon oxidation are accomplished in both aerobic and MBR tanks. Since nitrates are not present in the influent, they have to be brought into the anoxic reactor by internal recycling of the nitrified mixed liquor from the aerobic reactor when there is a need to reduce the effluent nitrate. This is done by tuning the external recirculation and playing with the position of the valve "C". The waste activated sludge (WAS) flow rate in MBR tank is controlled to reach a specific sludge retention time corresponding to different operation modes (full / Adaptive). The dissolved oxygen (DO) concentration in the aerobic and MBR tank are maintained above 2.0 mg/L, while the DO level of the anoxic tank is supposed to be 0 mg/L. The constant influent characteristics are available in the table I.

B. Model description

The biological reactions inside each reactor in the flexible platform are modeled using ASM1-2ND model [START_REF] Ostace | Agachi Cost reduction of the wastewater treatment plant operation by MPC based on modified ASM1 with two-step nitrification/denitrification model[END_REF]. It is a modified version of Activated Sludge Model 1 (ASM1) [START_REF] Henze | International Water Association[END_REF]. The main modification consists in the addition of a two-step process for nitrification and denitrification. While the ASM1 model consider only complete nitrification of ammonia (NH + 4 ) to nitrate (NO - 3 ) by autotrophic bacteria, followed by complete denitrification of nitrate to nitrogen gas, ASM1-2ND uses a 2-steps nitrification model: the nitrification is carried out in two sequential steps, the first one describing the ammonium (NH + 4 ) oxidation to nitrite (NO - 2 ) by Ammonia Oxidizing Bacteria (X AOB ) while the second one models the nitrite oxidation (NO - 2 ) into nitrate (NO - 3 ) by Nitrate Oxidizing Bacteria (X NOB ). The denitrification is performed following a parallel two-step scheme in which nitrate (NO - 3 ) or nitrite (NO - 2 ) is transformed into gas (N 2 ) by heterotrophic bacteria (X BH ).The processes describing the growth of aerobic bacteria are modeled using Monod kinetics and are presented in the equations ( 1) and ( 2). The aerobic growth kinetics of Ammonia Oxidizing Bacteria ρ AOB (mg L -1 d -1 ) is modeled by:

ρ AOB = µ AOB S NH K NH + S NH S 0 K OA + S O X AOB (1)
The aerobic growth kinetics of Nitrate Oxidizing Bacteria ρ NOB (mg

L -1 d -1
) is modeled by:

ρ NOB = µ NOB S NO2 K NO2 + S NO2 S 0 K OA + S O K NH S NH + K NH X NOB (2)
Where K OA ,K NO2 and K NH are half-saturation rates of oxygen, nitrite and ammonium, respectively. µ AOB , µ NOB are Maximum growth rates of Nitromonas and Nitrobacter, respectively. This model is used for simulating the biological processes that take place in the platform proposed above, in order to predict the growth of bacteria and the resulting consumption of carbon and nitrogen. This is done in taking into account nitrite as an intermediate state variable during the nitrification and denitrification processes.

C. Implementation of WRRF platform

Set-up of the 'simulation platform' was made under Matlab/Simulink-environment. By using three bioreactor objects, five 2-way flow combiner models, all implemented as C-code S-function objects. The membrane is supposed to be ideal (the effluent does not contain any particulate component). Furthermore, a hydraulic delay model was add (to avoid algebraic loops). and the controllers for internal recirculation flow rate recirculation flow rate with various connections and Valves are integrated.

D. The Solids Retention Time (SRT) control

The SRT is an important design and control parameter for the activated sludge process. It represents the average length of time that bacteria stay in the process before being wasted. The selected SRT directly affects the bacterial composition in an activated sludge system. In the literature, SRT is usually considered from a steady-state perspective. At steady-state, the static SRT is defined as the mass of microorganisms in the system divided by the mass of microorganisms wasted per time unit:

SRT = X r V r X w Q w + Q e X e (3)
Where:

V r : Total volume of aeration tanks, m 3 ; X r : Total suspended solids (T SS) concentration in the reactor, mg/L; X w : Total suspended solids (T SS) concentration in the waste activated sludge, mg/L; X e : Total suspended solids (T SS) concentration in effluent,mg/L; Q w : Waste activated sludge (WAS) flow rate, m 3 /d; Qe: effluent flow rate, m 3 /d X e = X r in completely mixed reactor:

SRT = V r Q w (4)
The first modeling step consists in determining the initial operating conditions of the WRRF associated with each operation mode: 1) Reuse Mode: It is necessary to find conditions in which the biological treatment will make it possible to preserve the nitrogen under its form NH + 4 . It means that the sludge age (SRT) must not be too high, so that nitrification will be limiting or it will not take place. The tests are done with a sludge removal rate (SRR), Q w varying between 10 m 3 /d and 40 m 3 /d. It is assumed that no oxygen limitation occurs and no nitrite accumulates.

2) Complete treatment Mode: The SRR applied in this mode must make nitrification possible to treat nitrogen. Simulations are made with Q w varying between 0.1 m 3 /d and 6 m 3 /d.

III. PLATFORM SIMULATION

In order to analyze the platform and to determine the best transition strategy from one mode of operation to another in terms of quality and time response, it is necessary to identify the scenarios for each mode of operation. Each scenario represents a mode of operation. To study the transition from one operating mode to another, we study the switch of environmental conditions (positions of valves, values of control signals). Different switches are simulated and the corresponding time responses -corresponding to the time required for the system to switch from one steady state to another one -are observed. It should be noted that the valves in the different configurations are supposed to be all or nothing valves. Each value thus represents a direction of the corresponding fluxes. The first two scenarios are used to model the adaptive treatment (useful for reuse), while the second two scenarios represent the full -or complete -treatment (conventional treatment).

1) Scenario S1: The anoxic and aerobic tanks (4 and 5) are not in operation, whereas the tank 6 is in operation, with the valves positioned at valve A = 1, valve B = 1. The internal and external recirculations are not activated Q ext =Q int =0 and sludge extraction from the tank 6 is Q w = 32.4 L/d. 2) Scenario S2: The anoxic tank ( 4) is not in operation, were the aerobic and MBR tanks (5 and 6) 4) Scenario S4: In this scenario, the configuration is the same for the scenario S3, except for the valve C, that results in position 1. It means that the returned sludge is oriented to the aerobic tank. As we will see, the transitions dynamics between the different modes of operation depend a lot on the initial conditions. Usually, water for reuse is needed during spring and summer times while complete treatment is required during fall and winter where water should be treated to comply with discharge of water in the environment. To simulate such conditions, we varied the time at which we switch from one operation mode to another one. During the various simulations carried out, different components of the effluent are followed, in particular the different forms of nitrogen, as well as the bacterial populations present in the bioreactor.

IV. RESULTS AND DISCUSSION

In the present work the results of simulation will be shown, focusing on the transition from the reuse mode to the full one or vice-versa, and describing the nitrogen dynamic during both modes. The alteration from the reuse to full mode had as goal the reduction of response time and to obtain NO - 3 in effluent. On the dissimilarity,when the alteration from the full to reuse one, the objective was to reach a concentration of NH + 4 less than 30 mg.N/L.

A. Influence of retention time on reuse mode

Figure 2 shows the transition from the scenario S1 (reuse mode) to the scenario S3 (complete treatment mode). Different initial conditions at the time of switching explain the observed difference in time responses: remember that the X NOB bacteria grow exponentially. At 200 days the transition occurs. At this instant the X NOB biomass concentration is lower than the one when the transition occurs at 100 day (the more the bacteria are subjected to higher wasted sludge flow Q w = 32.4 L/d for prolonged time, the more its concentration are diminished). It should be noted that in practice, such difference may not be observed: depending on the time during which X AOB and X NOB have not been grown under appropriate conditions, they can simply be considered to be absent of the system. In other words, while simulations may consider very small concentration (that may have no physical meanings), X AOB and/or X NOB may have definitely been washed out from the system. We can also note that the duration of partial nitrification is longer in the case where the transition was applied at 200 day because of the growth rate of X AOB bacteria is greater than the growth rate of X NOB bacteria, then the growth of X AOB is faster to X NOB . To simulate realistic situations, such considerations should be studied with care.

B. Influence of returned sludge direction

Figure 3 shows the comparison between two transition cases. The first one is the switch from scenario S1 to scenario S3 (dashed line) whilst the second one is the swith from scenario S1 to scenario S4 (continuous line). The first transition occurs 100 days slower than the second one. In the later one the returned sludge was directed to the aerobic tank at the instant of transition. This set-up promotes the nitrifying bacteria growth (X AOB , X NOB ), since it reduces the lag phase. On the other hand, in the first transition, the returned sludge was directed to the anoxic tank. In this case, the initial growth of nitrifying bacteria at the transition instant is not encouraged, It leads to an increase in the lag phase.

C. Effects of dilution on the returned sludge

The transition from S1 to S4 (from the reuse mode to the full treatment mode) was also simulated in considering two different values of the returned sludge, that were 100 L/d and 270L/d. In Figure 4 the profiles corresponding to this different values of returned sludge were plotted. When the flow of returned sludge equals 100 L/d (dashed line), the time response was shorter than 270 L/d (continuous line). In the same simulations, an increase in total nitrogen concentrations (NH + 4 , NO - 3 and NO - 2 ) was observed since low returned sludge flow means low dilution rate and more concentrated nutrients. So, in terms of time response the returned sludge equal to 100 L/d seems to be the most important but not in terms in terms of nutrient concentrations in the effluent. In addition, NO - 2 is important during partial nitrification, which is problematic given the toxic characteristics of nitrite.

D. Influence of biomass addition on the reduction of response time (bio-augmentation)

The next step is to focus on the use of bio-augmentation (addition of nitrifying bacteria) to minimize the time to switch from scenarios S1 to S4. From a simulation viewpoint, it is equivalent to change initial condition of nitrifying bacteria (X AOB and X NOB ) in the reactors according to the following rules:

• X AOB = 10 mg/L; X NOB = 10 mg/L (dashed line in fig 5). • X AOB = 1 mg/L; X NOB = 1 mg/L (continuous line in fig 5).

In figure 5, both profiles are represented when the added concentration of bacterial is 1 mg/L on both (X AOB and X NOB ). A short partial nitrification was observed because of an insufficient concentration of X NOB to obtain a sufficient growth and a full nitrification (NO - 2 to NO - 3 ). On the contrary, with the addition of 10 mg/L, a complete nitrification was observed. The addition of a more important bacterial Fig. 5. Influence of biomass addition. Ammonium (blue) and nitrate (green) and nitrite (red) concentrations in the effluent for two cases of addition of nitrification bacterial : 10 mg/L (dashed line), 1 mg/L (continuous line) concentration influenced positively the time response, reducing it from 50 days with the addition of 1 mg/L to 30 days with 10 mg/L. The Bio-augmentation is therefore a solution in order to decrease the time response of the system as well as the consequent formation of nitrites.

E. Two cycles simulation of transitions from Reuse to Full treatment

The figure 6 shows two transition cycles from the reuse mode to the complete mode. The first cycle was performed after 100 days of operation to the reuse mode to switch after that to full one again. The second cycle was started after 200 days in this cycle. The transition from the full to the reuse mode (from scenario S3 to S1) requires a limiting nitrification to obtain some NH + 4 in the effluent. This transition is fast since it depends only on the time necessary for the nitrifying bacteria to be washed out using the waste sludge flow. This transition is completed in only 15 days (figure 6). When comparing the observed time responses for all simulated transitions, the addition of biomass (X AOB and X NOB ) at the time of transition seems to be the most appropriate strategy. Indeed, autotrophic bacteria and in particular X NOB autotrophs develops very slowly. In addition, when working without bio-augmentation, the risk for nitrite accumulation in the transient is very high.

V. CONCLUSION

A new flexible wastewater treatment configuration was proposed in this paper. The basic objective underlying its design is to improve the transition operation between the resource recovery mode (Reuse) to the conventional treatment mode (full). In the present study various possible scenarios were discussed with the aim to analyze and design different transition situations. Suitable scenarios were selected for each operating mode in order to determine the best transition to minimize the time response and to ensure effluent quality Fig. 6.

Two cycles simulation of transitions between Reuse and Full treatment : ammonium (blue), nitrate (green) and nitrite (red) concentrations in the effluent in accordance with agriculture requirements. An improvement on the response time by an addition of nitrifying bacteria was observed. A future work will investigate extensions of the results to take in account a dynamics influent flow with manipulating in different control degree.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the flexible platform. Q in . influent; 2. Two position valve (position 1 or 0); 3. feed pump; 4. Anoxic tank; 5. Aerobic tank; 6. Aerobic tank; 7.Perfect membrane; E. Effluent; Q int . Internal sludge recycling; Q ext . external sludge recycling; 11. Air compressor; 12. Mechanical mixer; Q w . Waste sludge.

  are in operation and aerated, the valves position are: valve A = 1, valve B = 0, valve C = 0. The external recirculation is activated and Q ext = 270 L/d while the internal recirculation is not in activated Q int = 0 and sludge extraction from the MBR tank (6) Q w = 32.4 L/d. 3) Scenario S3: During this scenario all tanks (4, 5 and 6) are in operation, the valve A = 1, B = ∀ and C = 0, the flows of internal and external recirculation are active and are directed towards the anoxic basin and sludge extraction from the 6 tank Q w = 1.4 L/d.

Fig. 2 .

 2 Fig. 2. Influence of retention time. Dynamic ammonium (blue), nitrate (green) and nitrite (red) concentrations for transitions at 100 days (continuous line) and at 200 days (dashed line)

Fig. 3 .

 3 Fig. 3. Influence of returned sludge direction. Dynamic ammonium (blue), nitrate (green) and nitrite (red) concentrations in the effluent.

Fig. 4 .

 4 Fig. 4. Effects of dilution. Dynamic ammonium (blue) and nitrate (green) and nitrite (red) concentrations in the effluent for returned sludge directed to anoxic tank (dashed line) or to aerobic tank (continuous line).

TABLE I CONSTANT

 I

		INFLUENT CHARACTERISTICS
	Symbol Variable	Value Unit
	S I,in	Inert organic matter	21.75	mgCOD.L -1
	S S,in	Readily biodegradable substrate	98	mgCOD.L -1
	X I,in	Particulate inert organic matter	29	mgCOD.L -1
	X S,in	Slowly biodegradable substrate	112	mgCOD.L -1
	X BH,in	Active heterotrophic biomass	0	mgCOD.L -1
	X AOB,in	Active Nitrosomonas biomass	0	mgCOD.L -1
	X NOB,in	Active Nitrobacter biomass	0	mgCOD.L -1
	X P,in	Particulate product	20.22	mgCOD.L -1
	S O,in	Dissolved oxygen	0	mgO 2 .L -1
	S NO2,in	Nitrite nitrogen	0	mgN.L -1
	S NO3,in	Nitrate nitrogen	0	mgNL -1
	S N2,in	Nitrogen gas	0	mgN.L -1
	S NH,in	Ammonia nitrogen	36	mgN.L -1
	S ND,in	Biodegradable organic nitrogen	1.69	mgN.L -1

X ND,in Particulate biodegradable nitrogen 0 mgN.L -1


Mark Gowdy

Agnès Destrac-Irvine

Martina Haines

Gregory Gambetta

Philippe Pieri

Elisa Marguerit

Cornelis Van Leeuwen

Varietal responses to soil water deficit: first results from a common-garden vineyard near Bordeaux France

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Introduction

In wine producing regions around the world, climate change has the potential to decrease the frequency and amount of precipitation and increase average and extreme temperatures during the growing season. Understanding how vine water status responds to these changes across different Vitis vinifera varieties will help growers modify vineyard design and management practices to meet their quality and yield objectives [START_REF] Pellegrino | [END_REF]. Much has been published on how plants regulate water status, but not much to characterize the dynamics of this regulation across the wide range of Vitis vinifera varieties.

As soil water content decreases, the remaining water becomes more tightly bound with the soil matrix and the water potential in the plant must become more negative in order to remove the remaining water and maintain transpiration. In response to this, and other environmental factors, stomata begin to close to moderate water potential and protect the plant from hydraulic failure [2].

As stomata close there is a change in the carbon isotope composition of the carbohydrates and sugars generated by photosynthesis in the leaves at that time. Measurement of the isotope composition ( 13 C) in berry juice then provides a useful indication of the level of stomatal closure that existed during the important berryripening period [3]. Water potential measurements collected pre-dawn and mid-day during the berry ripening also provides an assessment of soil water conditions and plant water status during this same period. For estimating the soil water content corresponding to the above measurements, a water balance model was used. Such models can be useful tools when other more costly and intensive methods may be prohibitive [START_REF] Van Leeuwen | Methodologies and Results in Grapevine Research[END_REF].

This paper presents preliminary data and analysis characterizing and comparing the stomatal closure response (as indicated by  13 C) for 39 Vitis vinifera varieties; and the plant water status response (as measured by water potential measurements) for eight varieties; both in response to changes in soil water content as estimated by a water balance model. The data presented in this paper was collected from an experimental common-garden vineyard near Bordeaux France, which allows for collection of replicate data [START_REF] Destrac-Irvine | CLIMWINE Proceedings[END_REF]. Varietal responses to soil water deficits Vines are pruned double Guyot style with 100 cm plant x 180 cm row spacing and trained by vertical shoot positioning with a canopy height of roughly 160 cm. Varieties planted are from the Bordeaux region, as well as other European regions, and some hybrids that might be suitable in Bordeaux in the future as the climate changes. All varieties were planted on SO4 rootstock clone 761 in gravelly soil typical of the region [START_REF] Destrac-Irvine | CLIMWINE Proceedings[END_REF].

Methods

Experimental vineyard

Many studies of drought adaptation of plants are carried out in controlled conditions, such as greenhouses, allowing strict control over experimental variables. Fundamental differences between plants in controlled versus field conditions include root distribution and water availability in the profiles of the root zone [START_REF] Améglio | [END_REF], along with climatic conditions. Hence, experimentation under field conditions is necessary to put results obtained in controlled conditions into perspective [7].

Carbon isotope discrimination measurement

The relative concentration of 13 CO 2 to 12 CO 2 inside the intercellular space of leaves changes relative to that of the atmosphere because of preferential reaction during photosynthesis with 12 C. These ratios, however, are modified when stomata close and cut off diffusion of CO 2 into the intercellular space of the leaves from the atmosphere, causing the sugars formed and incorporated into various plant tissues at that time to contain a modified ratio of carbon isotopes [8]. This makes the measurement of carbon isotope concentration a useful measure of the level of stomatal closure that existed at the time the glucose building blocks of sampled plant tissue or berry juice were photosynthesized [START_REF] Van Leeuwen | Methodologies and Results in Grapevine Research[END_REF].

Berries were sampled weekly in 2012 through 2016 from all varieties in all blocks of the vineyard and analyzed for numerous parameters, including sugar concentration, berry weights, and acid content. Afterwards, one sampling date from each season for each variety was selected as representative of full maturity based on sugar content and titratable acidity of the juice and cluster sanitary status. Previously frozen samples from the selected date were then defrosted and one sample per variety per block was prepared for analysis. The ratio of 13 CO 2 to 12 CO 2 was determined in the laboratory by elemental analysis coupled to an isotopic mass spectrometer and reported as a ratio x10 -3 (‰) with less negative values indicating more stomatal closure [9].

Of those originally planted in 2009, 48 varieties are now consistently producing enough fruit for analysis each year, however, only 39 varieties produced enough fruit to perform the various analyses across all five years (2012 through 2016) in this analysis. Data from measurements collected in 2017 are currently being analyzed and measurements will be collected again in 2018.

Water potential measurements

In order for a vine to continue transpiring, its water potential must become more negative as it overcomes decreasing soil water potential and conductivity in a drying soil [9]. Predawn leaf water potential is considered a good measure of the equilibrium reached in the plant with this water potential in the root zone [START_REF] Choné | [END_REF].

Eight varieties in the experimental vineyard (Cabernet-Sauvignon, Merlot, Petit Verdot, Grenache, Tempranillo, Touriga nacional, Sauvignon blanc, and Semillon) were selected from the experimental vineyard for water potential measurement starting in 2017, with the intent of covering a range of varietal stomatal closure tendencies as described in various literature.

Midday leaf, midday stem, and predawn leaf water potential measurements were each made on leaves taken from two vines in each of four vineyard blocks for a total of eight vines per variety, totaling 64 for each water potential measurement. Measurements were performed using standard pressure chamber techniques [START_REF] Choné | [END_REF].

Water balance modeling

A water balance model [12] was used to estimate daily soil water content in the vineyard for years 2012 through 2017 for comparison against  13 C and water potential measurements. Inputs to the water balance model include phenology, canopy characteristics (dimensions, porosity, orientation), hedging dates, climate (solar radiation, reference evapotranspiration, temperature, precipitation), and an estimate of 160 mm for total transpirable soil water (TTSW) in the vine root zone. The model then outputs an estimate of soil water content as a fraction of total transpirable soil water (FTSW) on a daily time step. From this the average fraction of total transpirable soil water during the berry-ripening period (FTSW BR ) is calculated between dates of véraison and 75% of berry sugar accumulation for each variety. This averaging period was initially thought by the authors to correspond well with soil water content during berry sugar accumulation, but other averaging methods are being investigated as analysis continues.

Results

Water balance modeling

A box plot of average FTSW BR across all varieties for a given year shows 2012 and 2016 having significantly lower soil water content during the berry-ripening period than 2013 through 2015 (Figure 1). A plot of 2017 model output was not included as phenology and carbon isotope data are still under evaluation. FTSW estimates for 2017, however, were used for comparison against predawn water potential measurements. Absolute values of FTSW are likely underestimated due to difficulty estimating TTSW. Also, the model does not account for varietal differences in horizontal root system distribution. Overall, however, resulting relative differences between varieties remain valid.

Carbon isotope discrimination response

A box plot by year of the  13 C measurements on berry juice across all varieties for a given year shows significant difference between both 2012 and 2016 when compared to 2013 through 2015 (Figure 2). Similarly, a comparison back with Figure 1 finds overall less negative  13 C values (more stomatal closure) in the same years 2012 and 2016, which experienced lower FTSW BR (drier soil conditions during berry-ripening). Statistically significant differences were also found between some varieties in the response of  13 C to changes in soil water content (FTSW BR ). For example, a plot of  13 C versus FTSW BR for both Tinto Caõ and Petit Verdot shows similar response curves, but with consistently less negative values (more stomatal closure) seen across all FTSW BR values for Petit Verdot (Figure 3).

These curves also show the trend of less negative  13 C overall in response to decreasing soil water content, with the different varieties demonstrating a good non-linear fit using a power function regression; with all varieties having similar power coefficients as a formula used for estimating soil matrix potential as a function of soil water content [13]. This suggests a the link between increasing soil water potential and associated stomatal closure (vine stress) in response to decreasing soil water content.

The difference in  13 C between wet and dry FTSW BR conditions for the different varieties is also of interest as an indication of stomatal sensitivity to decreasing soil water content. In preliminary analysis many varieties demonstrated a difference in  13 C of between 3 to 4 ‰, similar to that shown in Figure 3 for Petit Verdot and Tinto Caõ. But before more definitive conclusions can be made about these varietal differences (and other responses described above), the authors are considering alternative calculations of FTSW BR and performing additional statistical analyses.

Water potential response

A plot of average predawn water potential measurements across blocks taken in 2017 versus corresponding water balance model estimates of FTSW shows low vine stress values (none less than -0.16 MPa) when FTSW was above 0.4 (Figure 4). After July 28 th (moving right to left) predawn water potential dropped significantly as soils dried further past FTSW = 0.4. This threshold is similar to values reported in the literature [START_REF] Krounbi | Encyclopedia of Agrophysics[END_REF]12]. There were also statistically significant differences between Semillon and three other varieties (Grenache, Petit Verdot, and Cabernet sauvignon) under drier soil conditions later in the year. The cause of this difference is uncertain and being investigated.

Also, a recently proposed method quantifies the relative stomatal closure tendency of different plant species by measuring the area covered by a plot of midday leaf water potential versus predawn leaf water potential across a range of water status, light, and VPD conditions. The relative size of these "hydroscapes" (as determined by a regression line using the most negative midday leaf water potential measurements) is indicative of the relative stomatal sensitivity of a given variety. A larger area indicating less stomatal closure and more variability in plant water potential and a smaller area indicating more stomatal closure and less variability of plant water potential [14]. Unlike other varieties, the water potential measurements for Semillon collected in 2017 began to fill out the hydroscape, although more data is still needed (Figure 5). More midday and predawn leaf water potential with more negative values are needed to fill out that important part of the hydroscape; although the results so far are promising.

Discussion and conclusion

Preliminary analysis of  13 C measurements from berry juice collected between 2012 and 2016 suggests greater stomatal closure for all varieties during years of lower average soil water content during the corresponding berry ripening period as estimated with a water balance model using phenology of the individual varieties. Some varieties appear to show significant differences from one another, with the response curves for individual varieties having a non-linear shape similar to those used for estimating soil matrix potential as a function of soil water content. Also, observed differences in  13 C between years with wet and dry conditions during berry-ripening ranged between 3 to 4 ‰, but more consideration is being given by the authors to different soil water content averaging methods that might better represent soil water conditions during the time of berry sugar accumulation. Data from subsequent years should also improve the power of statistical determinations of differences between varieties.

Water potential data collected in 2017 from eight varieties finds predawn measurements correlated well with soil water content estimates from a water balance model using the phenology of the individual varieties. The reason for the significantly different response of Semillon is uncertain and being investigated. Also, plots of midday leaf versus predawn leaf water potential measurements show the early development of hydroscape plots that might be useful in further characterizing the stomatal closure behavior of the different varieties in response to changing soil water content. Comparison of the different water potential measurements may give additional insight into stomatal closure response to decreasing soil water content. The authors will also give consideration to effects of vapor pressure deficit as data continues to be collected in subsequent years.

Figure 1 .

 1 Figure 1. Box plot of FTSW BR (mm 3 /mm 3 ) for all varieties that year for 2012 through 2016. Years grouped based Tukey Honest Significant Difference test at α = 0.05. Dark bar is median; box ends are 75th / 25th percentiles; upper/lower horizontal lines are maximum/minimum respectively, with circles being outliers.

Figure 2 .Figure 3 .

 23 Figure 2. Box plot of  13 C (‰) for all varieties measured for

E3SFigure 4 .

 4 Figure 4. Average predawn leaf water potential (MPa) versus corresponding water balance model estimate of FTSW (mm 3 /mm 3 ) for measurements taken in 2017 on Semillon, Cabernet sauvignon, Grenache, and Petit verdot.

Figure 5 .

 5 Figure 5. Plot of midday leaf versus predawn leaf water potential measurements (MPa) for Semillon in 2017, including hypothetical and calculated boundaries for the hydroscapes [14].

  The experimental vineyard used for this study was planted in 2009 at the Institute des Sciences de la Vigne et du Vin (ISVV) situated at the Institut National de la Recherche Agronomique (INRA) domain de la "Grande Ferrade" (33883 Villenave d'Ornon) near Bordeaux, France (44°47'23.83"N 0°34'39.3"W). It consists of 52 varieties laid out in a randomized block replicate design, with 10 plants of each variety in each of five blocks. This design allows the ability to account for variability in soil or other conditions within the vineyard.

	E3S Web of Conferences 50, 01043 (2018)	https://doi.org/10.1051/e3sconf/20185001043
	XII Congreso Internacional Terroir

Acknowledgments

These studies were carried out with support from the French National Research Agency (ANR) under the Investments for the Future Program, within Clusters of Excellence (ANR-10-LABX-45). The VitAdapt Project is also supported by the Conseil Interprofessionnel des Vins de Bordeaux and the Conseil Régional d'Aquitaine.