A Ciechocki
email: antoine.ciechocki@cea.fr

S Carnevali
email: sofia.carnevali@cea.fr

L Rossi

APPLICATION OF THE FSA SCALING METHOD TO THE LSTF ROSA 1.2 TEST AND COMPARISON TO AN APPLICATION OF THE H2TS METHOD

Keywords: SB-LOCA, Scaling, LSTF, FSA, H2TS

The general objective of this work is to compare scaling methods and some different ways to apply them. In the present work, an application of the Fractional Scaling Analysis (FSA) method is compared to an application of the Hierarchical Two-Tiered Scaling (H2TS) method done by the Polytechnic University of Valence (UPV). The ROSA 1.2 test performed on the Japanese LSTF-ROSA-IV facility is analyzed. It is a 1% Hot Leg break LOCA. This transient is divided into five phases. Important bifurcating events are identified. A scaling analysis is applied to the first and the fourth phases called respectively the blowdown and the high-quality mixture discharge. The blowdown starts at the break opening while high-quality mixture discharge starts when primary and secondary pressures cross. The FSA methodology is applied at system and component scale by writing mass and pressure equations for two different control volumes, the primary system and the pressurizer. The CATHARE system thermal hydraulic code is used to calculate the effect metrics, scaling criteria specific to this method, which are analyzed and compared with the π-groups of the H2TS, obtained by the application of the UPV with the TRACE code. Differences between the results of the two methods are discussed. Moreover, the relative weight of the processes that control the pressure are evaluated. Some conclusions are drawn on the scaling and possible distortions of the test facility.

INTRODUCTION

The "scaling'' activity is involved in several steps of a thermal hydraulic analysis of a nuclear reactor accidental transient [START_REF] Bestion | A state-of-the-art report on scaling in system thermal hydraulics applications to nuclear reactor safety and design[END_REF]. Experimental test facilities are necessary to simulate reactor transients. Experiments are performed to identify and understand the transient phenomena, and to support the validation of the numerical simulation tools. Since no full-size experimental test reactor is affordable, scaling methods are used to design reduced scale facilities and to define test conditions able to represent the behavior in the full-size reference reactor with minimal distortions. The scaling method uses a ranking of processes with respect to their impact on the Figures of Merit (FoM) of the transient in order to respect the similarity of the dominant (highly ranked) processes in the test design, and to estimate the distortions of the other processes. Scaling is also used to give requirements to code models, which should well describe dominant processes, and to define the validation matrix of the code. The code validation is done by comparing the results of the code predictions with experimental data. Codes should predict correctly dominant processes, and the scale effect of all influent processes to be able to do the transposition to the reactor transient. This is called the code scalability or code capability to do the "scaling-up''.

The majority of Integral Test Facilities (ITF) are scaled based on respecting power-to-volume ratio and respecting the full-height of the reactor (e.g. BETHSY, LSTF, LOBI, SPES). Then came the three-level scaling [START_REF] Ishii | The three-level scaling approach with application to the Purdue University Multi-Dimensional Integral Test Assembly (PUMA)[END_REF] with reduced height and a reduction of the time scale. With the need to design facilities with reduced height and pressure, the power-to-mass scaling method was developed [START_REF] Liu | Power-operated relief valve stuck-open accident and recovery scenarios in the Institute of Nuclear Energy Research integral system test facility[END_REF]. Later, other types of methods intended for scale transposition and to support system codes were developed. The Hierarchical Two-Tiered Scaling methodology (H2TS) introduced the first concepts of hierarchy and scaling groups [START_REF] Zuber | A hierarchical, Two-tiered Scaling Analysis, Appendix D of An Integrated Structure and Scaling Methodology for Severe Accident Technical Issue Resolution[END_REF]. Thereafter, the Fractional Scaling Analysis (FSA) was developed as an improvement of the H2TS [START_REF] Zuber | Application of fractional scaling analysis (FSA) to loss of coolant accidents (LOCA)[END_REF]. Finally, the recent Dynamical System Scaling method (DSS), similar to the H2TS and FSA was developed [START_REF] Reyes | The Dynaminal System Scaling Methodology[END_REF].

The H2TS method introduces the concept of top-down approach followed by the bottom-up approach [START_REF] Zuber | A hierarchical, Two-tiered Scaling Analysis, Appendix D of An Integrated Structure and Scaling Methodology for Severe Accident Technical Issue Resolution[END_REF]. The top-down step (also called stepwise design) starts the scaling analysis for an Integral Effect Test (IET) at the global system scale, in view of defining the scaling ratios for the core power, the total system volume, and the nominal flow rate. The analysis divides the whole system into individual parts and considering progressively smaller scales, i.e. the component scale, the sub-component scale, to the local process scale. Then, the bottom-up step is the reverse process, which brings together individual information about the IET design, from basic building blocks to the complex system. In the field of the study of nuclear transients, these two steps allow a comprehensive method for the scaling analysis. It was also claimed that this approach provides efficiency through cost-effective and timely resolution [START_REF] Zuber | An integrated structure and scaling methodology for severe accident technical issue resolution: Development of methodology[END_REF]. However, advanced methods such as H2TS and FSA require a high level of expertise in transient analysis and a deep understanding of two-phase thermal hydraulics. No handbook for scaling method exists and guidance on how to apply them is required. This work intends to clarify how they can be applied, to compare their merits, to identify the difficulties, and to see how system codes can support the scaling analysis.

The Small-Break Loss Of Coolant Accidents (SB-LOCAs) became important in reactor safety analysis after the Three Miles Island (TMI) accident. It is chosen to revisit a posteriori the scaling of an already existing SB-LOCA integral test using all the available knowledge and tools including the results of the test and the results of a system code simulation. A scaling analysis is performed using the FSA method and it is compared to an application of the H2TS method done by the Polytechnic University of Valence (UPV) [START_REF] Muñoz-Cobo | Top-down scaling methodology from the LSTF facility to a three loop PWR plant applied to a SBLOCA event -The ROSA 1.2 test[END_REF]. Some differences and similarities of these two methods will be identified to highlight the cause and importance of the choices that can be made during a scaling analysis.

The scaling analysis is carried out on the ROSA 1.2 test performed on the Japanese LSTF-ROSA-IV facility. In this study, the thermal hydraulic system code used is the CATHARE code [START_REF] Préa | CATHARE-3 V2.1: the new industrial version of the CATHARE code[END_REF][10] [START_REF] Bestion | The physical closure laws in the CATHARE code[END_REF]. The UPV uses the TRACE code.

BRIEF DESCRIPTION OF THE ROSA 1.2 TEST

The LSTF Facility

The ROSA 1.2 test was performed on the Large Scale Test Facility (LSTF) of the Japan Atomic Energy Research Institute (JAERI) [START_REF]Final Integration Report of OECD/NEA ROSA Project (2005-2009)[END_REF] [START_REF] The | ROSA-V large scale test facility (LSTF) system description for the third and fourth simulated fuel assemblies[END_REF]. LSTF is designed to simulate a Westinghouse four loops PWR with a full power of 3423 MWt -1100 MWe. As shown in Figure 1, LSTF has two loops, two steam generators and one pressurizer. The facility was designed using the same scaling factor for power and volume, a fullheight scaling, and a 1/48 scaled volume at full-pressure reference condition. The maximum core power is equal to 14% of the 1/48 nominal power of the reference reactor i.e. 10 MWt. This scaling respects the timing of events. The flow area is 1/48 scaled in the pressure vessel and 1/24 scaled in the cold and hot legs (HL & CL) and in the steam generators (SG) since each loop represents two reactor loops. This work will use the simulation of the ROSA 1.2 test using the CATHARE system code [START_REF] Préa | CATHARE-3 V2.1: the new industrial version of the CATHARE code[END_REF]. The model contains the primary and the secondary systems. The primary side contains the Reactor Pressure Vessel (RPV), two hot and cold legs with main coolant pumps, the pressurizer and the U-tubes of the two steam generators. The system also includes High Pressure Injection System (HPIS) and Accumulator Injection Systems (AIS). The nodalization of the LSTF primary system with CATHARE is shown in Figure 1.

The SB-LOCA Transient

The ROSA 1.2 test scenario consists of a SB-LOCA experiment. The break corresponds to 1% of the volumetrically scaled cross-sectional area of the reference reactor cold leg. The break is located on the hot leg of the loop without pressurizer.

The test starts at operating full-pressure of 15.5 MPa and at a nominal power of 10 MWt. The reactor SCRAM signal occurs at 12.97 MPa resulting in the core power decay. The main coolant pumps gradually stop as well as the pressurizer heaters and the secondary feed water. At a pressure of 12.27 MPa, the safety injection signal triggers the HPIS pumps with a 12 seconds delay. The accumulator injection system (AIS) is activated at 4.51 MPa.

Loop A Loop B break

At the break opening, the primary fluid is discharged first in liquid state. With decreasing pressure and decreasing mass inventory, the steam quality of discharged fluid at the break gradually increases. Throughout the transient, the main concern is the core cooling which may be challenged when a partially uncovered core induces a clad temperature excursion. This event mainly depends on the primary mass inventory, which itself depends a lot on the primary pressure evolution that influences the break flow rate and the safety injections. These two parameters are thus essential to describe the SB-LOCA transient.

The transient is divided into five chronological phases. These phases, also called "phenomenological windows", depend on the phenomena and on system parameters evolution. Bifurcating events that correspond to abrupt changes of phenomena, delimit the phases. In the literature, usually 3, 4 or 5 phase splits, characterized by physical processes and phenomena, are defined for SB-LOCA transients. In the current study, the SB-LOCA is divided into 5 phases following the UPV analysis [START_REF] Muñoz-Cobo | Top-down scaling methodology from the LSTF facility to a three loop PWR plant applied to a SBLOCA event -The ROSA 1.2 test[END_REF]. This choice allows separation of the dominant processes throughout the transient. They are presented in Figure 2 describing the evolution of the primary and secondary pressure as well as the primary mass inventory. The bifurcating events are summarized in Table I. After the opening of the break, the primary system depressurizes quickly from 15.5 MPa to 10.7 MPa in 119 seconds. The primary system remains liquid in the loops and in the pressure vessel up to 60 seconds. Flashing occurs first in the pressurizer due to its higher temperature. The SCRAM signal triggers at around 50 seconds and the core power will join a decay power curve, which is truncated at 14%. The primary pumps are stopped and the rotation speed gradually decreases depending on the flywheel, initiating the transition to the natural circulation. The end of the blowdown discharge is defined as the emptying of the pressurizer (less than 1% of the initial liquid mass).

The Natural Circulation phase:

The pumps being stopped, the fluid circulation is due to the differences in density between the ascending and descending parts of the system. The ascending part is the core, the upper plenum, the hot legs and the ascending part of the steam generators U-tubes. The descending part is the descending part of the steam generators U-tubes, the cold legs, and the vessel downcomer. The difference of gravity in the ascending and descending parts of the circuit are compensated by friction and singular pressure losses along the loop. During this phase, the primary pressure decreases slowly (from 10.7 MPa to 8.2 MPa) and stays above the secondary pressure to be able to transfer energy to the steam generator (SG). During this phase, the core provides more energy to the fluid than is lost at the break and the primary fluid needs to give the rest of power to the SG. This requires a higher temperature and higher pressure than inside the SG.

The Two-Phase Discharge phase:

The vapour created in the core condenses in the steam generators U-tubes and the condensate water is first entrained in co-current flow to the intermediate leg (IL) and the cold leg. Then the mass of water decreasing, the void fraction increases in SG tubes and the vapour is no more capable to entrain the condensate water to the top of the SG tubes. This is the end of the natural circulation (NC) and the beginning of the Reflux Condenser Mode (RCM). Some decay power is still removed by condensation in the SG, with the condensate water flowing back to the hot legs in the ascending part of SG tubes whereas the condensate in the descending part of SG tubes fills the intermediate legs. With the stop of NC, water tends to settle in the lower part of the circuit, the steam quality at the break increases and more power is evacuated at the break. Note that the Loop Seal Plugging (LSP) occurs without Loop Seal Clearing (LSC) during the 5 phases considered. During this Two-Phase Discharge, the primary pressure drops from 8.2 MPa to 7.7 MPa in 581 seconds.

4. The High-Quality Mixture Discharge phase:

The beginning of this phase is marked by the passage of the primary pressure below the secondary pressure. From this moment, the heat flux at the steam generators U-tubes is reversed. The gradual decrease of primary pressure increases the HPIS flow rate. The vapour condensation on the cold injected water increases the depressurization rate. During this phase, the pressure decreases significantly, from 7.7 MPa to 4.5 MPa in 1484 seconds. The liquid level in hot legs goes down until it reaches the height of the break. The discharge at the break is then almost vapour.

The Reactor Refilling phase:

This phase starts when the accumulator injection is triggered. A large flow rate of sub-cooled liquid is discharged into the primary system thanks to AIS and HPIS. From the beginning to the end of the accumulator discharge, the pressure decreases from 4.5 MPa to 1.8 MPa in 1148 seconds and the mass inventory starts increasing.

TOP-DOWN SCALING WITH FSA METHOD

The FSA methodology [START_REF] Zuber | Application of fractional scaling analysis (FSA) to loss of coolant accidents (LOCA)[END_REF] is first applied to the blowdown and high-quality mixture discharge phases. For the SB-LOCA transient, the Figure of Merit (FoM) is defined as the Peak Clad Temperature (PCT) [START_REF] Bestion | A state-of-the-art report on scaling in system thermal hydraulics applications to nuclear reactor safety and design[END_REF][5].

The most important Parameters of Interest (PoI) that influence the FoM are the mass inventory M and the primary pressure 𝑃 1 . Equations will then be written for the primary mass evolution and for the primary pressure evolution. The primary pressure is first controlled by the pressurizer until it is empty, and then it is controlled by the mass and energy exchanges in the whole primary circuit.

Mass Balance Equations in a Volume V

The mass balance equation in a constant volume V is given by:

dM dt ≜ M ̇ = M ̇in -M ̇out (1)
M ̇ is the time rate change of fluid mass in V. M ̇in and M ̇out are respectively the mass flow rates entering and leaving V through fluid boundaries A f . For a two-phase case, this equation can be split into liquid and vapour contributions. Mass balance equation becomes:

M ̇ = M ̇l + M ̇v = M ̇l,in -M ̇l,out + M ̇v,in -M ̇v,out (2)
The Eq. ( 2) applied to the pressurizer and the primary system gives respectively Eq. ( 3) and ( 4):

M ̇prz = -M ̇l,prz,out -M ̇v,prz,out (3)
M ̇1 = M ̇l,HPIS -M ̇l,break -M ̇v,break (4)
Each contributor to the mass rate of change M ̇ is normalized with time-averaged of variables Y ̅ during the phases. Although the FSA method recommends normalization with an initial value Y 0 [START_REF] Zuber | Application of fractional scaling analysis (FSA) to loss of coolant accidents (LOCA)[END_REF], it is chosen here to use this average calculated values by a code.

Y + (t) = Y(t) Y ̅ (5)
The Eq. ( 3) and (4) then writes respectively, introducing the Fractional Rate of Change (FRC) ω M ̇,j :

dM + dt | prz = -ω M ̇l,prz,out ⋅ M ̇l,prz,out + -ω M ̇v,prz,out ⋅ M ̇v,prz,out + (6) dM + dt | 1 = ω M ̇l,HPIS ⋅ M ̇l,HPIS + -ω M ̇l,break ⋅ M ̇l,break + -ω M ̇v,break ⋅ M ̇v,break + (7)
The normalized mass balance equation is obtained, function of the j Agents of Change ϕ M ̇,j + which define the processes acting on M ̇:

dM + dt | prz = -ω M ̇l,prz,out ⋅ ϕ M ̇l,prz,out + -ω M ̇v,prz,out ⋅ ϕ M ̇v,prz,out + (8) dM + dt | 1 = ω M ̇l,HPIS ⋅ ϕ M ̇l,HPIS + -ω M ̇l,break ⋅ ϕ M ̇l,break + -ω M ̇v,break ⋅ ϕ M ̇v,break + (9)

Pressure Equations in a Volume V

Let's now write the first principle of thermodynamics in a constant volume V with constant boundaries A to derive a total energy equation:

d dt ∫ ρ (e+ u 2 2 ) V dV = ∫ρ(F ⃗ ⃗ ⋅ u ⃗ +q ext ) V dV -∫ ρ (e+ u 2 2 ) u ⃗ ⋅ n ⃗ A f dA -∫q ⃗ ⋅ n ⃗ A dA-∫ (T ̿ ⋅ u ⃗ )⋅ n ⃗ A f dA (10)
Some simplifications are considered by neglecting the kinetic energy, the heat flux along fluid boundaries, the work of the gravity forces (F ⃗ ⃗ ⋅ u ⃗ ) and radiative sources q ext . Finally, the energy balance is expressed using enthalpy h instead of internal energy (h=e+ p ρ ). The two-phase energy balance equation for a volume V writes:

dM l ⋅ H l dt + dM v ⋅ H v dt -V dP dt = W w + M ̇l,in ⋅ h l,in -M ̇l,out ⋅ H l + M ̇v,in ⋅ h v,in -M ̇v,out ⋅ H v ( 11
)
With P the volume averaged pressure, h l,in and h v,in the liquid and vapour entering enthalpies and W w the thermal power received from the walls. It is considered that the phases leaves the volume with the mass weighted averaged enthalpy H k of the volume.

Taking into account the energy balance at the interfaces, the mass balance equation and a constant volume equation, is obtained [START_REF] Bestion | About phenomena identification in a PIRT[END_REF] a pressure equation expressed in terms of fluid volume rate of change:

(μ l ⋅ M l + μ v ⋅ M v )⋅ P ̇ = M ̇l,in ⋅ [ν l + ν l,h ' ⋅ (h l,in -H l )] + M ̇v,in ⋅ [v v + v v,h ' ⋅ (h v,in -H v )] -M ̇l,out ⋅ ν l -M ̇v,out ⋅ ν v + ν l,h ' ⋅ W w,l + ν v,h ' ⋅ W w,v + ν l,h ' ⋅ W i,l + ν v,h ' ⋅ W i,v - W i,v +W i,l ϖ + W w,i ϖ (12)
With:

ν' k,p ≜ ∂ν k ∂p | h k ; ν' k,h ≜ ∂ν k ∂h k | p ; μ k = -ν k,p ' -ν k ⋅ ν k,h ' ; ϖ ≜ h v -h l ν v -ν l
Each term of the equation is a volume rate of change (VRC, in m 3 /s). The term on the l.h.s is the fluid expansion in depressurization. The terms on r.h.s are the entering liquid and vapour volume flow rates and the associated thermal expansion when mixing with internal fluid, the VRC by exiting liquid and vapour flow rates, the VRC by liquid and vapour thermal expansion by wall heating and by interfacial exchanges, the VRC by vaporization or condensation, and the VRC by wall boiling or condensation. This formulation provides an easy interpretation of the impact of fluid volume changes on pressure. Fluid volume may change by source or sink of fluid at boundaries, by expansion/contraction by heat exchanges and by isentropic expansion/contraction.

In situations where both phases are close to the saturation, a simpler pressure equation assuming thermal equilibrium is written:

(λ l ⋅ M l + λ v ⋅ M v )⋅ P ̇ = M ̇l,in ⋅ (v ls + h l,in -h ls ϖ s ) + M ̇v,in ⋅ (v vs + h v,in -h vs ϖ s ) -M ̇l,out ⋅ ν ls -M ̇v,out ⋅ ν vs + W w ϖ s (13)
With:

λ k =

≜ dν ks dp

In this case, the l.h.s term represents all VRC due to expansion (or contraction) by pressure change; it includes isentropic expansion plus vaporization by flashing and thermal expansion following saturation curve.

The Eq. ( 13) applied to the pressurizer and the primary system gives respectively Eq. ( 14) and ( 15):

(λ l ⋅ M l + λ v ⋅ M v )⋅ P ̇prz = -M ̇l,prz,out ⋅ v ls -M ̇v,prz,out ⋅ v vs + W prz,heaters ϖ s + W prz,ow ϖ s ( 14
) (λ l ⋅ M l + λ v ⋅ M v )⋅ P ̇1 = M ̇l,HPIS ⋅ v ls + M ̇l,HPIS ⋅ (h l,HPIS -h ls ) ϖ s -M ̇l,break ⋅ v ls -M ̇v,break ⋅ v vs + W core ϖ s + W 1,ow ϖ s - W SG ϖ s (15)
To simplify the VRC due to the pressure change, the global expansion coefficient K s is defined:

K s = (λ l ⋅ M l + λ v ⋅ M v ) (16)
Each contributor to the pressure rate of change ̇ is normalized in the same way as above ( 5). The Eq. ( 14) and (15) then writes respectively, introducing the Fractional Rate of Change (FRC) ω P ̇,j :

dP + dt | prz = -ω P ̇,Q l,prz,out ⋅ M ̇l,prz,out + ⋅ v ls + K s + -ω P ̇,Q v,prz,out ⋅ M ̇v,prz,out + ⋅ v vs + K s + + ω P ̇,Q prz,heaters ⋅ W prz,heaters + K s + ⋅ ϖ s + + ω P ̇,Q prz,ow ⋅ W prz,ow + K s + ⋅ ϖ s + (17) dP + dt | 1 = ω P ̇,Q l,HPIS ⋅ M ̇l,HPIS + ⋅ v ls + K s + + ω P ̇,Q cond,HPIS ⋅ M ̇l,HPIS + ⋅ (h l,HPIS + -h ls + ) K s + ⋅ ϖ s + -ω P ̇,Q l,break ⋅ M ̇l,break + ⋅ v ls + K s + - ω P ̇,Q v,break ⋅ M ̇v,break + ⋅ v vs + K s + +ω P ̇,Q core ⋅ W core + K s + ⋅ ϖ s + +ω P ̇,Q 1,ow ⋅ W 1,ow + K s + ⋅ ϖ s + -ω P ̇,Q SG ⋅ W SG + K s + ⋅ ϖ s + (18)
The normalized pressure equation is obtained, function of the j Agents of Change ϕ P ̇,j + which define the processes acting on P ̇:

dP + dt | prz = -ω P ̇,Q l,prz,out ⋅ ϕ P ̇,Q l,prz,out + -ω P ̇,Q v,prz,out ⋅ ϕ P ̇,Q v,prz,out + +ω P ̇,Q prz,heaters ⋅ ϕ P ̇,Q prz,heaters + +ω P ̇,Q prz,ow ⋅ ϕ P ̇,Q prz,ow + (19) dP + dt | 1 = ω P ̇,Q l,HPIS ⋅ ϕ P ̇,Q l,HPIS + + ω P ̇,Q cond,HPIS ⋅ ϕ P ̇,Q cond,HPIS + -ω P ̇,Q l,break ⋅ ϕ P ̇,Q l,break + -ω P ̇,Q v,break ⋅ ϕ P ̇,Q v,break + +ω P ̇,Q core ⋅ ϕ P ̇,Q core + +ω P ̇,Q 1,ow ⋅ ϕ P ̇,Q 1,ow + -ω P ̇,Q SG ⋅ ϕ P ̇,Q SG + ( 20
)
From equations ( 8), ( 9), ( 19) and (20), the time scaling for relative importance of agents [START_REF] Zuber | Application of fractional scaling analysis (FSA) to loss of coolant accidents (LOCA)[END_REF] is carried out. The FRC are divided by the dominant one. The effect metrics are obtained thanks to this process, denoting the rate of change due to the contribution of Agents of Change:

Ω M ̇,χ j = ω M ̇,χ j |ω M ̇,χ D | ; Ω P ̇,χ j = ω P ̇,χ j |ω P ̇,χ D | (21)

Results of the a posteriori scaling analysis using the code simulation

Figure 3 displays the primary and secondary pressure evolution from experimental data and CATHARE simulation. The CATHARE code predicts the pressure evolution rather well. Figure 4 details the pressure evolution during the blowdown phase and during the primary-secondary pressure reversal. In Figure 5 the break mass flow rate is rather well predicted with a delay of the transition to HQMD and a higher flow at the end of accumulator discharge, which are not yet discussed in this work. The good predictions of pressure and break flow rate demonstrate the quality of the CATHARE code (less than 1% error for pressures and less than 4% for flow rate in BD and HQMD phases). It can then be used as a support to the scaling analysis. Main parameters of the study are shown in Table II. The values of the effect metrics for the mass equation Ω M ̇,j and the pressure equation Ω P ̇,j are displayed respectively for the blowdown and high-quality mixture discharge phases in Table III and Table IV.

Rate of mass change in V 1 due to the contribution by:

Liquid flow rate leaving the primary system through the break Ω M ̇l,break 1.000 Vapour flow rate leaving the primary system through the break Ω M ̇v,break 0.002

Rate of pressure change in V 1 due to the contribution by:

Liquid flow rate leaving the primary system through the break Ω P ̇,Q l,break 0.100 Vapour flow rate leaving the primary system through the break

Ω P ̇,Q v,break 0.002
Thermal expansion by vaporization due to the core heat generation Ω P ̇,Q core 1.000 Thermal expansion by condensation/vaporization due to SG tubes heat exchanges

Ω P ̇,Q SG 0.995
Thermal expansion by condensation/vaporization due to wall heat exchanges in the primary system

Ω P ̇,Q 1,ow 0.005
For the pressurizer control volume, the effect metrics that govern the mass and pressure change are those related to the flow rate leaving the pressurizer, which is close to the flow rate to the break. This was also observed by the UPV with the H2TS method. The scaling of the break flow rate should then respect the volume scale ratio. Since the choke flow mass flux is almost independent on the break diameter, this means that a good break scaling should apply the volume scaling factor to the break area. The effect metrics related to the pressurizer heaters and wall heat exchanges can be considered as influents. The UPV has instead taken into account a net heat term obtained from the subtraction of the pressurizer heat losses to the heat provided by the heaters [START_REF] Muñoz-Cobo | Top-down scaling methodology from the LSTF facility to a three loop PWR plant applied to a SBLOCA event -The ROSA 1.2 test[END_REF]. Their corresponding scaling criteria (from π-monomial) has a value of 0.160. An effect metric of 0.158 is obtained with the same net heat calculation and with their assumptions. As shown by the UPV when comparing LSTF data to reactor scale calculation, the distortion of this contribution of the external pressurizer wall is large. It could be then considered that a pressurizer scaling which would not respect the height (e.g. by adopting a linear scaling) would be better for this LOCA blowdown phase. Note that the design of the LSTF pressurizer has been revised to simulate that of the AP1000 and is relatively larger than that of the conventional Westinghouse PWR. It can also be noted that in the CATHARE code calculation, the vapour mass flow rate leaving the pressurizer is negligible but its volume flow rate is a significant contribution to pressure decrease. It may then be considered that a 0-D modelling of the pressurizer is not sufficiently precise to predict the entrainment of bubbles from pressurizer to the surge line (SL) and to the hot leg. This vapour is due to flashing of the water and it may be expected a gradient of void fraction from bottom to the level with a minimum value at the connection to the SL. A too simple 0-D modelling of the pressurizer may overestimate this void fraction and the vapour flow rate entrained in SL. This is an example showing how an a posteriori scaling analysis with a system code may help improving the modelling when necessary.

Regarding the primary system control volume, the dominant effect metric on the rate of mass change is the one related to the liquid flow rate at the break with a very small contribution of the vapour break flow. Although this repartition cannot be confirmed by measured data, this evaluation is easily accepted since almost only liquid water flows in the loops and pressure vessel. It can be observed on the rate of pressure change that the two processes related to heat exchange in the core and the SG tubes are dominant and almost compensate each other. The liquid volume flow rate at the break can be considered as influent. The vapour volume flow rate and wall heat exchanges are negligible during the blowdown.

Table IV. Effect Metrics for the LSTF ROSA 1.2 test during the HQMD phase

Rate of mass change in V 1 due to the contribution by: Liquid flow rate entering the primary system thanks to HPIS Ω M ̇l,HPIS 1.000 Liquid flow rate leaving the primary system through the break Ω M ̇l,break 0.802 Vapour flow rate leaving the primary system through the break Ω M ̇v,break 0.225

Rate of pressure change in V 1 due to the contribution by: Liquid flow rate entering the primary system thanks to HPIS system Ω P ̇,Q l,HPIS 0.062 Thermal expansion by condensation due to the liquid flow rate entering the primary system thanks to HPIS system Ω P ̇,Q cond,HPIS During the high-quality mixture discharge phase, the total contribution of the break mass flow rates to the primary mass is almost compensated by the HPIS.

The three most important effect metrics for the rate of pressure change are the one related to the HPIS, to the core heat generation and to the break (mainly the vapour flow rate). The UPV gives a contribution from HPIS of 0.498. Having assumed here a saturation state overestimate the condensation of HPIS water. Further analysis will be done using Eq. ( 12) to better estimate the condensation. The UPV analysis found the net heat term (heat balance between the core, the SG tubes and the losses [START_REF] Muñoz-Cobo | Top-down scaling methodology from the LSTF facility to a three loop PWR plant applied to a SBLOCA event -The ROSA 1.2 test[END_REF]) as the dominant term. It is actually observed that the SG tubes exchanges term is negligible. Contributions of the wall heat exchanges can be considered as influent. On the contrary, their value relative to the vapour flow rate is almost zero while the analysis here gives a value of 0.356. These differences could be explained by the nature of the corresponding terms in the pressure equations, which do not refer exactly to the same quantities.

An order of magnitude analysis of the terms of the pressure equation assuming saturation conditions (15) can estimates the expansion coefficient K s at the moment of the primary-secondary pressure reversal. It is found that the equation is not exactly balanced probably because the saturation assumption is not respected.

The comparative analysis of these methods carried out here shows that they are based on the same methodological approaches:

-The same normalization concept of the contributing terms of the balance equations are used. The H2TS and FSA methods suggest the use of boundary conditions for this normalization.

-A normalization via averaged quantities using the CATHARE calculation is made. This is more representative of the evolution of the contributing values over the phases, for example, by averaging the heat exchanges and flow rates terms.

-The two methods calculate scaling criteria but in different ways. The π-monomial in the H2TS and the effect metrics in the FSA. To calculate them, scaling methods take into account the ratio of the indicators of the intensities of the processes over the studied variable.

-To identify the scaling criteria and therefore distortions, H2TS performs time scaling through a characteristic time for the variable evolution. This characteristic time is the ratio between the control volume and the convective volumetric flow rate [4][8]. The equivalence in FSA corresponds to the time scaling for relative importance of agents. Since FRC takes into account the characteristic time of the processes, dimensionless effect metrics are directly linked to the dominant process [START_REF] Zuber | Application of fractional scaling analysis (FSA) to loss of coolant accidents (LOCA)[END_REF]. As mentioned in [START_REF] Bestion | A state-of-the-art report on scaling in system thermal hydraulics applications to nuclear reactor safety and design[END_REF] where a comparison of H2TS and FSA is also carried out: "The FSA determines scale distortion quantitatively from the ratios of fractional rate changes for each specific agent of change. H2TS computes scale distortion by taking ratios of dimensionless characteristic-time ratios." This addition to this newer method is perhaps a way to better scale the time.

CONCLUSIONS

An a posteriori scaling analysis is here applied to the ROSA 1.2 hot leg SB-LOCA test performed on the Japanese LSTF-ROSA-IV facility. The FSA method is applied here to selected equations. And for those equations, we evaluate the term values with the CATHARE code simulations. Attention is first focused on the blowdown and high-quality mixture discharge phases of this SB-LOCA. Mass and pressure equations on the pressurizer and primary system volumes are defined and applied to the methodology. This work intends to provide a comparison between the results of the FSA with the one obtained by the UPV with the H2TS and to see how these methods can be applied with the support of the system codes which have now a good maturity and a reasonable level of confidence.

The formulation of pressure equation for the pressurizer and for the primary circuit as a volume rate of change equation helps giving a clear physical meaning to each contributor. Any fluid volume source (resp. sink) contributes to pressurization (resp. depressurization). Any heating (resp. cooling) contributes to pressurization (resp. depressurization). Since the volume is constant, any imbalance between heating, cooling, fluid volume sources and sinks is compensated by a fluid volume expansion or contraction by pressure decrease or increase.

Several common aspects may be found in H2TS and FSA. The FSA may be seen as improvement of the older H2TS method [START_REF] Bestion | A state-of-the-art report on scaling in system thermal hydraulics applications to nuclear reactor safety and design[END_REF]. As shown by the application made here, the results obtained by the two methods are similar when the nature of the equations is equivalent. The most important is not the method itself but the choice of equations, the possible simplifications that are used, and the way to evaluate each term. Their choice is therefore essential for the qualitative study of a transient. In addition, the FSA improvements are likely to provide better temporal analysis. Moreover, it is here believed that the use of mature scaling code to better evaluate terms of the equation may be of interest.

From the first results of this scaling analysis, the hierarchy of the processes acting on mass and pressure change during the blowdown and the high-quality mixture discharge phases is established. All methods identify the same scaling criteria to apply to power and to break area. All methods identify distortions due to heat exchanges with solids (other than fuel rods and SG tubes). Surprisingly no ITF which used the power to volume scaling and full height scaling (i.e. BETHSY, LSTF, LOBI, SPES), adopted another scaling criterion for the pressurizer which would minimize the distortion by using a reduced height of this component only.

The use of a mature and extensively validated system code -here the CATHARE code -allows the user of the method to limit the assumptions about the values chosen for the study. The code actually provides additional precision by averaging these values rather than relying in estimations at boundary of the phases. System codes can now support scaling analyses. It may be of interest to apply the scaling first with the original FSA method and then to revisit it by re-evaluating the terms using a system code simulation.

Moreover, scaling methods can also support the improvement of the code modelling as shown in the pressurizer pressure equation analysis for the blowdown phase. This work will continue by analyzing the five phases and by using the pressure equation formulated without saturation assumption. It will be more precise but some instruments must be implemented in the system code to store the value of each term.

NOMENCLATURE

Figure 1 .

 1 Figure 1. Large Scale Test Facility (LSTF) used for the ROSA 1.2 test and the CATHARE nodalization of the primary system

Figure 2 .

 2 Figure 2. Primary and secondary evolutions during the phases of a SB-LOCA

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Primary and secondary pressures evolution for the LSTF ROSA 1.2 test

Table II . Main parameters of the BD and HQMD phases of the LSTF ROSA 1.2 test

 II

		1. BD	4. HQMD
	Initial primary mass M 1,0	5642	2044	kg
	Initial pressurizer mass M prz,0	508	-	kg
	Phase duration Δt	119	1484	s
	Primary mass variation ΔM 1	652	58.6	kg
	Pressurizer mass variation ΔM 𝑝𝑟𝑧	434	-	kg
	Pressure variation during the phases ΔP	4.80⋅10 6	3.34⋅10 6	Pa
	Averaged mass flow rate at the break M ̅ break	5.38	1.56	kg/s
	Averaged mass flow rate leaving the pressurizer M ̅ prz	3.59	-	kg/s
	Averaged mass flow rate of the HPIS M ̅ HPIS Averaged core power 𝑊 ̅ core Averaged pressurizer heaters power 𝑊 ̅ prz,heaters	-8.34⋅10 6 8.50⋅10 4	1.52 1.37⋅10 6 -	kg/s W W
	Averaged SG tubes heat exchanges 𝑊 ̅ SG Averaged primary wall heat exchanges 𝑊 ̅ 1,ow Averaged pressurizer wall heat exchanges 𝑊 ̅ prz,ow	-8.30⋅10 6 3.86⋅10 4 1.42⋅10 5	1.05⋅10 4 1.84⋅10 5 -	W W W

Table III . Effect Metrics for the LSTF ROSA 1.2 test during the BD phase Rate of mass change in V prz due to the contribution by:

 III

	Liquid flow rate leaving the pressurizer	Ω M ̇l,prz,out	1.000
	Vapour flow rate leaving the pressurizer	Ω M ̇v,prz,out	0.039

Rate of pressure change in V prz the due to the contribution by:

	Liquid flow rate leaving the pressurizer	Ω P ̇,Q l,prz,out	1.000
	Vapour flow rate leaving the pressurizer	Ω P ̇,Q v,prz,out	0.329
	Thermal expansion by vaporization due to the pressurizer heaters	Ω P ̇,Q prz,heaters	0.159
	Thermal expansion by condensation/vaporization due to wall heat exchanges in the pressurizer	Ω P ̇,Q prz,ow	0.265

Symbols


S Rasamimanana

G Lefèvre

Romain V H Dagnelie
email: romain.dagnelie@cea.fr

Various causes behind the desorption hysteresis of carboxylic acids on mudstones

Highlights

Desorption of acetate ortho-phthalate and citrate were studied on mudstone Soluble acetate was weakly adsorbed and degraded by bacterial activity Desorption hysteresis of phthalate vanished by porewater equilibration with mudstone Persistent hysteresis of citrate indicated a chemical reactivity with the solid Leaching protocol affects desorption hysteresis and thus environmental fate

Introduction

The retention of organic compounds in soils and sediments is an important issue in many fields, such as decontamination processes, transport of landfill leachates, and nuclear and hazardous waste management [START_REF] Luthy | Critical Review Sequestration of Hydrophobic Organic Contaminants by Geosorbents[END_REF]. Extensive studies illustrate that isotherm nonlinearity, adsorption-desorption hysteresis and slow-rate kinetics are characteristic of organic molecule transport in the environment [START_REF] Hamaker | Equilibrium models of aluminium and iron complexation with different organic acids in soil solution[END_REF]Brusseau et al., 1989;[START_REF] Kan | Adsorption /Desorption Hysteresis in Organic Pollutant and Soil Sediment Interaction[END_REF]. Soil organic matter (SOM) and its meso/microporosity plays a key role in the adsorption and desorption of nonpolar, hydrophobic compounds [START_REF] Oepen Von | Sorption of Nonpolar and Polar Compounds to soils: Processes, Measurements and Experience with the Applicability of the Modified OECD-Guideline 106[END_REF][START_REF] Pignatello | Mechanisms of Slow Sorption of Organic Chemicals to Natural Particles[END_REF][START_REF] Yang | Sorption behavior of 17 phthalic acid esters on three soils: Effects of pH and dissolved organic matter, sorption coefficient measurement and QSPR study[END_REF].

Hindered diffusion through SOM contributes to the slow desorption of hydrophobic compounds [START_REF] Brusseau | Non-equilibrium Sorption of Organic Chemicals: Elucidation of Rate-Limiting Processes[END_REF][START_REF] Weber | Hysteresis in the sorption and desorption of hydrophobic organic contaminants by soils and sediments 2. Effects of soil organic matter heterogeneity[END_REF]; this mechanism was investigated with respect to the remediation of contaminated soils [START_REF] Connaughton | Description of Time-Varying Desorption Kinetics: Release of Naphthalene from Contaminated Soils[END_REF]. The processes responsible for the adsorption of polar hydrophilic organic species differ from those for apolar compounds. The polar organic compounds and soluble fraction of SOM show strong affinity with (hydroxy)oxides [START_REF] Gu | Adsorption and Desorption of Natural Organic Matter on Iron Oxide: Mechanisms and Models[END_REF]Hees and Lundström, 2000). These polar organic compounds also exhibit desorption hysteresis in soils and sediments [START_REF] Gu | Adsorption and desorption of different organic matter fractions on iron oxide[END_REF]. Unfortunately, most recent studies dealing with the adsorption of polar organic compounds tend to lack details on the quantification and origin of desorption hysteresis. Yet this information is crucial for predicting the solid-liquid partitioning of sorbates in a given physicochemical condition. In this work, we have studied the reversibility of desorption for three polar compounds, focusing on the role of the bio-geo-chemical conditions on desorption hysteresis and the implications for the environment.

The chosen mudstone is the Callovo-Oxfordian (COx) clay-rich rock. Adsorption of several polar organic compounds has already been studied on COx mudstone in the past: phthalate, oxalate, isosaccharinate and EDTA by [START_REF] Dagnelie | Sorption and diffusion of organic acids through clayrock: comparison with inorganic anions[END_REF], polymaleic acid by [START_REF] Durce | Adsorption and transport of polymaleic acid on Callovo-Oxfordian clay stone: batch and transport experiments[END_REF], and Eu III -EDTA by [START_REF] Descostes | Adsorption and retarded diffusion of Eu III -EDTA through hard clay rock[END_REF]. In this study, desorption experiments were performed on this natural heterogeneous mudstone with three carboxylic acids. Acetate was chosen as reference for the carboxylate function. It is also know to enhance bacterial activity and used for bioremediation processes [START_REF] Abromaitis | Biodegradation of persistent organics can overcome adsorption e desorption hysteresis in biological activated carbon systems[END_REF]. The ortho-phthalate was chosen as a major degradation product of phthalate esters and amides. The retarded migration of phthalate is relevant during the bioremediation of river sediments [START_REF] Yang | Sorption behavior of 17 phthalic acid esters on three soils: Effects of pH and dissolved organic matter, sorption coefficient measurement and QSPR study[END_REF]2016). Finally, citrate was chosen as being representative of chelating agents. Citrate is especially used in the remediation processes of metals and PAHs from marine sediments and soils [START_REF] Hahladakis | Sequential application of chelating agents and innovative surfactants for the enhanced electroremediation of real sediments from toxic metals and PAHs[END_REF][START_REF] Ammami | Application of biosurfactants and periodic voltage gradient for enhanced electrokinetic remediation of metals and PAHs in dredged marine sediments[END_REF][START_REF] Dias-Ferreira | Ammonium citrate as enhancement for electrodialytic soil remediation and investigation of soil solution during the process[END_REF].

Recent spectroscopic investigations indicated that the adsorption of carboxylic acids on clay minerals is mostly dominated by outer-sphere complexes at neutral pH, with low energy bounds and fast reversible kinetics [START_REF] Kang | Adsorption of dicarboxylic acids by clay minerals as examined by in situ ATR-FTIR and ex situ DRIFT[END_REF]. However, natural samples display systematic desorption hysteresis, in contradiction with reversible adsorption. [START_REF] Kubicki | Attenuated total reflectance Fourier-transform infrared spectroscopy of carboxylic acids adsorbed onto mineral surfaces[END_REF] evidenced variable affinities of acetate, phthalate or citrate on illite kaolinite and montmorillonite. The authors suggested the irreversible chemisorption of carboxylic acids on clays is limited without the presence of Fe-hydroxides within the clay matrix. Similar assumptions were concluded by [START_REF] Li | Adsorption of phthalic acid and salicylic acid and their effect on exchangeable Al capacity of variable-charge soils[END_REF] with respect to phthalate adsorption on various soils, as concluded by [START_REF] Yeasmin | Influence of mineral characteristics on the retention of low molecular weight organic compounds: a batch sorptiondesorption and ATR-FTIR study[END_REF] for citrate on goethite and ferrihydrite. [START_REF] Ramos | Adsorption of lactate and citrate on montmorillonite in aqueous solutions[END_REF], however, observed the weak adsorption of lactate with a strong inner-sphere complex of citrate on montmorillonite. These contradictory results encouraged us to carry out further investigations focusing on desorption reversibility of carboxylates.

Several assumptions may explain hysteretic desorption. Among those, there is the distinction between "apparent" and "true" hysteresis (Brusseau and Rao, 1989). "Apparent" hysteresis is due to experimental artefacts such as slow kinetics, solid surface alteration by mixing, colloidal effects, or the dilution of competing adsorbate [START_REF] Huang | Hysteresis in the sorption and desorption of hydrophobic organic contaminants by soils and sediments, 1.A comparative analysis of experimental protocols[END_REF]. Such artefacts can be eliminated using a suitable experimental protocol (Sander and[START_REF] Sander | An isotope exchange technique to assess mechanisms of sorption hysteresis applied to naphthalene in kerogenous organic matter[END_REF]2007). In such cases, the assumption of reversible adsorption is legitimate in predictive transport modelling. "True" hysteresis appears when an irreversible process occurs, thus modifying the system undergoing adsorption or desorption. For example, non-reversible retention can result from kinetically blocked exchange, i.e. covalent grafting such as described in [START_REF] Wu | Influence of functional groups on desorption of organic compounds from carbon nanotubes into water: insight into desorption hysteresis[END_REF]. A well-known mechanism is the modification of solid hydrophobic domains caused by the adsorption of large amounts of non-polar sorbent, as demonstrated Accepted Manuscript on soils [START_REF] Xing | Dual-Mode Sorption of Low-Polarity Compounds in Glassy Poly(Vinyl Chloride) and Soil Organic Matter[END_REF] and other sediments [START_REF] Huang | Effects of organic matter heterogeneity on sorption and desorption of organic contaminants by soils and sediments[END_REF]. Another example on polar organic compounds is the effect of competitive adsorbents released from SOM discussed in the paper by [START_REF] Gu | Adsorption and desorption of different organic matter fractions on iron oxide[END_REF]. Our work has focused on the desorption hysteresis of the chosen acids on a natural mudstone. Several adsorption and desorption protocols were compared and their effect is discussed herein. These various protocols may be representative of the conditions during transport in sediments or of waste (bio)remediation. The objective of our study was to improve understanding of solid/solution partitioning of the adsorbed carboxylic acids in sediments. We focused on desorption hysteresis under various physico-chemical conditions, i.e. undisturbed as in the environment or disturbed by leaching. similar to the environmental conditions. A core sample, referenced EST40471, was argon-drilled in COx at a depth between 497.7 and 498.1 m in the Andra underground research laboratory (Meuse/Haute-Marne, France). This clay-rich rock formation is mainly composed of illite (17-21%), interstratified illite-smectite (20-24%), kaolinite (3-5%), chlorite (2-3%), quartz (17-21%), calcite (20-23%), dolomite (3-6%) and accessory minerals (5%) [START_REF] Gaucher | ANDRA underground research laboratory: interpretation of the mineralogical and geochemical data acquired in the Callovo-Oxfordian formation by investigative drilling[END_REF]. A bulk chemical analysis is provided in the supplementary material (table S1). A porosity of 18 ± 2% was measured by weighting before/after drying at 105°C. The content of carbonate contents were estimated by acidic dissolution to 30 ± 0.5% in mass. A total organic carbon content of 0.67% was measured by pyrolysis.

The cation-exchange capacity (CEC) measured by the addition of CsCl on potassic samples gave values of 14.6 ± 0.4 meq/100 g. Specific surface areas were measured by nitrogen adsorption at 77K with an ASAP2010 device from micromeritics (S BET (N 2 ) = 39 m 2 g -1 ) and by the ethylene glycol monoethylether method, EGME, (S EGME = 150 m 2 g -1 ). The sample was crushed and sieved at less than 125 µm. The synthetic poral solution is described in Section 2.2. [START_REF] Dagnelie | Sorption and diffusion of organic acids through clayrock: comparison with inorganic anions[END_REF]. The composition of the synthetic pore water is given in Table 2. It was prepared by the dissolution of salts before being mixed with mudstone powder with a solid/solution ratio m/V ~ 50 g.L -1 , for the equilibration of trace elements (Si, Al, Fe). It was then filtrated and bubbled with a gaseous N 2 /CO 2 mixture to reach pH ~ 7.0 ± 0.2.

Adsorption and desorption protocols

Numerous methods exist to quantify adsorption and desorption. Typical methods are batch experiments, flow suspension or flow column, with the benefits and drawbacks discussed in [START_REF] Green | An assessment of Methods for determining Adsorption-Desorption of organic Chemicals[END_REF]. A standard adsorption method in batches was involved in this study using sealed polycarbonate tubes of 10 mL (Nalgene PC 3138-0010, Thermo scientific). All experiments were operated at ambient temperature (21 ± 1 °C) in a sub-oxic glove box (P O2 /P 0 < 2 ppm). A carbon dioxide pressure was imposed (P CO2 /P 0 ~ 4000 ± 1000 ppm) in order to maintain the pH ~ 7.0 ± 0.2 by equilibrium with soluble carbonates. A mass of 1 g of dried clay was mixed with the synthetic pore water using a solution/solid ratio of V/m = 4 L.kg -1 . A pre-equilibration of mudstone was achieved [START_REF] Descostes | Definition of an equilibration protocol for batch experiments on Callovo-Oxfordian argillite[END_REF]) by twice performing three days of mixing, ultracentrifugation (50.000 g, 1h)

and solution renewal there after. Then, radiolabelled and non-labelled compounds were simultaneously added respecting the same V/m ratio. Two triplicates and a blank without clay were performed for each molecule. One triplicate had an initial concentration of organic: C(t=0) = 10 -3 M and the other: Blank experiments did not reveal any sorption on glassware. The distribution ratio, R d (L kg -1 ), defined by Eq. ( 1), was then calculated:

C(t=0) =
m V t n t n n X X t R solution adsorbed d             ) ( ) ( ] [ ] [ ) ( 0 (1)
with n (moles), the quantity of the adsorbate X after a given period (t), V the solution volume (L) and m the dry mass of clay (kg). V was recalculated after each sample to take into account the change of V/m ratio in Eq. ( 1). The theoretical error bars on R d values were calculated by propagating the uncertainties (2σ) of masses, volumes and concentrations. It is important to note that batch experiments only provide a distribution ratio, R d . The notion of K d assumes the reversibility of the adsorption processes. The notation of R d was thus preferred in this manuscript.

Determining R d from the batch method is usually considered as reliable, as soon as the pH (7.0 ± 0.2) and ionic strength (0.10 ± 0.01 M) are constant during the experiment. However, the addition of adsorbates may induce other perturbations. Chemical perturbation for example implies surface dissolution or surface grafting. Such conditioning effects modify the affinity for a sorbate in a second sorption cycle [START_REF] Lu | Demonstration of the "conditioning effect" in soil organic matter in support of a pore deformation mechanism for sorption hysteresis[END_REF]. In order to assess such phenomena, an isotopic exchange method was also used in this study.

Accepted Manuscript

Forward isotopic exchange

Radiolabelled molecules can be used to distinguish chemical perturbation from adsorption (Sander and[START_REF] Sander | An isotope exchange technique to assess mechanisms of sorption hysteresis applied to naphthalene in kerogenous organic matter[END_REF]2007). The Fig. 1 details the experimental protocol performed for adsorption and forward isotopic exchange (FIE). Forward isotopic exchange consisted in a first step to add organic (C(t=0) = 10 -3 M or C(t=0) = 10 -6 M ) to the tubes after the equilibration phase. Then the mixture was stirred for 6 weeks to reach macroscopic equilibrium. A small amount of 14 C-or 3 Hlabelled organic molecules (C(t=0) = 10 -8 to 10 -11 M) were only added in a second step. The isotopic ratio 14 C/ 12 C was then modified in the liquid phase and equalled to zero in solid. Monitoring started from this second step and was investigated for about 6 weeks by successive samplings. Reverse

Desorption protocols and hysteresis index

The Fig. 2 represents the experimental protocols evaluated for desorption. Adsorption experiments were investigated during 6 weeks (C(t=0)  10 -6 M; V/m  4 L.kg -1 ; pH = 7.0 ± 0.2).

Then tubes were centrifuged at 15,000 rpm for 1 hour to renew supernatants by different solutions.

Therefore, protocols #1 to #4 differed by the renewal solution. Protocol #1 used synthetic pore water.

Protocol #2 used pore water equilibrated with mudstone respecting the same V/m ratio than adsorption step. Protocol #3 used organic solution at the concentration measured after adsorption step. Protocol #4 used organic solution equilibrated with mudstone at the same concentration and V/m ratio as that used for the adsorption step. Classical desorption, DES (protocols #1 and #2) and RIE (protocols #3 and #4) were investigated during 6 weeks. The sampling and analysis methods used were the same as A desorption hysteresis index will be defined by equation ( 3):

Ce T ADS e ADS e DES e q q q I H , . .   (3)
where q e (mol kg -1 ) represents the quantity adsorbed by the solid, with the superscripts DES and ADS representing desorption and adsorption steps respectively, at constant temperature and equilibrium concentration (T, C e ). This index is equal to zero when adsorption becomes completely reversible. H.I. increases with the degree of irreversibility.

Results and Discussion

Adsorption isotherm and hysteresis kinetics

The adsorption isotherms of the carboxylic acids are shown in Fig. 3 (top). The data R d = f

[phthalate] eq. , is taken from [START_REF] Dagnelie | Sorption and diffusion of organic acids through clayrock: comparison with inorganic anions[END_REF] on similar COx samples. The adsorption isotherms of acetate and citrate were determined for the present study. The R d values for acetate were below detection limit, R d (acetate) < 2σ ~ 0.2 L kg -1 (discussed in section 3.3.1). In the case of phthalate and citrate, the isotherms displayed Langmuir shapes, which can be described using the following equation ( 2)

eq eq ads C K C K Q C      1 (2) Accepted Manuscript
where C ads is the concentration of the adsorbed molecule on the solid (mol kg -1 ); C eq , the equilibrium concentration in solution (M); K, the adsorption constant (L mol -1 ); and Q, the maximum amount adsorbed by the material (mol kg -1 ). Comparison between data from classical adsorption method and forward isotopic exchange. Initial concentration, C(t=0) = 10 -6 mol L -1 .

The parameters were obtained by the least square fitting of 1/C ads as a function of 1/C eq . We obtained Q(phthalate/COx) = 9.1 10 -4 mol kg -1 and K(phthalate/COx) = 1.7 10 3 L mol -1 with a coefficient of determination R 2 = 0.9992. At low concentrations, the ratio R d rose to a plateau of R d MAX (phthalate) = Q×K = 1.6 L kg -1 . The Langmuir adsorption parameters for citrate were Q(citrate/COx) = 9.1 10 - 3 mol kg -1 and K(citrate/COx) = 6.6 10 3 L mol -1 with a coefficient of determination R 2 = 0.9964 and R d MAX (citrate) = 60 L kg -1 . This result indicated that more adsorption sites are available for citrate than for phthalate and a stronger affinity of citrate.

A typical variation of R d MAX over time is shown for phthalate adsorption and desorption in Fig. 3 (bottom). In both cases, the system reached apparent equilibrium within a few hours. The corresponding R d values were 1.6 ± 0.5 L.kg -1 and 18 ± 7 L.kg -1 for the adsorption and desorption experiments respectively. These results are representative of the typical hysteretic behaviour of organic compounds in sediments [START_REF] Gu | Adsorption and desorption of different organic matter fractions on iron oxide[END_REF][START_REF] Chen | Irreversible sorption of pentachlorophenol to sediments: experimental observations[END_REF]. We applied isotopic methods in order to assess the origin of such hysteretic behaviour in COx mudstone.

Desorption hysteresis confirmed by isotopic exchange

To better understand these non-reversible aspects, four experiments were performed for each molecule, namely adsorption, desorption, forward isotopic exchange (FIE) and reverse isotopic exchange (RIE). Data obtained at low concentrations (C(t=0) = 10 -6 M) to avoid saturation effects is discussed herein. Results at higher concentrations (C(t=0) = 10 -3 M) are available in the supplementary material and discussed in section 4. The distribution ratios, R d , are presented in Fig. 4. The average of the triplicate results is represented as a function of time with (2σ) theoretical error bars. The results obtained by adsorption were consistent with results obtained by FIE for all three molecules (error bars overlapping). This indicated that there was no major modification in the solid during the first adsorption step. The results obtained by desorption were also consistent with results obtained by RIE for all three molecules (error bars overlapping). However, a significant difference was obtained between the adsorption and desorption results even after 42 days for all three molecules (error bars not overlapping).

Accepted Manuscript

Despite the systematic desorption hysteresis, some major differences appeared between the three acids. Firstly, the affinity of carboxylic acids for the mudstone strongly varied from 1.6 ± 0.2 L kg -1 for phthalate, up to 42 ± 5 L kg -1 for acetate, and 56 ± 12 L kg -1 for citrate respectively. The retention kinetics of phthalate and citrate were rapid and equilibrium was reached within 48 h (Fig. 4).

On the contrary, R d values for acetate increased between 2 and 42 days. The differences in the R d values between phthalate and citrate, and the slow kinetics for acetate indicated a specific retention mechanism for each molecule. Nevertheless, significant hysteresis was quantified for each species.

The corresponding indexes obtained with classical desorption (protocol #1) were H.I. = 3.9 ± 0.4, 3.3 ± 0.6, 3.3 ± 0.8 for acetate, citrate and phthalate respectively (Table 3). These values decreased to H.I.

= 2.4 ± 0.6, 2.3 ± 0.6, 1.0 ± 1.5 using another desorption protocol (#4) discussed in the following section.

Effect of desorption protocol

The variation in hysteresis was investigated using various desorption protocols starting from the same adsorption state (Fig. 2). The desorption data was compared with the adsorption data in Fig. 5. No effect of NaN 3

Case of acetate

We first discriminated adsorption on the solid from degradation due to bacterial activity. For this reason, adsorption experiments were performed in presence of sodium azide (10 -3 M) which is an inhibitor of bacterial activity. The values obtained were below uncertainties, 0 < R d (acetate) < 0.2 L kg -1 . Similar results were also reported on COx mudstone with the low affinity of monocarboxylic acids (R d (sorbate) ~ 0.7 L kg -1 , R d (tiglate) ~ 0.4 L kg -1 ; Vu-Do, 2013). This result evidenced low affinity between the sole carboxylic groups and the COx mudstone. The process driving acetate disappearance from the solution was bacterial activity, leading either to methane, carbon dioxide, carbonates or biomass [START_REF] Weber | Microbial acetate conversion to methane: kinetics, yields and pathways in a two-step digestion process[END_REF][START_REF] Abromaitis | Biodegradation of persistent organics can overcome adsorption e desorption hysteresis in biological activated carbon systems[END_REF] and apparent desorption hysteresis.

This sole biodegradation phenomenon was in agreement with the observations made of a slow kinetics (Fig. 4). The R d value of phthalate and citrate stabilised within 48 hours without effect from NaN 3 , in consistency with fast adsorption processes.

Case of phthalate

In the case of phthalate, desorption hysteresis that was observed with protocols #1 and #3 vanished when using protocols #2 and #4. These protocols, #2 and #4, included an equilibration of the renewal solution with a sacrificial sample of mudstone. This equilibration step lowered the chemical perturbation during supernatant renewal. The observations on phthalate suggested that trace elements contained in mudstone strongly affected the desorption step. Synthetic pore water is typically used to perform adsorption and desorption experiments. This protocol was efficient with most cations, but was found to be inadequate here despite an equilibration step of the synthetic pore water with the mudstone based on a solid/solution ratio of 50 g L -1 . The fact that protocol #2, without phthalate in renewal pore water, also suppressed the hysteresis phenomenon, indicated that chemical perturbation induced by phthalate does not influence the retention capacity of mudstone. The key fact remains that adsorption reversibility was demonstrated for phthalate with desorption protocols #2 and #4. This is in agreement with results of [START_REF] Conkle | Competitive sorption and desorption behavior for three fluoroquinolone antibiotics in a wastewater treatment wetland soil[END_REF] on the reversible adsorption of fluoroquinolones (aromatic acids) in a wastewater treatment wetland soil.

Case of citrate

A specific behavioural pattern was observed for citrate. The use of protocols #2, #3 and #4 only partially reduced the desorption hysteresis phenomenon. Hysteresis indexes decreased from 3.3 ± 0.6 (protocol #1) to 3.1 ± 0.5 (protocol #2), 2.7 ± 0.6 (protocol #3), and 2.3 ± 0.6 (protocol #4). The most efficient protocol was #4, which minimised chemical perturbation by using a renewal solution equilibrated with both clay and organic molecules. Nonetheless, such protocols could not fully suppress desorption hysteresis. This result suggested the existence of "true" hysteresis, meaning that the adsorption process modifies the sorbing phase or incorporates sorbate in a solid phase. Similar hysteresis was reported for citrate on several model phases by [START_REF] Yeasmin | Influence of mineral characteristics on the retention of low molecular weight organic compounds: a batch sorptiondesorption and ATR-FTIR study[END_REF]. Based on this data, the adsorption of citrate rather tends to occur on minor iron-rich phases such as ferrihydrite or goethite with R d values above 10 4 L kg -1 . Assuming a percentage P = 99.9 % of citrate retained during desorption (99 < P < 100 % detailed in the study), the hysteresis indexes H.I.(citrate) = 3.3 ± 0.4 and 5.0 ± 3.4 were calculated on ferrihydrite and goethite respectively at equilibrium concentrations ranging from 2 10 -8 and 2 10 -6 M (Table 3). Similar values were also reported for another dicarboxylic acid, namely oxalate, with a better level of confidence on the retained percentage: H.I.(oxalate) = 3.2 ± 0.2 and 3.6 ± 1.3 on ferrihydrite and goethite respectively.

Discussion

This section focuses on the origin of desorption hysteresis for small soluble carboxylic acids and the environmental implications.

Concerning acetate, the low affinity of the carboxylic group with the solid surfaces of minerals was confirmed. This is in agreement with adsorption data of acetate on alfisol and sediments reported by [START_REF] Oepen Von | Sorption of Nonpolar and Polar Compounds to soils: Processes, Measurements and Experience with the Applicability of the Modified OECD-Guideline 106[END_REF] After 24 hours, the increase in R d is representative of the disappearance of molecules from the solution. The apparent hysteresis of acetate was due to bacterial degradation.

Several bacteria in these media are able to degrade molecules like acetate, ethanol or carbohydrates [START_REF] Abromaitis | Biodegradation of persistent organics can overcome adsorption e desorption hysteresis in biological activated carbon systems[END_REF]. Complementary experiments would be useful in order to quantify the fractions of carbon from acetate which is metabolised in "living" organic matter, incorporated in calcite or released as soluble carbonates or into the gas phase.

Ortho-phthalate displayed a higher affinity with mudstone than expected. [START_REF] Oepen Von | Sorption of Nonpolar and Polar Compounds to soils: Processes, Measurements and Experience with the Applicability of the Modified OECD-Guideline 106[END_REF] reported lower values, R d < 0.5 L.kg -1 , for benzoate, phthalate and phenylacetate in alfisol and sediment with similar pH levels and amounts of natural organic matter and clay minerals. The adsorption of ortho-phthalate seems related to the presence of two carboxylic groups [START_REF] Gu | Adsorption and desorption of different organic matter fractions on iron oxide[END_REF].

Outer-sphere complexes with low-energy bonds were also evidenced on oxides by [START_REF] Hwang | Surface complexes of phthalic acid at the hematite/water interface[END_REF] and on clay minerals by [START_REF] Kang | Adsorption of dicarboxylic acids by clay minerals as examined by in situ ATR-FTIR and ex situ DRIFT[END_REF]. These low-energy interactions are consistent with the reversible adsorption of phthalate on COx mudstone. A striking result was that desorption hysteresis occurred because of pore water renewal. One assumption is that the leaching of a fraction of natural organic leads to the releases of adsorption sites. Unfortunately, the ATR-FTIR signal of adsorbed phthalate was too low on COx mudstone to confirm or deny such an assumption. Regardless of this mechanism, significant differences in R d DES values were expected for phthalates, depending on the desorption protocol. Some soil science guidelines recommend desorption protocol in CaCl 2 , NaClO 4

or NaCl to impose the pH and ionic strength. This could lead to the overestimation of desorption in natural samples. The use of a synthetic solution may also be inadequate for phthalate removal from sediments. Nonetheless, the reversible adsorption of phthalate consolidates the estimations of solid liquid partitioning and retarded transport in soils and sediments.

Citrate displayed persistent desorption hysteresis on COx mudstone. [START_REF] Yeasmin | Influence of mineral characteristics on the retention of low molecular weight organic compounds: a batch sorptiondesorption and ATR-FTIR study[END_REF] quantified citrate and oxalate adsorption on several pure phases. Based on this data, an additive model would predict the adsorption of dicarboxylic acids driven by iron-rich phases. Hees and Lundström (2000) also suggested that Fe and Al oxyhydroxides drive the adsorption of citrate in natural systems.

However, in both papers, hysteresis is reported without further interpretation on solid solution partitioning. [START_REF] Yang | Mobilization of soil organic matter by complexing agents and implications for polycyclic aromatic hydrocarbon desorption[END_REF] demonstrated that complexing agents, such as citrate, EDTA or oxalate, mobilise cations (Fe, Al) in the aqueous phase, leading to the dissolution of natural organic matter. However, the authors suggested that this mechanism would rather enhance the desorption of apolar compounds. Another assumption studied by [START_REF] Lee | The role of citrate and phthalate during Co(II) coprecipitation with calcite[END_REF] is the incorporation of citrate in solid phases. The authors suggested that citrate co-precipitates with calcite whereas phthalate does not. Our mudstone samples contained 30% of carbonates which are a potential trap for citrate.

We thus performed adsorption experiments on decarbonated samples (Fig. S2 in the supplementary material). The adsorption of citrate on decarbonated sample was similar to adsorption on raw samples.

This invalidates the assumption of co-precipitation in carbonate phases. This result is supported by the shape of citrate's adsorption isotherm which differs from typical adsorption/precipitation shapes (case of oxalate illustrated in Fig. S2). Citrate is also known to be sensitive to reduction processes and ferric citrate is used as an oxidant to measure the reducing capacity of organic matter in mudstones [START_REF] Bleyen | Assessing the oxidising effect of NaNO 3 and NaNO 2 from disposed Eurobitum bituminised radioactive waste on the dissolved organic matter in Boom Clay[END_REF]. It would be worth studying the immobilisation of citrate in COx mudstone, which is a reductive medium, to see if it is affected by oxydo-reduction processes.

The case of citrate remains the most puzzling. The desorption hysteresis indicated a chemical perturbation of the solid. This is contradictory to the similar values obtained by adsorption and forward isotopic exchange at low concentrations, C(t=0) ~ 10 -6 M (Fig. 3). The pre-equilibration of mudstone with higher concentrations (C(t=0) ~ 10 -3 M), led to a very slight decrease in adsorption, i.e. (Fig. 3). These effects were not significant in comparison with uncertainties and concentration dependency. Nevertheless, such behaviour would be in agreement with the capping of non-reversible adsorption sites leading to a lowered but reversible second step of adsorption/desorption (RIE). A similar effect is expected during the transport of chelating agents in sediments. It would be worth performing transport experiments before/after equilibration with citrate in order to assess if the irreversible capping of sorbing phases does indeed lead to a reduction in the retardation factor. Similar systems, chelating agents/sediments, warrant such studies in order to improve the prediction of solid/solution partitioning with focus on desorption modelling.

Conclusions

The adsorption and desorption of acetate, ortho-phthalate and citrate was studied on a Callovo-Oxfordian mudstone. The mudstone displayed equal parts of carbonates, clay minerals, quartz and may be considered as being similar to many sedimentary rocks. The three species displayed similar desorption hysteresis evidenced by a classical method and reverse isotopic exchange. Yet, this behaviour was shown to result from three different mechanisms.

Acetate was mainly removed from the liquid phase by bacterial activity. Adsorption and desorption of acetate were isolated by adding sodium azide and found to be negligible compared with other carboxylic acids. In the case of phthalic acid, the equilibration of the renewal pore water with mudstone suppressed desorption hysteresis. This result confirmed the significant and reversible adsorption of phthalate in the mudstone. It also highlighted the impact of trace elements on desorption.

The use of a synthetic solution may prove to be inadequate for phthalate removal from sediments.

Finally, a high level of adsorption and persistent desorption hysteresis was evidenced for citrate. This result suggested the existence of a third adsorption mechanism. Citrate may involve chemical perturbation such as surface capping, dissolution or oxydo-reduction processes. At low concentrations, adsorption is constant before/after equilibration with citrate and desorption remaining hysteretic in both cases. At high concentrations, adsorption and desorption hysteresis were slightly reduced by preequilibration with citrate but in proportions lower than saturation effects. Such systems (chelating agents/sediments) warrant further studies in order to improve the prediction of solid/solution partitioning. It would be worth performing transport experiments before/after equilibration with citrate in order to evidence the irreversible capping of sorbing phases and a decrease in the retardation factor.

  10 -6 M. The pH of the solution was controlled inside the glovebox at the beginning and the end of the experiments. Variations in the pH below 0.2 were measured during adsorption or desorption. Sorption and desorption experiments were investigated for about 6 weeks by successive Accepted Manuscript samplings. Desorption experiments started after the last adsorption sampling. It consisted in renewingsupernatant by synthetic pore water using the same water/rock ratio as that used for the adsorption experiments. Before sampling, tubes were centrifuged at 15,000 rpm for 1 hour. An aliquot of 100 µL was taken, weighted and analysed by beta liquid scintillation counting (Perkin-Elmer Tricarb 2700).

Figure 1 .

 1 Figure 1. Scheme of adsorption and desorption experimental setup. Comparison of classical

Figure 2 .

 2 Figure 2. Scheme of the desorption protocols #1 to #4, with various renewal solutions. 12C-

  isotopic exchange (REI) experiments consisted in renewing supernatant by organic solution at the concentration measured after the FIE step. Adsorption-desorption equilibrates the 14 C/ 12 C ratio between solid and liquid at constant macroscopic concentrations in both phases. Both methods, classical and isotopic give different information. The classical method quantifies simultaneous effects of adsorption and chemical perturbation by adsorbates. Forward isotopic exchange isolates adsorption at a constant macroscopic concentration on the system which is already perturbed by the first step of adsorption/equilibration. A "conditioning effect" during this first step would be evidenced by different distribution ratios measured by classical adsorption, R d ADS , and forward isotopic exchange R d FIE .

  the chemical perturbation during solution renewal. It assesses the reversibility of adsorption with low sorbate concentrations, close to far-field environmental conditions. Protocols #1 and #2 were more standard desorption protocols. They were representative of leaching or decontamination processes using synthetic solutions.

Figure 3 .Figure 4 .

 34 Figure 3. (Top) Adsorption isotherm for 14 C-phthalate (triangles , taken from[START_REF] Dagnelie | Sorption and diffusion of organic acids through clayrock: comparison with inorganic anions[END_REF]

  desorption hysteresis, i.e. R d DES and R d REI values almost on adsorption isotherm

Table 1 .

 1 Properties of the organic molecules used as adsorbates.

	Acid	Formula	M (g moL -1 )	pKa	Log K ow
	Acetic C 2 H 4 O 2		60.1	4.76	-0.17
	Ortho-phthalic C 8 H 6 O 4		166.1	2.94 / 5.43	0.79
	Citric C 6 H 8 O 7		192.1	3.13 / 4.76 / 6.40	-1.6

2. Materials and Methods 2.1 Mudstone characterisation and preparation

	Adsorption desorption experiments were carried out on a sample extracted from the Callovo-
	Oxfordian (COx) formation (East of Paris basin), which is currently studied as a potential host rock for
	the deep underground radioactive waste repository called Cigéo project. The sample was preserved
	from air (i.e. O 2 ) to prevent mineral oxidation (Fe being mostly in pyrite form, FeS). Neither the
	oxidative removal of SOM nor the reductive antibacterial treatment was performed in order to remain

OH O HOOC COOH OH O OH OH O O O H

Table 2 .

 2 Composition of synthetic pore water used for this work. Theoretic data are reported in the literature (10 -3 mol L -1 ) and measurements after desorption experiments.

			Measured in Vinsot et al. 2008	Calculated in Appelo et al. 2008	Synthetic pore water	After desorption protocol #1	After desorption protocol #2
	pH	7.2	7.05	7.15	7.2±0.2	7.2±0.2
	Na +	52.3	54.0	46.0	51.1	54.3
	K +	0.6	0.65	1.0	1.4	2.1
	Mg 2+	5.0	5.0	7.4	8.8	11.1
	Ca 2+	6.3	6.5	6.7	9.1	13.4
	Sr 2+	1.1	0.2	0.2	n.d.	n.d.
	Fe 3+	n.d.	n.d.	0	0.006	0.032
	Cl -	41.0	46.8	41.0	51.4	53.7
	SO 4	2-	15.6	13.5	15.6	19.4	26.9
	Alk.	3.0	4.5	3.3	n.d.	n.d.
	H 4 SiO 4	0.4	0.18	0.05	n.d.	n.d.
	D.O.C.
	(mg/L)	6.5	0	0	0.6 -1.1	1.4 -2.2

.2 Chemicals and radiolabelled compounds The

  properties of the carboxylic acids are shown in Table1. Stable solutions were prepared from high purity salts: sodium acetate (Merck, purity > 99%), phthalic acid (Merck, purity > 99.5%) and citric acid (Merck, purity > 99.7%). The pH was adjusted to 7.0 ± 0.2 with concentrated NaOH or HCl standard solutions. Experiments were performed with radiolabelled compounds: acetic acid [ 3 H] (ARC, 150 Ci/mol), acetic acid [ 14 C] (ARC, 58 Ci/mol), citric acid [1,5-14 C] (ARC, 112 Ci/mol), phthalic acid [2,3,4,5-3 H] (ARC, 60 Ci/mol) and phthalic acid [ 14 C] (homemade synthesis detailed in

Table 3 .

 3 Experimental results from adsorption and desorption of acetate, citrate and phthalate on COx mudstone. The different renewal pore waters used in protocols #1 to #4 are detailed in Fig.2.

			Acetate	Phthalate	Citrate
			< 0.2	1.6 ± 0.2	60 ± 12
	Rd ADSORPTION	(with NaN 3 )	(with or without NaN 3 )
		(L kg -1 )	42 ± 5	1.4 [1.2 -2.0]	[10-40] C>10 -3 mol L -1
			(without NaN 3 )	(Dagnelie et al., 2014)	(Durce, 2011)
	Rd DESORPTION (L kg -1 )	143 ± 6	2.0 ± 0.3 (protocol #2 & 4)	183 ± 8 (protocol #4)
		Protocol #1	3.9 ± 0.4	3.3 ± 0.8	3.3 ± 0.6
	H.I.	Protocol #4	2.4 ± 0.6 on ferrihydrite (Yeasmin et al., 2014) 1.0 ± 1.5	2.3 ± 0.6 3.3 ± 0.4
			on goethite (Yeasmin et al., 2014)	5.0 ± 3.4

Figure 5. Effect of desorption protocol on the release of acetate (top), citrate (middle) and phthalate (bottom) from mudstone after 42 days.

Acknowledgments

This work was partially financed by the "Agence Nationale pour la gestion des Déchets Radioactifs" (Andra). We would like to thank Dr Jean-Charles Robinet for his useful discussion and input on the manuscript. This paper is dedicated to Dr Eric Giffaut. We are grateful for his unwavering support while co-advising this work.

Accepted Manuscript