Maxime Touffet

Mohamed Hatem Allouche

Mostapha Ariane

Olivier Vitrac
email: olivier.vitrac@agroparistech.fr

Coupling between oxidation kinetics and anisothermal oil flow during deep-fat frying

Keywords: Heat and mass transfer, oil oxidation, CFD, anisothermal oil flow, frying

 description of oil flow and temperature distribution in kitchen appliances during heating and holding times;  simplified descriptions of the generation and decomposition of hydroperoxides from triacylglycerols as a mixture of fatty esters in frying conditions;  coupling between Eulerian and Lagrangian descriptions to analyze the effects of residence times at different temperatures;  validation experiments with a modified and instrumented deep-fryer and reference decomposition kinetics in anoxia.

I. Introduction

Deep-frying consists of immersing in a rich-water product in a liquid with a high boiling point liquid, usually a vegetable oil (denoted oil in the rest of the article). Oil is an ingredient incorporated within the finished product during cooling [START_REF] Touffet | Revisiting the mechanisms of oil uptake during deepfrying[END_REF][START_REF] Vauvre | Multiscale modeling of oil uptake in fried products[END_REF] and the heat transfer fluid. Oil is a high Prandtl number fluid enabling intense heat transfer coefficients by natural convection (i.e., in the vertical direction). As a result, fried products (French-fries, potato chips, chicken pieces and fingers, fish fillets, shrimp) are hot and crispy on the outside and cooked safely in the center (low risk of residual bacteria contamination). Due to the cycle of production of fried products in kitchen appliances, the residence time of oil within batch deep-fryers (several hours after repeated heating cycles) is much longer than frying time (indicatively from five to twenty minutes). Oil abuse by thermo-oxidation due to oxygen exposure at high temperatures can lead to various toxic compounds, with possible health concerns [START_REF] Schneider | Routes to 4-hydroxynonenal: Fundamental issues in the mechanisms of lipid peroxidation[END_REF]. A similar matter exists for Maillard reactions with the apparition of carcinogenic acrylamide in contact with oil at high temperature [START_REF] Mottram | Acrylamide is formed in the Maillard reaction[END_REF]. Though no regulations have been established in the United States, the apparition of oxidation products is strongly regulated in the EU [START_REF]2158 of 20 November 2017 establishing mitigation measures and benchmark levels for the reduction of the presence of acrylamide in food[END_REF].

Vegetable oils are mixtures of triacylglycerols (TAG), which can be described as three fatty acids esterified onto the same glycerol backbone. Fatty esters (FE) naturally exhibit a variable number of carbon-carbon double bonds (unsaturations), which depend on plan variety [START_REF] Gunstone | Vegetable Oils in Food Technology[END_REF].

Prevalent mono-, di-, and tri-unsaturated FE are the oleate (C18:1), linoleate (C18:2), and linolenate (C18:3). Unsaturated fatty esters are desirable in the diet with the following preference ratios 1:4:1 for FE with unsaturation in the n-6, n-3, n-9 position (also called ω-6, ω-3, ω-9 FE), respectively [START_REF] Asif | Health effects of omega-3,6,9 fatty acids: Perilla frutescens is a good example of plant oils[END_REF][START_REF] Simopoulos | The importance of the ratio of omega-6/omega-3 essential fatty acids[END_REF]. The positions are counted from the methyl end. These unsaturations are brought in common frying oils by linoleate (n-6), linolenate (n-3), and oleate (n-9), respectively. For example, these three fatty esters represent ca. 90% of FE in sunflower oil. They are also the most sensitive to thermo-oxidation. The complex chemistry of thermooxidation leads to a dramatic loss of nutritional value by a displacement of double bonds (conjugated hydroperoxides), the apparition scission products (e.g., volatile aldehydes and esters [START_REF] Frankel | Volatile lipid oxidation products[END_REF]) responsible for off-flavors and toxic compounds (e.g., 4-HNE [START_REF] Schneider | Routes to 4-hydroxynonenal: Fundamental issues in the mechanisms of lipid peroxidation[END_REF]), the production of cross-linked polymers responsible of fouling.

The first stages of the thermo-oxidation process are controlled by the kinetics of oxygen dissolution and intermediary products' transport through the oil bath [START_REF] Touffet | Online reconstruction of oil oxidation kinetics and reaction schemes during deep-frying by deconvolution of ATR-FTIR spectra[END_REF][START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]. Similar effects have been shown in thermoplastics, where oxidation and polymer damages start from the external surfaces and the most exposed to atmospheric oxygen [START_REF] François-Heude | Impact of oxygen transport properties on the kinetic modeling of polypropylene thermal oxidation. II. Effect of oxygen diffusivity[END_REF]. In simple words, mineral oxygen from the atmosphere is converted at the oil-air interface into chemically bounded oxygen (hydroperoxides). Hydroperoxides are thermally unstable products whose endothermic decomposition (i.e., activated by increasing temperatures) generate either scission products (products responsible for the typical smell of fried products) or alkyl radicals, very reactive with oxygen. When the latter dominate, the oxidation process propagate in chain between the surface, the bulk, and the heating elements. In most of the EU member states, it is considered by law that frying oil should be discarded as soon as total polar compounds (TPC) exceed 25% in mass or a value similar. Polar compounds are defined as any species incorporating oxygen atoms beyond the initially present esters. Therefore, the legal oil lifetime depends on the rate of renewal of radicals responsible for the production of hydroperoxides and their subsequent decomposition in hot regions. As fried foods represent a significant fraction of the diet in Western and Asian countries, there is an urgent need to identify the possible strategies to minimize oil abuse and reduce oil wastes. Prior research focused on the reduction of oil uptake in fried products [START_REF] Touffet | Revisiting the mechanisms of oil uptake during deepfrying[END_REF][START_REF] Vauvre | Multiscale modeling of oil uptake in fried products[END_REF]. While improving the dietary balance, this strategy reduces consumer exposure to oxidized products but does not explicitly address the loss of nutritional value in fried products. Little work has been done to relate the design of batch deep-fryers with oxidation kinetics. Pioneer works [START_REF] Rock | Factors Affecting the Rate of Deterioration in the Frying Qualities of Fats. 2. Type of heater and method of heating[END_REF][START_REF] Rock | Factors Affecting the Rate of Deterioration in the Frying Qualities of Fats. 1. Exposure to air[END_REF] in the sixties showed that substantial gain could be achieved by updating the surface-to-volume ratio of oil and oil flow distribution in the bath.

The current study addresses the lifetime hydroperoxides transported by the natural convection in the cavity of a kitchen appliance deep-fryer. We chose as reference a deep-fryer design including a cold zone below the submerged heating electrical resistor. Their decomposition is highly endothermic and is the root cause for all the inconveniences encountered during deepfrying: fouling, bad smells, reduced lifetimes, the apparition of toxic compounds. Coupling an anisothermal flow with a chemical reaction requires special treatment, as the molecular carrier is a TAG carrying different chemical functions with variable sensitivity to oxidation. This work

shows as a Lagrangian description can reconcile the models based on mass balance on chemical functions and transport by an anisothermal flow.

The paper is organized as follows. Since deep-frying involves temperatures above the boiling point of pure water, section 2 justifies and details the oxidation model of edible oil between 80°C and 180°C and the formulation of hydroperoxide decomposition along streamlines. Our simulations cover the heating and holding periods, which is 50% to 90% of the deep-fryer's operation time. Indeed, the heat flux density between the oil and the frying product is so high (up to 150 kW⋅m -2 [START_REF] Vitrac | A method for time and spatially resolved measurement of convective heat transfer coefficient (h) in complex flows[END_REF][START_REF] Vitrac | Continuous measurement of convective heat flux during deep-frying: validation and application to inverse modeling[END_REF]) that a batch deep-fryer cannot operate a steady-state. Most of the operation time is used to recover the set temperature (maximum temperature 175°C according to [START_REF]2158 of 20 November 2017 establishing mitigation measures and benchmark levels for the reduction of the presence of acrylamide in food[END_REF]). The experimental conditions used to parameterize the hydroperoxides' decomposition model in isothermal conditions and to validate it in anisothermal ones are detailed in Section 3.

Simulated results and the coupling between the transport and the degradation are analyzed in Section 4. Finally, the last section summarizes the main findings and the directions to keep oil healthy during deep-frying.

II. NUMERICAL METHODS

A.

Combinatorial oxidation reaction kinetics

Principles

Triplet dioxygen is the common allotrope of oxygen in our environment. Because it carries two unpaired electrons, occupying two  molecular orbitals, it is a stable diradical, denoted OO R in Eq. ( 1)). This step between a radical and a stable species preserves free-valence and propagates it from oxygen to carbon atoms. The first oxidation products of fatty esters are hydroperoxides LOOH and radicals L  .

In the context of deep-frying, the chain reaction's initiation occurs mainly by the continuous regeneration of L  radicals at high temperatures [START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]. Hydroperoxides are, indeed, unstable above 100°C [START_REF] Tobolsky | Low Temperature Autoxidation of Hydrocarbons: the Phenomenon of Maximum Rates1,2[END_REF] and decompose endothermically by homolytic scission of LO-OH bond (bond energy between 100 and 160 kJ⋅mol -1 ) [START_REF] Schaich | Challenges in Elucidating Lipid Oxidation Mechanisms. When, Where, and How Do Products Arise ?[END_REF].

Bolland and Tobolsky [START_REF] Tobolsky | Low Temperature Autoxidation of Hydrocarbons: the Phenomenon of Maximum Rates1,2[END_REF][START_REF] Bolland | Kinetic studies in the chemistry of rubber and related materials. VII. The mechanism of chain propagation in the oxidation of polyisoprenes[END_REF][START_REF] Bolland | Kinetic studies in the chemistry of rubber and related materials. II. The kinetics of oxidation of unconjugated olefins[END_REF][START_REF] Bolland | Kinetic studies in the chemistry of rubber and related materials. III. Thermochemistry and mechanisms of olefin oxidation[END_REF] pioneered the first general kinetic description of hydroperoxides in unsaturated materials (lipids, thermoplastic polymers). Depending on whether hydrogen bonds stabilize a cage mechanism (see Ref. [START_REF] Śliwiok | The influence of hydrogen-bond association on the destruction of hydroperoxides in the autoxidation process of oleyl alcohol, oleic acid, and methyl oleate[END_REF]) or not, their decompositions are considered either unimolecular [START_REF] Nishiike | Stability of Linoleic Acid Hydroperoxide in the Oil System[END_REF][START_REF] Kawano | Degradation of fats and oils by autoxidation VI : Production and decomposition rates of hydrperoxide during the autoxidation of trioleil[END_REF][START_REF] Hatate | Autoxidation rates of oleic-acid and oleates[END_REF][START_REF] Shin | A Mechanism for The hydroperoxyde Decomposition in a Soybean oil during Thermal Oxidation[END_REF][START_REF] Erkilla | Study of the accelerated oxidation of low and high erucic rapeseed oil[END_REF][START_REF] Pokorny | Lipid Oxidation. Part.1. Effetc of Free Carboxyl Group on the Decomposition of Lipid Hydroperoxide[END_REF][START_REF] Privett | Autoxidation and autoxidative polymerization[END_REF][START_REF] Stannett | Autoxidation of 1,4-Dimethylcyclohexane[END_REF][START_REF] Thomas | The Thermal Decomposition of Alkyl Hydroperoxides1[END_REF][START_REF] Bateman | Hydroperoxide decomposition in relation to the initiation of radical chain reactions[END_REF][START_REF] Bell | Reactions of free radicals associated with low temperature oxidation of paraffins[END_REF][START_REF] Stannett | The Kinetics of the Decomposition of Tertiary Hydroperoxides in Solvents1,2[END_REF] or bimolecular [START_REF] Stannett | Autoxidation of 1,4-Dimethylcyclohexane[END_REF][START_REF] Bateman | Hydroperoxide decomposition in relation to the initiation of radical chain reactions[END_REF][START_REF] Stannett | The Kinetics of the Decomposition of Tertiary Hydroperoxides in Solvents1,2[END_REF][START_REF] Bateman | The thermal decomposition of cyclohexenyl hydroperoxide in hydrocarbon solvents[END_REF]. The molecularity of the decomposition remains an open question (see the discussion in Ref. [START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]) in frying conditions, but first-order decay has been mainly observed [START_REF] Nishiike | Stability of Linoleic Acid Hydroperoxide in the Oil System[END_REF][START_REF] Kawano | Degradation of fats and oils by autoxidation VI : Production and decomposition rates of hydrperoxide during the autoxidation of trioleil[END_REF][START_REF] Hatate | Autoxidation rates of oleic-acid and oleates[END_REF][START_REF] Kawano | Autoxidation Rate of Methyl Oleate Diluted by Methyl Laureate[END_REF][START_REF] Hatate | Degradation of fats and oils by autoxidation III -Production and decomposition rates of hydroperoxide during the autoxidation of ethyl oleate[END_REF][START_REF] Mochida | Effects of Metal Chlorides of Periodic Group V, VI, and VII on the Autoxidation of Methyl Oleate and the Decomposition of Methyl Oleate Hydroperoxide[END_REF][START_REF] Mochida | Effects of oxides and Chlorides of Metal of Periodic Groups 5, 6 and 7 on the Autoxidation of Methyl Oleate abd the Decomposition of Hydroperoxides of Methyl Oleate[END_REF]. In this work, unimolecular decomposition was chosen accordingly. The detailed mechanism reads [START_REF] Schaich | Challenges in Elucidating Lipid Oxidation Mechanisms. When, Where, and How Do Products Arise ?[END_REF][START_REF] Privett | Autoxidation and autoxidative polymerization[END_REF][START_REF] Bateman | Olefin oxidation[END_REF]: They are not equivalent as their abstraction energies vary from 313 kJ•mol -1 down to 209 kJ•mol -1 for mono-(i.e., adjacent to one single unsaturated carbon atom) and bi-allylic (i.e., adjacent to two unsaturated carbon atoms) hydrogen respectively [START_REF] Choe | Chemistry of Deep-Fat Frying Oils[END_REF]. The relative oxidability of C18:0 (stearate, no unsaturation), C18:1, C18:2 and C18:3 is ranked 1:11:114:179 [START_REF] Brodnitz | Autoxidation of Saturated Fatty Acids. A Review[END_REF].

Combining Eq. ( 1) and transport during deep-frying can be performed either by introducing a mass balance on the different types of labile protons and subsequent products or by introducing a mass balance on the different chemical functions held by the different unsaturated FE. Both balances are equivalent when the formation of di-, tri-, and cyclic hydroperoxides is discarded.

This assumption was considered acceptable in the normal conditions of frying. Discarding FE oxidized twice or three times facilitated the reconstruction of the profile of hydroperoxides from real experiments and the design of a thermal activation model. This level coarse-graining process was chosen in this study. . codes for C18:2). For the same FE carrier, possible hydroperoxides differ from the position of the hydroperoxide function (OOH) and the configuration of the substituent groups (cis or trans, see [START_REF] Frankel | Hydroperoxide formation[END_REF][START_REF] Frankel | Stereochemistry of olefin and fatty-acid oxidation .3. The allylic hydroperoxides from the autoxidation of methyl oleate[END_REF][START_REF] Frankel | Stereochemistry of olefin and fatty-acid oxidation .1. Autoxidation of hexene and hepta-2,5-diene isomers[END_REF]). j is the isomer index in the delta nomenclature system [START_REF] Rustan | Fatty Acids: Structures and Properties[END_REF] [45]: autoxidation at 95°C, [START_REF] Frankel | Stereochemistry of olefin and fatty-acid oxidation .1. Autoxidation of hexene and hepta-2,5-diene isomers[END_REF]: autoxidation at 75°C, [START_REF] Frankel | Hydroperoxide formation[END_REF]: autoxidation at 80°C. † Roman numbers represent the carbon numbers according to the delta nomemclature system 216 [START_REF] Rustan | Fatty Acids: Structures and Properties[END_REF].

Coding the lineage of hydroperoxides

12

The proposed coarse-graining level recasts Eq. ( 1) into four populations of FE, denoted

  1 1 1 ,, i jk ,   2 2 2 ,, i jk ,   3 3 3 ,, i jk ,   4 4 4
,, i jk ) :

d i,j,k b 1 1 1 1 1 1 b 2 2 2 2 2 2 o p i , , 3 3 3
3 3 3 k 01 i,j,k i,j,k k 02 i,j,k i ,j ,k i,j,k i ,j ,k k' 03 i ,j ,k 2 i ,j ,k k 04 i,j,k i,j,k k 05 i,j,k i ,j ,k R : L OOH L O + OH R : L O +L H L OH+L R : HO +L H H O+L R : L + OO L OO R : L OO +L H jk                   3 3 3 t1 t2 t3 i,j,k i ,j ,k k • 06 i,j,k i,j,k k 07 i,j,k i,j,k k 08 i,j,k i,j,k L OOH+L R : L OO +L OO Termination products R : L OO +L Termination products R : L +L Termination products           (2)
The combinatorial description propagates the lineage iteratively from initial FE to subsequent products. Reaction rates constants

,, i j k d k and ,, i j k p k
need to be identified from carefully chosen experiments, accounting for the thermal stability of each considered hydroperoxide [START_REF] Frankel | Hydroperoxides[END_REF] and the lability of hydrogens in considered unoxidized FE [START_REF] Fatemi | Analysis of oleate, linoleate and linolenate hydroperoxides in oxidized ester mixtures[END_REF]. Though TAG carriers were discarded in the model, preserving FE enables direct comparison with experiments with pure FE, pure oils (TAG mixtures), and oil blends.

B. 3D unsteady CFD simulations of turbulent natural convection in a batch deepfryer 1. Governing equations

The buoyancy-driven oil flow in the deep-fryer was simulated under the Boussinesq approximation by assuming a constant isobaric thermal expansion coefficient:

1 P T          ( 3
)
where  is the oil density and T is the oil temperature.

The equations of continuity, momentum, and energy are:

        0 0 1 p p t T dT C k T t T d                                v vv v vg (4)
where v , p and T are the velocity, pressure, and temperature fields, respectively. The dynamic viscosity,  , the heat capacity, p C , and the thermal conductivity,  , were allowed to vary with temperature as tabulated in Table 1. g is the gravitational acceleration. No-slip was set for the flow at all boundaries: 0  v [START_REF]2158 of 20 November 2017 establishing mitigation measures and benchmark levels for the reduction of the presence of acrylamide in food[END_REF] At the walls of the tank (1.2 mm thick stainless steel for the vertical walls) 2  and in agreement with temperature and flux measurements, an effective heat loss by conduction, convection, and radiation (linearized temperature dependence) was applied:

  2 2 T h T T         n (6)
At the free surface with air ( 1  ), a slip boundary condition was applied. Erasing the vertical component of velocity is an acceptable approximation, but, in the real-life, heating plumes at the vertical of the heating elements make the surface rough and turbulent even in the absence of food products. These effects increase the oil surface area exposed to air, and therefore oxygen. A heat loss by convection was enforced at the free surface. Eqs. ( 6) and ( 8) assuming that the surrounding temperature, T  , is kept at 25°C.

 

1 1 1 0, 0 p                v n v t n (7)
 

1 1 T h T T         n (8)
Heat is transferred to oil via a three-coil sheathed tubular heating element (heating surface area:

          Φ n (9)
The frying temperature

Numerical resolution

Eq. ( 3)-( 9 for first-order kinetics, the variation of hydroperoxide content is also related to the distribution of residence times at different temperatures for considered trajectories:

  ,, i,j,k i,j, , k i,j,k , ,, 1 L OO L H L OOH OOH p p i j k p pp d x y z x y z d t dt d k d d T d                    U θ (10) where ,
    ,, ,, i,j,k ,, ,, , , i,j,

D.

Validation of an activation model for the decomposition of ,,

i j k

L OOH in mixtures in isothermal conditions

According to Table 1 and Eq. ( 2), sixteen different hydroperoxides may be decomposed according to sixteen different rate constants. All individual values cannot be determined experimentally because the different i,j,k L OOH cannot be produced independently, even from a pure FE. Reconstructing their concentrations from TAG or FE initial composition is an option but at the expense of adding unknown production or renewal rates.

A feasible approach was followed by tracking only the likeliest hydroperoxide isomers of each FE, as supported by qualitative experiments on pure FE. Since hydroperoxides from oleate, linoleate, and linolenate exhibit different thermal stability [START_REF] Frankel | Hydroperoxides[END_REF][START_REF] Morales | Olive Oil Oxidation[END_REF], the index i was kept.

Among stereoisomers, trans-hydroperoxides, conjugated or not, are more stable and prevail at high temperatures, so that only the case

        3 i 1 LOOH L OOH tt total i   (13)
Since hydroperoxides were obtained during or after oxygen exposure, the corresponding   k were fitted from experimental data in the presence of oxygen (no fitting required in anoxia) [START_REF] Mallégol | Long-term behavior of oil-based varnishes and paints. Fate of hydroperoxides in drying oils[END_REF]. Predictions were highly acceptable at 80°C and above, i.e. in the frying temperature range.

At 40°C, the time to reach a steady-state (maximum of     LOOH t total ) was underestimated, and the effective decomposition rate was overestimated (see Figure 3e-f). At ambient temperatures, hydroperoxides' stabilization by hydrogen bonds would favor bimolecular decompositions (cage-effect mechanism) instead of unimolecular ones.

The preliminary validation in isothermal conditions generalized to mixtures oxidation schemes proposed for thin films or homogeneous conditions (see Ref. [START_REF] Mallégol | Long-term behavior of oil-based varnishes and paints. Fate of hydroperoxides in drying oils[END_REF]). The remainder of the study will extrapolate previous results from isothermal to anisothermal conditions by introducing an activation energy for the decomposition of each i L OOH . As the corresponding activation entropies capture the decomposition's molecularity, only reference kinetics above 80°C were subsequently considered.

III. EXPERIMENTAL METHODS

A.

Oil and fatty acid methyl esters

Fresh sunflower oil (SO) was supplied directly by the Lesieur's refinery production plant of Coudekerque (France). Methyl oleate (Alda Aesar, purity of 96%), methyl linoleate (Acros Organics, purity of 99%), and methyl linolenate (Sigma Aldrich, purity of 99%) were purchased.

B.

Deep-fryer

The reference deep-fryer used in this study was a commercial kitchen appliance (model Semi-Pro, SEB, France, oil capacity 4 L, 2052W) including an immersed heating element (surface area: 222.7 cm²). A second deep-fryer with an alternative design was also considered to highlight the effect of design on the decomposition of hydroperoxides and indirectly on oil lifetime.

C. Bubbling reactor

Reaction rates of hydroperoxide decomposition of pure FE were determined in a miniaturized bubbling reactor (vertical tube reactor of capacity 30 mL), based on the principles as the larger one described in Ref [START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]. The FE temperature was controlled by immersing the glass reactor within a large secondary oil bath (0.5 L, heating power 600 W) equipped with a PID temperature-controller. Dry air or nitrogen was injected from the bottom of the test tube with the help of a Pasteur pipette to reproduce aerated or anoxia conditions, respectively.

D. Determination of hydroperoxide content

The total hydroperoxide content was determined experimentally by differential scanning calorimetry (DSC) according to the method described in Ref. [START_REF] Mallégol | Thermal (DSC) and chemical (iodometric titration) methods for peroxides measurements in order to monitor drying extent of alkyd resins[END_REF]. A small oil aliquot (10-20 mg) was heated from 25°C to 250°C at a rate of 3°C/min. Since the endothermic decomposition of hydroperoxides in the pan occurs in anoxia, the formed radicals recombine together exothermically into stable products. The total amount of hydroperoxides was quantified by assuming an effective enthalpy of decomposition of 320 kJ/mol.

E.

Temperature measurements

Local temperatures in the oil bath were recorded with thermocouples (type K, TC SA, France) connected to a data acquisition system (Ni cDAQ-9178 and module NI 9213, National Instruments, Austin, Texas, USA) at a frequency of 3 Hz. Temperatures at the surface of the oil bath were imaged with a thermal camera (model Ti9, Fluke, USA) with a emissivity of 0.95.

IV. Results and Discussion

A.

Parametrization of the hydroperoxide decomposition kinetic model

The assumption of first-order decomposition kinetic of all i L OOH was tested on pure FEs, including seven kinetics of oleate hydroperoxide (ranging from 80°C and 180°C), five kinetics of linoleate hydroperoxides (ranging from 120°C to 180°C), and seven kinetics of linolenate Decomposition rate constants were fitted with an Arrhenian activation model:

  0 exp with i=1,2,3 i i di Ea k T k RT      (14)
where 0 i k is the pre-exponential factor, i Ea the activation energy, R the gas constant, and T is absolute temperature. The distributions of activation energies and pre-exponential factors due to uncertainties were estimated by bootstrap (1000 samples). It was particularly noticeable that all hydroperoxides' decomposition rates looked similar at 180°C whereas they were different at 80°C. It is, therefore, hinted that all FE hydroperoxides propagate oxidation at the highest frying temperature, whereas only the most labile ones do during initial heating.

B.

CFD results

1.

Validation during the initial heating period

Simulated 3D temperature and velocity fields during the heating step were recorded with a temporal resolution of ~54 s. They are plotted along three principal planes of the deep-fryer in Figure 5. Along the smallest vertical cross-section XZ, the four heating element branches generate four ascensional plumes starting as shown at t=54s. Two convections cells are fully developed after one min and are maintained throughout the whole heating period (up to 542 s).

The circulation operates mainly within the plane XZ without a significant mixing in the plane YZ. The ascensional sections are small and associated with ascensional velocities up to 2.8 cm•s -1 . Descent regions are located along the walls are thicker with velocities up to 4 mm•s -1

The region below the heating element remains stagnant with delayed heating. Once the set temperature was reached after ca. 9 min, surface heterogeneities decreased down to 5°C. During 10 min of simulation, steady-state (flow and temperature) was reached above the heating element, but it would have required 2 hours to reach steady-state below the heating element.

Analysis of the turbulence

Oil flew as smooth plumes at the vertical of the heating coils before spreading in multiple whorls and eddies. Large-scale turbulent patterns were visually revealed by the change of the refraction index and the displacements of solid particles (see, for example, Figure 9b). The flow was asymmetrically structured between ascending and descending flows. Hot plumes rose rapidly in an almost laminar fashion before spreading rapidly along the shortest cross-section.

The descent was conversely more turbulent and dissipated progressively in the entire volume.

Turbulence on small scales was analyzed in Figure 8 from the temporal fluctuations of velocities and temperatures at four typical positions in the deep-fryer (reported in Figure 2). Although the entire system was out of equilibrium during heating, velocity fluctuations were thought ergodic enough on short and intermediate length scales so that Taylor's hypothesis of , in agreement with stratified turbulence [START_REF] Bolgiano | Turbulent spectra in a stably stratified atmosphere[END_REF][START_REF] Basu | Kolmogorov or Bolgiano-Obukhov scaling: Universal energy spectra in stably stratified turbulent fluids[END_REF] The stratification was exemplified by fluctuations of velocities and temperatures three magnitude orders lower below the heating element (bottom position).

The flow described with a deep-fryer under constant heating differed from the classic configuration of Rayleigh-Bénard natural convection cells [START_REF] Kaczorowski | Analysis of the thermal plumes in turbulent Rayleigh-Bénard convection based on well-resolved numerical simulations[END_REF][START_REF] Trias | Spectrally-Consistent Regularization of Navier-Stokes Equations[END_REF]. In particular, fluctuations did not decrease at the vicinity of the free surface and did not increase at the deep-fryer center.

The differences arose from the heating element's geometry covering only a small fraction of the deep-fryer's bottom, the heat loss at the surface. The presence of a cold zone but not isothermal below the heating element also affected the gap between the vertical walls and the heating element. The contribution of heating on temperature fluctuations at the centre of the deep-fryer were analysed by plotting the windowed mean square of experimental fluctuations at different time scales, 1/ f  

, in Figure 8c. The passage of plumes is visible on short time scales with an external envelope decreasing exponentially according to two times scales: 28 s and 127 s. The fastest decay was associated with the oscillations of plumes and the longest to vertical mixing. Fluctuations were lower than ± 2.1°C at steady state and were independent  .

C. Description of oil flow at steady state

At steady state, the natural convection is fully developed in the entire oil bath, as shown in Figure 9. The spatial distribution of residence times was estimated on a regular grid of elementary volumes r V from mass flow considering a compressible flow according to the following equation:

            1..6 ,, ,, , , , , H , , r ff f
f V r f t x x y z xy y z dx y z x y z x z y d d dz                U n U n (15)
where  

,, x y z U is the velocity field at positions   ,,

x y z ,   Local residence times shown in Figure 9a were calculated with a spatial resolution of 1.7 mm.

They highlighted the presence of a central chimney were oil transited rapidly. Its presence was confirmed by the direct observation of the flow (Figure 9b). The tracked particle shows that the chimney compartmented recirculation loops on both sides of the cross section. Dwelling times below the heating element reaches several minutes up to almost one hour with temperatures 80°C lower than those observed above. As first approximation and in agreement with the overrepresentation of cold surfaces, residence times were significantly greater in cold regions than in cold ones. Non-uniform residence times correlated with non-uniform temperature distributions would extend the lifetime of hydroperoxides and confirmed our initial intuition about the likely relationship between deep-fryer design and oil abuse.

D. Coupling between anisothermal oil flow, transport and hydroperoxide decomposition in the deep-fryer

The transport of hydroperoxides from one region to another one was simulated at steady state using a Lagrangian formulation. This description enables to simulate transport on time scales (several hours) much longer than those use to simulate the flow velocity at steady state ( ,, steady x y z

U

). The trajectory of fluid particle " " containing hydroperoxides was integrated explicitly over time, where the displacement α Δr during t  is given by:  depending on lag but not on time t . From Eq. 12.148 of Ref. [START_REF] Kundu | Fluid Mechanics[END_REF] it reads:

  , , , , 2 ,
  2 ,, , 22 ,, , 0 21
y x y z x y z t xz t d t              u Δr (17) where 2 ,
In the central part of the deep-fryer where the flow rate was the highest, correlation time 0,0,0  was about 24±8 s. As a result, choosing

0,0,0 0.1 s t   
justified that the contribution of velocity fluctuations could be described as random ballistic trajectories (i.e., proportional to time). The decomposition of hydroperoxides was simulated using a root mean square

1/2 2 , , , , 0.025
x y z x y z  uU (i.e., 95% of fluctuations ±5%), enabling smooth trajectories of particles as experimentally observed in Figure 9b, while enabling particles to explore the entire bath.

As a scalar field, temperature fluctuations are passively affected by the turbulent velocity field.

They decorrelated, therefore, almost similarly as velocity on time scales, t  (see Figure 9b), where L is the largest eddy scale, typically half-width. On time scales commensurable to t  , several complications complicated the sampling of thermal fluctuations as reported in Refs.

[ [START_REF] Mestayer | Local anisotropy in heated and cooled turbulent boundary layers[END_REF][START_REF] Gibson | Structure of sheared turbulent fields[END_REF][START_REF] Holzer | Turbulent mixing of a passive scalar[END_REF][START_REF] Pumir | A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient[END_REF] In liquids with high Prandtl numbers as oil, intense sources and intermittently rising plumes were responsible for strong memory effects, including the persistence of temperature fronts and profiles mimicking the shape of heating elements (see Figures 5 and9). In the sake of simplicity, turbulent variations of temperatures were introduced in our Lagrangian formulation by authorizing the visited temperature The outside branches followed the shape of the oil bath and the symmetry planes while avoiding the inner part of the loops. This configuration maximizes the exposure to high temperature (path along the heating resistor) and oxygen (path along the free surface). Mixing times were evaluated using the concept of random walks in Figure 10m and defined as the average mixing time ( ) required to get a mean-square displacement (MSD) equal to the surface area of the explorable section.

Mixing was said isotropic when MSD grew similarly along the main axes of the cross-section.

By analogy with diffusion on a line of length L , the mixing length along direction d is given by   6 max along d

MSD

. Along direction X and Z, MSD increased proportionally with vising time (parabolic with  ) in a very similar manner in less than 200 s. This behavior was typical of uncorrelated displacements and corresponded to a maximum visited distance of 67±3 mm, which was periodically achieved approximately every 460±20 s. Along Y, displacements followed ballistic trajectories proportional to  . Forty-minutes were required to visit half- length of the deep-fryer without reaching a perfect mixing, while it was obtained in almost 3 minutes along the half-section of XZ.

The incursions under the heating element from the upper part of the bath remained exceptional and brief (see Figure 10c). The particles under the heating element move essentially horizontally at a very low speed due to pressure fluctuations. The difference in viscosity on both sides prevents mixing. The zone is stagnant (Figure 10i-l) and does not exhibit any significant turnover under simulated conditions (first ten minutes of heating). These conditions are responsible for the long duration (two hours) needed to reach a steady thermal regime under the heating resistor. Lower temperatures during the first ten minutes exacerbated dead volume effects. The vertical distributions of particles on short and long-time scales are plotted in Figure 11, based on the sampling of more than 3×10 3 trajectories starting from the top, median, and bottom positions. On short time scales, the vertical dispersion was faster from the median position than from top and bottom ones. Residence times were indeed much longer at the free surface and bottom (see Figure 10). On longer time scales, dispersions were similar from top and median positions almost uniform, but with large fluctuations in time. Vertical mixing times were above ca. 10 2 s and beyond 10 4 s above and below the heating element, respectively. As a result, only 2±1% of particles initially present in the cold zone could reach the region above (Figure 11f).

The risk of accumulation of hydroperoxides in the cold zone by convection was low during initial heating and temperature holding. On the corollary, hydroperoxides present at the beginning of heating would decompose very slowly in the stagnant region without escaping.

Without mixing and in extreme anoxia, polymerization and cyclization termination reactions (exothermic) will dominate, whereas scission reactions (endothermic) would dominate immediately above the heating element due to much higher temperatures and rapid renewal.

The production of volatile compounds far from the surface, where they are finally desorbed, has been previously characterized by local sampling [START_REF] Touffet | Online reconstruction of oil oxidation kinetics and reaction schemes during deep-frying by deconvolution of ATR-FTIR spectra[END_REF].

as shown in Figure 12a. The separation between upper and lower regions was artificially imposed by the starting position, but the exchange between regions was permitted.

For the upper volume, single-particle tracking (denoted PT) and residence time (denoted RT) approaches are theoretically equivalent for durations longer than mixing time (~3 min along XY). In this case, the exploration of regions at different temperatures is independent of the order in which they are visited. Residence times depicted in Figure 12a were sampled from random walks initiated from the upper region to avoid sampling biases due to hydroperoxides decomposing faster than the time of passages from one region to the next (see justification in the general discussion of Ref. [START_REF] Metzler | Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking[END_REF] ). The distribution for particles originating from arbitrary positions is shown in Figure 9c. Only the PT approach gives access to the variability within the For the lower region, thermal equilibration times (~2 hours) much longer than simulated heating times (10 min) make simulated temperatures (~70°C) unsuitable for predicting the decomposition of hydroperoxides. A constant temperature of 135°C close to experimental measurements (135±3°C) was applied instead. Since the region was uniform in temperature, no PT results are shown. It is nevertheless underlined that the displacements are slow in the stagnant region (see Figure 10l). Since the exchange between upper and lower regions was minimal, the proposed approximation led to excellent predictions below the resistance during the two first hours. Beyond two hours, their amount was overestimated due to a significant mass transfer loss by diffusion and advection. E.

Effect of deep-fryer design

Since the lifetime of hydroperoxides is longer than vertical mixing times, the distribution of residence times and temperatures govern the kinetics of decomposition of hydroperoxides.

Figure 13 shows the oil flow structure and residence times within a deep-fryer directly heating from a rounded bottom and a limited cold zone. Heating elements are located at the top-end and at the bottom of the rounded part of the tank. Depicted simulations showed similar temperatures averaged over the entire oil volume but with different distributions (Figure 13b). The theoretical effects of the decomposition of hydroperoxides is shown in Figure 13c. In de depicted configuration, the new design without cold zone doubled the effective lifetime of hydroperoxides. The effects could not be guessed without simulation as some effects are antagonist. The modified configuration introduced hot spots, but also accelerated the dispersion in a larger accessible volume. The higher efficiency originates from the rounded bottom performing the recirculation and the creation of three independent plumes. This study demonstrates the strong spatialization of oxidation radical reactions. The upward velocities although fast in the central part of the fryer are insufficient to justify a transport of highly unstable alkoxyl, hydroperoxyl and alkyl radicals (lifetimes of the order of microseconds) from the heating elements to the free surface. The decomposition of hydroperoxides in depth and in anoxia produces essentially scission compounds in competition with the polymerization reactions responsible for the fouling of the heating surfaces. The coupling between the production of voluble organic compounds by scission, their transport and their desorption at the surface will be detailed in a companion paper.

Appendix A: Simplified oxidation model in mixture

As the propagation via alkoxyl and hydroxyl radicals and oxygen addition are several magnitudes faster than the hydrogen abstraction via peroxyl radicals and hydroperoxide decomposition [START_REF] Schaich | Challenges in Elucidating Lipid Oxidation Mechanisms. When, Where, and How Do Products Arise ?[END_REF], the lumped mass balance reads:

          2 2 [] [ ] , [ ] [] , [ ] i i i i d i p i i pi d L OOH k T L OOH k T O L H dt d L H k T O L H dt     (19)
where     Simplified model (Eq. ( 7)) is applicable to any isothermal dataset in anoxia and in oxygenated conditions if i p k constants are determined independently. Following Refs. [START_REF] Howard | Absolute rate constants for hydrocarbon autoxidation. VI. Alkyl aromatic and olefinic hydrocarbons[END_REF][START_REF] Stirton | Oxygen Absorption of Methyl Esters of Fat Acids, and the Effect of Antioxidants[END_REF], the equality

  1) is realistic, but it remains a simplified description of the full thermo-oxidation mechanism. It involves chemical functions and does not consider their carriers: either TAG or attached FE. Besides, the notation LH is ambiguous as it represents a generic unsaturated FE with several labile hydrogens, whose number and type depend on the considered FE. Labile hydrogens are in α position of a double bond (hold by the first carbon after the double bond).

Figure 1

 1 Figure 1 summarizes the multiscale problem with the following key steps: the oil anisothermal

Figure 1 :

 1 Figure 1: Coarse-grained description of the oxidation process at the scale of the deep-fryer. (a) a coarse-grained molecular description is proposed with building blocks containing two to three heavy atoms. The plotted oxygen profile corresponds to the vertical density of oxygen atoms as calculated on the elementary volume. (b) Illustration of oxygen addition subsequently to a random abstraction of labile hydrogens (i.e., on allylic sites). (c) Pro and cons associated with the loss of molecular details.

Figure 2 .

 2 Figure 2. Design of the reference deep-fryer used in this study: (a) overview of the deep-fryer equipped with three windows to image the flow, (b) top view of the tank, (c-d) overview of the mesh. Positions indicated by 1, 2, 3 and 4 in blue circles correspond to the center (X=0, Y=0, Z=0), the top(X=0, Y=0, Z=0.0514 m), the lateral (X=0.08, Y=0, Z=0) and the bottom (X=0, Y=0, Z=-0.041 m) respectively.

2. 23 • 3 Φ

 233 10 -2 m 2 ) placed above the tank's bottom. Two bulb and capillary thermostats placed in the middle of the first coil control temperature, as shown in Figure2a. The longest and thinnest bulb positioned near the heating element's bottom plane prevents oil overheating above 200-210°C. The second bulb is positioned at the mid-height of the heating element and controls the frying temperature. Although very basic, the second sensor's mechanical thermostat offers a somewhat frequency proportional control close to the set temperature. The heating periods are long, far from the setpoint, and shorter otherwise. A similar on-off control was implemented numerically by imposing the heat flux, , at the external wall of the tubular heating element (effective heat flux density 3  = 9.2•10 4 W⋅m -2 ). The skin temperature of the heating element, skin T , was given by:

  frying T was defined by frying temperature at position 1 (see Figure 2 X=0, Y=0, Z=0). The set temperature set T was chosen equal to 165°C. In the real and simulated oil baths, the skin temperature reached values up to 40°C above frying T . All simulations started with an initial temperature of 25°C equal to the ambient one T  .

  Figure1illustrates the likely coupling between the transport of chemical species after exposure

  , x y z U is the local velocity vector oil and p  θ is the unitary vector tangent to the streamline p  .In anoxia conditions (i.e., without hydroperoxide regeneration), the local concentration in any location of the deep-fryer is obtained by averaging the decomposition over a representative set of trajectories starting from a reference region (e.g., surface) and joined in the same point  ,,x y z . By denoting   the N trajectories in the set, the variation of hydroperoxide content becomes for an arbitrary initial distribution of hydroperoxides:

N

  is the number classes uses to discretizes the considered temperature range.

iL

  OOH kinetics were reconstructed from     LOOH t total ones using a simplified version of Eqs. (2), as detailed in Appendix A. Typical reconstructed kinetics during the decompositiondominant phase are shown in Figure 3. Aerated and anoxia conditions reproduced those encountered at the surface and in the oil bulk volume without temperature gradient. Decomposition rate constants,

Figure 3 :

 3 Figure 3: Comparison between predicted (lines) and experimental (symbols) kinetics of total hydroperoxide content in oxygenated conditions (filled symbols) and in anoxia (empty symbols). Experimental data in poppyseed oil (PO) and linseed oil (LO) are from Ref. [61]. The different subpopulations of hydroperoxides are presented in the inlets. When index i has been dropped, the value represents the total concentration in hydroperoxides.

  hydroperoxides (ranging from 100°C to 180°C). Decompositions were monitored in the bubbling reactor in strict anoxia. Normalized kinetics are shown as semi-log plots in Figure4ac with corresponding Arrhenius plots depicted in Figure4dalong with reaction rates reported in the literature. All linearized plots confirmed unimolecular decomposition in frying conditions. Strong activation of decomposition by temperature was evidenced for all hydroperoxides with a lifetime reduction varying from 50 to 400 from 80°C to 180°C. Oleate hydroperoxides exhibited the highest stability with temperature and the highest activation energy around 78 kJ⋅mol -1 .

Figure 4 .

 4 Figure 4. Kinetics of decomposition of hydroperoxides of pure esters (a) methyl oleate (MO), (b) methyl linoleate (ML), (c) methyl linolenate (MLn) in frying conditions, and (d) their corresponding decomposition rate constants and activation energies. Empty symbols are from literature: A from Ref. [38] (ethyl oleate), B from Ref. [39] (MO), C from Ref. [40], E from Ref. [63], F from Ref. [26], G from Ref. [25], H from Ref. [64], I from Ref. [65], J from Ref. [24] and K from Ref. [66].Filled symbols are from this work and produced in anoxia (nitrogen flow rate of 0.5 L⋅min -1 ) in the bubbling reactor.

Figure 5 :

 5 Figure 5: Temperature and velocity distributions during initial heating in three principal planes of the deep fryer Measured temperature kinetics and surface temperatures are shown in Figure 6. Simulated

Figure 6 .

 6 Figure 6. a) Thermal imaging of the oil bath surface during the heating period and b) the corresponding temperature distribution and c) the comparison between measured (continuous lines) and simulated (symbols) local temperature.

Figure 7 :

 7 Figure 7: a) Kinetics of temperature profile along Z during heating and b) zoom near the surface region

  frozen turbulence held. Normalized spectra of velocity autovariance are plotted vs. natural frequency, f in Figure 8a. Its interpretation within the Kolmogorov turbulence phenomenology requires to relate temporal fluctuations with spatial ones. By assuming that the convected eddy pasts the probe with a mean convection velocity U and unchanged characteristics, the frozen turbulence assumption associates the wavelength / Ln with f nU L  . L represents the largest length scale and n any number greater than 1. All Power spectra were similar and exhibited power-law decays as ranging from 0.4 and 8 Hz. Spectra invariant with position and 5/3's scaling signed locally isotropic turbulence in the inertial subrange, where turbulent kinetic energies cascaded from production scales (low frequencies) down to dissipation scales (high frequencies) [69]. This result, never shown in a deep-fryer, suggests an efficient mixing down to the millimeter scale within the deep-fryer due to turbulence. On intermediate time scales, the power spectra of temperature fluctuations decreased rapidly with f , but not isotropically or in a self-similar manner. Simulated and measured spectra were quite similar and were approximately associated with an exponent 7 / 5 m 

Figure 8 .

 8 Figure 8. Analyses of fluctuations of (a) velocities and (b-c) temperatures. PSD. (a-b) Temporal power spectrum densities at different positions and comparison with its scaling as



  is the oil density at the same positions calculated from the local temperature.   1..6 f f  Ω are the faces of the element with normals f  n and H the Heaviside function.

Figure 9 .

 9 Figure 9. (a) Temperature (iso-contours), velocity (arrows) and residence time (color) fields at steady state in the principal planes of the deep-fryer (resolution of 1.7 mm) and (b) experimental observations of scattered light at 90° through the plane XZ at steady state. The amount of scattered light characterized temperature gradients and the presence advected particles (~20 flat wood chip particles with a diameter of ca. 3-5 mm were dispersed in oil bath). The red circle tracks the position of these particles. (c.) Residence time at different temepratures of two trajectories in the bottom and in the top region.

1

 1 is a 3×3 diagonal matrix whose diagonal elements are random drawn from the standard normal distribution. -square displacement due to the random contribution of turbulence, where angular brackets denote an ensemble average. Following Taylor's theory, fluctuations of particle velocities were assumed to be governed by a stationary random process so that

, x y zu

  is the variance of fluctuations. By choosing an integration time step sufficiently small comparatively to the total correlation time

0 Y

 0 a random white noise normally distributed around 0 with a standard deviation proportional to ,, proposed noise scaling reproduced minimal fluctuations within the stagnant region and maximal in heated ones as experimentally assessed.Though hydroperoxides may be present at the beginning of the heating cycle in the oil bath, they are continuously regenerated at the extreme surface. As a result, the trajectories of fluid particles emitted from the surface are more relevant to envision their fate in the oil bath. Typical fluid particle trajectories originating from the surface, in the median plane ( 0 Z  ), and below the heating element are shown in Figure10. They correspond to a convective transport longer than 1200 s. The rapid recirculation from the heating element back to the free surface confirms that an oil particle transporting hydroperoxides may be rapidly exposed again to atmospheric oxygen, facilitating their renewal. The recirculation above the heating element much faster than the decomposition rate of hydroperoxides (between 5•10 -4 and 1•10 -3 s -1 at 160°C) justifies that they are decomposed at a steady rate at frying temperature.In detail, recirculation loops were similar whether the particles came from the surface (Figure10a-d) or the bulk oil above the heating element (Figure10e-h). Particles close to the vertical symmetry plane  remained almost confined within the plane XY . Outside the central region, particles were oriented along the main cross-sections (diagonal planes), exploring all corners. The merging of ascending plumes into one single central chimney forced the convergence of trajectories towards the symmetry axis 0 XY .

Figure 10 :

 10 Figure 10: Simulated trajectories originated from (a-d) the surface (5 trajectories), (e-h) the median plane (Z=0, 5 trajectories) and (i-l) the bottom of the deep-fryer (40 trajectories): XZ (b,f,j), YZ (c,g,k) and XY (d,h,l) views. For each trajectory, starting and final points are plotted with triangle and square symbols, respectively. The plane of the heating element is materialized in red. (m) mean-square displacement (MSD) versus the square root of mixing time  .

  deep-fryer due to dead volumes or overheated regions close to the heating element. More than 60% of PT values were indiscernible from RT determinations. Only three values were significantly above and corresponded trajectories reaching the bottom region underneath the heating element. Lower values were comparatively more frequent and were observed as soon as the trajectory approached the heating element (with temperatures above 160°C), leading to much faster decomposition. The detailed reconstruction for each hydroperoxides showed that, though hydroperoxides from oleic (L1OOH) and linoleic fatty esters (L2OOH) were produced in almost similar amounts in the bubbling reactor, the second decompose four times faster.L2OOH were all depleted in one hour whereas only half of L1OOH was. During a typical production cycle of one hour (heating, temperature holding, frying, cooling), these results would demonstrate that L2OOH would responsible for most of the inconvenience (smell, fouling, loss of nutritional value) whereas L1OOH would certainly reinitiate the oxidation of lipids during the next production cycle.

Figure 12 :

 12 Figure 12: Residence time at each temperature calculated for 1200 trajectories (a) and comparison between measured and predicted values of LOOH concentration in the top and bottom regions (b)

Figure 13 .

 13 Figure 13. (a) Anisothermal oil flow within a 4 L deep-fryer without cold zone and equipped with four heating coils (two at the bottom and two at the transition between the rounded bottom and the vertical walls). (b) Comparison of temperature residence times with the reference design (see Figure 9) at steady-state and (c) corresponding hydroperoxides decomposition.



  are the temperature dependent reaction rates of decomposition and apparent propagation, respectively. By definition, the apparent reaction rate 0

•

  -glycerol from: 1-linoleoyl-2-linoleoyl-3-oleoyl-glycerol from: 1-oleoyl-2-oleoyl-3-linoleoyl-glycerol oxidability and prevalence in sunflower oil approximately 2balance of labile hydrogens • average reactivity • no specific thermal activation • no effect of TAG composition • no ability to predict the profile of oxidation products • specific reactivity based on FE composition • specific thermal activation • effect of TAG composition • lineage of oxidation products • no correlated transport for FE attached to the same glycerol backbone • realistic description • correlated transport of FE attached to the same glycerol • profile of oxidized TAG. • possibility to add mono-and diacylglycerols.

  throughout the manuscript, as justified in Ref.[START_REF] Borden | Dioxygen: What Makes This Triplet Diradical Kinetically Persistent?[END_REF]. Its unusual electronic structure explains why this abundant molecule is encouraged to react in the deep-fryer with an alkyl radical L

		

 centered on the carbon by forming bonds. Both radicals (di-and mono-) stabilize themselves, but the extra valence of dioxygen creates a new radical centered on oxygen, so-called peroxyl radicals, and denoted LOO  . At this stage, it is crucial noticing that the abundant OO   diradicals cannot directly attack a fatty ester LH . The attack involves an intermediate step of hydrogen abstraction between LOO  and LH (reaction 05

of them may exhibit potentially different stabilities with temperature. Fatty ester Fatty ester structure † Hydroperoxide isomers Hydroperoxide structure i j k Prevalence 100×(kgisomers/ kgLOOH) Refs. Oleate (cis)

	and be produced from k the stereoisomer index. In shorts, 8, 4, and 4 different hydroperoxides can 1, 2,3 i  respectively. It is worth noticing that the proposed coarse-graining reduced the complexity dramatically to consider. In the real-life, each TAG carries 0 to 3 hydroperoxide functions, evolving to one of the         32 (8 1) 2 4 1 (8 1) 2 4 1 6498           non-palindromic sequences of trans-10-OOH trans-8-OOH trans-11-OOH cis-8-OOH cis-11-OOH cis-10-OOH hydroperoxides. Each trans-9-OOH cis-9-OOH	1 1 1 1 1 1 1 1	1 2 3 4 3 4 2 1	22.5 22 19.5 19 5.4 6.1 2.8 2.7	45

Linoleate (cis-cis)

	trans-trans-9-OOH trans-trans-13-OOH cis-trans-9-OOH cis-trans-13-OOH	2 2 2 2	1 2 1 2	90 10	46

Linolenate (cis-cis-cis)

	cis-cis-trans-16-OOH	3	1	43.3
	trans-cis-cis-9-OOH cis-trans-cis-13-OOH	3 3	2 1	34.2 11.5	44
	cis-trans-cis-12-OOH	3	2	11

Table 2 . List of parameters used to simulate the anisothermal heat flow within the deep-fryer.

 2

	Parameters	Symbols Value					Units	Refs.
										sunflower oil
	density			  0.64 1101.1 TT   	with	T in	K	3 kg m  	[50]
	thermal
	expansion								4 7.0 10  	1 K 	-
	coefficient
	dynamic viscosity			  T		6 4.37 10 exp  	6    23.4 10 RT   	Pa s 	[51]
										with	T in K
	specific heat capacity	C	p	  C T p		 T in K 3 273.15 2.566  T    with 3.477 10 	11 kJ kg K  	[51]
	thermal conductivity	k							  k T with 0.149 10  K T in	4 	T	11 W m K  	[52]
					deep-fryer (technical specifications)
	oil tank length	e L							2 26.51 10  	m	-
	oil tank width	W							2 17.82 10  	m	-
	oil height	H							2 9.14 10  	m	-
	oil volume	V								4	L	-
		oil
	oil initial temperature	0 T							25	°C	-

heat transfer coefficients (validated experimentally at steady state, >9 min after initial heating)

	free surface				1 h	8 from the correlation with LW  LL 1/3 0.14 Nu Ra 	21 W m K  	[53]
	walls				2 h		2 (experimental determination)		21 W m K  	-
							from the correlation	Nu		0.145	Ra	0.29	with
												LL
	heating element			3 h	L =0.07 m (oil height above the heating	21 W m K  	[54, 55]
											element)
	where	Nu	L		i hL 	for	1,3 i  ,	Ra			 3 g TL / rho   	2 Pr	and Pr			k C	p	are Nusselt, Rayleigh and
	Prandtl dimensionless numbers, respectively.

  present at 75°C. Additionally, since isomers 13-OOH and 9-OOH hydroperoxides of linoleate have similar lifetimes at high temperatures[START_REF] Tokita | Different Rates of Geometric Isomers of Linoleate Hydroperoxide in Acid-catalyzed Decomposition[END_REF], the proposed mass balance did not separate them ( or 2 j  ), and the less probable isomer 11-OOH was neglected (see the justification in Ref.[START_REF] Bergström | Autoxidation of Linoleic Acid[END_REF]). As a result, indices j and k were dropped and i

					L OOH
	represented the likeliest mixture of i,j=1 or 2,k=2 L	OOH , with	i 	1, 2,3	.
	The rationale for the transposition of constant rates from pure FE to complex TAG mixtures
	(with 3 FE attached to glycerol) was tested on two oils from 40 to 120°C: poppyseed (13%
	oleate, 75% linoleate, and 0.6% linolenate, from Ref. [59]) and linseed (22% oleate, 13%

2

k  was considered. For instance, 83% of trans oleate hydroperoxides are present at 95°C, and 90% of conjugated trans-trans linoleate hydroperoxides are linoleate and 54% linolenate, from Ref.

[START_REF] Mallégol | Long-term behavior of oil-based varnishes and paints. Photo-and thermooxidation of cured linseed oil[END_REF]

). All oxidation kinetics were obtained in isothermal conditions under mechanical mixing. Reported experimental peroxide values, expressed in milli-equivalents peroxide or O2 per 1 kg of fat (see interpretation Figure

1c

), were related to the total amount of hydroperoxides (i.e., regardless of the FE or TAG carrier) as:

position of mono-allylic and bi-allylic hydrogens, respectively.

Acknowledgements

The authors would like to thank the project Investissements d'Avenir FUI-AAP17 "Fry'In", which contributed to fund a part of the study. Mathilde Blondel (SEB) and Laurent Berthier (Lesieur) are particularly thanked for supplying the deep-fryers and the vegetable oil, respectively.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The decomposition of hydroperoxides in anoxia was simulated during four hours, starting from a uniform initial distribution in the deep-fryer, mimicking a real production cycle using sunflower oil. The effective decomposition for the regions above and below the heating element are compared with experimental values inFigure 12. The initial hydroperoxides were generated by bubbling air at 140°C (this temperature was shown to maximize hydroperoxide production in a reasonable time according to Ref. [START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]). The initial composition in hydroperoxides was reconstructed from Ref. [START_REF] Patsioura | Effects of oxygenation and process conditions on thermo-oxidation of oil during deep-frying[END_REF]. Experimentally, only the total amount of hydroperoxides was measured by FTIR [START_REF] Touffet | Online reconstruction of oil oxidation kinetics and reaction schemes during deep-frying by deconvolution of ATR-FTIR spectra[END_REF]. The concentration in hydroperoxides was integrated along 1.2•10 3 trajectories initiated from equispaced points distributed uniformly via Eq.( 11) and calculated independently with Eq.( 12) from the distribution of residence times in temperature in the bath

Declaration of interest

The authors declare no conflict of interest.


Nicola Schneider

INALCO) "From state feminism to individual projects: some theoretical and empirical reflections on female emancipation in Tibet"

My paper will first trace the advent of state feminism in Tibet during the early 1950s with the foundation of the 'Tibetan Patriotic Women's Association' (Bod ljongs rgyal gces bud med tshogs pa), closely linked to the introduction of the Chinese communist regime with its mass organizations. The strong mobilization of Tibetan women during the uprising in 1959 in Tibet led to a split among its members and then to a temporary interruption of state feminism; only many years later, state feminism has been reactivated by two different movements: the 'Tibetan Women's Federation' in Tibet (Bod kyi bud med mnyam 'brel lhan tshogs) and the 'Tibetan Women's Association' (Bod kyi bud med tshogs pa) in exile, both with their respective political agendas.

I will then present several examples from Tibet and exile of more individually led projects and actions in favor of women and gender equality since the last twenty years such as the 'Tibetan Nun's Project' (Bod kyi btsun ma'i las 'char), which successfully introduced the geshema degree (dge bshes ma) for nuns and female WeChat groups seeking to inform themselves about and distribute feminist ideas. Finally, my paper will ask if state feminism was, and somehow is always, a hindrance to the development of collectively led feminist movements and political empowerment in Tibetan communities.

Introduction

First of all, I would like to welcome all the participants and listeners to our panel "What is 'feminism' about in Tibet?. We are pleased about the interest you have shown in this subject and also a bit sad for those who have finally not been able to come. Let me first say some words on the title and subject we have chosen, which is voluntarily formulated as an interrogation rather as an affirmation. By doing so, we wanted to invite researchers on Tibet to think about a rather disputed term or concept, which has been applied by some Western researchers, especially those interested in Buddhism and gender, but with which Tibetan women generally speaking do not seem to identify very much. Of course, there are some exceptions, like for instance the female artist Monsal Pekar on which I have presented a paper at the last IATS and which will soon come out on the AsianArt.com website. She is one of the rare Tibetan women who considers herself to be a feminist and openly says that she is "fighting for women's rights" (to quote her own words). Another one is no other than the Dalai Lama, who said at several occasions in interviews that he considers himself to be a feminist and furthermore that he might come back in the body of a woman. But the question remains open: are we talking here about exceptions that confirm somehow a rule (that feminism does not exist in Tibet)? Or can we detect some changes going on concerning the notion of feminism and its application in Tibetan society? As for me, I came to the conclusion that the latter is the case: diapo. Let me explain how I came first to this conclusion or to the question why it is necessary to do research about feminism in Tibet. For my PhD and some follow up research, I studied Tibetan nuns, a population, which can be considered as being rather conservative. Like you might all know, nuns were (and always are in some regards) relegated to the second rank in the monastic community: they are not represented in the ecclesiastic hierarchy; they do not have the same monastic status (that is they cannot receive full ordination); they do not have the same economic support and their education was, until recently, only basic. However, in the recent three decades, the situation for nuns has changed dramatically: thanks to the efforts of many people from different backgrounds, nuns have now access to the highest Buddhist education and can become geshe (or geshema) or khenpo (or khenmo) -even though this is of course not the case everywhere. They are also on the way to obtain full ordination -a subject to which Darcie will come back later. As for their access to the geshema-degree, which I have studied in the long run and in detail, they had to undergo a long journey with multiple barriers. To make it short here, they had to negotiate hard and wait for each small step further. In anthropological words, they had to "bargain with patriarchy", a notion, which I borrow from Deniz Kandiyoti, Bina Agarwal and others 1 . The "bargain" approach wants to throw light on how gender asymmetries are constructed and contested in the household and outside, and the links between both. Tibetan nuns also had to contest gender asymmetries and they did so in a silent way that is instead of claiming gender equality with loud voices, they were doing (learning how to debate and so forth) and proving thus that they are capable to do so. In this way, they succeeded to draw attention and finally they were allowed to pass their examination to become geshema. Diapo Not only that, by the time they were ready for the first final examination, in May 2016, the term geshema, a feminized form of the word geshe that was not in use in the past, was already introduced into Tibetan vocabulary, showing its acceptance by the society in large. But is contesting or challenging gender roles synonymous with being a feminist? I myself don't think so. And I have not met any Tibetan nun telling me openly that she considers herself to be a feminist -even though I have met nuns from Indian Himalayas who told me that they consider themselves to be feminists! Be that like it is. Let us have a short look on what the definition of 'feminism' in the West is: diapo. Without going further into detail (and polemics or differences even in Western cultures), we have the following short definitions in English Read here the ppt. With these few and short definitions, we can already see that there is no clear cut meaning behind the word/concept of feminism in our own cultural areas. And if many people can identify with the first definition mentioned here, this is probably not the case with the last one, which also presumes some kind of active political action from the person who considers herself or himself to be a feminist. As for Tibetan women I have questioned and who rarely say that they consider themselves to be a feminist, once I mentioned the first definition given here, they agreed with it.

Let us now turn to some Tibetan translations of the term "feminism" (diapo) We have here quite diverging notions which can translate the Western term "feminism", reaching from "gender equality" to "women's rights" and "women's power". In my view, this is one of the reasons why Tibetans have problems to identify with it. Not only can words like "right" or "power" be interpreted in very different ways, but it is also difficult to add an agentive suffix in order to translate the word "feminist". We are in a situation of "lost in translation" and it also reminds us of not putting our ideas and concepts on other cultures, taking them as granted. Instead we need to look for what really happens in other cultures. At the same time, we have to keep in mind that ideas also circulate and even more so today in a more and more globalized world. Let us now turn to Tibetan notions of gender and gender inequality. These tend to turn around the gender distinction of pho mchog and mo dman (diapo) -the excellent or superior male and the inferior female,… and the highly derogatory derived term skyes dman (diapo) -inferior birth, which is used (among other words) to refer to "woman". According to Tibetan scholars 2 1 Kandiyoti, Deniz, « Bargaining with Patriarchy », Gender and Society, 2 (3), September, 1988, p. 274-290. Bina Agarwal, « 'Bargaining' » and gender relations : within and beyond the household », Feminist Economics, 3 (1), 1997, p. 1-51. 2 Tshe ring chos mtsho, 1995, « Bod rigs bud med skor gyi zhib 'jug rgya mtsho'i chu thigs / Preliminary Research on Women in Tibetan Society », in G.yu mtsho / Journal of Tibetan Women's Studies, vol. 3, n° 1, p. 3-13 & Tashi Tsering Josayma, « Bod kyi rig gzhung nang bud med kyi nus pa dang bzhad yams skor kha bsngogs tsam byas pa », The Tibet Journal, vol. 42 (1), 2017, pp. 111-141. (discussed by Tsering Chötso and recently by Tashi Tsering Josayama), this distinction came most probably with the arrival of Buddhist religion and is linked to the ideology of female pollution/defilement (grib), current in Indian and Chinese society in ancient time. According to findings, the Tibetan terminology was more equal before that with neutral words like pho / mo; skyes pa / skyes ma and so forth being in use to refer to men and women. Even though we can always find the term skyes dman in use today, diapo the honorific term in Tibetan for woman, bud med3 , has replaced it in Tibet as well as in exile, at least in official language.

Let us now come to the question: when and how did feminism reach Tibet? According to my tracing it was first introduced by the communist party, in 1952, when Lhasa saw the foundation of its branch of the All-China Women's Federation: the Tibetan Patriotic Women's Association (Bod ljongs rgyal gces bud med tshogs pa) (diapo). It is one of the two mass organizations founded by the Chinese Communist Party (the other being the Patriotic Youth Association) and considered to be one of the United Front organizations. The initial idea came from some wives of Chinese Generals who brought it to the wives of Tibetan ministers during a dinner party (among them Rinchen Dolma Taring, Ngapö Tseten Dölkar and Thangmé Könchog Palmo, to name just a few). Once the Dalai Lama had given his consent, the Tibetan ladies were ready to found the association. Right from the beginning, Chinese communists considered Women's liberation (funü jiefang) to be a priority goal for "New China". Thus general emancipation was to be achieved with women's large-scale involvement in public production. 4 In order to realize this aim, the All-China Women's Federation (Quanguo funü lianhehui5 ) was entrusted with mobilizing women to take public roles in politics, urban and village government, production, education, and campaigns for marriage. 6 The Lhasa PWA being a branch of the Federation followed the same model. In the years following the foundation of the PWA in Lhasa, meetings and lectures were arranged regularly with the purpose to study about attacks from imperialists, to learn about mother and children's healthcare, to praise the fatherland. The association also aimed to teach illiterate women how to read and to write and how to take responsibility in the society. Some of the association's activities were more concrete: like digging and planting trees, helping in the fields or carrying baskets. It seems that many members of the women's association appreciated the so-called Chinese way. I quote Mrs Surkhang: "They would say that today we women have a lot to be appreciative about. The Communist Party has given women equality, and it is just great that we can meet in an office. . . . They wanted women to be respected and have equal rights." 7 "They said that in the old days we Tibetan women were hopeless. Even regarding school, it was felt women didn't need education. They were just kept at home. So today women are on par with the men and can be educated." 8 Being a member of the women's association had also some advantages, the most important being travelling around. Thus many of the Tibetan women got for the first time the occasion to visit Beijing and other places in China, and even to go abroad (Congress of the People for Peace" in Vienna, 1952 -Tsering Dolma). But when on March 10, 1959, the Tibetan Uprising took place, a group of Tibetan women organized one day later a meeting to dissolve the PWA and founded a new association: the Tibetan Women's Association. Joint by ordinary women, as well as by many nuns, they organized three separated demonstrations between 12 th and 18 th of March diapo. Because of the historical developments, both associations had to be stopped, but what can be seen is that the PWA introduced a new concept: organized political activity by women. This concept was mostly state driven, that is relying on communist ideology. The TPWA was newly founded and renamed in June 1960 under the auspices of the Women's Federation of China and has several branches in Tibetan inhabited areas. In exile, the revival of the Tibetan Women's Association took more time and was re-launched in 1984 after the advice of the DL. The latter is considered to be an NGO, but most of its agenda is closely linked to the one of the Central Tibetan administration and, until recently, was doing little with regard to particular gender issues (like sexual harassment, awareness rising on gender inequality etc.). Thus, like the first one, it might be seen as a form of state feminism, albeit with a different political ideology. Both do not seem to reach out to women in a very efficient way: in exile, the TWA had for long time no particular gender issues -even though the TWA counts as one of the two largest NGOs; in Tibet also, several women told me that the Women's Federation does not sufficiently reach out to the community (it seems that it is difficult to know what they exactly do in Jyekundo for example). However, both count also very devoted members who do or have done their best to improve the condition of women.

Individual Projects

Meanwhile, in the last two decades, a couple of Tibetan women have come out to speak in favor of feminism or have started to run their own projects directed towards women in particular. Among the former, we find mostly intellectuals and artists (diapo) like the artist Monsal Pekar I mentioned earlier, the writer and blogger Jamyang Kyi or the female rapper Tenzin Seldon. All have spoken for more gender equality and have criticized customs and habits, which are depreciative for women. Several NGOs have been created and run by women for women, like Machik, Shem or Phende (most having been forced to shut down). They are mostly involved in women's education and women's health issues but have helped also in other areas like building of water pipelines. New technologies and social media are creatively used by Tibetan women to share and discuss ideas on gender and female emancipation, while at the same time exchanging books, articles and other information material. Thus we find WeChat groups like "Bod pa'i bud med" or "Great Women" with more than a hundred members. The latter has also organized several gatherings at Charu, a co-working space in Chengdu, around particular topics like for ex. "Defining the key terminologies (of feminism)" (what is feminism? the difference with menhating; what needs to be done to achieve equal participation?) or "Looks aren't everything" (a book by Cameron Russell). The sessions run like this: several women volunteer to present short films with interviews of famous women or books and share their reflections with the public. In exile, similar initiatives have been taken, like the launching of the blog/website "Tibetan feminist collective" diapo by three Tibetan women, two from the US and one from India, who form the editorial board. Authors can submit their ideas through articles, which then can be discussed by other people. Articles reach from critical discussions of Western feminist ideologies to more Tibetan orientated problems like the cancellation of the election of Mrs. Dardon Sharling as a Kalon because of age limit (she had not passed her 35 th birthday at the moment of election; see "Exile Tibet: Where Is the Representation for Women?"). One of the founders is a young Tibetan lady, Kaysang, who had a formal education in gender studies. Very engaged in informal discussion groups on gender issues and empowerment, she is also one of the rare women who considers herself to be a feminist. In 2017, she participated as a staff member in the revision of the Women's Empowerment Policy in Eight Points formulated by the Women's Empowerment Desk and adopted at the first ever CTA-organized Tibetan Women's Empowerment Conference in Dharamsala in February 2017. The Women's Empowerment Desk is a project-funded office founded in 2008 by the Central Tibetan Administration. It (I cite):

… aims to ensure gender perspectives are incorporated in the design, implementation and monitoring of all of CTA's program activities. It works to ensure women and men participate in and benefit equally and strengthen CTA capacity to address key issues facing Tibetan woman and girls. A major responsibility of WED is to ensure that the Tibetan Women's Empowerment Policy is implemented to the fullest extent possible. (Women Status Report 2018) Right after the conference, the WED organized with the help of some Indian experts a largescale study on the status of Tibetan women. Some of the results are: there is a need for awareness of gender equality issues; there is need of programs that inform women about different legislations that protect them from sexual harassment; economic opportunities have to be strengthened; women are always underrepresented in the political sphere, in settlements like in the Parliament; there is lack of awareness of Sexual and Gender-based violence (mostly considered to be domestic violence by respondents) -hesitancy to report cases to Higher authorities (be they Indian or Tibetan); on one side women claim they are equal -on the other side society does not think so. On the one hand, the Women's Empowerment Desk can also be considered as a kind of state feminism (being linked closely to the administration), but on the other hand it is a donor dependent project. It stands in concurrence/contrast to the TWA: which is registered as a NGO; but whereas the latter is well known in the community, the former is near to unknown (over 90% of the study sample did not know of its existence). Most of the projects I have presented so far are public or publically available; let me finish with 2 less public ones. The first is initiated by an official, serving in a department for poverty alleviation among nomads. In the frame of her work, she has noticed that uneducated single women with children face special hardships. And in order to help them, she has decided to teach them Chinese language (through WeChat) in her free time. She also organizes meetings once a while where she explains to the female leaders of different nomadic communities how to cope best with future in her view. She underlines in particular the necessity to educate the children in order to enable them to find better jobs in town. The next and last example is a female entrepreneur, who started her career as a simple tailor and who has build up her own factory with 20 fixed employees over the years. Being from a nomadic family and having learnt tailoring at the age of 7 from her mother, she has slowly expanded her know how and creates her own designs, a mix between traditional Tibetan chuba and modern designs. Having started with a small shop in town, she has now settled her factory at the outskirts. Most of her employees are young women from nomadic communities, which she recruits on voluntary basis and to whom she teaches tailoring. If they like the job and learn quickly, she employs them after a period of 6 weeks, thus helping these women to become professionals and settle in town. Her artistic work, but also her social enterprise have brought her a lot of recognition and prizes, among other from the bureau of the Women's federation.

In conclusion:

To conclude, I would like to come back to the question: can we detect some changes going on concerning the notion of feminism and its application in Tibetan society? I think we can and I further think that we can start to discern some features of feminism made by Tibetans. Even though most Tibetan women are always shy to express their feelings (I would say mostly because of the bad reputation feminism has as causing social disorder, being anti-Buddhist, fighting between sexes, too Westernized…) they see the need to empower women and are willing to help other women to do so. They are not feminist activists, like we can find in Western countries or in Present-day China, but they thrive towards more gender equality and thus meet the definition of feminism mentioned. For instance they meet always with lack of awareness (like the WED remarks in its report on Woman status), but this might be only a question of time and generation.

Already in use to indicate religious women and women of high rank in the past, see Chokyi Dronma, Diemberger, p. 13.

Pong, 2009: 84 et 89.

The Federation was closed down during the first part of the Cultural Revolution(1966)(1967)(1968)(1969).

For example, a mayor change in Chinese marriage law took place in 1953 (see Croll).

Mrs Surkhang interview.

Goldstein, 2009: 406; Mrs Surkhang interview.

Feminism some English definitions

• Lexico online: The advocacy of women's rights on the ground of the equality of the sexes. • Oxford English: The belief and aim that women should have the same rights and opportunities as men; the struggle to achieve this aim. • Merriam Webster (American):

the theory of the political, economic, and social equality of the sexes organized activity on behalf of women's rights and interests Feminism some possible Tibetan translations