Mourad Aklouf
email: mourad.aklouf@l2s.centralesupelec.fr

Marc Leny
email: mleny@ektacom.com

Michel Kieffer
email: michel.kieffer@l2s.centralesupelec.fr

Frédéric Dufaux
email: frederic.dufaux@l2s.centralesupelec.fr

INTERFRAME-DEPENDENT RATE-QP-DISTORTION MODEL FOR VIDEO CODING AND TRANSMISSION

Keywords: HEVC, x265, inter-frame dependency, Rate-QP model, video coding

In this paper, we propose a new inter-dependent Rate-QP-Distortion model. This model predicts the size of the picture after compression based on the distortion (D) of the reference frame and the current Quantization parameter (QP). This model is particularly useful when adjusting the QP of the picture according to the allocated bitrate budget. Simulation results demonstrate that the proposed model outperforms other models in the literature. In the video sequence Tango, up to 90% of all prediction errors are inferior to 8.6% when using constant QP encoding, and 90% of all prediction errors are inferior to 12% when using variable QP encoding. One application of this model is in low latency video streaming, where each frame of the video sequence needs to be coded with a specific target bitrate, due to variations of the instantaneous transmission rate.

INTRODUCTION

In video coding, the control of the encoding rate is of paramount importance to meet strict rate constraints imposed by band-limited transmission channels. This is useful when encoded videos have to be transmitted using, for instance, 5G evolved Multimedia Broadcast & Multicast Services (eMBMS) [START_REF] Guo | Roads to multimedia broadcast multicast services in 5G new radio[END_REF], although buffers at the transmitter and at the receiver may mitigate encoding and transmission rate jitter. When ultra-low latency live streaming is required, such as in remote driving [START_REF] Kang | Augmenting self-driving with remote control: Challenges and directions[END_REF] or remote surgery applications [START_REF] Khire | Region-of-interest video coding for enabling surgical telementoring in low-bandwidth scenarios[END_REF], rate control becomes of paramount importance. In such applications, a significant part of the delay is introduced by the video encoder and decoder, whereas only very small buffers are allowed to limit the delay to a minimum. Consequently, any mismatch between the encoding rate and available bandwidth on the transmission channel may lead to a unacceptable increase of the delay.

Video encoding rate control usually relies on a model of the dependency between the size of the bitstream resulting from the encoding of a single or a group of frames, and the video encoding parameters. The quantization parameters QP at the frame level or at the smallest coding unit level are usually considered to adjust the encoding rate.

Parametric models between the rate and QP have been proposed in the literature, some of them are reviewed in Section 2. Usually, a trade-off has to be found between the accuracy and the complexity of the model, i.e., the number of parameters involved in the model structure. The model complexity impacts directly that of the parameter estimation process which has to be performed while encoding the video sequence. These models also differ by the type of information they use to infer the rate (and quality) of the resulting encoded frame. Some information are relatively high-level and do not need a precise knowledge of the behavior of the encoder, e.g., the distortion of the reference frame. Some others are evaluated within the encoder during the encoding process, e.g., the Lagrangian multiplier λ and the percentage ρ of zero-valued transformed coefficients.

Preliminary experiments allow us to observe that the relation between the rate Rn and the quantization parameter QPn of the n-th frame depends significantly on the distortion of the corresponding reference frames. In this paper, assuming a low-delay encoding profile, where the reference frame used to encode the n-th frame is assumed to be the previous one, we propose a new model of the relation between Rn and QPn depending on the Mean Square Error (MSE) distortion Dn-1 for the reference frame n -1. This model, denoted as R-(QP,D), is especially suitable for video coding and low-latency rate adjustment.

After reviewing some related models to predict the encoding rate of video frames in Section 2, Section 3 introduces the proposed frame-level R-(QP,D) model. We especially give some insights on the way the structure of the model has been obtained. Then, Section 4 compares the performance of the proposed approach with state-of-the-art models in terms of accuracy of the encoding rate prediction, especially when the encoding parameters change significantly with time, as required by ultra low-latency streaming applications. Finally, we draw some conclusions in Section 5.

RELATED WORKS

The relation between the picture bitrate and the QP is widely discussed in literature. Models can be classified into two categories. The first category assumes that the coding units are independent of each others, whereas the second category considers dependencies among the coding units.

In [START_REF] Ding | Rate control of mpeg video coding and recording by rate-quantization modeling[END_REF], a model between the rate R of a frame and the quantization step size Q is proposed. This model has been refined in [START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF] to predict the encoding rate of the H.265/HEVC frames and CTUs, accounting for the Mean Absolute Difference (MAD) between the original p k (i, j) and the reconstructed k-th coding unit

p k (i, j) of size M × N , M AD k = M i=1 N j=1 | p k (i, j) -p k (i, j)| R k = M • N • M AD k • p1 Q 2 k + p2 Q k , (1)
where

Q k = 2 QP k -4 6
. The MAD of CTU k is predicted using that of the CTU k -1 : M AD k = p3M AD k-1 + p4. This model involves thus 4 parameters, p1, p2, p3 and p4.

In the same spirit, [START_REF] Ma | Rate-distortion analysis for H.264/AVC video coding and its application to rate control[END_REF] and [START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF] present a model involving the Sum of Absolute Difference (SAD) to describe the rate of the k-th frame

R k = p1 SAD k Q k + p2, (2)
where

SAD k = M •N •M AD k . Additionally, SAD k is inferred as SAD k = SAD org k,k-1 + p3 √ D k-1 + p4
, where SAD org k,k-1 denotes the SAD between the original frame k and the original reference frame k -1. D k-1 is the MSE of the reference frame after encoding, 2 . This model also involves 4 parameters.

D k-1 = 1 M •N M i=1 N j=1 ( p k-1 (i, j) -p k-1 (i, j))
The models introduced in [START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF] and [START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF] only account for the quality of the reference CTU k -1 via its SAD or its MAD. This bring us to the second category of R-Q 1 models [START_REF] Yang | Rate control for H.264 with two-step quantization parameter determination but single-pass encoding[END_REF], where the temporal dependency between the H.264/AVC macroblocks is more explicitly taken into account to get the model

R k = p1 • M • N σ 2 k Q 2 k ( 3
)
where σ k is the standard deviation of the motion-compensated residue of the k-th macroblock. This model has a single parameter. All above models have been designed to adjust the QP on the block level, but they may be extended at the whole frame level. Lin and Ortega [START_REF] Lin | Bit-rate control using piecewise approximated rate-distortion characteristics[END_REF] propose a piecewise cubic model of the relation between R k of frame k, QP k and the QP of the reference frame. However, Zhang et al. [START_REF] Xi | Constant quality constrained rate allocation for FGS-coded video[END_REF] experimentally show that the piecewise cubic model is inaccurate in fine-granular scalability (FGS) coders.

Two other types of models have also been proposed. The Rρ model [START_REF] Liu | Lowcomplexity rate control based on ρ-domain model for scalable video coding[END_REF] predicts the rate R using the percentage ρ of zerovalued transformed coefficients. Finally, the R-λ [START_REF] Tang | A generalized rate-distortion-λ model based HEVC rate control algorithm[END_REF] predicts the rate R using the Lagrangian multiplier λ. These two approaches model the rate using low-level encoding parameters. Thus, they do not provide a straightforward control of the encoder behavior as the R-Q models. This can represent a weakness in applications such as low latency video streaming. Consequently, only R-Q models are compared against our proposed model in this paper.

INTER-DEPENDENT R-(QP,D) MODEL

This section introduces the proposed R-(QP,D) model for rate control at the frame level. The proposed model has been derived based on experiments with H.265/HEVC [START_REF] Gary J Sullivan | Overview of the high efficiency video coding (HEVC) standard[END_REF] (using the x265 encoder [START_REF]x265 software documentation[END_REF]), Nevertheless, the methodology is generic and can be extended to VVC [START_REF]Versatile video coding (VVC)[END_REF] or AV1 [START_REF]AV1[END_REF].

In order to motivate our model, we show experiments considering two typical frames, with indexes n = 79 and n = 131 of the video sequence ParkScene 2 encoded with x265 [START_REF]x265 software documentation[END_REF]. Figure 1a shows the rate Rn of frame n as function of the distortion of the reference frame Dn-1, for different values of QPn. These results have been obtained by encoding frame n with six different QPn ∈ {QP (1) = 20, QP (2) = 24, . . . , QP (6) = 40}.

All previous frames, including frame n-1 have been encoded with seven QPn-1 = QPn + ∆QP , where ∆QP ∈ {∆QP (1) = -7, . . . , ∆QP (7) = 5}. The resulting rates are denoted as Rn,i,j, where QPn = QP (i) , i = 1, . . . , 6 and ∆QP = ∆QP (j) , j = 1, . . . , 7. Each curve for a given value of QPn consists of two almost linear parts as a function of log (Dn-1). Conversely, for small values of Dn-1, Rn increases slowly with Dn-1. For larger values of Dn-1, the increase is steeper. We observe that the dependance 1 This notation covers both R-Q and R-QP models, as there is a direct relationship between the quantization step size Q and QP. 2 Experiments performed on other sequences show similar results but are not shown due to space limit.

where g1 (QPn) describes the bitrate Rn for small values of Dn-1, and R 0 n (QPn, Dn-1) = Rn (QPn, Dn-1) -g1 (QPn) = g2 (QPn) (tanh (g3 (QPn) log(Dn-1) -g4 (QPn)) + 1) , [START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF] describes the bitrate Rn for large values of Dn-1.

Figure 1-b represents R 0 n,i,j = Rn,i,j -g1(QP (i) ) as a function of Dn-1 for different values of QPn = QP (i) . For each value of QPn = QP (i) , i = 1, . . . , 6, a least-squares estimation of g2, g3, and g4 is performed using R 0 n,i,j , j = 1, . . . , 7 to get g2(QP (i) ), g3(QP (i) ), and g4(QP (i) ).

Finally, Figure 2-d illustrates the relation between the square root of g4 and QP (i) , which justifies the following quadratic model for the dependency of g4 in QPn, depending on the parameters p6 and p7 g4 (QPn) = (p6QPn -p7) 2 . (9) Consolidating the previous results, the proposed model ( 4) involves a vector of 7 parameters p = (p1, . . . , p7) , which value is frame-dependent and needs to be determined to accurately predict Rn as a function of QPn and Dn-1.

EVALUATION OF THE PROPOSED MODEL

The performance of the proposed model to predict Rn as a function of QPn is compared to the reference models (1) [START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF], (2) [START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF], and (3) [START_REF] Yang | Rate control for H.264 with two-step quantization parameter determination but single-pass encoding[END_REF], used at a frame level.

Experimental setup

Three JCT-VC test sequences, namely Tango, Racehorses, and ParkScene [START_REF] Suzuki | JVET-d1002 : Work plan for assessment of test material[END_REF], are selected for the experiments, as well as a recording from the inside of racing car (Magnycours). The encoding is performed with the x265 software [START_REF]x265 software documentation[END_REF], configured in low delay mode and with an intra-refresh cycle of one second.

The parameters for the three reference models and the proposed models are only estimated for every four frames of the video sequences. The parameters are then assumed to remain constant for the next three frames. For that purpose, 42 coding trials are done with QPn ∈ {QP (1) , . . . , QP (6) } and QPn-1 = QPn + ∆QP with ∆QP ∈ {∆QP (1) , . . . , ∆QP (7) } to get Rn,i,j, i = 1, . . . , 6, and j = 1, . . . , 7. A weighted least-square estimation of the model parameters is then performed considering the following cost function

Cn (p) = 6 i=1 7 j=1 1 Rn,i,j
Rn,i,j -Rn(QP (i) , D

n-1 ) 2 , (j)
where Rn(QP (i) , D

n-1 ) is given by the proposed model in (4), or by the models (1), ( 2) or (3), and

D (j) n-1 is the distortion for frame n -1.
To compare the performance of the four models, in a first set of experiments all frames of the video sequences are encoded at constant QP ∈ {QP (1) , . . . , QP (6) }. In a second set of experiments, QP may vary from frame to frame as the realization of a firstorder Markov process such that with a probability P = 0.6, QPn = QPn-1, and with a probability 1 -P , QPn is uniformly distributed in the set Qn = {QPn-1 -5, . . . , QPn-1 + 5} ∩ {20, . . . , 40}.

The ability to predict the actual encoding rate is evaluated using the relative rate error

En = 100 (Rpred -Ractual) /Ractual, (11)
where Ractual is the actual size of the encoded frame n and Rpred is the predicted one obtained from the rate model.

Results

We first compare the performance of the four models in a set of experiments when encoding with constant QP. Figure 3 shows the histogram of the prediction errors obtained with the proposed model in (4), and by the models in (1), ( 2) and (3), used on the Tango sequence coded at high bitrates, i.e., QPn = 20, 24. We notice that our proposed model provides the best performance in this case. The errors of model ( 4) are mainly between -13.6% to 14%, with a peak at 3.4%. The prediction errors with the models (1) and ( 2) are between -11% and 19%, with a peak near 11% and -6.5% respectively. Model (3) has the worst performance, with prediction errors spreading between -50% and 64%.

Figure 4 shows the error histograms of the four models on the Tango sequence coded at low bitrates, i.e., QPn = 36, 40. The performance of our proposed model slightly decreases but significantly outperforms the three other ones. Prediction errors are between -28.6% and 18.3%, with a peak around 4.9%. Both models in ( 1) and ( 2) lead to errors between 0% and 81% with a peak near 48% and 27.5% respectively. The model in (3) largely underestimates the rate, with prediction errors distributed between -90% and -58%. Figure 5 shows the error cumulative distribution functions (CDF) when using the four QPn = 20, 24, 36 and 40 for the Magnycours sequence. The proposed model achieves the lowest prediction error compared to the other models. For instance, with QPn = 20, 90% of the prediction errors are less than 12.2% with the proposed model, compared to 16.7%, 22.6%, and 51.8% for the models in (1), (2), and (3), respectively.

Figure 6 shows the error CDF for RaceHorses sequence. Here, we see close performance of models ( 4), [START_REF] Guo | Roads to multimedia broadcast multicast services in 5G new radio[END_REF], and (2) with QPn = 20 and 24, and close performance between model ( 4) and ( 1) with QPn = 40. The proposed model shows a significant advantage in the other cases.

Figures 5 and6 both show that the gains with our model tend to be more significant at low bitrates (i.e., high QPs). This can be explained by the fact that we incorporate the distortion of the reference frame in our model, which is especially important at low bitrates.

Figure 7 illustrates the average error CDF when coding the four sequences with a constant QP. Our proposed model achieves the best performance for all test sequences.

Next, we present results of the second set of experiments using time-varying QPs based on a first-order Markov process. Figure 8 shows the average error CDF with time-varying QPs for the four test 4), (b) (1) from [START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF], (c) (2) from [START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF], and (d) (3) from [START_REF] Yang | Rate control for H.264 with two-step quantization parameter determination but single-pass encoding[END_REF] sequences. The proposed model outperforms the other ones for all sequences. For Magnycours, ParkScene, and Tango, the improvements are significant, whereas for RaceHorses, the models in (4), [START_REF] Guo | Roads to multimedia broadcast multicast services in 5G new radio[END_REF], and (2) reach close performances.

CONCLUSION

This paper presents a new inter-dependent Rate-QP-Distortion model to predict the size of the n-th encoded frames based on the distortion Dn-1 of the reference frame and the current quantification parameters QPn. The performance of the proposed model is evaluated in two coding scenarios: encoding at constant QP and encoding with frame-dependent QP. In both cases, the proposed model outperforms other models in the literature. The gains are especially significant at low bitrates (high QPs), showing the benefits of accounting for the distortion of the reference frame in our model. Finally, the accuracy of the prediction relies on the quality of the parameters fitting and on the characteristics of the video sequence.

In our future work, we will focus on the integration of the proposed model in a rate control algorithm for low-delay video streaming.

Fig. 1 .Fig. 2 .

 12 Fig. 1. (a) The bitrate Rn for the frames n = 79 and n = 131 of ParkScene as a function of the distortion Dn-1 of the reference frame for different values of QPn; (b) R 0 n for frame n = 79 of ParkScene as a function of Dn-1 for different values of QPn

Figure 2 -

 2 a represents Rn as function of QPn for the smallest values of Dn-1 obtained for frame n = 79. In this regime, Rn decreases exponentially with QPn according to g1 (QPn) = p1 exp (-p2QPn) .

Figure 2 - 7 )Figure 2 -

 272 Figure 2-c illustrates g3 as function of QP (i) , which is adequately represented by the linear model depending on p5 g3 (QPn) = p5QPn. (8)

Fig. 3 .

 3 Fig. 3. Histogram of prediction errors for Tango at high bitrates, (a) proposed model (4), (b) (1) from[START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF], (c) (2) from[START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF] and (d) (3) from[START_REF] Yang | Rate control for H.264 with two-step quantization parameter determination but single-pass encoding[END_REF].

Fig. 4 .

 4 Fig. 4. Histogram of prediction errors for Tango at low bitrates, (a) proposed model (4), (b) (1) from[START_REF] Choi | Pixel-wise unified rate-quantization model for multi-level rate control[END_REF], (c) (2) from[START_REF] Li | Inter-dependent rate-distortion modeling for video coding and its application to rate control[END_REF] and (d) (3) from[START_REF] Yang | Rate control for H.264 with two-step quantization parameter determination but single-pass encoding[END_REF].

Fig. 5 .Fig. 6 .

 56 Fig. 5. CDF of prediction errors for Magnycours sequence.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Average error CDF with constant QP for each sequence.


Nadia Mrad
email: nadia.mrad@mines-nantes.fr

Fethi Aloui
email: fethi.aloui@univ-valenciennes.fr

Mohand Tazerout
email: mohand.tazerout@mines-nantes.fr

PERFORMANCE AND EMISSIONS OF DIESEL ENGINE USING BIO-FUEL DERIVED FROM WASTE FISH OIL

In the present work, waste fish fat from fish processing industry is considered as an energy source for diesel engines. In this regard, catalytic cracking process is considered for this present study. The physical and chemical properties of biofuel are very close to diesel fuel. The experiments were conducted in a single cylinder diesel engine to study the performance, emission and combustion characteristics of biofuel. As a result, fuel undergoes good combustion and hence there is significant improvement in performance and reduction in emissions. Experimental results indicate a marginal increase in brake thermal efficiency at all loads compared to diesel fuel. The results show that despite of high NOx and CO 2 , the engine has lesser UHC, CO and PM than standard diesel fuel. The premixed and diffusion combustion duration is decreased with biofuel compared to diesel fuel. The engine was running smooth at all load conditions with biofuel. It is concluded that the biofuel derived from waste fish fat can be consider as a substitute for diesel fuel.

INTRODUCTION

The depletion of world petroleum reserves and the increased environmental concerns have stimulated the search for alternative sources for petroleumbased fuel, including diesel fuels. Biofuel have been used as an alternative fuel for diesel engines [START_REF] Agarwal | Biofuels (alcohols and biodiesel) applications as fuels for internal combustion engines[END_REF][START_REF] Vollebergh | Environmental externalities and social optimality in biomass markets: waste-to-energy in The Netherlands and biofuels in France[END_REF]. It derived from biomass (generally vegetable oil) using various technologies. The methods can be divided into biochemical conversion and thermochemical conversion. The thermochemical conversion is performed using some processes such as: direct combustion, gasification, liquefaction, and pyrolysis (thermal cracking) [START_REF] Ucar | Characterization of products from the pyrolysis of rape seed oil cake[END_REF]. Pyrolysis process appears to be a simple method for biomass conversion and it is the subject of many research studies; it isthe technique of applying high heat to organic matter under the inert atmosphere [START_REF] Ucar | Characterization of products from the pyrolysis of rape seed oil cake[END_REF]. The products of this decomposition are: a mixture of non-condensable gases or bio-gas (CO 2 , CO, H 2 and CH 4 ), liquid (bio-oil or biofuel), and coke (solid carbon).The liquid pyrolysis product is composed generally of a complex mixture of hydrocarbons with different chemical groups and carboxylic acids. The large scale utilization of bio-oil directly as biofuel resources has been quite limited due to its high carboxylic acids content (acidity). The acidity is the main reason that restricts the direct use of biooil as a fuel for diesel engine [START_REF] Guozuo-Gang | Separation of acid compounds for refining biomass pyrolysis oil[END_REF]. The upgrading of bio-oil becomes an important issue. The recent upgrading technologies contain molecular distillation [START_REF] Guozuo-Gang | Separation of acid compounds for refining biomass pyrolysis oil[END_REF] catalytic hydrotreatment, emulsification, and catalytic esterification [START_REF] Mahfud | Hydrogenation of fast pyrolysis oil and model compounds in a two-phase aqueous organic system using homogeneous ruthenium catalysts[END_REF] and catalytic craking [6 -7], which is the best way to upgrade the bio-oil to biofuel. An increase in world population as well as industrialization, massive amounts of waste material is thrown away from the industries. It was suggested that the production of biofuel from this waste are the key areas for developing bioenergy for the future [START_REF] Wu | The development of bioenergy technology in China[END_REF].

According to the FAO (Food and Agriculture Organization), the world fish production in 2009 was 144 million tons of fish, and around 50% of the total fish material processed becomes waste material. Most of the known technologies for the use of grease waste from fish industries are not economically attractive, and sanitary landfills and effluent treatment ponds are not recommended due to the unpleasant odour they produce. In this regard, catalytic cracking can be used as an alternative to convert this grease waste to biofuel. Many studies have been reported that biofuel derived from waste can be used directly in diesel engine without any modification in the engine [START_REF] Arpa | Experimental investigation of the effects of diesel-like fuel obtained from waste lubrication oil on engine performance and exhaust emission[END_REF][START_REF] Rasim | Performance and emission study of waste anchovy fish biodiesel in a diesel engine[END_REF]. Most of the researchers quantified the effects of different biofuels with performance and emission characteristics of diesel engines. It has been reported that some of the biofuel have higher and others have slightly lower brake thermal efficiency and gives lesser CO, HC, PM and higher NOx compared to base diesel fuel [START_REF] Lapuerta | Effect of biodiesel fuels on diesel engine emissions[END_REF][START_REF] Ndayishimiye | Use of palm oil-based biofuel in the internal combustion engines: Performance and emissions characteristics[END_REF]. This change in performance and emission characteristics of different biofuels is due to the physico-chemical properties of the biofuels. The aim of the work is to produce biofuel from catalytic cracking of fish waste industries and evaluate the performance, emission and combustion characteristics of this biofuel, in a diesel engine.

BIOFUEL PREPARATION

WASTE FISH OIL

The feedstock used in our work is an industry waste and it is obtained from, SIRH group specialized in vegetable, animal and marine oils located in north of France. To produce an edible omega 3 fish oils, the fish was treated by a series of processes, starting with cooking it thoroughly in boiling water, pressure it and centrifugation of the product to separate the crude oil and the water and impurities. The crude oil undergone a filtration by winterization to make it limpid by removing the waxes or fats naturally contained. Indeed, the oil is slowly cooled down to obtain a homogeneous crystallization of waxes and fats. It is then filtered through cloths that retain these waxes or fats. The Fat, residue of this treatment, is brown in color and was used in this study without any special purification treatment. The main components of this organic matter are triglycerides. The typical fatty acid composition has been analyzed by gas chromatography analysis (GC/FID). The saturated acids C14:0, C16:0 and C18:0 are identified. The major fatty acids found are the unsaturated acids C18:1 and C18:2 responsible for 45.6 % and 20.6 % respectively of the total composition.

CATALYTIC CRACKING PROCEDURE

Catalytic cracking experiments were carried out at temperatures ranging from 320 to 480°C with a slow heating rate using a laboratory scale reactor. The schematic diagram of the process is shown in Fig. 1. The fat was introduced in the reactor and then heated by an external electric resistance. The catalyst is placed just above the fat on a bed with small holes. When the temperature inside the reactor achieved 320°C, the wax fish was cracking. The generated vapors were passed directly over the catalyst surface, before leaving through the top of the reactor and they enter in a water-cooled, counter flow, heat ex-change which water was kept at 15°C. As a result, two fractions were collected in the flask, which the first is the pyrolysis water, and the other represents the bio-oil fraction.

Figure 1. Schematic diagram of catalytic cracking process

The bio-oil fraction was collected in two steps (from 320 to 400°C and from 400 to 480°C). The first fraction with a high acid (28 0.8mg KOH /g oil ) value and the second with a low acid fraction (0.8mg KOH /g oil ) and this later which consists mainly of hydrocarbons, was analyzed and used as a biofuel for diesel engine. The properties and the major chemical compounds of biofuel and diesel are given respectively in Table 1 andTable 2. The flash point was measured by NPM 440 model (PENSKY-MARTENS). The acid value was determined by titration with KOH/C 2 H 5 OH solution using phenolphthalein as an indicator. The density of the bio-oil was estimated with a pycnometer. The gross heating value was measured using an oxygen bomb calorimeter (model 6200, Parr Instruments Company). The dynamic viscosity was measured with a SV 10 Fibro viscometer. The compositions of the main organic elements (C, O, H, S, and N) were determined using an Elemental Analyzer (Flash EA 1112, CE Instruments). The chemical compounds of the biofuel, was identified using a CG/MS (Gas chromatography / mass spectrometry) analysis. For this purpose a Perkin Elmer Turbo Mass Gold Mass Spectrometer coupled with a gas chromatograph CLARUS 500 was used. The column was SBLTM-5ms Capillary type, 30 m in length and 0.25 mm in internal diameter. 2 shows the schematic diagram of the experimental set-up. The technical specifications of the engine are given in Table 3. The engine was mounted on a fixed table and coupled with an eddy current dynamometer that converts mechanical energy generated by the engine power directly to the net work.

DATA ACQUISITION SYSTEM

Two systems modes are used to manage the control and acquisition of measured signal:

• Slow acquisition (0.5 Hz): The data concerned by this type of acquisition are the air and fuel flow rates, torque, engine speed, pressure and temperature in the collectors. • Quick acquisition (90 kHz): this type of acquisition is used for the intake and in-cylinder pressures. It is also used for the crank angle measures. The in-cylinder pressure is measured using a piezoelectric sensor AVL QH32D installed on the cylinder head. The crank angle determination is realized with an angular encoder (AVL 364C), installed on the crankshaft.

A differential pressure transmitter type LPX 5481 measured the flow of intake air. For temperature measurements, the test engine was equipped with a series of thermocouples type K. Ambient temperature was measured by an active transmitter for humidity and temperature, type HD 2012 TC/150. The fuel flow was measured using a Coriolis mass flowmeter. A bay of analysis (Crystal COSMA 500) placed on the line of engine exhaust gas was used to analyze the main pollutant gases. The emissions of hydrocarbons (HC) were measured by FID flame ionization using a heated hydrocarbon analyzer (model GRAPHITE 52M), the emissions of nitric oxide (NO) and nitrogen oxides (NOx) were measured via a chemiluminescence nitrogen oxide analyzer TOPAZE 32M. Emissions of carbon monoxide (CO), carbon dioxide (CO 2 ) and oxygen (O 2 ) were measured by absorption of infrared radiation using a 2M MIR analyzer. Particulate emissions were measured using a dust analyzer in real time (TEOM model 1105), for measurement and continuous weighing of the mass concentration of particulate exhaust.

RESULTS AND DISCUSSION

The engine performance and emission characteristics of the biofuel, was analyzed and compared using a commercial diesel. The results were as follows.

Brake thermal efficiency

The influence of brake power on brake thermal efficiency for diesel and biofuel is presented in Fig. 3. It can be observed from the figure that as the brake power increases, the brake thermal efficiency increases to the maximum at 80% load and then decreases for the two type of fuel tested. It is observed too that the brake thermal efficiency of biofuel is always higher compared to diesel fuel at various load conditions. This is mainly due to good compound of hydrocarbons are present in the biofuel compared to diesel. The brake thermal efficiency is mainly due on how much amount of energy released during the initial phase of combustion.

Emissions

Nitrogen Oxides

The formation of nitrogen oxides (NOx), is affected by the peak flame temperature, the residence time of the high burning gas temperature, ignition delay, and the content of nitrogen and oxygen available in the reacting mixture [START_REF] Lin | Engine performance and emission characteristic sofa three-phase emulsion of biodiesel produced by peroxidation[END_REF]. The variation of NOx emissions with brake power for the diesel and the biofuel is shown in Fig. 4. Oxides of nitrogen (NOx) in the exhaust emission increase from 20% load to 80%. NOx emissions for the neat biofuel operation are 857 ppm at full load, which is maximum, compared to the diesel. The increase in NOx with biofuel is due to the higher intensity of premixed combustion. The higher premixed combustion is due to higher centane number of biofuel, which initiates the combustion early. Also the fuel is burned quickly due to its higher oxygen content which is 5.34 %.

Unburned Hydrocarbon

Fig. 5 shows the variation of unburned hydrocarbons (UHC) with brake power. The UHC in the exhaust gas is due to the fuel droplets unable to reach the ignition temperature leads to incomplete combustion. It is also due to insufficient quantity of oxygen to oxidize the fuel present in the combustion chamber. It is seen that neat biofuel operation emits lower UHC compared to diesel. The UHC emission for biofuel and diesel at full load is 502 ppm and 575 ppm respectively. The ignition is stared earlier and proceeding good combustion is achieved with biofuel and diesel due to good fuel quality and high cetane number. It is lowest with biofuel, due to sufficient quantity of oxygen present in the fuel itself (5.34 %).

Carbon Monoxide Emission (CO)

Lin and Wang [START_REF] Lin | Diesel engine performance and emission characteristics using three-phase emulsions as fuel[END_REF] indicated that a lower burning gas temperature would inhibit the conversion rate of CO to CO 2 . The variation of carbon monoxide for biofuel and diesel fuel with brake power is shown in Fig. 6. The neat biofuel gives slightly lower CO emission compared to diesel. Indeed, at full load, the CO emission with diesel is 0.59 % and it is 0.39 % with biofuel. Due to the good spray characteristics as a result of low viscosity of biofuel, all the fuel droplets are mixed with air in the end part of the compression stroke, oxidation reaction of the fuel occurs which results complete combustion. Also, biofuel itself has higher oxygen content tends to increase the oxidation process.

Particulate Matter

The results of particulate matter (PM) emissions of the biofuel and diesel are presented in Fig. 7. PM consists of a solid portion (carbon and ash), soluble organic fraction and sulfates (mostly heavy hydrocarbons) that have condensed on the carbon particles. The formation of the PM resulted from the incomplete burning of the hydrocarbon fuel and the partially reacted carbon content in the liquid fuel [START_REF] Heywood | Internal Combustion Engine Fundamentals[END_REF]. This occurs at low air-fuel ratios such as high load and during transient events when boost pressure is limited. High cylinder temperatures and the availability of oxygen increase the oxidization of the solid carbon particles and hydrocarbons to carbon monoxide and carbon dioxide. The PM emission for biofuel is very low compared to diesel fuel, are less than the PM emitted during engine operation with diesel. The PM emission is about 0.028 g/h and 0.03 g/h at full load with biofuel and diesel respectively. The presences of oxygen in the fuel and intake air improve the air fuel mixing rates which help to prepare more quantity of fuel for premixed combustion. The high cetane number of biofuel starts the combustion early. Also, the combustion duration is very less with biofuel. All the above mentioned reasons will lead to reduce the PM in the exhaust.

Combustion parameters

Cylinder Pressure

The variation of cylinder peak pressure with brake power at maximum load is shown in Fig. 8.The cylinder peak pressure with neat biofuel is higher compared to diesel fuel. The chemically bound oxygen in biofuel is the reason for the increased Crank angle (degree)

Cylinder pressure (bar)

cylinder pressure. This is mainly due to fast and enhanced combustion of biofuel inside the combustion chamber. The fuel inside the combustion chamber reacts with oxygen and releases energy, which is used to increase the cylinder pressure.

Rate of Heat Release

The rate of heat release with crank angle for biofuel and diesel at maximum efficiency point (80 % load) is shown in Fig. 9. The premixed combustion with biofuel is very sharp and the diffusion combustion is low which reflects on the rate of combustion. This indicates that most of the fuel injected during the delay period is prepared for ignition and it is participated during premixed combustion process, which is reflected on the formation of NOx. Higher oxygen concentration, high cetane number and more group of hydrocarbons present in the biofuel increase the rate of combustion. The premixed combustion is higher with biofuel followed by diesel, which shows that the combustion with biofuel is very good when compared to diesel.

ANALYSIS OF UNCERTAINTY

In measuring any quantity, the results will always differ from the true value even with careful experimentation. This error in measurement may be either random or systematic. By adding a correction value, the systematic error can be removed. Random error can only be estimated statistically and cannot be predicted in advance. Its presence can be detected only when the same quantity is measured again and again under the same conditions and with the same care.The uncertainty was estimated based on Gaussian distribution method.From the measured parameters, the uncertainty is computed based on Kline and McClintock method [START_REF] Kline | Describing Uncertainties in Single-Sample Experiments[END_REF].

The estimated uncertainty values at different operating conditions are: -Brake power: 0.4-1.7% -Brake thermal efficiency: 0.6-1.9%.

CONCLUSION

The engine performance and emission characteristics of biofuel produced from catalytic cracking of waste grease fish were investigated and compared with diesel fuel in the study reported herein. The two fuels (diesel and biofuel) were tested in a single-cylinderdirect-injection diesel engine at a constant speed of 1500 rpm. The experimental results of the study can be summarized as follows.

-Compared with the diesel fuel properties, biofuel have a higher kinematic viscosity, oxygen content and cetane number. -Heat release rate with neat biofuel indicates higher premixed burning and lower diffusion burning rates as compared to diesel. The higher premixed combustion is due to higher oxygen content of biofuel which increase the air fuel mixing rate and high cetane number results the combustion starts early. -NOx levels are higher with biofuel compared with neat diesel due to the higher premixed heat release rate. -Lower CO and UHC emission is observed with biofuel is due to high combustion rate. -There is a large reduction in PM emission of engine with biofuel operation. The reduction in PM emission is due to the better combustion of injected fuel in the hotter combustion chamber by the early combustion of biofuel.

Figure 2 .

 2 Figure 2. Schematic diagram of the engine setup (1) Test Engine, (2) Biofuel Tank, (3) Diesel Fuel Tank, (4) Exhaust Gas Analyzer, (5) Eddy Current Dynamometer, (6) Particulate Matter Analyzer, (7) Low Frequency Data Acquisition System, (8) Charge Amplifier, (9) High Frequency Data Acquisition System, (10) Crank Angle Encoder/ Speed Sensor, (11) Injection Pressure Signal, (12) Cylinder Pressure Signal.

Figure 3 .

 3 Figure 3. Brake thermal efficiency for different brake power

Figure 4 .

 4 Figure 4. Variation of the NOx emissions

Figure 5 .

 5 Figure 5. Unburned hydrocarbon variation

Figure 7 .

 7 Figure 6. Variation of CO emissions

Figure 8 .

 8 Figure 8. Variation of cylinder pressure crank angle at maximum load

Figure 9 .

 9 Figure 9. Variation of heat release rate at maximum load

Table . 1: Properties of biofuel and diesel Properties Biofuel Diesel

 .

	Density, at 20°C (Kg/m 3 )	825	830
	LHV (MJ/kg) Dynamic viscosity (Ns/m 2 ), at 40°C	42.74 43.36 2.30 2.52
	Acid value (mg KOH /g oil )	0.80	-
	Flash point (°C)	57	56
	Auto ignition temperature	230	220
	Cetane Number	57	52
	Composition
	H	11.59 13.30
	C	72.33 86.00
	N	0	0.20
	O	5.34	0.20

Table . 2: Chemical composition of biofuel and diesel

 .

	Biofuel	Diesel
	Alkene	Iso-alcane
	Alcane	Alcane
	Cycloalcane	Aromatique
	ketone	Cycloalcane
	3. ENGINE TEST
	3.1. EXPERIMENTAL SETUP
	Fig.

Table 3 . Specifications of engine

 3

	Make	Lister Petter
	Number of Cylinders	One
	Type	Direct injection, air
		Cooled
	BorexStroke (mm)	95,5 x 88,94 mm
	Compression ratio	18:1
	Rated power	4.5 kW
	Rated speed	1500rpm
	Start of injection	20°b TDC

ACKNOWLEDGMENTS

This research was supported by the GEPEA Laboratory and the Department of Energetic and Environmental Systems at the École des Mines de Nantes.

These supports are gratefully acknowledged.


Emna Berrich
email: emna.berrich@mines-nantes.fr

Fethi Aloui
email: fethi.aloui@univ-valenciennes.fr

Jack Legrand
email: jack.legrand@univ-nantes.fr

EXPERIMENTAL STUDY ON OSCILLATORY COUETTE -TAYLOR FLOWS BEHAVIOUR

Keywords: Oscillatory Couette-Taylor flows, Reversing and Non-Reversing Taylor Vortex Flow RTVF & NRTVF, Mass Transfer rate, Wall Shear Stress, Inverse method

vortices appear. Comparing to a steady velocity case, oscillatory flow accelerate the instability apparition, i.e. the critical Taylor number corresponds to the transition is smaller than that of the steady case. For high oscillation amplitudes of the inner cylinder rotation, the mass transfer time-evolution has a sinusoidal evolution with non equal oscillation amplitudes. If the oscillation amplitude is large enough, it can destabilize the laminar Couette flow, Taylor vortices appears. The vortices direction can be deduced from the sign of the instantaneous wall shear rate time evolution.

ABSTRACT

In the simplest and original case of study of the Taylor-Couette TC problems, the fluid is contained between a fixed outer cylinder and a concentric inner cylinder which rotates at constant angular velocity. Much of the works done has been concerned on steady rotating cylinder(s) i.e. rotating cylinders with constant velocity and the various transitions that take place as the cylinder(s) velocity (ies) is (are) steadily increased. On this work, we concentrated our attention in the case in which the inner cylinder velocity is not constant, but oscillates harmonically (in time) clockwise and counter-clockwise while the outer cylinder is maintained fixed. Our aim is to attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case. If the modulation amplitude is large enough to destabilise the circular Couette flow, two classes of axisymmetric Taylor vortex flow are possible: reversing Taylor Vortex Flow (RTVF) and Non-Reversing Taylor Vortex Flow (NRTVF) [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF][START_REF] Lopez | Modulated Taylor-Couette flow: Onset of spiral modes[END_REF]. Our work presents an experimental investigation of the effect of oscillatory Couette-Taylor flow, i.e. both the oscillation frequency and amplitude on the apparition of RTVF and NRTVF by analysing the instantaneous and local mass transfer and wall shear rates evolutions, i.e. the impact of vortices at wall. The vortices may manifest themselves by the presence of time-oscillations of mass transfer and wall shear rates, this generally corresponds to an instability apparition even for steady rotating cylinder. On laminar CT flow, the time-evolution of wall shear rate is linear. It may be presented as a linear function of the angular velocity, i.e. the evolution is steady even if the angular velocity is not steady. At a "critical" frequency and amplitude, the laminar CT flow is disturbed and Taylor

INTRODUCTION

Taylor-Couette flows are frequently re-encountered on engineering processes (such on rotor-stator systems) and medical applications. It is widely used for the depleted uranium enrichment, as a membrane or filter, on bioreactors and for blood detoxification. Most of the studies were interested to the Couette-Taylor simple case where the fluid is contained between a fixed outer cylinder and a concentric inner cylinder which rotates at constant angular velocity. Much of the works done has been concerned on steady rotating cylinder(s) i.e. rotating cylinders with constant velocity and the various transitions that take place as the cylinder(s) velocity (ies) is (are) steadily increased. On this work, we concentrated our attention in the case in which the inner cylinder velocity is not constant, but oscillates harmonically (in time) clockwise and counter-clockwise while the outer cylinder is maintained fixed. Our aim is to attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case.

If the modulation amplitude is large enough to destabilise the circular Couette flow, two classes of axisymmetric Taylor vortex flow are possible: Reversing Taylor Vortex Flow (RTVF) and Non-Reversing Taylor Vortex Flow (NRTVF). For Reversing Flow, the inner cylinder is rotating anti-clockwise for the first part of the cycle; the vortices respond by rotating in the same direction of the cylinder motion. For the second part of the cycle, the inner cylinder rotates in the opposite direction, the vortices respond by changing their rotation direction. For Non-Reversing Flow, the inner cylinder is rotating anti-clockwise and the vortices rotate in the same direction of the cylinder motion. However, when the inner cylinder reverses its rotation direction, the vortices continue to rotate in the same direction as in the first half-cycle. [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF] [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF] studied numerically the two flow types. They used a combination of second-order accurate Crank-Nicolson and Adams-Bashforth methods. The velocity components are represented by potentials which were expanded spectrally over Fourier modes in the azimuthal and axial directions and over Chebyshev polynomials in the radial direction. More details on numerical code are available on Willis & Barenghi [START_REF] Willis | Hydromagnetic Taylor-Couette flow: numerical formulation and comparison with experiment[END_REF]. Their code has been tested in the linear and nonlinear axisymmetric regime by using Barenghi [3] and Jones [START_REF] Jones | Numerical methods for the transition to wavy Taylor vortices[END_REF] results. Youd et al. [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF] demonstrated that for relatively high oscillation frequencies, the Taylor vortices direction does not change every half-cycle, i.e. presence of NRTVF. Youd et al., [START_REF] Youd | Non-reversing modulated Taylor-Couette flows[END_REF] proved that, for Couette-Taylor system characterized by a radial ratio η = R int /R out = 0.75, there was a transition to a new kind of axisymmetric timemodulated flow, i.e. the Non-Reversing Taylor Vortex Flow (NRTVF) in which the rotation direction of the Taylor vortices is independent on the inner cylinder rotation direction. They illustrated that if the oscillation amplitude is large enough, it is possible that the resulting time-dependent Taylor vortex rotates in a direction which does not depend on the direction of the azimuthal motion which drives it. The transition to this "nonreversing" Taylor vortex flow is examined as a function of oscillation frequency and radial ratio η. Taylor vortex pairs always rotate in the same direction, despite the inner cylinder driving the flow in the opposite direction. NRTVF takes place at sufficiently high modulation frequencies for which there is not enough time for the toroïdal motion to vanish to sufficiently small values, i.e. these frequencies are small enough so that δ s, δ s = (2υ/ω) 1/2 , where υ is the kinematic viscosity and ω the oscillation frequency, is the thickness of the Stokes layer generated when the oscillating boundary induces a damped viscous wave which penetrates into the fluid, is comparable to the length scale δ = R 2 -R 1. NRTVF can be caused by linear instability [START_REF] Lopez | Modulated Taylor-Couette flow: Onset of spiral modes[END_REF] or finite-amplitude effects [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF]. NRTVF may be affected by the weak Ekman circulation (absent in the calculation of Youd et al., [START_REF] Youd | Reversing and nonreversing modulated Taylor-Couette flow[END_REF]) which is necessarily induced by the fixed top and bottom ends of the Taylor-Couette System. Carmi and Tustaniwskyj [START_REF] Carmi | Stability of modulated finitegap cylindrical Couette flow: linear theory[END_REF] numerically studied oscillatory Couette flow. They did not detect the existence of NRTVF. Their approach was based on studying the effects of infinitesimal perturbations over a cycle, i.e. the Floquet theory. The stability of the basic state can be determined by Floquet analysis, where the Navier-Stokes equations are linearized about the basic state, the linear system with a complete basis for general perturbations can be evolved for a period. More details about the theory are available on [START_REF] Youd | Non-reversing modulated Taylor-Couette flows[END_REF][START_REF] Lopez | Modulated Taylor-Couette flow: Onset of spiral modes[END_REF]. It is possible that NRTVF is due to finite-amplitude effects, whereas in RTVF, the radial velocity becomes negative for part of the cycle, NRTVF decays only to a certain order of magnitude level, at the end of a cycle, never becoming infinitesimal. However, nonreversing solutions have been found by Lopez & Marques [START_REF] Carmi | Stability of modulated finitegap cylindrical Couette flow: linear theory[END_REF] using Floquet analysis for oscillation of the outer cylinder, so NRTVF could be due to a linear instability too. Our work presents an experimental investigation of the effect of oscillatory Couette-Taylor flow, i.e. both the oscillation frequency and amplitude on the apparition of RTVF and NRTVF by analysing the instantaneous and local mass transfer and wall shear rate evolutions, i.e. the impact of vortices at wall.

EXPERIMENTAL INSTALLATION

The experimental installation is composed of an inner cylinder with a radius R 1 =97±0.2 mm and an outer cylinder with a radius R 2 =100±0.2 mm. The corresponding radius ratio is η=0.97 and the aspect ratio is Г=150. The whole apparatus is made in Plexiglas (figure 1a). The inner cylinder was driven by a α-Step motor (marketed by the Omeron company). The electrical motor imposed to the mobile inner cylinder a maximum speed of 120 rpm. It was supplied by a signal generator. The revolutions were controlled by a speed controller. The engine assures to impose sinusoidal motion to the inner cylinder. Polarography technique, known as the limiting diffusion current method, has been used. This requires the use of Electro-Diffusion ED probe which allows the determination of the local mass transfer rate from Limiting Diffusion current measurement delivered by the probe while it is polarized by a polarization voltage. In addition, a series of platinum probes have been mounted flush to the inner surface of the outer cylinder of the Couette Taylor System. ED results are presented for the probe G (see Figure 1a and1b), for different oscillation amplitudes as well as for high and low oscillation frequencies. The wall shear stress has been determined using different solutions mostly used in polarography, such as the Lévêque solution [START_REF] Lévêque | Les lois de transmission de la chaleur par convection[END_REF] and the quasisteady solution of Sobolik [START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids[END_REF] and the inverse method [START_REF] Rehimi | Inverse method for electrochemical diagnostics of flows[END_REF].

POLAROGRAPHY TECHNIQUE

In our tests, we have used a ferri-ferrocyanure of potassium as electrochemical solution with a concentration of 25 mol/m 3 , with a dispersion of 2.5% of Kalliroscope particles to visualize the flow, an access of the sulfate of potassium K 2 SO 4 (130 mol/m 3 ) as supporting electrolyte and 40% of Glycerin to relay the instability apparition (viscosity effect). To attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case, we begin by characterizing the flow for the case where the inner cylinder rotates at constant angular velocity i.e. on steady case. The Taylor number is generally used to characterize modes transition. It describes the competition between the viscous dissipation and the centrifugal force. It is defined as:

int int int Re i cent R Ta R            (1)
where the Reynolds number describes the competition between the viscous dissipation and the inertia effect:

Re a R      (2)
The time-evolution of the Taylor number is presented on figure 2. It shows that the Taylor number is constant and that the average Taylor number is equal to 8 ± 0.4. For this Taylor number, we studied the time-evolution of the mass transfer and the wall shear rates. This allows flow instability detection because the vortices manifest themselves by the apparition of variations on mass transfer and wall shear rates. The dimensionless mass transfer rate is almost constant (figure 3). The average value is equal to 1±0.02. The flow is thus steady. It corresponds to Couette flow. The time-evolution of dimensionless wall shear rate for Ta =8 is presented on figure 4. The dimensionless wall shear rate remains constant. The average value is equal to 1±0.05. The flow is thus stable and no vortices appear. To conclude, on laminar CT flow, the timeevolution of wall shear rate is linear. It may be presented as a linear function of the angular velocity.

For the same average inner cylinder velocity, we studied the effect of oscillations on flow for different amplitude and frequencies modulations.

MODULATION AMPLITUDE EFFECT ON RTVF AND NRTVF

Protocols and Taylor number evolutions

As the Reversing Taylor Vortex Flow RTVF and the Non Reversing Taylor Vortex Flow RTVF strongly depend on the inner cylinder rotation direction, we studied three principle protocols:

 CASE A: The first one corresponds to non-stop advancing inner cylinder i.e. the inner cylinder always advances while oscillating. In this case, the Taylor number remains strictly positive and the modulation amplitude β is strictly lesser than the unit.  CASE B: The second protocol corresponds to advancing -stopped inner cylinder. The Taylor number time-evolution presents then a zero Taylor number when the inner cylinder is stopped and strictly positive values when it advances. In this case, the modulation amplitude β should be equals to the unit. As we can see, for the different oscillatory cases, the time evolution of the Taylor number presents a sinusoidal evolution characterizing the oscillatory flow presence. The average Taylor number is equal to 8 which correspond to Taylor number for the steady case studied on the first section. For example, the Taylor number varies between 12 and 4 (case A). It oscillates between 0 and 16 (case B) and between -31 and 47 (case C.1). So, the average Taylor number is equals to 8 which correspond to Taylor number for the steady laminar Couette flow case studied on the first section. As we have proved that for Taylor number equal to 8, the flow corresponds to laminar Couette Flow. The question now is if the oscillation disturbs the flow enough and accelerates the transition apparition. We propose an answer by analyzing the mass transfer and the wall shear rates. The Sherwood number is defined by:

Mass transfer rates analysis

0 il Sh n F C A D  ( 3
)
Where i is the Limiting Diffusion current delivered by the Electro-Diffusion ED probe, l the probe length, n the ions number, F the Faraday number, C 0 the initial concentration, A the probe area and D the coefficient of Diffusion.

Wall shear rates analysis

Basing on mass transfer evolutions, we have found that, if the oscillation amplitude is large enough, laminar Couette flow can be destabilized, Taylor vortices appears. The vortices direction can be deduced from the sign of the instantaneous wall shear rate time evolution. Even if the limiting diffusion current sign is positive thus the mass transfer sign still positive, the sign of the wall shear rate can be negative which may corresponds to RTVF. In all the oscillatory cases studied, the average Taylor number is equal to 8 which correspond to Taylor number for the steady case corresponding to laminar Couette flow. The impact of vortices, if they appear, on the inner wall of the outer cylinder can be studied via the local and instantaneous mass transfer rate and wall shear rate. The wall shear rate was determined using three approaches. The first one is the "Lévêque solution" [START_REF] Lévêque | Les lois de transmission de la chaleur par convection[END_REF] which relate the local and instantaneous mass transfer to the local and instantaneous wall shear rate:

This method is valid on steady regime, for high Péclet numbers and when the axial diffusion can be neglected. The second approach is the Sobolik et al. [START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids[END_REF] method which is a correction of the Lévêque [START_REF] Lévêque | Les lois de transmission de la chaleur par convection[END_REF] solution:

() 2 ( ) ( ) ( ) 3 q Sob q St S t S t t t        (5)
where:

3 2 () 0.807 q Lev D Sh t SS l     (6)
and:

2 1 2 3 3 3
( ) 0.486 ( ) q t l D S t     (7)
Sobolik et al. [START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids[END_REF] added a term obtained from the unsteady response of the ED probe multiplied by the time derivative of the mass transfer rate. Rehimi et al. [START_REF] Rehimi | Inverse method for electrochemical diagnostics of flows[END_REF] applied the inverse method using simulated data to determine wall shear rate. They tested the sequential estimation for a known wall shear rate of the form:

*( ) 1 cos(2 * ) S t ft       (8)
The instantaneous Sherwood number can be calculated, by integration using the Simpson method, of the equation: The vortices manifest themselves by the presence of timeoscillations on wall shear rate evolution. For low modulation frequency (F = 0.1 Hz), comparing the results of modulation amplitudes β = 0.53 (i.e. nonstop advancing CT inner cylinder) to those of β = 1.08 (i.e. advancing -stop advancing CT inner cylinder), we proved that if the oscillation amplitude is "small" enough, the resulting time-dependent Taylor vortex rotates in a direction which depend on the direction of the azimuthal motion which drives it.

( ) ( ) ( ) ( ) () num num num i num i i i i i St Sh t Sh t S t S t Sh t S              (10) where *

MODULATION FREQUENCY EFFECT ON RTVF AND NRTVF

In this section, we analyze the modulation frequency effect on RTVF and NRTVF apparition by comparing the mass transfer and wall shear rates evolutions for relatively high oscillation frequencies (F = 1000 mHz and F = 2000 mHz) to those of the first section where the modulation frequency is equal to F = 100 mHz. The results presents the Taylor number, the mass transfer and the wall shear rate evolutions for different oscillation amplitudes; for example for the modulation amplitude β = 1.08 (figure 15 For a mean Taylor number equals to 8 and for relatively high modulation frequency (F = 1000 Hz), for a modulation amplitude β = 1.08, even if the inner cylinder stops i.e. the Taylor number is equal to zero in such moments (figure 16), the mass transfer rate remains positive (figure 17) and the wall shear rate (figure 18) is not equal to zero on the appropriate moment, thus the vortices do not disappear, contrarily, they continue to advance. The Taylor vortices direction does not change. Comparing the results for F = 0.1 Hz, F = 1 Hz and F = 2 Hz, we demonstrated that for relatively high oscillation frequencies, the Taylor vortices direction change every half-cycle, i.e. presence of RTVF. There is apparition of Reversing Taylor Vortex Flow (RTVF) in which the rotation direction of the Taylor vortices is dependent on the inner cylinder rotation direction. According to the numerical study of Youd et al., [START_REF] Youd | Non-reversing modulated Taylor-Couette flows[END_REF] for Couette-Taylor system characterized by a radial ratio η=R int /R out =0.75, NRTVF takes place at sufficiently high modulation frequencies for which there is not enough time for the toroïdal motion to change its direction. Thus, Taylor vortex always rotates in the same direction, despite the inner cylinder driving the flow in the opposite direction. However, in our experimental investigation for the study of Couette-Taylor system characterized by a radial ratio η=R int /R out =0.97; we proved experimentally that for frequencies modulation lesser than F=2 Hz and for modulation amplitude lesser than β=9.73; Taylor vortices follow the direction of the inner cylinder i.e. RTVF. The high amplitude accords to vortices sufficient time to follow the direction of the inner cylinder despite the relatively high frequency which seems to be not enough to have NRTVF.

CONCLUSIONS

This paper presents an experimental investigation on oscillatory Couette-Taylor flows. Two experimental techniques were used: visualizations using Kalliroscope particles and the polarography technique known also as the Electro-Diffusion method. The vortices may manifest themselves by the presence of timeoscillations of mass transfer and wall shear rates, this generally corresponds to an instability apparition even for steady rotating cylinder. The wall shear rate was determined using three approaches: the "Lévêque solution" [START_REF] Lévêque | Les lois de transmission de la chaleur par convection[END_REF], Sobolik et al. [START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids[END_REF] and the inverse method [START_REF] Rehimi | Inverse method for electrochemical diagnostics of flows[END_REF]. The results illustrate that if the modulation amplitude is large enough, it can destabilize circular Couette flow. The instantaneous and local mass transfer rates evolutions demonstrate the apparition and development of Taylor vortices which manifest themselves by the presence of oscillations having constant amplitudes for β = 0.53 and β = 1.08; and non constant amplitudes for sufficiently high amplitudes. For frequencies modulation lesser than F = 2 Hz and for modulation amplitude lesser than β = 9.73; Taylor vortices follow the direction of the inner cylinder i.e. Reversing Taylor Vortex Flow (RTVF). The high amplitude accord to vortices sufficient time to follow the direction of the inner cylinder despite the relatively high frequency which seems to be not enough to have Non-Reversing Taylor Vortex Flow (NRTVF).

:

  Inner cylinder radius (m) : Surface of the probe (m 2 ) : Concentration (mol/m 3 ) : Coefficient of diffusion (m 2 .s -1 ) : Frequency of oscillation (Hz) : Height of the CTS : Current (A) : Probe length : Péclet number : Radius (m) : Reynolds number : Wall shear stress (s -1 ) : Sherwood number : Time (s) : Taylor number : Velocity (m.s -1 ) : Coordinates (m) : Amplitude of the oscillation : Angular velocity of the rotating plate (rad.s -1 ) : Kinetic viscosity (m 2 .s -1 ) : δ = R 2 -R 1 (m)

Figure 1 .

 1 Figure 1. (a) Experimental installation; b) Electrochemical probes disposition on the inner wall of the outer cylinder; b) Notations and coordinates of the electrochemical probes used4. STEADY TAYLOR VORTEX FLOW CHARACTERIZATIONTo attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case, we begin by characterizing the flow for the case where the inner cylinder rotates at constant angular velocity i.e. on steady case. The Taylor number is generally used to characterize modes transition. It describes the competition between the viscous dissipation and the centrifugal force. It is defined as:

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Evolution of the Taylor number for steady flow case



  CASE C: The third case corresponds to the case where the inner cylinder advance then moves back then advances again. The Taylor numbers presents thus positives values for a half cycle and negative values for the other half cycle. The modulation amplitude β should be bigger than the unit. We studied several cases depending on modulation amplitudes (bigger than the unit). The time-evolutions of the Taylor number are shown respectively on figure 5 (case A), figure 6 (case B), and figures 7 and 8 (cases C) respectively for modulation amplitudes β β = 4.96 (case C.1) and β = 9.73 (case C.2).

Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 5678 Figure 5. Evolution of the Taylor number for F=100 mHz and β=0.53 (case A)

  Mass transfer rates time-evolution i.e. the Sherwood number for the different oscillatory flow are presented respectively on figure 9 (case A), figure 10 (case B), and figure 11 and 12 (cases C) respectively for modulation amplitudes β = 4.96 (case C.1) and β = 9.73 (case C.2).

Figure 9 .Figure 12 .

 912 Figure 9. Evolution of dimensionless mass-transfer rate for Ta m =8, F=100 mHz and β=0.53

  Sherwood number exp ( *) Sh t calculated via the limiting diffusion current exp () It , allows the numerical determination of the wall shear stress [10]:

  needs a long calculation time for the resolution of the inverse problem in quasi-steady regime, which introduces a difference between the numerical solution and the experimental one at the beginning of the calculation.  For high modulation amplitudes, the inverse method needs sufficiently smoothed signals to converge.The inverse method was validated in previous works. It gives satisfactory results for reversing flows contrary to linear approaches of Lévêque[START_REF] Lévêque | Les lois de transmission de la chaleur par convection[END_REF] and Sobolik et al.[START_REF] Sobolik | Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids[END_REF]. The experimental mass transfer signals were smoothed (filtered) before using it to determine the wall shear rate because the different approaches are very sensitive to noise. The temporal evolutions of wall shear rates, for relatively low modulation frequency F = 100 mHz, are presented respectively for modulation amplitude β = 0.53 and β = 1.08 on figure 13 (case A; β = 0.53) and Figure14(case B; β = 1.08).

Figure 13 .

 13 Figure 13. Evolution of dimensionless mass-transfer rate forTa m =8, F=100 mHz and β=0.53

Figure 14 .

 14 Figure 14. Evolution of dimensionless wall shear rate for Ta m =8, F=100 mHz and β=1.08

  to 17); then for β = 4.96 (case C.2); β = 9.73 (case C.4) for F = 1000 mHz and F = 2000 mHz. This allows us to analyze also the effect of modulation amplitude for high modulation frequencies.

Figure 16 .

 16 Figure 15. Evolution of the Taylor number for Ta m =8, F=1000 mHz and β=1.08 (case B)

Figure 17 .

 17 Figure 17. Evolution of dimensionless wall shear rate for Ta m =8, F=1000 mHz and β=1.08 (case B) The evolution of the Taylor number for Ta m = 8, F = 1000 mHz, β = 4.96 (case C.1) is presented on figure 18. While the evolution of the mass transfer rate and the wall shear rate are respectively presented on figures 19 and 20. The inner cylinder

Figure 19 .

 19 Figure 18. Evolution of the Taylor number for Ta m =8, F=1000 mHz and β=4.96 (case C.1)

Figure 20 .Figure 21 .

 2021 Figure 20. Evolution of dimensionless wall shear rate for Ta m =8, F=1000 mHz, β=4.96 (case C.1)

Figure 22 .

 22 Figure 22. Evolution of dimensionless wall shear rate for Ta m =8, F=1000 mHz and β=9.73 (case C.2)

  Figure 23. Evolution of the Taylor number for Ta m =8, F=2000 mHz and β=9.73 (case 2)

Figure 24 .

 24 Figure 24. Evolution of dimensionless wall shear rate for Ta m =8, F=2000 mHz and β=9.73 (case 2)

  The instantaneous and local mass transfer rates evolutions demonstrate the apparition and development of Taylor vortices which manifest themselves by the presence of oscillations having constant amplitudes for β = 0.53 (case A) and β = 1.08; and non constant amplitudes for sufficiently high amplitudes (cases C); despite that the mean Taylor number corresponds to a steady case i.e. Couette flow. So, we can deduce that oscillations accelerate the transition apparition and thus the critical Taylor number corresponding to the transition from laminar Couette flow to Wavy Vortex flow is lesser than that of the steady case. For high modulation amplitudes, the mass transfer time-evolution has a sinusoidal evolution with non equal oscillation amplitudes. If the oscillation amplitude is large enough, it can destabilize the laminar Couette flow, Taylor vortices appears. The mass transfer rate remains always positive in all the studied cases. The vortices direction can be deduced from the sign of the instantaneous wall shear rate.

ACKNOWLEDGMENTS

This work was supported by the laboratory GEPEA (University and École des Mines of Nantes) and the "Agence Nationale de la Recherche (ANR), France", by the grant n° ANR-08-BLAN-0184-02. These supports are gratefully acknowledged.


domain: nlin spi

text:
B Ducharne

P Tsafack

Y A Tene Deffo

B Zhang

G Sebald

Fractional operators for the magnetic dynamic behavior of ferromagnetic specimens: an overview

Keywords: Magnetic hysteresis, frequency dependence, fractional derivative, anomalous diffusion

This paper reviews the use of the fractional derivative operators for the dynamic magnetization of ferromagnetic specimens. Magnetic behaviors in ferromagnetic specimens are strongly nonlinear and frequency dependent. Magnetism has an atomic origin but the magnetic behavior as observed at the human scale is highly affected by phenomena occurring at larger scales. Under the influence of an external magnetic field, the homogeneity of a ferromagnetic sample magnetization is linked to the excitation dynamics. Models and simulations in this domain are strongly needed, they provide theoretical explanations and allow to anticipate complex phenomena, difficult to observe in a practical way.

On the one hand, such multi-scale dynamical behaviors can hardly be taken into account with the usual mathematical operators, on the other hand, correct simulation results on large frequency bandwidths can be obtained using fractional derivative operators. The use of fractional derivatives can be envisaged through different approaches: Lump models based on time fractional differential equations is one option, fractional anomalous diffusion equations is another. In this manuscript, these two methods are detailed and compared. Theoretical results are compared to experimental ones, and conclusions and perspectives are drawn such as possible improvements.

-Introduction:

Ferromagnetic behaviors and properties have been studied since more than a century but ferromagnetism is extremely complex [START_REF] Bitter | On inhomogeneities in the magnetization of ferromagnetic materials[END_REF]- [START_REF] Bozorth | Ferromagnetism[END_REF]. The majority of electromagnetic devices rely on magnetic conversions, in this domain progresses are continuous. Such understanding is facilitated by numerical simulations based on theoretical concepts [START_REF] Gyselinck | Harmonic-balance finite-element modeling of electromagnetic devices: a novel approach[END_REF] [START_REF] Bouissou | Study of 3D formulations to model electromagnetic devices[END_REF].

The influence of a magnetic excitation dynamics on the magnetization behaviors manifests itself though the frequency dependence of the hysteresis cycle [START_REF] Zirka | Dynamic hysteresis modelling[END_REF]- [START_REF] Petrun | Iron-Loss and magnetization dynamics in non-oriented electrical steel: 1D excitations up to high frequencies[END_REF].

The hysteresis cycle is the standard magnetic signature, it displays the evolution of the magnetic state M as a function of the magnetic excitation H, assuming both these vector quantities in a collinear situation [START_REF] Fiorillo | Measurement and characterization of magnetic materials[END_REF].

Below a given frequency known as the quasi-static threshold, the frequency dependence of the hysteresis cycle is insignificant and the hysteresis shape remains unchanged. Beyond the quasi-static threshold under maximal H imposed conditions, the hysteresis area (also known as the hysteresis losses) grows first up to a maximum value then decreases continuously until a full disappearance under extreme frequency levels when a shield of macroscopic eddy currents prevents magnetic penetration in the tested specimen. On the one hand, the sequence of this two phases is always true, on the other hand due to the complexity of hysteresis and ferromagnetism, this trajectory is always a little different from one sample to another [START_REF] Broner | Dynamical hysteresis without static hysteresis: scaling laws and asymptotic expansions[END_REF].

Under the quasi-static state, it is common to separate the magnetization into a reversible and an irreversible contribution. When the reversible contribution manifests itself through the anhysteretic curve, the irreversible one is linked to the hysteresis area, i.e. the irreversible domain wall motions and the irreversible magnetization rotation. In [START_REF] Hauser | Energetic model of ferromagnetic hysteresis: isotropic magnetization[END_REF], author called these contributions reversible and irreversible work.

The frequency dependence of hysteresis finds its origin in two phenomena: _ the effect of a dynamic excitation on the domain wall motions. Beyond the quasi-static threshold, the varying magnetic excitation induces a ripple effect on the domain wall motions which become larger and faster, generating a surplus of microscopic eddy currents and their corresponding Joules losses. According to the Bertotti's separation losses theory, this surplus can be assimilated to the "Excess losses".

_ the macroscopic eddy currents due to the magnetic excitation variations through the tested specimen [START_REF] Raulet | The magnetic field diffusion equation including dynamic hysteresis: a linear formulation of the problem[END_REF]. This contribution is theoretically expressed through the magnetic field diffusion derived from the Maxwell equations:

2 . dB H dt    (1)
B is the magnetic induction field and σ the electrical conductivity. This frequency dependent contribution is not limited to ferromagnetic materials and can be generalized to all electrical conductive materials.

The simultaneous simulation of these two contributions as observed experimentally for ferromagnetic materials is not straight forward. The kinetic of the magnetic domains interferes with the diffusion process forcing Eq. 1 resolution to divergent results, far from the experimental observations. This distorted magnetic diffusion exhibits unusual answer and can be classify as anomalous.

In [15][16], authors proposed to use alternative mathematical operators found in the framework of the fractional derivatives and adapt Eq. 1 to the simulation of such unfamiliar diffusion processes. By adjusting the fractional order, this alternative simulation method allows to match precisely the experimental observations. But this is not the only way to simulate hysteresis precisely on a large frequency bandwidth, another solution consists on a phenomenological approach, based a lump model: a quasistatic hysteresis model extended to the frequency dependence through the adjunction of a fractional viscous-type dynamic effect. In [START_REF] Zhang | Dynamic magnetic scalar hysteresis lump model, based on JilesAtherton quasi-static hysteresis model extended with dynamic fractional derivative contribution[END_REF][18], this method have been tested with success for the well-known Preisach and Jiles-Atherton (J-A) quasi-static hysteresis models.

In both methods fractional derivative operators bring flexibility in the simulation process. The fractional orders act as additional degrees of freedom which once adjusted provide simulation results of improved accuracy.

In this manuscript, we will review sequentially both methods, providing physical insights and additional perspectives. Descriptions and details of both the simulation methods will be described successively in the second section, with comparisons to experimental measurements in the third one, followed by conclusions and perspectives.

-Simulation methods

The magnetic core loss prediction is of large importance, exhaustive solutions are still absent but results obtained using time fractional derivative operators look promising [START_REF] Ducharne | Anomalous fractional magnetic field diffusion through the cross-section of a massive toroidal ferromagnetic core[END_REF]- [START_REF] Zhang | Preisach's model extended with dynamic fractional derivation contribution[END_REF].

2-1) Fractional anomalous diffusion equation

The concept of anomalous diffusion focuses on specific random processes, which have nonlocal properties of the corresponding dynamics or have a memory wait-time [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF]. For the magnetic field diffusion in ferromagnetic specimens, we are facing the second case: the physical properties behind the magnetic dynamic are spatially homogeneous but the interactions between the microscopic flux variations due to the domain wall kinetic and the macroscopic one due to the magnetic stimulation generate additional lag which can be interpreted as a long-time heavy tail decay. Therefore the adapted diffusion equation is obtained by replacing in Eq. 1 the first order time derivative with a fractional one:

2 . d B H dt      (2)
The additional lag limits the diffusion process which becomes sub-diffusive, α is limited to the 0 < α < 1 interval. Illustration of the resolution of Eq. 2 for different experimental situations:

Cross-section of a ferromagnetic toroidal core (left), lamination (center), can be found in [START_REF] Ducharne | Anomalous fractional magnetic field diffusion through the cross-section of a massive toroidal ferromagnetic core[END_REF] [START_REF] Ducharne | Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination[END_REF]. For all these situations, a space discretization of Eq. 2 left term is done and a time fractional differential equation is solved for each node of the mesh.

Fractional calculus is defined as the branch of mathematical analysis dedicated to study the possibilities to extend derivatives or integrals to noninteger (real or complex) orders.

Fractional derivatives and fractional integrals are hardly distinguishable since unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral. Several definitions can be found in the litterature to describe fractional derivatives. Liouville [START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF]- [START_REF] Machado | Recent history of fractional calculus[END_REF] (Eq. 5), Grünwald-Letnikov [START_REF] Grünwald | Über "begrenzte" derivationen und deren anwendung[END_REF][23], Caputo [START_REF] Caputo | Linear models of dissipation whose q is almost frequency independent-ii[END_REF], are some of them, but the Riemann-Liouville [START_REF] Riemann | Versuch Einer Allgemeinen Auffassung der Integration und Differentiation[END_REF] is the classical one:

  ( ) 1 ( ) ( ) ( ) , (1 )
t d f t d D f t t f d t dt dt                     (3)
Eq. 3 can be understood in terms of convolution between f(t) and t α /Γ(1-α), where Γ(α) is the gamma function. In the Fourier domain, where every signal is harmonic and where a first order derivative is a jω multiplication, fractional derivatives become (jω) α multiplications [START_REF] Guyomar | Time fractional derivatives for voltage creep in ferroelectric materials: theory and experiment[END_REF]- [START_REF] Ducharne | A unique fractional derivative operator to simulate all dynamic piezo ceramic dielectric manifestations: from aging to frequency dependent hysteresis[END_REF].

For a numerical resolution, Eq. 3 is expressed in its discrete form which lead to the direct Grünwald-Letnikov derivative definition:

0 0 ( ) 1 ( ) lim ( 1) ( ) ( 1) ( 1) ( 1 ) k h k d f t D f t f t kh k dt h k k k                                     (4)
The α derivative of a function f(t) at a given time t is a local time property only when α is an integer; this is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional derivative at t of a function f(t) depends only on values of f very near t, in the way that integer-power derivatives certainly do. Eq. 4 denotes weight coefficients which are expressed by:

, ( 1) k k k            (5)
As illustrated in Fig. 1a-e below, these weight coefficients depend strongly on the fractional order. For sub-diffusive time fractional derivative equation (0 < α < 1), the difference between the weight of the first coefficients with the remaining ones increased as α is getting closer to 0 meaning a larger influence of the past in the calculus of a given t time derivative. Eq. 6 below gives the expression of the fractional derivative diffusion equation in its Grünwald-Letnikov expression:

2 0 0 1 .lim ( 1) ( ) ( 1) ( 1) ( 1 ) k h k H B t kh k h k k k                                   (6)
The magnetic state of a ferromagnetic specimen and its variations are strongly dependent on its history. Using fractional derivative operators in the numerical method is coherent as it increases the influence of the past in the simulation of the magnetic behavior at a given time t.

In Fig. 1f is depicted an illustration of the fractional order influence on the balance of the frequency effect. The evolution of the hysteresis area is plotted as a function of the frequency for different values of fractional derivative order.

A decrease of α can be interpreted as a stretch of the hysteresis area <A> vs frequency f curve in the x direction but not only, it also affects the average permeability which becomes lower and reduces the losses maximum values.

2-2) Lump model combined to a fractional viscous-type dynamic effect

A lump-element model simplifies the description of a spatially distributed physical systems into a topology consisting of discrete entities. In our case, the whole ferromagnetic specimen is considered as a magnetic single lump-element, its inputs are external stimuli which can be of magnetic, mechanic or thermal nature and its output is the element magnetic state.

Although the hysteresis cycle is a local information depending of space because of the eddy currents, a lump model (from spatial distribution to discrete entities) is still able to simulate the hysteresis cycle as measured experimentally (i.e. spatially averaged of B as a function of the surface tangent H). In a one hand, the advantages of using a lump model are clear: it limits the simulation approach to the replica of the measured quantities and it excludes any inconvenient due to the conformity of the magnetic state distribution. On the other hand, the physical interpretation is limited since the simulation results are limited to macroscopic average quantities and prohibit access to local information.

A lump approach is proposed by Zirka and al. in [START_REF] Zhang | Simulation tool for the Eddy Current Magnetic Signature (EC-MS) non-destructive method[END_REF], to consider hysteresis in a ferromagnetic specimen:

0 2 2 ( )( ) ( ) 1 m s dB r B H H dt B r B R B                  (7)
r(B) is introduced as a dynamic magnetic resistivity which can supposed to be constant or B dependent, H0 is a threshold field. But this approach gives limited accuracy in a restrained frequency bandwidth, then to improve it, authors proposed the following extension:

  ( ) ( ) ( ) ( ) v st dB t r B H t H B dt   (8)
Where Hst(B) is a quasi-static contribution calculated from a quasi-static hysteresis model (Preisach, Jiles-Atherton …) and v a real number which is supposed to be controlling the model dynamics (i.e. its temporal and frequency properties). In [START_REF] Ducharne | Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination[END_REF][17], authors use similar equation but the balance of the model dynamic is maintained through the use of a fractional derivative operator:

( ) ( ( ))

dyn static d B t H t f B t dt       (9)
Just like the anomalous diffusion equation described in the previous section, adjusting the fractional order gives latitude in the simulation method and allows to match precisely with the experimental results on large frequency bandwidths.

-Experimental validation

Different experimental measuring set-up have been developed to confront the simulation results to experimental ones. Whatever the shape of the specimen tested, the basic experimental principles remain always the same:

_ a large turns number coil supplied with a current controlled power amplifier ensures the magnetic excitation (a magnetic yoke can also be associated to drive the magnetic excitation up to the tested area if necessary).

_ a noise shielded radiometric linear Hall probe positioned ideally as close as possible to the magnetic sample measured the surface excitation magnetic field Hsurf.

_ a wounded search coil ensures the magnetic flux variations monitoring and the specimen local magnetic state evaluation.

On the left hand side of Fig. 2 below is depicted the single sheet tester experimental device we have been using to characterized ferromagnetic laminations. On the right hand side is displayed a micro Helmholtz coil system designed specifically to characterize cylindrical, low diameter specimens. In [17][18] experimental Hysteresis cycles measured on ferromagnetic laminations are compared with success to simulation results obtained with the fractional lump model. Similar comparisons also successful with results coming from the resolution of the fractional anomalous diffusion equation can be found in [START_REF] Ducharne | Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination[END_REF]. The fractional diffusion equation is solved using polar coordinates. The details of the numerical resolution will be the object of a future communication. For both simulation method, the J-A model has been used as quasi-static contribution. The anhysteretic curve is simulated using Eq. 10:

1 tan e anh s e H M M bH a          (10)
The simulation parameters (Fig. 3) have been set using the optimization methods described in details in [15][29]. A complete hysteresis cycle is obtained in less than 4 seconds for the fractional lump model (1 10 6 samples.period -1 ), 14 seconds are necessary for the fractional diffusion equation, with a mesh consisting of 80 nodes. The accuracy of the fractional diffusion equation is higher, in particularly for the magnetic behavior close to the inflection point, just before saturation. It is worth mentioning here that in a recent article [START_REF] Liu | Analytical prediction model of energy losses in soft magnetic materials over broadband frequency range[END_REF], Liu & al. obtained improved results with the lump method by adding a second term, the fractional contribution is restricted to the classic losses (first term) and the excess losses are taken into account according to the Bertotti separation losses theory (second term).

-Conclusions

In this manuscript is reviewed the use of fractional derivative operators for taking into account the frequency dependent magnetization in ferromagnetic specimens. In the first method called fractional magnetic field diffusion equation, the time derivative term of the classic diffusion equation is replaced by a fractional one. The sub-diffusive resulting equation exhibits an additional lag in the diffusion process which can be adjusted by fixing the fractional order.

A space discretized method is used for the resolution of this equation which gives access to local information. In the second method, a lump approach is proposed and the dynamic contribution is considered through the product of a constant ρ to the time fractional derivative of the induction field B. This method is faster in terms of simulation time, but for the cylindrical shape low carbon steel specimens tested in this study the accuracy is lower. Future works and perspectives in this domain include to test the methods on other experimental situations (like a flat surface excited by a pancake coil) to extend the dynamic consideration to alternative excitation sources (mechanical, thermal …) and to progress again with the physical interpretation and meaning of the fractional derivative operators.

Fig. 1 a

 1 Fig. 1 a-e, Weight coefficients. Fig. 1 f, Hysteresis area vs frequency for different fractional orders.

Fig. 2 -

 2 Fig. 2 -Experimental setup for the magnetic characterization of ferromagnetic lamination (left) and cylindrical specimens (right).

Fig. 3

 3 Fig. 3 below shows the comparison of the experimental hysteresis cycles obtained using the

Fig. 3 -

 3 Fig. 3 -Comparison simulation/measurement, for both the fractional diffusion equation and the fractional lump model.

  Tab. 1 -Chemical composition of the low carbon steel specimens tested.

	Element	Fe	Mn	Cu	C	Cr	Ni	Al	Si	P	S	Ti
	Composition (Wt%)	99.539	0.181	0.082	0.075	0.036	0.032	0.023	0.014	0.009	0.007	0.002

Data availability statement:

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