domain: sde chim chim.anal

text:
Weilu Yang

Nihal Oturan

Jialin Liang

Mehmet A Oturan
email: mehmet.oturan@univ-eiffel.fr

Synergistic mineralization of ofloxacin in electro-Fenton process with BDD anode: Reactivity and mechanism

Keywords: Electro-Fenton, Anodic oxidation; BDD anode, Synergistic catalysis, Hydroxyl radial, Pharmaceutical wastewater

This study aimed at exploring the performance of the electro-Fenton (EF) process with a borondoped diamond (BDD) anode and an electrochemically exfoliated graphene-based cathode (EEGr), for removal of antibiotic ofloxacin (OFLO) from contaminated water. Compared with anodic oxidation (AO) process with BDD anode, the EF process with BDD anode (EF-BDD) could improve the decay kinetics of OFLO by 1.4-2.6 times and reduce the disposal cost by 28%-41%. Removal efficiency of TOC in EF-BDD process under 4.2 mA cm -2 was better than in AO process under 8.3 mA cm -2 , pointing out that the cost effectiveness of the EF process for mineralization of OFLO under low current density conditions. This advantage is due to the simultaneous formation of homogeneous ( • OH) and heterogeneous (BDD( • OH)) hydroxyl radicals in the process. The synergistic factor of • OH and BDD( • OH) for mineralization of OFLO decreased with the current density rising from 4.2 (0.28) to 16.6 mA cm -2 (-0.08). Decay kinetics of OFLO in the EF-BDD and AO were more converging under higher current densities owing to the production of large amount of BDD( • OH). In the EF-BDD, a trade-off should be considered between accumulated BDD( • OH) on the surface of BDD anode and H2O2 generation on the cathode. Absolute rate constant for oxidation of OFLO by • OH (kOFLO) was calculated to be 3.86 × 10 9 M -1 s -1 . Evolution of carboxylic acids and inorganic ions as well as toxicity assessment were followed during the treatment of OFLO solutions. Results obtained proved that the EF-BDD process is outstanding for oxidative degradation of OFLO thanks to the synergistic effect of •OH and BDD( • OH). These very strong oxidants being non-selective, this result can be extended to the treatment of antibiotic organic pollutants.

Introduction

Antibiotics are largely used throughout the world. The majority of discharged antibiotics and their metabolites escape from being eliminated in the traditional wastewater treatment plants due to slight solubility and strong resistance to degradation by conventional biological or chemical processes [START_REF] Hapeshi | Drugs degrading photocatalytically: kinetics and mechanisms of ofloxacin and atenolol removal on titania suspensions[END_REF][START_REF] Dirany | Study of the toxicity of sulfamethoxazole and its degradation products in water by a bioluminescence method during application of the electro-Fenton treatment[END_REF]. Types of antibiotics and their incomplete transformation products were detected in multiple aquatic ecosystems and have drawn close attention from researchers and public authorities [START_REF] Jurado | Occurrence, fate and environmental risk assessment of the organic microcontaminants included in the Watch Lists set by EU Decisions 2015/495 and 2018/840 in the groundwater of Spain[END_REF]. These pollutants pass through the municipal water treatment plants, since they are not designed to retain this kind of pollutants. Removal of these pollutants from aquatic environments, as eco-toxicological threats to humans and aquatic animals, has been a great challenge. In contrast to the conventional treatment processes, the electrochemical advanced oxidation processes (EAOPs) have attracted great interest for removing efficiently persistent/toxic organic pollutants from water via in situ generated oxidant free radicals [START_REF] Sirés | Electrochemical advanced oxidation processes: today and tomorrow[END_REF][START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF][START_REF] Du | Nanostructured electrodes for electrocatalytic advanced oxidation processes: From materials preparation to mechanisms understanding and wastewater treatment applications[END_REF][START_REF] Nidheesh | Recent advances in electro-Fenton process and its emerging applications[END_REF].

As one of the most popular EAOPs, the electro-Fenton (EF) process have gained prominence owing to its economic feasibility and remarkable mineralization efficiency [START_REF] Zhao | Catalytic activity of MOF (2Fe/Co)/carbon aerogel for improving H2O2 and OH generation in solar photoelectro-Fenton process[END_REF][START_REF] Zhao | Continuous bulk FeCuC aerogel with ultradispersed metal nanoparticles: an efficient 3D heterogeneous electro-Fenton cathode over a wide range of pH 3-9[END_REF]. This promising technology has been successfully applied for complete mineralization of various refractory organic pollutants including pharmaceutical compounds [START_REF] Olvera-Vargas | Electro-Fenton beyond the degradation of organics: Treatment of thiosalts in contaminated mine water[END_REF][START_REF] Cheng | Hierarchical iron phosphides composite confined in ultrathin carbon layer as effective heterogeneous electro-Fenton catalyst with prominent stability and catalytic activity[END_REF][START_REF] Camcioglu | Fast and complete destruction of the anti-cancer drug Cytarabine from water by electrocatalytic oxidation using electro-Fenton process[END_REF][START_REF] Trellu | Clofibric acid removal at activated carbon fibers by adsorption and electro-Fenton regeneration -Modeling and limiting phenomena[END_REF]. In the EF process, homogeneous • OH radicals (E. 0 OH/H2O = 2.8 V/SHE) are generated electrocatalytically in the bulk solution through Fenton's reaction [START_REF] Sirés | Electrochemical advanced oxidation processes: today and tomorrow[END_REF][START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF][START_REF] Nidheesh | Recent advances in electro-Fenton process and its emerging applications[END_REF][START_REF]Electro-Fenton Process: New trends and scale-up[END_REF]. On the other side, the anodic oxidation (AO) process provides heterogeneous • OH radicals on the surface of anodes with high O2 evolution overvoltage [START_REF] Ma | Development of Ti/TiOx foams for removal of organic pollutants from water: Influence of porous structure of Ti substrate[END_REF][START_REF] Hu | Anodic oxidation of organic pollutants: anode fabrication, process hybrid and environmental applications[END_REF] . The AO processes belong to direct electrooxidation and usually involves the mass transfer limitation [START_REF] Yang | Electrocatalytic generation of homogeneous and heterogeneous hydroxyl radicals for cold mineralization of anticancer drug Imatinib[END_REF][START_REF] Chen | Heterogeneous Fenton chemistry revisited: Mechanistic insights from ferrihydrite-mediated oxidation of formate and oxalate[END_REF]. The use of an appropriate anode (M) can produce heterogeneous radicals (M( • OH)) via oxidation of water on the surface of anode (Eq. ( 1)) [START_REF] Zhu | Strategies for enhancing the heterogeneous Fenton catalytic reactivity: a review[END_REF][START_REF] Sheng | Stable and selective electrosynthesis of hydrogen peroxide and the electro-Fenton process on CoSe2 polymorph catalysts[END_REF].

M + H2O  M( • OH) + H + + e - (1)
The contribution of M( • OH) to EF process is strongly related to the nature of anode material, as well as in the overall performance of AO [START_REF] Hien | Comparison of homogeneous and heterogeneous electrochemical advanced oxidation processes for treatment of textile industry wastewater[END_REF]. Anodes characterized with high O2-evolution overpotential are regarded as "non-active" anodes, while those with low O2-evolution overpotential are of the "active anodes" type. In the first case, M(OH) are barely adsorbed on the surface (physisorption) while in the second case they are strongly adsorbed (chemisorption). Boron-doped diamond (BDD), as typical "non-active" anode, is the most potent anode for AO due to its high O2 evolution overpotential (around 2.2 V/SHE), high stability and available of physisorbed BDD( • OH) [START_REF] Özcan | Propham mineralization in aqueous medium by anodic oxidation using boron-doped diamond anode: influence of experimental parameters on degradation kinetics and mineralization efficiency[END_REF][START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF]. In contrast to the chemisorbed M( • OH) on the surface of active anodes such as Pt and DSA anodes, a large amount of physically adsorbed BDD( • OH) enhance the effectiveness of the EF process, especially for the complete destruction of aliphatics [START_REF] Oturan | Outstanding performances of the BDD film anode in electro-Fenton process: Applications and comparative performance[END_REF]. In EF processes with BDD anode (EF-BDD process), the oxidative degradation of organics can be regarded as a synergistic effect of heterogeneous BDD( • OH) (formed at the surface of BDD anode) and homogeneous • OH generated in solution [START_REF] Wang | Efficiently degradation of perfluorooctanoic acid in synergic electrochemical process combining cathodic electro-Fenton and anodic oxidation[END_REF][START_REF] Olvera-Vargas | Near-neutral Electro-Fenton Treatment of Pharmaceutical Pollutants: Effect of Using a Triphosphate Ligand and BDD Electrode[END_REF]. We previously showed that BDD anode provides outstanding performances for mineralization of drug imatinib solution compared to other three anodes (DSA, Pt and Ti4O7), with complete mineralization of imatinib solution (corresponding to 24.11 mg L -1 initial TOC) in EF-BDD process under 12.5 mA cm -2 [START_REF] Yang | Electrocatalytic generation of homogeneous and heterogeneous hydroxyl radicals for cold mineralization of anticancer drug Imatinib[END_REF].

This study focuses on the contribution of BDD( • OH) in the EF-BDD process, at different stage and under different operating conditions during the treatment of pharmaceutical wastewater.

Indeed, there are very few studies devoted to this topic [START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF] and further research is needed to clarify the synergistic effect of BDD(•OH) on the EF-BDD process during wastewater treatment. Therefore, the oxidative degradation and mineralization of a synthetic solution of a fluoroquinolone antibiotic, the ofloxacin, largely used for the treatment of a number of bacterial infections, was studied by electro-Fenton process with BDD anode. The primary aims of the study were (i) identify the outstanding performance of BDD-EF for removal of OFLO, (ii) evaluate relative contributions of homogeneous

Preparation of EEGr-CF cathodes

The graphene used in this study was synthesized via electrochemical exfoliation of graphite foil (for simplification, it was noted as EEGr). The graphene-based cathode (EEGr-CF) was prepared with EEGr, carbon black, PTFE, deionized water and ethanol as described in reference [START_REF] Yang | Ultrahigh yield of hydrogen peroxide on graphite felt cathode modified with electrochemically exfoliated graphene[END_REF]. A certain amount of EEGr and carbon black powder were mixed at a mass ratio of 1:4, with PTFE, ethanol and deionized water forming a semifluid upon rapid shaking for 15 s. The milky mixture was coated on both sides of the pre-treated CF. After drying at ambient temperature; CF was placed in a tube furnace under nitrogen atmosphere and annealed at 360 o C for 30 min, which was marked as EEGr-CF cathode.

Degradation experiments

Degradation of OFLO in EF-BDD, AO and EF-DSA processes were carried out in 250 mL single-chamber electrolytic cells. The effective surface area of BDD and DSA anodes was 4 × 6 cm 2 , and that of the prepared EEGr-CF cathodes was 2 × 3 cm 2 . These electrodes were used both in EF and AO processes. The BDD anode was replaced with DSA in order to exclude the heterogeneous • OH produced in EF-DSA, because this anode generate only a few DSA( • OH).

The distance between the anodes and cathodes was 2 cm. Aeration was provided to the solution at a flow rate of 0.75 L min -1 to maintain it a O2-saturated level. Aqueous solution of 0. The degradation efficiency of a target pollutant was calculated by the following equation:

R% = (C0 -Ct) / C0 × 100 (1)
The pseudo-first-order rate constant (kapp) for oxidative degradation of OFLO was determined by using Eq. ( 2):

Ln (Ct / C0) = -k × t (2)
where C0 and Ct were concentrations of the target pollutant at the beginning of electrolysis and at time t, respectively.

The total organic carbon (TOC) removal efficiency was calculated according to TOC decay measurement as follow:

TOC(%) = (Δ(TOC)exp / TOC0) × 100

Total organic carbon (TOC) of OFLO (0.1 mM solution corresponding to an initial TOC of 21.6 mg L -1 ) was measured with an Analytik jena multi N/C 3100 TOC analyzer (Jena, Germany). The concentration of H2O2 was measured by potassium titanium (IV) oxalate method using an UV759 UV-Vis spectrophotometer (Shanghai, China). Carboxylic acids generated during the mineralization of OFLO were identified with an Agilent 1260 HPLC (CA, USA) equipped with a Carbomix H-NP10 column (6.0 μm, 7.8 × 300 mm) at wavelength of 210 nm. Mobile phase and flow rate were H2SO4 (0.25 mM) and 0.6 mL min -1 . Inorganic ions generated during EF process were monitored with a Dionex ICS-900 ion chromatograph (CA, USA). Ofloxacin transformation products (OTPs) were detected with HPLC and GC-MS (Text S1-1). Acute toxicity (LC50), developmental toxicity, bioaccumulation factor were evaluated by quantitative structure-activity relationship using Toxicity Estimation Software Tool.

The relative contributions of homogeneous • OH and heterogeneous BDD( • OH) for TOC removal were calculated according to the Eq. ( 2) and (3):

R% ( • OH) = 100 (TOCEF-BDD -TOCAO) / TOCEF-BDD (2)
R% (BDD( • OH)) = 100 (TOCAO) / TOCEF-BDD

where TOCEF-BDD and TOCAO were TOC decay values obtained in the EF-BDD and AO processes, respectively.

The mineralization current efficiency (MCE) and energy consumption (EC) of OFLO solution were calculated according to following equations:

MCE % = 100 n F Vs Δ(TOC)exp / 4.32 ×10 7 m I t (4) C18H20FN3O4 + 32H2O  18CO2 + F -+ 3NH4 + +72H + + 74 e - (5)
EC (kWh (gTOC) -1 ) = U I t / Vs Δ(TOC)exp [START_REF] Du | Nanostructured electrodes for electrocatalytic advanced oxidation processes: From materials preparation to mechanisms understanding and wastewater treatment applications[END_REF] where n is the number of electron (which was taken as 74 assuming that the majority of N atom contained in OFLO molecule was released as NH4 + in eq. ( 5)) consumed per mol of OFLO in the electrochemical mineralization reaction. F is the Faraday constant (96485 C mol-1), (TOC)exp means the TOC decay at t time, 4.32×107 represents a conversion factor to homogenize units (3600 s h-1 × 12000 mg of C mol-1). U, I, t are the average cell voltage (V), the current applied (A) and the electrolysis time (h), respectively.

Synergistic factor (SF) was calculated to confirm the synergistic effects of homogeneous • OH and heterogeneous BDD( • OH) for oxidation of organics.

SF = [kEF-BDD -(kEF-DSA + kAO)] / (kEF-DSA + kAO) (7)
where kEF-BDD, kEF-DSA and kAO were kinetic constants of OFLO oxidation in EF-BDD, EF-DSA and AO processes, respectively.

Results and discussion

Comparative oxidation of OFLO by EF process

In EF process, degradation of organics is carried out by homogeneous • OH through Fenton reaction (Eq. ( 2)) for which the Fenton's reagent (H2O2 + Fe 2+ ) is electrochemically generated in the treated solution (Eqs. ( 8) and ( 9)):

H2O2 + Fe 2+  Fe 3+ + OH -+ • OH (8) O2 + 2H + + 2e - H2O2 (9)
Fe 3+ + e - Fe 2+ (10)
To evaluate the comparative efficiency of different EAOPs (EF-BDD, AO and EF-DSA), oxidative degradation 0.1 mM OFLO was studied under current density of 8.3 mA cm -2 . First, preliminary experiments were performed to determine the better concentration of catalyst (Fe 2+ ), since it affects the generation of homogeneous • OH in the solution [START_REF] Yang | Electrocatalytic destruction of pharmaceutical imatinib by electro-Fenton process with graphene-based cathode[END_REF][START_REF] Garcia-Rodriguez | Mineralization of electronic wastewater by electro-Fenton with an enhanced graphenebased gas diffusion cathode[END_REF]. For this, the solution pH was set to 3 as this value is well-known to be suitable for EF process [START_REF] Brillas | Electro-Fenton process and related electrochemical technologies based on Fenton's reaction chemistry[END_REF][START_REF] Nidheesh | Recent advances in electro-Fenton process and its emerging applications[END_REF][START_REF] Camcioglu | Fast and complete destruction of the anti-cancer drug Cytarabine from water by electrocatalytic oxidation using electro-Fenton process[END_REF][START_REF]Electro-Fenton Process: New trends and scale-up[END_REF][START_REF] Oturan | Outstanding performances of the BDD film anode in electro-Fenton process: Applications and comparative performance[END_REF][START_REF] Moreira | Electrochemical advanced oxidation processes: a review on their application to synthetic and real wastewaters[END_REF]. The complete degradation of 0.1 mM OFLO reached in 60, 50, 25 and 20 min for catalyst concentration of 0.05, 0.1, 0.2 and 0.3 mM Fe 2+ , respectively (Fig. 1). The apparent rate constants (kapp) was 0.07 and 0.09 min -1 with 0.05 and 0.1 mM Fe 2+ , and fluctuated little with 0.2 (0.15 min -1 ) and 0.3 mM (0.17 min -1 ) Fe 2+ for OFLO removal in EF process. However, the concentration of 0.2 mM Fe 2+ was chosen as the proper concentration of catalyst for the following EF experiments. Low concentration of Fe 2+ will guarantee the continuous reaction via reduction of Fe 3+ (Eq. ( 10)) generated in Fenton reaction [START_REF] Casado | Towards industrial implementation of Electro-Fenton and derived technologies for wastewater treatment: A review[END_REF][START_REF] Barhoumi | Kinetics of oxidative degradation/mineralization pathways of the antibiotic tetracycline by the novel heterogeneous electro-Fenton process with solid catalyst chalcopyrite[END_REF] and avoids the wasting of generated • OH through the reaction given in Eq. [START_REF] Cheng | Hierarchical iron phosphides composite confined in ultrathin carbon layer as effective heterogeneous electro-Fenton catalyst with prominent stability and catalytic activity[END_REF] which is favored at high Fe 2+ concentrations. Besides, Fe 2+ can also be regenerated through other chemical reactions in the solution with slower kinetics (Eqs. ( 12) and ( 13)).

Fe 2+ + • OH  Fe 3+ + OH - (11)
Fe 3+ + H2O2  Fe 2+ + HO2 The comparative oxidation efficiencies of different systems are given in Fig. 2. The oxidative degradation efficiency of OFLO was 100%, 90% and 76% in EF-BDD, AO and EF-DSA systems under current density of 8.3 mA cm -2 , upon electrolysis time of 40 min, respectively (Fig. 2(a)). In the EF-BDD process, OFLO could be completely disappeared at 30 min of electrolysis due to large amounts of BDD( • OH) generated on surface of anode (Eq. ( 1)) and homogeneous • OH generated in solution (Eq. ( 7)). The solution TOC was also completely removed upon electrolysis time of 6 h, which was much higher than that in the EF-DSA (53%) (Fig. 2(b)). The considerable performance of TOC removal in AO (87%) process was due to remarkable BDD( • OH) generated on surface of BDD anode. The mineralization current efficiency (MCE) values (Eq. ( 4)) increased to peak values at early stages (1 h) of electrolysis in three systems (Fig. 2(c)), which can be attributed to quick degradation/mineralization of OFLO and intermediates by • OH. Afterwards, the MCE decreased gradually with formation of recalcitrant by-products hardly oxidizable such as short chain carboxylic acids in solution and decrease in organic matter concentration. The values of energy consumption (Eq. ( 5)) for removal of TOC increased with treatment time and reached to 0.8, 0.9 and 1.1 kWh (g TOC) -1 in EF-BDD, AO and EF-DSA systems under 8.3 mA cm -2 , respectively (Fig. 2(d)). These results point out that EF-BDD is an outstanding process for oxidative degradation and mineralization of organic pollutants thanks to simultaneous production homogeneous • OH and heterogeneous BDD( • OH).

Effect of current densities on electrocatalytic activity of EF-BDD process

The formation of homogeneous • OH in solution and heterogeneous BDD( • OH) on the surface of BDD anode are strongly related to applied current which monitor the formation of hydroxyl radicals through Eqs. ( 1), ( 7) and ( 8) [START_REF] Sirés | Electrochemical advanced oxidation processes: today and tomorrow[END_REF][START_REF] Oturan | Comparative performance of ten electrodes in electro-Fenton process for removal of organic pollutants from water[END_REF]. The effect of current on degradation kinetics of OFLO is shown in Fig. would suffer a predominant reduction when the current densities were higher than 16.6 mA cm -2 . This can be explained with the competing reactions occurred at high current densities [START_REF] Moreira | Electrochemical advanced oxidation processes: a review on their application to synthetic and real wastewaters[END_REF][START_REF] Ghanbari | Insights into paracetamol degradation in aqueous solutions by ultrasoundassisted heterogeneous electro-Fenton process: Key operating parameters, mineralization and toxicity assessment[END_REF].

H2O2 + 2H + + 2e - 2H2O (14)
2H Fig. 4 Effect of current densities on the apparent rate constants for oxidation of OFLO.

Operating conditions: [Fe 2+ ]: 0.2 mM, [Na2SO4]: 50 mM, pH: 3.

Synergistic effects of • OH and BDD( • OH) on oxidation of OFLO

Effect of current densities on the apparent rate constants for oxidative removal of OFLO in EF-BDD, AO and EF-DSA processes are presented in Fig. 5. The kapp values in EF-BDD (0.23 min -1 ), AO (0.16 min -1 ) and EF-DSA (0.09 min -1 ) processes were greatest under 16.6 mA cm - 2 , which were much higher than those obtained under 4.2 (0.09, 0.04 and 0.03 min -1 ) and 8.3 (0.15, 0.08 and 0.05 min -1 ) mA cm -2 , respectively) (Fig. 6). The values of the synergistic factor (SF) were determined for oxidation of OFLO under different current densities. Interestingly, the value of SF was 0.28 under current density of 4.2 mA cm -2 , and gradually decreased to 0.15 (8.3 mA cm -2 ) and -0.08 (16.6 mA cm -2 ), respectively. The results demonstrate the synergistic effects between homogeneous • OH and heterogeneous BDD( • OH) in EF-BDD under 4.2 and 8.3 mA cm -2 . The simultaneous generation of H2O2 on cathode and BDD( • OH) on the surface of anode contribute to the enhanced removal efficiencies of OFLO [START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF]. The SF value less than zero demonstrates no synergistic effects exist between • OH and BDD( • OH) in EF-BDD under 16.6 mA cm -2 . This can be explained by the fact that the rates of decay kinetics of OFLO in EF-BDD and AO were more convergent under 16.6 mA cm -2 with great amount of BDD( • OH) generated and dominated the degradation process.

As can be seen in Fig. 7, TOC removal efficiency is high during early stage of electrolysis (in first 2 h), afterwards, it becomes slower due to the diminution of organic matter in the solution and formation of refractory products, such as short-chain carboxylic acids [START_REF] Yang | Enhanced activation of hydrogen peroxide using nitrogen doped graphene for effective removal of herbicide 2, 4-D from water by iron-free electrochemical advanced oxidation[END_REF][START_REF] Ganiyu | Sub-stoichiometric titanium oxide (Ti4O7) as a suitable ceramic anode for electrooxidation of organic pollutants: a case study of kinetics, mineralization and toxicity assessment of amoxicillin[END_REF][START_REF] Lin | Removal of artificial sweetener aspartame from aqueous media by electrochemical advanced oxidation processes[END_REF]. The TOC removal efficiencies were 100%, 92.4% and 69% under 16.6, 8.3 and 4.2 mA cm -2 in EF-BDD processes, upon mineralization time of 240 min. These results were much higher than those obtained in AO (92%, 60.3% and 45%) and EF-DSA (61%, 40.2% and 26%) processes (Fig. 7(a)-7(c)). The performance of TOC removal in EF-BDD process under 4.2 mA cm -2 was better than in AO process under 8.3 mA cm -2 , demonstrating the cost effectiveness of the EF process for mineralization of OFLO under low current densities [START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF]. This advantage is due to the simultaneous formation of homogeneous ( • OH) and heterogeneous (BDD( • OH)) hydroxyl radicals in the EF process.

Current density / mA cm -2 The values of EC increased with the rising current densities. Under current density of 4.2 mA cm -2 , the value of EC in AO (0.6 kWh (gTOC) -1 ), EF-DSA (0.7 kWh (gTOC) -1 ) and EF-BDD (0.5 kWh (gTOC) -1 ) processes was almost similar (Fig. 7(d)). In contrast, the value of EC in EF-DSA (4.5 kWh (gTOC) -1 ) was more than two times in EF-BDD (2.1 kWh m -3 ) under 16.6 mA cm -2 , demonstrating the cost effectiveness of BDD anodes for organics removal under high current densities. It seems that the contribution of homogeneous • OH is not significant at high 301 current densities since oxidation efficiencies in two processes presented few differences at 16.6 302 mA cm -2 (Fig. 7(c)). However, the value of EC in AO (1.3 and 3.6 kWh (gTOC) -1 ) was much 303 higher than in EF-BDD (0.93 and 2.1 kWh (gTOC) -1 ) under 8.3 and 16.6 mA cm -2 . Energy 304 consumption for OFLO removal from wastewater could be reduced by 28%-41% in EF-BDD 305 compared to AO process. Therefore, a balance need to be found between removal efficiencies 306 and cost of disposal with BDD anode. 307 To test the efficacy of the processes under examinations for the treatment of real wastewater, mineralization of a second wastewater effluent (SWE) collected from Guangzhou Liede sewage treatment plant (Table 1), was examined (Fig. 8). The TOC removal efficiencies were 72%, 64% and 37% for EF-BDD, AO and EF-DSA processes under 16.6 mA cm -2 , respectively, upon 480-min treatment time. Compared with the outstanding mineralization performance of synthetic OFLO solution (Fig. 7c), TOC removal efficiency was diminished for treatment of SWE. The decrease in mineralization efficiency during the treatment of SWE can be explained by (i) the presence of natural organic matter in SWE which could scavenge hydroxyl radicals that reduce the amount of • OH available for oxidation of OFLO and (ii) consumption of • OH in reactions with inorganic ions such as Cl -and HCO3 -present in SWE, with generation of less reactive radicals such as ClOH •-and CO3 •- [START_REF] Hien | Comparison of homogeneous and heterogeneous electrochemical advanced oxidation processes for treatment of textile industry wastewater[END_REF], [START_REF] Yang | Surface-catalyzed electro-Fenton with flexible nanocatalyst for removal of plasticizers from secondary wastewater effluent[END_REF].

Table 1. Quality parameters of secondary wastewater effluent. The relative contribution of homogeneous • OH under 4.2 (54%) and 8.3 (57%) mA cm -2 was higher than BDD( • OH) upon electrocatalysis time of 1 h (Fig. 9). Subsequently, the contribution of • OH to the mineralization gradually slowed down and mineralization rate of OFLO is dominated by BDD( • OH) after electrocatalysis time of 4 h. This result can be explained by two reasons: (i) formation of a large amount of BDD( • OH) on the BDD anode, especially under high current densities and (ii) acceleration of electrons transfer rate leading to the mineralization of carboxylic acids [START_REF] Garcia-Segura | Mineralization of the recalcitrant oxalic and oxamic acids by electrochemical advanced oxidation processes using a boron-doped diamond anode[END_REF], which are resistant to oxidation by • OH. When the current density increased to 16.6 mA cm -2 , the relative contributions of BDD( • OH) reached to 72%, 85%, 92% and 100% upon electrocatalysis time of 1, 2, 4 and 6 h, respectively. These results proved again that high current densities are advantageous to BDD( • OH) production in EF-BDD process. These results were consistent to the similar TOC removal efficiencies in EF-BDD and AO processes under 16.6 mA cm -2 (Fig. 7), where the mineralization of OFLO was dominated by BDD( • OH). The second order rate constant for oxidation of OFLO by • OH is then determined using Eq. ( 17) [START_REF] Camcioglu | Fast and complete destruction of the anti-cancer drug Cytarabine from water by electrocatalytic oxidation using electro-Fenton process[END_REF][START_REF] Mbaye | Rapid removal of fungicide thiram in aqueous medium by electro-Fenton process with Pt and BDD anodes[END_REF].

ln [ ] [ ] = × ln [ ] [ ] (17)
The rate constant for OFLO oxidation by OH can be determined from the slope of the linear was two-fold of the one between 4-HBA and • OH (0.04 min -1 ) and kOFLO was calculated to be 3.86 × 10 9 M -1 s -1 (Fig. 11). Fig. 11 Determination of the absolute (second order) rate constant of oxidation of OFLO by

• OH (kOFLO) using 4-HBA as standard competitor under current of 2.1 mA cm -2 .

3.5. Formation and evolution of short-chain carboxylic acids and inorganic ions in EF-

BDD process

During the mineralization of 0.1 mM OFLO solution by EF-BDD process, six short-chain carboxylic acids were identified and followed upon electrolysis time of 8 h (Fig. 12(a)). Acetic, formic, oxamic and oxalic acids appeared at retention time of 9.2, 12.9, 14. [START_REF] Olvera-Vargas | Electro-Fenton treatment of real pharmaceutical wastewater paired with a BDD anode: reaction mechanisms and respective contribution of homogeneous and heterogeneous OH[END_REF]. These results are in agreement with the TOC decay values obtained in EF-BDD process (Fig. 7) and relative contributions of BDD( • OH) in Fig. 9.

Toxicity assessment of OFLO and intermediates in EF-BDD process

Generally, some intermediates formed during AOPs treatment would be more toxic than the initial organic pollutants, leading to types of environmental problems [START_REF] Dirany | Electrochemical treatment of the antibiotic sulfachloropyridazine: kinetics, reaction pathways, and toxicity evolution[END_REF][START_REF] Brillas | Electrochemical destruction of chlorophenoxy herbicides by anodic oxidation and electro-Fenton using a boron-doped diamond electrode[END_REF]. The transformation products (OTPs) identified during oxidation of OFLO, using HPLC and GC-MS analyses were presented in Table 2. Two cyclic compounds (OTP5 and OTP6) were detected in addition of four main short chain carboxylic acids. In EF-BDD process, defluorination reaction occurred at the first stage of electrooxidation as shown in Fig. 12b.

Meanwhile, piperazinyl ring was attacked by • OH/BDD( • OH) causing dealkylation. Shortchain carboxylic acids generated through progressive oxidative cleavage of aromatic rings and further oxidation of aliphatics structures. Values of LC50, developmental toxicity and bioaccumulation factor of OTPs were calculated by quantitative structure activity relationship predication using Toxicity Estimation Software Tool (T.E.S.T.) software [START_REF] Jiang | Porous 0D/3D NiCo2O4/g-C3N4 accelerate emerging pollutant degradation in PMS/vis system: Degradation mechanism, pathway and toxicity assessment[END_REF]. Acute toxicity expressed in LC50 value of OTP1, OTP2, OTP3 and OTP4 was 60.8, 11.8, 18.1 and 85 times that of OFLO (17 mg L -1 ), respectively, and these OTPs were considered as "not harmful" (Fig. 13(a)). The developmental toxicity of OFLO (1.03) was higher than that of OTP1, OTP2, OTP3 and OTP6 was considered "developmental toxicant" (Fig. 13(b)). Values of developmental toxicity proved that OTP4 and OTP5 were "developmental non-toxicant". The bioaccumulation factor of OTPs (OTP1 (0.6), OTP2 (0.53), OTP3 (0.12), OTP4 (0.17), OTP5 (0.67) and OTP6 (0.57) decreased more than 80% compared with that of OFLO (3.53) (Fig.

13(c))

. In short, the toxicity of OTPs was assessed as much lower than that of OFLO.

Conclusions

The present study explored the reactivity and mechanism of EF-BDD process for removal of the antibiotic ofloxacin from water. High current densities were found to be beneficial to generation of BDD( • OH) because of more converging decay kinetics rates of OFLO in EF-BDD and AO processes. EF-BDD process was more efficient in removing solution TOC compared to AO process due to the simultaneous formation of homogeneous ( • OH) and

Fig. 1 :

 1 Fig. 1: Effect of concentration of catalyst Fe 2+ on oxidation of OFLO by EF-BDD process and corresponding apparent rate constants. Conditions: Current density: 8.33 mA cm -2 , [Na2SO4]: 50 mM, pH: 3.

Fig. 2 :

 2 Fig. 2: (a) Oxidation of OFLO; (b) TOC removal efficiency; (c) Mineralization current efficiency and (d) Energy consumption in EF-BDD, AO and EF-DSA systems. Conditions: Current density: 8.3 mA cm -2 , [Fe 2+ ]: 0.2 mM, [Na2SO4]: 50 mM, pH: 3.

  [START_REF] Jurado | Occurrence, fate and environmental risk assessment of the organic microcontaminants included in the Watch Lists set by EU Decisions 2015/495 and 2018/840 in the groundwater of Spain[END_REF]. The complete oxidative degradation of 0.1 mM OFLO requires 60, 50, 30 and 10 min electrolysis time for 2.1, 4.2, 8.3 and 16.6 mA cm -2 current densities (Fig.3(a)). These results are in agreement with Fig.[START_REF] Hapeshi | Drugs degrading photocatalytically: kinetics and mechanisms of ofloxacin and atenolol removal on titania suspensions[END_REF], highlighting the increase in kapp values of OFLO oxidative degradation along with higher current densities (Fig.4). When the current density increased to16.6 mA cm -2 , the kapp value (0.46 min -1 ) of OFLO oxidation was more than three time that at 8.3 mA cm -2 (0.15 min -1 ). High current densities have the advantage of generating BDD( • OH), while being detrimental to production of H2O2 on cathode due to side reactions. The generation of H2O2 increased with current densities from 2.1 to 8.3 mA cm -2 , while further increased current density (from 8.3 to 16.6 mA cm -2 ) led to a drop of H2O2 accumulation ((Fig.3(b)). The generation of H2O2 reached to 180 mg L -1 under 8.3 mA cm -2 , upon electrolysis time of 100 min, except for those have been activated by • OH. The average rates of H2O2 accumulation were 4.1, 5.8, 6.7 and 5.4 mg h -1 cm -2 under current densities of 2.1, 4.2, 8.3 and 16.6 mA cm -2 , respectively. It can be expected that the accumulation of H2O2

Fig. 5

 5 Fig. 5 Effect of current densities on the apparent rate constants for oxidative removal of OFLO in EF-BDD, AO and EF-DSA processes. Operating conditions: [Fe 2+ ]: 0.2 mM, [Na2SO4]: 50 mM, pH: 3.

Fig. 6

 6 Fig. 6 Synergistic factor (SF) of • OH and BDD( • OH) in EF-BDD and pseudo-first-order rate constant for OFLO removal under 4.2, 8.3 and 16.6 mA cm -2 . Conditions: [Fe 2+ ]: 0.2 mM (EF-BDD and EF-DSA), [Na2SO4]: 50 mM, pH: 3.

Fig. 7 (

 7 Fig. 7 (a-c) Removal efficiency of TOC and (d) values of EC for OFLO removal in EF-BDD, AO 309 and EF-DSA processes under 4.2, 8.3 and 16.6 mA cm -2 . Conditions: [Fe 2+ ]: 0.2 mM (EF-BDD 310 and EF-DSA, [Na2SO4]: 50 mM, pH: 3. 311

2 Fig. 8

 28 Fig. 8 Removal efficiencies of TOC in SWE with EF-BDD, AO and EF-DSA processes under 16.6 mA cm-2. Operating conditions: [Fe 2+ ]: 0.2 mM, [Na2SO4]: 50 mM, pH: 3.

Fig. 9 3 . 4 .

 934 Fig. 9 Relative contributions of homogeneous • OH and heterogeneous BDD( • OH) under 4.2, 344 8.3 and 16.6 mA cm -2 upon electrolysis time of (a) 1 h, (b) 2 h, (c) 4 h and (d) 6 h in EF-BDD 345 processes. Operating conditions: [Fe 2+ ]: 0.2 mM, [Na2SO4]: 50 mM, pH: 3. 346 347

10 and 11 .

 1011 plot of ln ([OFLO]0/[OFLO]t) = f (ln ([4-HBA]0/[4-HBA]t). Electrolysis were carried out under low current (2.1 mA cm -2 ) and short time (30 min) to avoid the interference of the intermediate formed, using same concentration (0.1 mM) of OFLO and 4-HBA. Competitive degradation kinetics and determination of kOFLO are presented in Figs. The apparent rate constant for oxidative degradation of OFLO with • OH (0.08 min -1 )

Fig. 12

 12 Fig. 12 Time-course of identified carboxylic acids (a) and inorganic ions (b) generated during mineralization of 0.10 mM OFLO solution by EF-BDD process. Conditions: [Fe 2+ ]: 0.2 mM, current density: 16.6 mA cm -2 , [Na2SO4]: 50 mM, pH: 3.

Fig. 13

 13 Fig. 13 Determination of acute toxicity (a); developmental toxicity (b) and bioaccumulation factor (c) of OFLO and OFLO transformation products (OTPs). Numbers 1 to 6 represent OTP1 to OTP6, respectively.

  • OH and heterogeneous BDD( • OH) during oxidation of OFLO by EF-BDD process, (iii) explore the synergistic effect of homogeneous

	• OH and heterogeneous BDD( • OH) hydroxyl radicals for mineralization of OFLO in EF-BDD
	process, (iv) elucidate the possible pathway of OFLO degradation and (v) assess acute toxicity
	(LC50), developmental toxicity, bioaccumulation factor and mutagenicity by quantitative
	structure-activity relationship using the Toxicity Estimation Software Tool.
	2.	Materials and methods

2.1. Chemicals and reagents

Carbon felt (CF) and graphite foil were purchased from Beijing Jinglong Special Carbon (Beijing, China), polytetrafluoroethylene was obtained from Shanghai Hesen Electric (Shanghai, China). Methanol, ethanol, sulfuric acid, potassium hydroxide, anhydrous sodium sulfate (Na2SO4; 99 100% purity), H2O2 (30% by weight) and iron (II) sulphate heptahydrate (99.5%) were purchased from Tianjin Kemiou Chemical Reagent (Tianjin, China). Analytical grade OFLO (C18H20FN3O4, MW: 361.1 g mol

-1

, CAS n o : 82419-36-1, 99.9% purity, with chemical structure given in Table

2

), phosphoric acid, acetonitrile, 4-hydroxybenzoin acid, tertiary butyl alcohol and benzoquinone were supplied by Aladdin Chemical Reagent (Shanghai, China). Dimethyl sulfoxide, potassium titanium (IV) oxalate, and 2,4dinitrophenylhydrazine were purchased from Beijing Puyihua Science and Technology (Beijing, China).

Table 2 .

 2 OFLO transformation products (OTPs) generated in EF-BDD process.

	Compound	Molecular mass (g mol -1 ) Molecular structure Retention time (min)
	OFLO	361.1	27.2
	OTP 1 (Acetic)	60	9.2
	OTP 2 (Fumaric)	116	14.2
	OTP 3 (Oxamic)	89	15.5

μm) and with a diode array detector. Analysis was carried out under following conditions:

Acknowledgements

The present study was financially supported by National Natural Science Foundation of China (Nos. 22006053) and Guangdong Basic and Applied Basic Research Foundation (2019A1515110797).

heterogeneous (BDD( • OH)) hydroxyl radicals in the former one. For example, TOC removal efficiency of EF-BDD process under 4.2 mA cm -2 was better than that of AO process under 8.3 mA cm -2 . In EF-BDD process, the synergistic factor of • OH and BDD( • OH) decreased with current density rising from 4.2 (0.28) to 16.6 (-0.08) mA cm -2 . The value of EC in EF-DSA (4.5 kWh (gTOC) -1 ) was more than two times of that in EF-BDD (2.1 kWh m -3 ) under 16.6 mA cm -2 . Evolution of carboxylic acids, inorganic ions (NH4 + , NO3 -and F -) and toxicity assessment in treated solution prove the complete mineralization of OFLO in EF-BDD process.

This study show that EF-BDD process could provide great potential to treatment of antibiotics wastewater, and a balance need to be found between removal efficiencies and energy consumption with BDD anode.

CRediT authorship contribution statement

Declaration of Competing Interest

The authors declare no competing financial interest.


Hervé Morel

Bruno Allard

Sabrine M'rad

Cyril Buttay

Kaicar Ammous

Anis Ammous

Bond Graph Modeling of Current Diffusion in Magnetic Cores

Keywords: Bond Graphs, Variational approximation, Diffusive Representation, Current Diffusion, Maxwell's Equation

The paper presents different modeling approaches that yield a same bond graph representation structure. The first one is based on a variational approximation of the Magnetic Induction Diffusion Equation and supplies various degrees of modeling starting from the simple standard electrical inductor. The second approach is based on the diffusion representation and may be simply connected to numerical simulation of the Maxwell's Equations in Frequency Domain. Finally such approaches should be applied to advanced modeling of Integrated Power Systems that operate at high switching frequency.

INTRODUCTION:

Electrical transformers and inductors are mainly based on magnetic cores. The design of such systems depend mainly on the losses they generate. These losses may be divided into hysteresis losses and current diffusion losses (induced currents), also known as Eddy's currents or Foucault's currents. Only current diffusion losses are addressed in this paper. Current Diffusion [Pérez 1997], [START_REF] Morel | Skin Effect And Bond Graphs[END_REF]] occurs in an electrical conductor or magnetic-conductive material. Under the assumption that the displacement currents are negligible, Maxwell's Equations combined with,

 J =  E (1) yields,   B-  ∂  B ∂ t =0 (2)
where  is the electrical conductivity,  E is the electric field,  B is the magnetic induction field,  J is the current density vector and  is the magnetic permeability.

We have shown [START_REF] Morel | Skin Effect And Bond Graphs[END_REF]] that an order-2 variational approximation of equation (2) yields to a bond graph of the skin effect in a rectangular-section conductor. Equation (2) is the same, but the system is different because here, the magnetic core has not the role to conduct a current: it is only a parasitic effect.

The classical solution consists in solving the Maxwell's equations using numerical methods like finite element methods. An alternative method is the Diffusive Representation [START_REF] Montseny | Diffusive Representation Of Pseudodifferential Time-Operators[END_REF]]. It is a part of the Pseudo-differential operator theory that may be seen as an extension of transfer functions and Laplace's transform for systems of infinite order. Recent studies [START_REF] Laudebat | Modélisation et identification sous représentation diffusive de comportements dynamiques non rationnels en génie électrique[END_REF]] have shown the ability to apply diffusive representation in the modeling of the skin effect, or of a self-inductance having a conductive-magnetic core. In the latter cases, the transfer function is not rational, i.e. it is proportional to the square root of the frequency.

In this paper, a variational approximation of the current diffusion is constructed. Then it is compared with the diffusion representation. Both approaches yield a same bond graph representation.

MODELING CURRENT DIFFUSION IN A MAGNETIC CORE

PDE Modeling

The standard modeling approach is a local formulation of a Partial Differential Equation, PDE.

The Classical solution of equation ( 2) exists for a semiinfinite space. Practical cases correspond to parallelepiped volume as in Fig. 1. The assumption of semi-infinite space is satisfied if the geometrical domain are much larger than skin depth. So, assuming a long and thin magnetic core, i.e. h,a >> b (Fig. 1), the magnetic field does not depend neither on x nor on y. Therefore, the magnetic field depends only on z and on time. Moreover, edges effects are neglected [START_REF] Laudebat | Modélisation et identification sous représentation diffusive de comportements dynamiques non rationnels en génie électrique[END_REF]]. The (oxy) and (oxz) planes are two planes of antisymetry for the distribution, thus the magnetic field is in the direction of the x-axis.

Hence, Maxwell's equations in the magnetic core may be written as

∂ 2 B ∂ z 2 -  ∂ B ∂ t =0 (3)
Fig. 1 The self inductance geometry

F =N I t = ∫ 0 h H dx= h  B 0 (4)
Where F is the magnetic potential difference of the magnetic core, N is the number of turns and I(t) the current flowing through the turns. Faraday's Law on electromagnetic induction enables to write the e.m.f. in one turn as

= v N = d  dt (5)
Where λ is the magnetic flow variable, v is the voltage drop and f is the magnetic flow, given by

=a ∫ -b 2 b 2 B  z dz (6)
Taking into account equation ( 3) and ( 5), the derivation with respect to time of equation ( 6) yields

= d  dt =a ∫ -b 2 b 2 ∂ B ∂ t dz= a   ∫ -b 2 b 2 ∂ 2 B ∂ z 2 dz= a   [ ∂ B ∂ z ] b 2 -b 2 (7)
Equations ( 3) and ( 7) define a boundary value problem. A variational approximation may be applied.

Variational Approximation

Our modeling approach is based on the variational approximation of "Galerkin" [START_REF] Shawalter | Hilbert Space Methods for Partial Differential Equation[END_REF]]. The advantage of this numerical modelling method, known as "meshless", is to avoid the meshing of the structure what allows to decrease the simulation cost.

The idea is applied on the Magnetic Induction Diffusion Equation ( 2) what yields a bond graph model of Eddy's currents in the magnetic core. Indeed, the method consists to build a finite state space model from the variational equation of relation (3). The variational equation is obtained by multiplying the two sides of equation ( 3) by a trial function s(z) then by integrating the result over the range [-b/2, b/2] where b is the thickness of the magnetic core. This yields the following variational equation,

∫ -b/2 b/2 ∂ 2 B ∂ z 2 s dz-  ∫ -b/2 b/2 ∂ B ∂ t s dz=0 (8)
The objective is to seek an expression of the magnetic field B using functions that verify the variational formulation (8) for any trial function s(z). This is equivalent to verify the differential equation (3) against the magnetic field B and thus describes the physical phenomenon considered here.

Spatial distribution of the magnetic field B may be assumed to be symmetrical with respect to z because of the symmetry of the geometry. Notice that this assumption is no longer valid if it exists several magnetic core or other electrical conductive elements in the system. Fortunately, in the later case only second order effects occur. Function s is also assumed to be symmetrical with respect to z, so (7) implies

[ ∂ B ∂ z s  z  ] b /2 -b /2 =s  b 2 [ ∂ B ∂ z ] b /2 -b/2 =s  b 2    a (9)
so using ( 9), ( 8) may be rewritten as

  d dt ∫ -b/2 b/2 B s dz ∫ -b/ 2 b/2 ∂ B ∂ z ∂ s ∂ z dz=s  b 2    a (10)
Which is a variational equation of the boundary value problem (3,7).

In order to resolve the problem (10), we apply a variational approximation to the magnetic field B which consists in a projection of the solution of the problem (10) on a finite-dimension space, i.e,

 B t , z = ∑ j=0 n  j t  w j  z  (11)
A difficult task is to define the decomposition functions, wj. A good choice yields an accurate approximation with a small number of functions.

We have chosen eigen-functions of the steady state operator in (3) as

w i  z =cosh  2iz b  0 ≤i≤n (12)
Indeed, the hyperbolic cosine functions answer correctly to the constraints of the spatial distribution of the magnetic field B: they are symmetrical with respect to z. Classically in variational approximation techniques, the same base of functions wj are chosen for the trial functions s(z). Finally, combining equations (10-12) yields,

2 b ∑ j=0 n  j i j ∫ -1 1 sinh iusinh  judu   b 2 ∑ j=0 n  j ∫ -1 1 cosh iucosh  judu= cosh i     a (13)
Moreover, equation (4) gives the output relation of the model,

F =N I t = h  B 0= h  ∑ j=0 n  j cosh 0= h  ∑ j=0 n  j (14)
Defining,

F i = h   i (15)
Equation ( 14) becomes

F = ∑ j=0 n F j (16)
Equation ( 13) may be rewritten as,

2 b ∑ j=0 n  j M ij d   b 2 ∑ j=0 n j M ij s =cosh i    a ( 17
)
where,

M ij s = ∫ -1 1 cosh iucosh  judu (18) M ij d =i j ∫ -1 1 sinh iusinh  judu (19) defining s 0 =1 , s k = sinh k  k k 0 (20)
the matrix elements may be written as

M ii s =1 s 2 i , M ij s =s i j s i-j i≠ j (21)
and

M ii d =i 2 -1s 2i  , M ij d =ij s i j -s i-j  i≠ j (22)
Notice that M s is symmetrical and

M 0 j d =0 0≤ j≤n and M i0 d =0 0≤i≤n (23)
The first values of the elements of these matrices are

M s = [ 1s 0 2 s 1 2 s 2 ... 2 s 1 1s 2 s 1 s 3 ... 2 s 2 s 1 s 3 1s 4 ... ... ... ... ... ] (24)
and

M d = [ 0 0 0 ... 0 s 2 -1 2s 3 -s 1  ... 0 2s 3 -s 1  4 s 4 -1 ... ... ... ... ... ] (25)
Equations ( 14) and ( 17) define a state space model of the magnetic core. Equation ( 17) may be rewritten as

2 r 0 N d F  P 0 2 N s Ḟ = u 1 (26)
Where

r 0 = a  b h (27)
is a parameter that has the physical dimension of a resistance.

P 0 =  ab h
is a parameter having the physical dimension of a permeance or an inductance. N d is a matrix of coefficients:

N ij d = M ij d cosh i  (28)
N s is a matrix of coefficients:

N ij s = M ij s cosh i  (29)
u1 is the vector of dimension n,

u 1 = [ 1 1 ... 1 ] (30)
Finally F is the state vector whose components Fi are defined in equation ( 16).

F = [ F 0 F 1 ... F n ] (31)
In the case of the self inductance (Fig. 1), variational approximation yields the following finite state space model:

Ḟ = P -1 . u 1 - P -1 . r . F (32) I t = 1 N ∑ k =0 n F k (33)
where :

P= P 0 2 N s r=2 * r 0 N d
This state space model is analyzed in next section.

Bond Graphs of the Obtained Models

The Simplest Model, n=1

If n=1, equation ( 26) yields

P 0 Ḟ 0 =u 1 (34)
Equation ( 34) is the equation of a standard electrical inductor having a permeance P0 . However, such a model does not take into account any induced current effect.

A Simple Model with Current Diffusion, n=2

If n=2, equation ( 17) yields

r⋅F P Ḟ = u 1 ( 35
)
where

P=P 0 [ 1 s 1 s 1 cosh 1 1s 2 2 cosh 1 ] (36) and r=r 0 [ 0 0 0 2 s 2 -1  cosh 1 ] (37)
Equations (35-37) are clearly represented by the bond graph in Fig. 3 Fig. 3 Associated bond graph.

A n-Order Model

The n-order model is defined by equation ( 26). The third order model bond graph is given in Fig. 4. The generalization to a n-order model requires the use of non-standard bond graph representations like those in Fig. 5.

The junction 1 has 2 one-port bonds and a multi-port bond. To satisfy equation ( 26), it means that the flow variable is the same in the 2 one-port bonds and in the multi-port bond.

SABER Simulation Of The developed Variational Models: Temporal Analysis

While considering equations (32-33), we analyse the time behavior of the developed models with n=2 and n=3, for iron and ferrite magnetic cores.

Using SABER simulator [START_REF] Saber | Mast Reference Manual, Analogy, Inc[END_REF]], we simulate the obtained variational models in this simple electrical circuit:

Fig. 6 The electrical circuit of simulation.

E MAGNETIC CORE 'S MODEL Rch i vL C λ GY N F i v C GY:N i v λ F 1 0 F 0 F 1 Rs:s C GY:N i v λ F 1 0 F 0 F 1 Rs:s2 0 Rs:s1 C GY:N i v λ F 1 0 F 0 F 1

Rs:s

The following figures shows the results of the simulations: 0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 0,0040 0,000 0,002 0,004 0,006 0,008 0,010

Ferrite magnetic core: N=3 Ferrite magnetic core: N=2 Iron magnetic core: N=3 Iron magnetic core: N=2

Current [A] Time [s]
Fig. 7 Simulation of the current waveform in iron and ferrite magnetic cores for models with n=2 and n=3. As it is shown in figures 7 and 8, Foucault's currents phenomena are well described by the developed variational model.

The impact of Foucault's currents is visible by comparing the current waveform in the ferrite magnetic core model with the current waveform in the iron magnetic core model. The same thing, if we compare the voltage drop waveform across these two magnetic cores.

Indeed, the magnetic phenomenon is described by a fast dynamics of the voltage drop across the magnetic core at the beginning, due to the appearance of the Eddy currents. These currents, opposed to the magnetic flow, push the magnetic field outside the magnetic core. When the dynamic phenomena slow down. There is a progressive disappearance of the Foucault's currents that allows a progressive and slow diffusion of the magnetic induction inside the core. Thus, a phenomenon of " long memory " appears. Apparent inductance of iron is lower than in the case of ferrite where in the absence of Eddy currents, induction is distributed uniformly trough the section.

Moreover, it is clear on figures 7 and 8 that if we increase the order of the variational model, current and voltage drop waveforms across the iron magnetic core are improved.

Frequency Analysis:

The Fig. 9 shows the comparison of the normalized inductance solution of the variational model with n=2 and the classical solution given by [START_REF] Laudebat | Modélisation et identification sous représentation diffusive de comportements dynamiques non rationnels en génie électrique[END_REF]]

L r = L L 0 = tanh   i  r   i  r (38)
where " i " is the complex operator,

L 0 = N 2 ab  h =N 2 P 0 (39)
is the self-inductance of the magnetic core at lowfrequency, and

 r =   0 (40)
is the normalized frequency with

 0 = 4 b 2   =4 r 0 P 0
It is clear that the self-inductance decreases when the frequency increases. This is due to the Eddy's current.

Fig. 9 Modeling the magnetic core -normalized inductance with and without current diffusion modeling.

Diffusive Representation Model

In the case of the self inductance (Fig. 1), diffusive representation yields the following infinite order. This is an exact model [START_REF] Laudebat | Modélisation et identification sous représentation diffusive de comportements dynamiques non rationnels en génie électrique[END_REF]].

d  k dt = -4 k 2  k b 2   v K = 0,1,2, ... ∞ (41) I t = h N 2  a b ∑ k =0 ∞  k (42) Defining F k = h N  ab  k N ij d =i 2  ij (43) and N ij s = ij (44)
Equation ( 41) and ( 42) are equivalent to (32) and ( 33). So, the diffusive representation yields to the same bond graph (Fig. 5). Differences exist in the matrix values. They are due to different assumptions, i. e. equations ( 41) and ( 42) are based on the choice of residue values of the diffusive representation symbol [START_REF] Laudebat | Modélisation et identification sous représentation diffusive de comportements dynamiques non rationnels en génie électrique[END_REF]]. The main interest of the diffusive representation, is the ability to build a finite-order approximated state space model from a frequency domain representation. The frequency domain representation may be obtained numerically, by classical FEM simulations.

CONCLUSION

The paper presents different modeling methods that yield to the same bond graph structure of the model of the current diffusion in magnetic core. The first approach yields a first order model, the standard electrical inductor model. The second order takes into account current diffusion (induced currents). The N-order model equations have the same structure than the diffusive representation model. However diffusive representation enables an efficient coupling with powerful numerical simulators based on FEM and should yield an accurate bond graph model of the current diffusion. The approach may be extend to any phenomenon based on a diffusion equation.

  's Theorem on the loop represented in mixed line in Fig.1 yields,

Fig. 4

 4 Fig. 4 Bond Graph of the 3 rd order model.

Fig. 2

 2 Fig. 2 Standard model of the electrical inductor

Fig. 8

 8 Fig.8Simulation of the voltage waveform across an iron and a ferrite magnetic cores for models with n=2 and n=3.