F M Schmidt

N Reuge

Y Le Maoult

J P Arcens

B David

F Abbé

Inflation technique optimisation for an elastomer biaxial characterisation

It can be extremely useful to know biaxial properties of materials like elastomers used for mechanical applications or thermoplastics for the simulation of injection stretch/blow moulding or thermoforming processes. The biaxial stress-strain data of such materials can be obtained using a bubble-inflationtechnique. A circular membrane is clamped at the rim and inflated by a pressure applied to one of its side. The inflation of the circular membrane is recorded using a CCD video camera and the blowing pressure by a pressure sensor. This technique was applied to the determination of an elastomer rheological behaviour, and will be adapted for thermoplastics (including a heating device). First, an experimental setup was carried out, using a thickness measurement magnetic probe, in order to validate rheometer results. Circonferential biaxial Cauchy stress-elongation curve of the elastomer studied was compared with the uniaxial stress-elongation curve.

Introduction

The inflation of uniform circular membrane has been of interest to scientists and engineers for many years. The earliest experimental programs used thin sheets of rubber, both uncured and vulcanised. [START_REF] Treolar | Strains in inflated rubber sheet and the mechanism of bursting[END_REF] used vulcanised latex sheets to study the distribution of principal strains as well as the axial profiles of the inflated sheets and the mechanism of bursting. [START_REF] Rivlin | Large elastic deformation of isotropic materials -VII. Experiments on the deformation rubber[END_REF] used the Neo-Hookean and Mooney forms of the strain-energy functions in the elasticity relations and numerically solved the equilibrium and geometric relations for the inflation of a circular membrane. [START_REF] Alexander | Tensile Instability of Initially Spherical Balloons[END_REF] inflated membranes in order to model tensile instabilities observed with atmospheric balloons.

More recently, viscoelastic sheets have been investigated. [START_REF] Schmidt | Biaxial Stretching of Heat-Softened Plastic Sheets: Experiments and Results[END_REF] developed numerical models using non-linear constitutive equation issued from the rubber-like materials theory for the membrane-inflation problem and compared their results with experimental values obtained with highimpact polystyrene and cellulose acetate butyrate above their respective glass-transition temperatures. [START_REF] Denson | A rheometer for measurement the viscoelasticity response of polymer melts in arbitrary planar and biaxial extensional flow fields[END_REF] studied ABS membranes close to the forming temperature (138°C) using oil for inflation. The incompressibility of the oil allows to proceed to stress retardation experiments where the pressure is imposed. [START_REF] Thomas | Développement et mise au point d'une méthode d'essai de bullage de flans en P.E.T avec mesure vidéométrique[END_REF] inflated circular PET membranes and used a video camera device to study the biaxial elongation of PET above the glass transition temperature. [START_REF] Verron | Contribution expérimentale et numérique au procédés de moulage par soufflage et thermoformage[END_REF] developed a numerical procedure using the bubble height to compute the rheological parameters for a given constitutive equation during the membrane inflation. This numerical approach avoids a complex video camera device and to mark the membrane but the method supposes to know a priori the rheological behaviour.

Our goal is to develop and optimise a rheometer based on an inflation technique similar to [START_REF] Thomas | Développement et mise au point d'une méthode d'essai de bullage de flans en P.E.T avec mesure vidéométrique[END_REF]. For that, we have proceeded to comparisons with the measured thickness of the inflated membrane using the inflation device and a magnetic probe thickness measurement. The inflation rheometer was then applied to the determination of the biaxial rheological properties of an elastomer at ambient temperature (no heating stage).

Bubble inflation assumptions

An uniform circular membrane is clamped at the rim and inflated by a pressure applied to one of its side in order to produce an inflated bubble (fig. 1).

Fig. 1 -Inflated bubble in biaxial extension

The inflation of the circular membrane results in a biaxial extension. The assumption of spherical symmetry allows us to equate   to   at the pole of the bubble. So, the Cauchy stress tensor  expressed in spherical coordinates reduces to:

   rr 0 0 0   0 0 0               (1)
If the thickness of the inflated membrane e is much smaller than the curvature radius R c [START_REF] Schmidt | Modélisation numérique et étude expérimentale du soufflage de bouteilles en P.E.T[END_REF], the thin shell assumption leads to neglect the radial stress  rr in front of the circonferential stress   . In addition, we equate   to the thickness-average circonferential stress which leads to:

   P R c 2e (2)
assuming that the thickness e is uniform at the pole of the bubble. P is the differential inflation pressure. Consequently, we are now able to calculate the circonferential stress by measuring the curvature radius and the thickness.

Bubble inflation technique

The biaxial stress-strain data of an elastomer can be obtained using a bubble-inflation technique. A general view of the bubble inflation rheometer is presented in figure 2.

Fig. 2 -View of the bubble inflation rheometer

A circular membrane is mounted between two metal disks containing a circular aperture (40 mm in diameter) and clamped on a support. Compressed air is then used to inflate the membrane. The inflation of the membrane is recorded using a CCD SONY video camera (752X582 pixels) and a MATROX image grabber. The inflation pressure is recorded using a pressure sensor located close to the back side of the membrane. The curvature radius is measured using video camera images analysis. Details of the experimental technique and the measurement made on the bubbles can be found in [START_REF] Reuge | Mise au point et optimisation d'un rhéomètre équibiaxial pour la caractérisation d'un élastomère[END_REF].

Elastomer membranes were successfully inflated beyong the hemispherical stage to large bubbles (fig. 3). Hemispherical deformation of the inflated membrane occurs only for low circonferential stretch   . Furthermore, the membrane thickness is non-uniform. A preprinted made of a silicone coat provides a reference mark which localises a pole zone where the thickness is assumed to be uniform. Using the volume balance at a local level, the thickness may be calculated using the following relationship:

Applications

First, an experimental set-up was carried out using a thickness measurement magnetic probe, in order to validate rheometer thickness measurement (fig. 4). Fig. 4 -Magnetic probe in contact with the pole The clamping system is reversed. Then, a ball magnet is placed upon the membrane. The membrane is inflated for given pressure values. For each pressure value, the magnetic probe which has been fixed on a rack is bringing in perfect contact with the pole of the reversed bubble. We have proceeded to five measures for each pressure value. The bubble height H is measured using the video camera images analysis. In figure 5, the comparison between the thickness measured using the magnetic probe and the inflation rheometer thickness is presented. The agreement is fair except for large elongation i.e.   4.18 (H≥ 42 mm). In that domain, the uncertainty of the magnetic probe measurement increases due to local variations of the thickness (the sample slides between the clamps). Nevertheless, the weak discrepancy (15%) between the two methods at low and medium elongation allows to validate the inflation rheometer measurement.

We note the existence of an instability zone for 25 mm  H 42 mm. This phenomenon is related to the stress-induced crystallisation of the elastomer during the biaxial elongation [START_REF] Mullins | Effect of stretching on the properties of rubber[END_REF].

Fig. 5 -Comparison between magnetic probe thickness and inflation rheometer thickness versus bubble height

Preliminary results obtained for static inflation are presented now. The pressure is imposed at a given value. The different parameters are calculated using the inflation rheometer (thickness of the inflated membrane, curvature radius, circonferential stress and stretch) The procedure is repeated until the elastomer limit of elongation is reached (before bursting). Circonferential biaxial stress-elongation curve of the elastomer studied is plotted in figure 6 and compared with the uniaxial stress-elongation curve at ambient temperature. Uniaxial and biaxial curves are very closed. This is not the case with generalised stress-strain data (fig. 7). Depending on the type of elongation, the equivalent strain and stress are defined bellow:

   zz ,    zz (uniaxial elongation) (5)   2  ,     (biaxial elongation) (6)

Conclusion

The biaxial rheological properties of an elastomer have been characterised using an inflation rheometer. For this elastomer in particular, the biaxial behaviour is very close to the uniaxial behaviour. But the comparison depends whether we use the equivalent stress and strain or the uniaxial and circonferential stress and strain.

Adding an air convection oven to the present bubble-inflation rheometer will allow to heat polymers above the glass transition temperature and to obtain the biaxial stress-strain data of polymers.

  flat circular membrane thickness. The circonferential stretch is the ratio between the current length of the mark l and the initial length of the mark l o : 288 bar (  = 15 %) P = 0.598 bar (  = 150 %) P = 0.605 bar (  = 325 %) P = 0.799 bar (  = 485 %) Fig. 3 -Intermediate bubble shapes

Fig. 6 -

 6 Fig. 6 -Comparison between biaxial and uniaxial stress-elongation for an elastomer (T= 25°C)


S Monteix

F Schmidt

Y Le Maoult

EXPERIMENTAL STUDY AND NUMERICAL SIMULATION OF PREFORM OR SHEET INFRA-RED RADIATIVE HEATING

Keywords: Contour integration method, control-volume method, heating stage, infrared camera, stretch-blow moulding, thermoforming, view factor computation. 1

A critical stage of both the injection-stretch blow moulding and thermoforming processes is the heating stage, as the final part thickness distribution is strongly dependent on the preform or sheet temperature distribution prior to forming.

Optimisation of the infrared oven is therefore necessary. Experiments have been conducted in order to characterise the heat source of the infrared emitter and the interaction between the heaters and a semi-transparent PET sheet. An 880 LW AGEMA infrared camera has been used to evaluate the surface distribution of the transmitted heat flux by measuring the temperature distribution on the surface of the thermoplastic sheet. In addition, numerical simulations using control-volume method have been carried out and compared with experimental data.

INTRODUCTION

In both the injection stretch-blow moulding and thermoforming processes, a tube-shaped or sheet-shaped thermoplastic preform (PET, PVC, PP,...) must be heated before forming. In order to increase the temperature of the preform above the glass transition, infrared ovens are used [START_REF] Lindenlaub | Thermoformage[END_REF]. Infrared heating permits a rapid heating with an important heat flux (proportional at the fourth power of the source temperature), even on the inner surface far from the incident radiative source.

As the final thickness distribution of the part is dependent on the initial temperature distribution inside the preform, it is very important to investigate influent parameters describing the heating stage. They depend both on the geometrical and spectral interactions between lamps and irradiated plastic materials.

Consequently of the specific interaction, the ratio between electric power and absorbed energy in the PET tubular preform is approximately 20%. Optimisation of the infrared oven is then necessary to allow for a control of the temperature distribution and also for a minimisation of the energy costs. In this way, the following methodology has been applied on the heating of P.E.T sheets :  Characterisation of infrared heaters ;  Measurement of the surface temperature distribution ;  Development of a 3D control-volume model of the heating stage.

CHARACTERIZATION OF THE HEAT SOURCE

The spectral properties of the infrared heaters have already been determined in previous papers [START_REF] Le Maoult | Measurement and Calculation of Preform Infrared Heating , 4 th Int. Workshop on Adv. Infrared Tech. and App[END_REF][START_REF] Monteix | Profiles Infrared Radiative Heating in blow molding and Thermoforming[END_REF]. The infrared lamps used are sketched in Figure 1. They are composed of a coiled tungsten filament, contained in a quartz tubular enclosure and a reflector in order to increase the heat flux efficiency to the product.

The lamp called respectively, L300 and L400 present a nominal electric power of 300W and 700W. The L300 is used with a cylindrical reflector made in polished aluminium. The second one L400 commonly plugged in industrial oven, is fitted with a reflector made of a ceramic coating on the back side of the quartz tube in Figure2.

With a characteristic electrical power in range, and a characteristic resistivity the tungsten filament reaches in steady state, for the L300 at 300W, a temperature about 2300K, and for the L400 used at 400W a temperature about 1700K.

An experimental set-up (see Figure 3), using a thermopile detector (remote sensing heat flux), was developed in order to measure the spatial directivity of the lamp. The flux sensor moves on a half circle around the filament, in order to measure the spatial directivity of each heater. Then it intercepts the radiation, in each direction described by the angle . The thermopile presents a bandwidth defined between 8-14 m. In this spectral range, the quartz tube presents a full absorption of the filament radiation [START_REF] Le Maoult | Measurement and Calculation of Preform Infrared Heating , 4 th Int. Workshop on Adv. Infrared Tech. and App[END_REF]. We assume that the heaters directivity is the same as the quartz tube because of their concentric geometry (the cylindrical filament in the quartz tube).

In Figure 4, the normalised intensity is plotted versus the angle . For all the measurement, the solid angle of the thermopile is centred on the middle of the lamp filament. In comparison with a Lambertian source (diffuse radiation), the Lamp L400 and the L300 used without reflector is similar to an isotropic source (except some border's effects). As expected, we observe that the lamp L300 fitted with a specular reflector exhibits a strong directionality in front of the reflector.

TEMPERATURE DISTRIBUTION MEASUREMENT

Experimental Procedure

Figure 5 shows the experimental set-up, which has been developed in order to measure the surface temperature evolution, when a PET sheet is heating using previous infrared lamp.

Fig. 5-Experimental heating set-up

An 880 LW AGEMA infrared camera (8-12 m bandwidth) is used to evaluate in a remote sensing, the spatial and transient temperature distribution. The surface dimension of the sheet is 20cm x 20cm and the thickness is 1.5 mm. The distance between heaters and sheets is 5 cm. The frequency of analysis is 25 frames per second and the device is connected to a 12 bit data acquisition board drive with a real time software. The camera's objective is oriented normally to the sheet's surface. However, due to the spectral absorbance of the PET [START_REF] Hartwig | Simulation of the Stretch-Blow Molding Process and Characterization of the Relevant Material Behavior[END_REF], we have to take care of this participating media, in order to secure the surface temperature measurement [START_REF] Lallemand | Mesure de Température dans les milieux semi-transparents[END_REF]. To do this, the spectral absorbance of the PET must be characterised previously to determine the bandwidth where the material is assumed to be opaque (see Figure 6). For which, only the radiation emitted from the material surface is intercepted by the camera's sensor, although all the other radiation from the bulk or the sheet environment are absorbed, before reaching the viewed surface [START_REF] Haji | Radiation Pyrometry on Semitransparent Sheets. I: Gray Media[END_REF].

(µm) A (m -1 ) Transparent (1)
Opaque ( 2)

Thickness (2) (1)
Fig. 6-The PET spectral absorbance Previous work deals with PET transmittivity [START_REF] Shelby | Effects of Infrared Lamp Temperature and Other Variables on the Reheat Rate of PET[END_REF], in the spectral range of the heaters intensity, defined by the Wien's law for a used temperature about 2300K. In the bandwidth of the camera (8-12 m) they are no available data. Then in order to observe the PET behaviour, we have assessed reflectivity and transmitivity coefficients using the infrared camera and a laboratory blackbody.

Tab 1-PET's transmittance [8 ;12] m (Black-Body temperature = 150°C) Thickness (mm) sheet 0,7 mm 5 10 -3 1,5 mm 2 10 -3

Tab 2-Normal PET reflectance [8 ;12] m.

Black Body Temperature (°C) n 70 0,06 110 0,07 150 0,07

Measurements have been processed for 0.7 and 1.5 mm PET sheet thickness. Tab 1 and Tab 2 show a very low average transmittivity on this spectral band. The average emissivity is then calculated from the reflectivity measurement.

Taking into account these characteristics, temperature are measured during heating and cooling stage, from the back and the front surfaces of the sheet. It takes 3 seconds to bring the front face in front of the camera, during wich a cooling stage due to the natural convection occurs.

During this transfer a relative difference less than 0.7 % appears in temperature measurements similar to a cooling with a transfer coefficient h equal to 10 Wm -1 K -1 .

In Figure 7, two infrared thermograms are presented for a heating time of 35 seconds. .a shows the surface temperature distribution measured on the back face of the sheet for the heat source L400. The surface temperature distribution is relatively isotropic due to the presence of the diffuse reflector. This results confirms that the Lambertian source assumption (diffuse radiation) is valid for the lamp L400. As expected, the surface temperature distribution for the lamp L300 (specular reflector) is strongly anisotropic (see Figure 7.b).

Results

In this part, thermogams for 1.5 mm PET sheets thickness are examined. Thermograms recorded from the front and the back faces with the same experimental heating conditions provides (Figure 8), the measurement of the temperature gradient, reached across the thickness after an heating time about 35s. This gradient is related to the location of the measured point on the sheet surface in association with the temperature level at this point. The highest gradient (about 8 o C) is reached at the sheet centre (the same as the heater's centre) corresponding to the highest temperature level about 85 o C. The temperature versus time is presented in figure 9, from different z-values (see Fig. 7) on the sheet. The dotted line indicates the end of the heating. These measurements during the heating and the cooling stage, reveal a delay between the heater switch-of and the record of the maximum temperature level from the back face. This delay (about 8s) is less important than the time of the diffusion phenomenon in the semi-infinite sheet submitted to a surface temperature Ts. In this case, the time tdiff required to reach half the initial temperature difference (between Ts and the temperature at a depth e) is equal to

 92 . 0 2 e diff t 
, [START_REF] Agassant | La mise en forme des matières plastiques[END_REF].

The PET diffusivity,  is equal to 1.4.10 7 m 2 s -1 which corresponds to tdiff equal to 17 s. This discrepancy reveals the relative influence of the natural convection, who tends to accelerate the equilibrium of temperature levels across the sheet thickness. In contrary , the spatial temperature distribution after the same equilibrating time presents a larger difference on the z axis. This means the ascendant of the radiative 1D transfer across the thickness on the conduction transfer in 3D.

CONTROL -VOLUME FORMULATION

A 3D control-volume model has been developed for computing radiative heat transfer during the infrared heating stage. The sheet-shape domain sketched in Figure 10, is discretised into cubic elements, called control volumes [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF].

V t dt dV T C                      V r q c q t p  (1)
The temperature balance (Eq.1) including radiative transfer (thermal power absorbed by the polymeric sheet from the infrared heaters) is integrated over each control volume and over the time from t to t+t. V= x y z is the control-volume definition, qc the Fourier's flux

( T k c q    
), qr the internal radiative heat flux. The unknown temperatures are computed at the cell centres. The heat conductivity k, the specific mass  as well as the heat capacity Cp are assumed to be constant, in agreements with previous works [START_REF] Esser | Infrared Radiation in the Processing of Plastics: Precise Adjustment-the key to Productivity[END_REF].

Internal radiative heat flux

Previous works [START_REF] Lebaudy | Etude et Simulation de la répartition des températures dans un cylindre creux de PET soumis a un rayonnement infra-rouge[END_REF][START_REF] Denis | Etude expérimentale et modélisation du chauffage par rayonnement infrarouge de polymères semi-cristallins: influences des centres diffusants[END_REF] dealt PET sheet's heating with a radiant heat flux, uniform all around the radiated face, or assumed an arbitrary intensity without consideration about heaters dimension. Here, the specific lamp geometry is considered, taking into account from a view factor computation. We assume a constant filament temperature, and then an uniform source temperature. Thus the amount of incident radiation, reaching each front elements of the irradiated sheet, is performed with the contour integration method (Eq.2) using Stoke's theorem [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors for Radiation[END_REF] :

s s r H r r d H r d r H S s r H F       ln 2 1  (2)
where the subscripts H refers to the heater and rs to the sheet front element under consideration ; r is the distance between two elemental linear elements, issued from the discretisation of the entire heater border and  rs for each surface of control element's border reaching the irradiated sheet's face. This contour integration method was chosen between different numerical view factor method (Monte-Carlo or geometric method). This method allows a good accuracy at a very low computational time and a simple way of use. An efficient numerical method based on Gaussian quadrature has been applied in order to perform the contour integration.

In order to take into account the participation of the diffuse reflector in the amount of the incident radiation, a coefficient of efficacy krf is introduced. krf is calculated, assuming a strip surface for both heater and his reflector, and considering all the flux emitted from this equivalent surface equal to the flux emitted from the entire filament surface [START_REF] Harmand | Contribution a la modélisation des échanges radiatifs entre un champ d'émetteurs infrarouges et un corps en déplacement[END_REF], i.e. Sfil = d Lfil, where d represents the coiled tungsten diameter equal to 4mm and Lfil is the length about 64mm. Using these assumptions krf is equal to .

Amount of incident Radiation

Thus, the amount of incident radiation (rs) on each surface element is(Eq. 3) :

                      d fil T o Fil T S , 0 , L r H S r H F rf k z y x s r s s (3)
where () is the spectral tungsten emissivity, T F the filament temperature, L o the black body intensity and  a given wavelength between [0,2-10] m.

In a first approximation, internal radiative transfer is assumed to be mono-dimensional across the sheet thickness. The PET bulk temperature ( 400K) is very low in comparison to the source temperature (Ts=1700K). So the assumption of "cold material" is convenient. This leads to express the transmitted flux qr across the thickness (y axis) of the sheet, for an homogenous amorphous PET, using the Beer-Lambert's law in the y direction (Eq.4):

    y r z y x q      -A e z 0, y x, , ,    (4)
where A() is the spectral absorption coefficient of the material (Eq.5) :

A() = -  L   ln (5)
L is the thickness of the PET film used to obtain an intrinsic transmission coefficient () and consequently an accurate A() dependence with wavelength.

The internal source Sp on the principal P control volume is (Eq.6):

Sp =              t d S r q dt t d V y dt S v .  (6)
S are the surfaces of the control volume for which normal are collinear to the assumed main incident flux direction, see Figure 11. Finally the Sp formulation is expressed by (Eq.7):

Sp =               L y L y z y x n s      , 0 , (7)
In a next step, the integration of (Eq.1) over all the control-volumes leads to a linear equations system, which is solved iteratively using the line by line Gauss Seidel method in the y direction of range to obtain a three-dimensional system resolved by the TDMA algorithm.

Validation of the numerical model :

The numerical model is validated for the analytical case of a semi-infinite partially transparent solid [START_REF] Wriendt | La transmission de la Chaleur[END_REF], submitted to an uniform incident heat flux (qo). Its value is evaluated using (Eq.3), considering the entire spectrum and all the irradiated sheet surface (Eq.8).

          4 fil T moy R H F R S H S rf k o q   (8)
In these specific conditions, the temperature distribution across the thickness (y axis) versus time is computed using Laplace's transforms in order to solve the one-dimensional energy equation with the term source per unit volume, qo  (= qo  A e -Ay ) and considering adiabatic borders. A corresponds to an average absorption coefficient (Eq.9), evaluated from (Eq.5) with an average PET's transmittivity [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF] :

                  d fil T o M d fil T o M (9)
The solution is (Eq.10):

                                                t y t A erfc Ay t A e kA o q t y t A erfc Ay t A e kA o q t y ierfc t o q Ay e o q ini T t y T     ( 10
)
where Tini is the initial temperature. Analytical and numerical solutions are compared for a 100x100 mm 2 sheet of 200 mm thickness. Its front surface is subjected to an incident radiation (9) considering a filament temperature about 1700K, corresponding to an average tungsten emissivity about 0.26 and an absorption coefficient of 2660 m -1

, for a wavelength delimited between 0.2 and 10 m. The view factor in this case, between the strip heater and the sheet is evaluated to 0.53.

In Figure 13, comparisons for the temperature versus thickness are shown. The agreement is fair. The error is less than 2%. After the validation of the model, heating of thin sheets is now performed. The predicted temperature using numerical simulations are compared to experimental temperature measurements in the case of a PET sheet of 1.5 mm thickness. The heating time is 35s, corresponding to typical heating times for industrial processes.

Boundary conditions are given below (Eq.14):

  0 , 0 , 0 k - 0 k -                                               H z z T l x x T T S T h e y y
T y y T [START_REF] Harmand | Contribution a la modélisation des échanges radiatifs entre un champ d'émetteurs infrarouges et un corps en déplacement[END_REF] An uniform convection coefficient is applied to the back and the front faces (estimated to 10 W m -2 K -1 ) [START_REF] Sacadura | Initiation aux Transferts Thermiques[END_REF].

The numerical modelisation leads to the following temperature distribution across the sheet thickness, in particular at the sheet centre (see Figure 14). Then, as soon as the heating is switch off, the temperature difference between the front and the back faces rapidly decreases. Five seconds are sufficient to observe the establishment of an equilibrium temperature level all across the thickness. It provides a confirmation of the temperature increase, recorded experimentally from the back face as the natural cooling starts.

The comparison between experimental and numerical results, for the temperature profile versus z axis is plotted in Figure 15. Points localised near the sheet centre and in front of the heater (in a circle of 2cm diameter), the difference between the two temperature results is less than 2%. For the other part of the sheet, the discrepancy is more important. This observation is valid for both front and back faces of the sheet. This difference is firstly due, to the incident flux distribution on the irradiated face, more than the assumption, permitting the flux absorption only normally to the incident surface. Several reasons could explain this inaccurate incident front conditions. In particular the numerical geometry of the strip surface assumed to be equivalent to the radiative cylindrical heater surface. Previous workers [START_REF] Turner | Numerical and experimental Analyses of the Radiant Heat Flux Produced By Quartz Heating Systems[END_REF], have shown a non-negligible quartz tube effect; denoting a refraction of the radiative beam from the tungsten filament centred in the quartz tube. This will lead an increase of the dispersion angle of the radiant heat flux.

In figure 16, the temperature profile versus time is plotted. The same remark is still available with the temperature localisation on the sheet. During the heating stage, numerical temperature evolutions are linear, which do not represent the experimental measurement. Comparisons during the cooling stage, exhibits a more important natural convection transfer for computed results.

CONCLUSION

Preliminary investigations of the infrared heaters properties (spatial and spectral) fitted with diffuse reflectors, allows to consider them as an isotropic radiative source. The interaction between these lamps and PET sheet has been observed, using an infrared camera. The temperature distribution from the back and the front face of the sheet has permit to estimate the gradient temperature at the heating switch off. Advantages of the infrared camera must be emphasise here, offering the capability to record spatial and transient surface temperature distribution, during heating and cooling stage. The heating stage of the PET sheet, submitted to an industrial infrared heater, is taking account with the one dimensional Beer-Lambert law, computed with a control volume model. Considering an efficiency reflector coefficient, the comparison between experimental and numerical results presents discrepancies. Further developments will permit, to take better into account the real interaction between PET preform and infrared heaters and the oven's environment :

 defining the reflector efficiency  observing transfer convection  modelling tubular PET preforms

Fig. 1 Fig. 2 -

 12 Fig. 1-Infrared heaters

Fig. 3 -Fig. 4 -

 34 Fig. 3-Experimental set-up for infrared heaters characterisation.

Figure 7 -

 7 Figure 7-a : Th. with diffuse reflector.Figure 7-b : Th. with specular reflector

Figure 7 -

 7 Figure 7-a : Th. with diffuse reflector.Figure 7-b : Th. with specular reflector Figure 7.a shows the surface temperature distribution measured on the back face of the sheet for the heat source L400. The surface temperature distribution is relatively isotropic due to the presence of the diffuse reflector. This results confirms that the Lambertian source assumption (diffuse radiation) is valid for the lamp L400. As expected, the surface temperature distribution for the lamp L300 (specular reflector) is strongly anisotropic (see Figure 7.b).

Fig. 8 -Fig. 9 -

 89 Fig. 8-Measurement of the temperature gradient after heating stop

Fig. 10 -

 10 Fig.10-The sheet shape mesh

YFigFig. 11 -

 11 Fig Bilan radiatif local s n

Fig. 13 -

 13 Fig. 13-Comparison between numerical and analytical temperatures versus thickness

Fig. 14 -

 14 Fig.14-Numerical Temperature distribution across the thickness During the heating time, a constant temperature gradient across the thickness, equal to 7 o C appears. Then, as soon as the heating is switch off, the temperature difference between the front and the back faces rapidly decreases. Five seconds are sufficient to observe the establishment of an equilibrium temperature level all across the thickness. It provides a confirmation of the temperature increase, recorded experimentally from the back face as the natural cooling starts.The comparison between experimental and numerical results, for the temperature profile versus z axis is plotted in Figure15.

Fig. 16 -

 16 Fig.15-Gradient Temperature

Acknowledgements

This work is supported by PERRIER-VITTEL COMPANY. Special thanks go to G. Denis, A. Contal and J-P. Arcens.


S Monteix

F M Schmidt

Y Le Maoult

G Denis

ANALYSIS OF INFLUENT PARAMETERS DURING INFRARED RADIATIVE HEATING OF PET PREFORM

The processing parameters during radiative heating of PET preform are:

 The number of infrared heaters N h ,  The temperature T i of each infrared heater L i1, N h  The heating time t h ,  The cooling time t c ,  The coordinate vector x i1,N h of each infrared heater L i1, N h ,  The heat transfer coefficient h c between cooling air and PET preform.

In previous papers [START_REF] Monteix | Experimental Study and Numerical Simulation of Preform or Sheet Infrared Radiative Heating[END_REF][START_REF] Monteix | Profiled infrared radiative heating in blow moulding and thermoforming[END_REF], the spectral emissivity and directivity of different halogen lamps (Philips 300-700W electric nominal power made of a coiled tungsten filament, contained in a quartz tubular enclosure and a diffuse reflector made of a ceramic coating) have been reported. Recent measurements using a 0.6-25 m thermopile sensor on similar halogen lamps of 1000W (used in an industrial oven of injection-blow moulding machine) have been processed. The maximum temperature of the tungsten filament is 2400 K. As shown in fig. 1, these infrared heaters exhibit a slightly different behaviour from a lambertian source (diffuse radiation) due to the reflector. The average discrepancy is about 8 %. In order to take into account the participation of the diffuse reflector in the amount of the incident radiation, a coefficient of efficacy k rf is introduced. Fig. 2 shows the experimental set-up that has been developed in order to measure the surface temperature distribution (front face and back face), when a PET sheet is heating using previous infrared heaters. An 880 LW AGEMA infrared camera (8-12 m bandwidth) is used to measure the spatial and transient temperature distribution. The surface dimension of the PET sheet is 20cm  20cm cm and the thickness is 1.5 mm. A 3D control-volume model has been developed for computing radiative heat transfer during the infrared heating stage. The sheet-shape domain sketched in fig. 3 is discretised into cubic elements, called control-volumes [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF]. The energy equation including radiative transfer integrated over each control volume V e  x y z and over the time from t to t  t leads to:

* Corresponding author c p T t d dt  k  T.n    e  t   e  t  d dt   r .n ddt  e  t  (1)
Where  is the specific mass, c p the heat capacity, k the heat conductivity.  e is the surface of control-volume V e . The assumption of "cold material" is convenient, as the PET bulk temperature  120C



 is very low in comparison to heaters temperature 1500 2000 C  . Consequently, the net radiative transfer over each control yields:

 r .n d  e   k rf F  i  e   T i   L  0 T i       y 1 e   
Where e is the thickness of the PET film used to obtain an intrinsic transmission coefficient   ,   is the spectral tungsten emissivity, L  0 is the black body intensity. The diffuse view factor F is computed using the contour integration method [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors for Radiation[END_REF].

The integration of equation ( 1) over all the control-volumes leads to a set of linear algebraic equations, which is solved iteratively using the line by line Gauss-Seidel method in the y-direction to obtain a 3D-system computed by the TDMA algorithm.

An example of PET sheet heating has been processed. The dimensions of the sheet as well as the halogen lamps are sketched in fig. 4. The thickness of the sheet is 3mm. The processing parameters for the infrared heaters are summarised in the table below. It is to be noted that in this simplified case, the distance between the lamps and the front face of the sheet is constant and equals to 50mm.

T i (K) x i (mm)

  t h is 35 s and the cooling timet c is 25s. A constant heat transfer coefficient of 10 W m 2K 1 is applied to the back and front faces of the sheet. The temperature distribution of the front face of the sheet is plotted as shown in fig.5. In addition, fig.6shows the temperature distribution throughout the thickness versus time.The temperatures of the central lamps L i 2,3,4 have been decreased in order to improve the temperature distribution on the front face. Simulations results will be further discussed during the podium presentation.

Figure 1 :Figure 2 :

 12 Figure 1: Halogen lamps (1000 W) directivity

Figure 3 Fig. 4 :Fig. 5 :Fig. 6 :

 3456 Figure 3: 3Dcontrol-volumes


K Langueh

G Zheng

T Floquet

Impulsive fixed-time observer for linear time-delay systems

published or not. The documents may come

Impulsive fixed-time observer for linear time-delay systems

Kokou Langueh, Gang Zheng, Thierry

Introduction

Time-delay systems describe a wide range of dynamic processes arising often in chemical, biological and economic applications. The observer design problem of linear system without delays has already been solved in [START_REF] Luenberger | Observer for multivariable systems[END_REF], [START_REF] Darouach | Full-order observers for linear systems with unknown inputs[END_REF], [START_REF] Hostetter | Observing systems with unmeasurable inputs[END_REF], [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF], [START_REF] Kudva | Observers for linear systems with unknown inputs[END_REF], [START_REF] Yang | Obervers for linear systems with unknown inputs[END_REF] and [START_REF] Wang | Observing the states of system with unmeasurable disturbances[END_REF]. However, for time-delay systems which are used sometimes to model practical applications [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF], [START_REF] Zheng | Secure data transmission based on multi-input multi-output delayed chaotic system[END_REF], the observer design problem becomes not trivial when the system states and the output contain commensurate delays.

Observability and identifiability are analyzed in [START_REF] Bejarano | Observability of linear systems with commensurate delays and unknown inputs[END_REF] for linear time-delay systems and in [START_REF] Zheng | On observation of time-delay systems with unknown inputs[END_REF] and [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF] for nonlinear one. Concerning the observer design for time-delay systems, different techniques have been proposed in the literature, such as infinite dimensional approach [START_REF] Salamon | Observers and duality between observation and state feedback for time delay systems[END_REF], polynomial approach based on the ring theory [START_REF] Sename | Unknown input robust observer for time delay system[END_REF]; [START_REF] Emre | Regulation of split linear systems over rings: Coecient-assignment and observers[END_REF]; [START_REF] Bejarano | Observability of singular systems with commensurate time-delays and neutral terms[END_REF]; [START_REF] Zheng | Observer design for linear singular time-delay systems[END_REF], Lyapunov function based on LMI [START_REF] Darouach | Linear functional observers for systems with delays in state variables[END_REF]; [START_REF] Seuret | A sliding mode observer for linear systems with unknown time varying delay[END_REF] and so on. Recently, some new inequality techniques have been addressed in Chen et al. (2017a) and Chen et al. (2017b) which can be also used to design observers.

All those mentioned observers in the literature provide only asymptotic (or exponential) estimation. However, in some applications it is desired to have a faster estimation, for example in a prescribed finite time [START_REF] Shi | Stochastic finite-time state estimation for discrete time-delay neural networks with markovian jumps[END_REF], which is named as non-asymptotic convergence. When designing state feedback control for nonlinear systems, the non-asymptotic estimation enables us to avoid the closed loop analysis for the observer and controller, even if the separation principle is generally not satisfied for nonlinear systems. Non-asymptotic estimation has a long history, and many different approaches have been proposed in the literature, such as algebraic method [START_REF] Sira-Ramrez | Algebraic Identification and Estimation Methods in Feedback Control Systems[END_REF] and sliding mode technique [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF]. Usually the nonasymptotic convergence, when applying sliding mode method, depends on initial conditions [START_REF] Zheng | A simple finite-time observer for linear time-delay systems[END_REF]. Some extensions have been done to provide the uniform non-asymptotic convergence with respect to the initial conditions [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], named as 'fixed-time'. Inspired by the works of [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF]; [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF] for linear systems, where the authors combined two simple Luenberger observers to construct a fixed-time observer, where the fixed time can be arbitrary assigned, this paper utilizes this idea to treat linear time-delay systems with known inputs. As an extension of [START_REF] Langueh | Finite-time observer for linear system with time delay[END_REF], the fixed-time convergence of the proposed observer is achieved by updating the observers with an impulsive manner.

This paper adopts the method based on ring theory since it enables us to use some useful techniques developed for linear systems without delays [START_REF] Zheng | Unknown input observer for linear time-delay systems[END_REF]. The following notations will be used in this paper. R is the field of real numbers. The set of positive integers is denoted by N. I p means p × p identity matrix. R[δ ] is the polynomial ring over the field R and R n [δ ] is the R[δ ]-module whose elements are vectors of dimension n and whose entries are polynomials. By R q×s [δ ] we denote the set of matrices of dimension q × s, whose entries are in R

[δ ]. For a matrix M(δ ), rank R[δ ] M(δ ) means the rank of the matrix M(δ ) over R[δ ].
The maximal order of δ in all polynomials in the matrix M(δ ) is denote deg δ M(δ ) We denote Inv S [M(δ )] = {Ψ i (δ )} 1≤i≤r as the set of invariant factors of the Smith form of M(δ ).

Problem statement

In this paper, we consider the following class of linear systems with commensurate delays:

ẋ(t) = ∑ k a i=0 Āi x(t -ih) + ∑ k b i=0 Bi u(t -ih) y(t) = ∑ k c i=0 Ci x(t -ih) (1)
where the vector x(t) ∈ R n , the input vector u(t) ∈ R m , and the output vector y(t) ∈ R p , the initial function ϕ(t) is a piece-wise continuous function ϕ(t) :

[-kh, 0] → R n (k = max{k a , k b , k c })
where h represents the basic commensurate delay; thereby x(t) = ϕ(t) on [-kh, 0]. Āi , Bi and Ci are matrices of appropriate dimension.

In order to simplify the analysis, let us introduce the delay operator δ such that x(tkh) = δ k x(t), k ≥ 0. Let R[δ ] be the polynomial ring of δ over the field R. After having introduced the delay operator δ , system (1) can be written as follows:

ẋ(t) = A(δ )x(t) + B(δ )u(t) y(t) = C(δ )x(t) (2)
where

A(δ ) = ∑ k a i=0 Āi δ i , B(δ ) = ∑ k b i=0 Bi δ i and C(δ ) = ∑ k c
i=0 Ci δ i . As for x(t, ϕ, u), we mean the solution of system (2) with the initial functions equal to ϕ(t) and the inputs equal to u. In the same way, we define y(t, ϕ, u) = C(δ )x(t, ϕ, u), which is the output of system (1) when x(t) = x(t, ϕ, u).

When designing observers for time-delay system (2), we can only use y(t) and its delayed values to estimate the current x(t). Otherwise, the designed observer is not causal. In this sense, it is desired to use actual and past information of measurement. For this, let us firstly recall the definition of backward observability stated in [START_REF] Bejarano | Observability of linear systems with commensurate delays and unknown inputs[END_REF].

Definition 1. System (1) is said to be backward observable on [t 1 ,t 2 ] if and only if for each τ ∈ [t 1 ,t 2 ], there exist t 1 < t 2 ≤ τ such that for every initial condition ϕ, y(t, ϕ, 0) = 0 for all t ∈ [t 1 ,t 2 ] implies x(τ, ϕ, 0) = 0.
Concerning systems with time delays, we have also the following definition of asymptotic and non-asymptotic observers [START_REF] Zheng | Finite-time-observer design for nonlinear impulsive systems with impact perturbation[END_REF].

Definition 2. The dynamics

ξ = f (y, ξ , δ , u) x = g(y, ξ , δ , u)
with y being the output of (2), ξ ∈ R ñ, x ∈ R n where ñ ≥ n and some user-chosen functions f and g, is called as an asymptotic observer of (2) if

lim t→∞ || x(t) -x(t)|| = 0.
It is said to be a finite-time observer if there exists T s (e 0 ), depending on the initial observation error e 0 = x(t 0 )x(t 0 ), such that

|| x(t) -x(t)|| = 0, ∀t ≥ T s (e 0 ).
In addition, this dynamics is named as a fixed-time observer if

|| x(t) -x(t)|| = 0, ∀t ≥ T s
where T s is independent of any initial observation error.

For system (2), [START_REF] Hou | An observer design for linear time-delay systems[END_REF] proposed a simple Luenberger-like asymptotic observer and sufficient conditions were deduced to calculate the gains of the proposed observer. To the best of our knowledge, no results on fixed-time observer for system (2) have been studied in the literature. As an extension of [START_REF] Bejarano | Observability of linear systems with commensurate delays and unknown inputs[END_REF], the main contribution of this paper is to extend the methods proposed in [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF] and [START_REF] Hou | An observer design for linear time-delay systems[END_REF] to treat system (2), and to deduce some sufficient conditions which enable us to present a constructive procedure to design a simple impulsive fixed-time observer.

Definition and preliminary results

Consider system (2) which is described by the polynomial matrices over R[δ ], let us firstly give some useful definitions of unimodularity and change of coordinates over R[δ ].

Definition 3. For a given polynomial matrix A(δ

) ∈ R n×q [δ ], it is said to be left (or right) unimodular over R[δ ] if there exists A -1 L (δ ) ∈ R q×n [δ ] with n ≥ q (or A -1 R (δ ) ∈ R n×q [δ ] with n ≤ q), such that A -1 L (δ )A(δ ) = I q (or A -1 R (δ )A(δ ) = I n ). A square matrix A(δ ) ∈ R n×n [δ ] is said to be unimodular over R[δ ] if A -1 L (δ ) = A -1 R (δ ). Let us recall that for any polynomial matrix D(δ ) ∈ R p×m [δ ] with rank R[δ ] D(δ ) = k ≤ min{p, m}, it is known (see Hou et al. (2002)) that D(δ ) ∈ R p×m [δ ] is left unimodular over R[δ ] if and only if rank R[δ ] D(δ ) = m ≤ p and Inv S [D(δ )] ⊂ R. Definition 4. A change of coordinates z = T (δ )x with T (δ ) ∈ R n z ×n [δ ] is said to be bicausal if T (δ ) is left unimodular over R[δ ].
Then, we can define the following polynomial matrix over R[δ ]:

O l (δ ) =      C(δ ) C(δ )A(δ ) . . . C(δ )A l-1 (δ )      ∈ R pl×n (3)
where l ∈ N, and make the following assumption.

Assumption 1. It is supposed that there exists a least integer l

* ∈ N such that O l * (δ ) defined in (3) is left unimodular over R[δ ], i.e., rank R[δ ] O l * (δ ) = n and Inv S O l * (δ ) ⊂ R.
It is clear to see that if Assumption 1 is satisfied, then system (2) is backward observable [START_REF] Hou | An observer design for linear time-delay systems[END_REF]. In other words, the above assumption guarantees that the studied system (2) is backward observable, then we can design a fixed-time observer to estimate x(t), which will be presented in the next section.

Impulsive fixed-time observer

This section proposes an impulsive fixed-time observer based on two simple coupled Luenberger-like observers. Before this, let us recall a useful result stated in [START_REF] Hou | An observer design for linear time-delay systems[END_REF].

Lemma 1. [START_REF] Hou | An observer design for linear time-delay systems[END_REF] There exists a bicausal change of coordinates z = T (δ )x which transforms system (2) into the following observable normal form:

ż = A 0 z + F(δ )y + T (δ )B(δ )u y = C 0 z (4)
where the matrices

F(δ ) = [F T 1 (δ ), . . . , F T l * (δ )],
and

A 0 =       0 I p 0 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • . . . 0 0 0 • • • 0       ∈ R pl * ×pl * C 0 = [ I p 0 0 • • • 0 ] ∈ R p×pl * (5)
if and only if there exists a least integer l * ∈ N such that O l * (δ ) defined in (3) is left unimodular over R[δ ]. Moreover, the bicausal change of coordinates z = T (δ )x with T (δ ) = col{T 1 (δ ), . . . , T l * (δ )} is defined as follows:

T 1 (δ ) = C(δ ) T i+1 (δ ) = T i (δ )A(δ ) -F i (δ )C(δ ), for 1 ≤ i ≤ l * -1 (6)
with F i (δ ) being determined through the following equation:

[F l * (δ ), . . . , F 1 (δ )] = C(δ )A l * (δ )[O l * (δ )] -1 L . (7)
The above lemma shows that if Assumption 1 is satisfied, then (2) can be transformed into (4) via z = T (δ )x where T (δ ) is defined in (6). For the obtained observer normal form (4), consider now the following dynamics:

ż = Lẑ + P(δ )y + T (δ )B(δ )u x = T -1 L (δ )ẑ (8)
where L ∈ R pl * ×pl * and P(δ ) ∈ R pl * ×p [δ ] will be determined hereafter. Then the estimate error ε z = zẑ is governed by:

εz = ż -ż = A 0 z + F(δ )y + T (δ )B(δ )u - ż = A 0 z + F(δ )C 0 z -[Lẑ + P(δ )C 0 z] = [A 0 + F(δ )C 0 -P(δ )C 0 ]z -Lẑ
Since (A 0 ,C 0 ) is observable, thus there exists a constant matrix J such that [A 0 -JC 0 ] is Hurwitz. By choosing P(δ ) = J + F(δ ) and L = A 0 -JC 0 , then we obtain

εz = [A 0 -JC 0 ]ε z (9)
which is asymptotically convergent to zero. We would like to emphasize that this convergence is independent of delay. Note that ε x = T -1 L (δ )ε z , then ε x converges to zero as well asymptotically. According to Definition 2, the dynamics ( 8) is an asymptotic observer of system (2).

The following will use two simple observers of type (8) to construct an impulsive fixed-time observer for system (2). For this, let us choose two different constant matrices J i such that both [A 0 -J i C 0 ] are Hurwitz for i = 1, 2. Then we note

L i = A 0 -J i C 0 ( 10
)
and

P i (δ ) = J i + F(δ ) (11)
for i = 1, 2.

Define d max = deg δ P i (δ ) for 1 ≤ i ≤ 2 as the maximum degree of polynomials P i (δ ) with respect to δ . With those matrices, for a prescribed positive constant σ > h × d max where h represents the basic commensurate delay in (1), we can then design the following two coupled observers:

           żi = L i ξ + P i (δ )y + T (δ )B(δ )u, ẑ = Γ(δ ,t) ẑ1 ẑ2 ẑ1 (t) = ẑ2 (t), ∀t ∈ [-h × d max , 0] x = T -1 L (δ )ẑ (12)
where

Γ(δ ,t) =      [ 1 2 I pl * , 1 2 I pl * ], if t = σ I pl * -e (L 2 -L 1 )t -1 [-e (L 2 -L 1 )t , I pl * ], if t = σ (13)
Theorem 1. Suppose that Assumption 1 is satisfied for system (2). For a given prescribed constant σ > h × d max , the dynamics ( 12) is an impulsive fixed-time observer of system (2) with the prescribed settling time σ , i.e. ||x(t)x(t)|| = 0 for all t ≥ σ .

Proof 1. It has been shown in the above that if Assumption 1 is satisfied for system (2), then one can always find two constant matrices J 1 and J 2 such that both

[A 0 -J 1 C 0 ] and [A 0 -J 2 C 0 ] are Hurwitz. Note ε z 1 = z -ẑ1 and ε z 2 = z -ẑ2 , then according to (9) we obtain εz 1 = [A 0 -J 1 C 0 ]ε z 1 = L 1 ε z 1 ( 14
)
and

εz 2 = [A 0 -J 2 C 0 ]ε z 2 = L 2 ε z 2 (15)
By solving ( 14) and (15), we get

ε z 1 (t) = [z(t) -ẑ1 (t)] = e L 1 t ε z 1 (0) ε z 1 (t) = [z(t) -ẑ2 (t)] = e L 2 t ε z 2 (0) (16)
Due to the fact that the initial functions for ẑi (t) for all t ∈ [-h × d max , 0] are the same, thus ẑ1 (0) = ẑ2 (0) which implies ε z 1 (0) = ε z 2 (0). Therefore, by eliminating ε z 1 (0) and ε z 2 (0) in ( 16), we obtain

e L 2 t [z(t) -ẑ1 (t)] = e L 1 t [z(t) -ẑ2 (t)] which gives z(t) = [I pl * -e (L 2 -L 1 )t ] -1 [-e (L 2 -L 1 )t , I pl * ] ẑ1 (t) ẑ2 (t)
When t = σ , the above equation is equal to

z(σ ) = Γ(δ , σ ) ẑ1 (σ ) ẑ2 (σ ) .
According to the impulsive behavior defined in (14), we have

ẑ1 (σ ) = ẑ2 (σ ) = z(σ ) thus x(σ ) = T -1 L (δ )z(σ ) = x(σ )
, which implies that the exact value of x(t) has been obtained at time σ . Finally, we always have ||x(t)x(t)|| = 0 when t ≥ σ .

Therefore, according to Definition 2, system (12) is an impulsive fixed-time observer for (2) with the prescribed settling time σ independent of any initial condition.

Remark 1. Clearly, the result presented in Theorem 1 is based on the results stated in [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF] and [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF]. In [START_REF] Raff | An impulsive observer that estimates the exact state of a linear continuous-time system in predetermined finite time[END_REF], two simple Luenberger observers were coupled together to achieve the finite-time estimation for linear time-invariant systems, while [START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF] proposed a Luenberger observer to achieve asymptotic estimation of linear time-delay systems. Compared to those results, Theorem 1 presented in this paper introduces a constructive way to achieve finite-time estimation for a class of time-delay systems.

If Assumption 1 for a certain l * ∈ N is satisfied for system (2), then the following procedure presents how to design the proposed fixed-time observer:

Step 1: Transform system (2) into (4) and deduce A 0 and C 0 defined in (5);

Step 2: Calculate O l * (δ ) defined in (3) and its left unimodularity

[O l * (δ )] -1 L ;
Step 3: Deduce T (δ ) and F(δ ) according to ( 6) and ( 7);

Step 4: Choose two different constant matrices J 1 and J 2 such that L 1 and L 2 defined in (10) are Hurwitz, then calculate P i (δ ) according to (11);

Step 5: Finally we obtain Γ(δ ,t) according to (13) for a prescribed σ ≥ h × d max , and observer ( 12) can be then designed.

Illustrative example

Let us consider the following example of the form (2) with

A(δ ) =   δ 2 1 δ δ δ 1 + δ 1 δ δ 2   , B(δ ) =   1 δ 0 1 1 + δ 2 1   and C(δ ) = 1 0 0 δ 0 1 .
It can be checked that there exists l * = 2 such that Assumption 1 is satisfied. Therefore, the studied system is backward observable. In order to design the proposed impulsive fixed-time observer, we can follow the proposed procedure.

Step 1: According to Lemma 1, we obtained

A 0 =     0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0     and C 0 = 1 0 0 0 0 1 0 0
Step 2: According to (3), we compute :

O l * (δ ) = C(δ ) C(δ )A(δ ) =     1 0 0 δ 0 1 δ 2 1 δ δ 3 + 1 2δ 2δ 2    
and its left unimodular matrix over R[δ ] is equal to:

[O l * (δ )] -1 L =   1 0 0 0 0 -δ 1 0 -δ 1 0 0   .
Step 3: According to (7), we can deduce F(δ ) as follows:

F(δ ) =     2δ 2 + δ 0 3δ 3 + 2δ 2 + 1 0 -δ 4 -δ 2 + δ -δ 2 + δ + 1 -2δ 2 (δ 3 + δ -1) 2δ (-δ 2 + δ + 1)    
which gives us, based on (6), the following T (δ ):

T (δ ) =     1 0 0 δ 0 1 -δ 2 -δ 1 δ -2δ 3 -2δ 2 2δ δ 2     whose left unimodular matrix over R[δ ] is: T -1 L (δ ) =   1 0 0 0 2δ 2 + δ -δ 1 0 -δ 1 0 0   .
Step 4: Then, we can freely choose J 1 and J 2 such that L 1 and L 2 defined in (10) are stable. In this example, we take

J 1 =     6 0 0 12.5 8 0 0 37.5     and J 2 =     2.4 0 0 5 1.28 0 0 6     which set
the eigenvalues of L 1 as [-7.5, -5, -4, -2] and those of L 2 as [-3, -2, -1.6, -0.8].

Finally we have:

P 1 (δ ) =     2δ 2 + δ + 6 0 3δ 3 + 2δ 2 + 1 12.5 -δ 4 -δ 2 + δ + 8 -δ 2 + δ + 1 -2δ 2 (δ 3 + δ -1) 2δ (-δ 2 + δ + 1) + 37.5     and P 2 (δ ) =     2δ 2 + δ + 2.4 0 3δ 3 + 2δ 2 + 1 5 -δ 4 -δ 2 + δ + 1.28 -δ 2 + δ + 1 -2δ 2 (δ 3 + δ -1) 2δ (-δ 2 + δ + 1) + 6     which yields d max = deg max P i = 4.
Step 5: With a prescribed positive constant σ > 4h, we can easily design an impulsive fixed-time observer of type ( 12) to achieve a fixed-time estimation of x(t). For the simulation, the sampling time is set as 0.0005s and the basic delay h = 0.001s. The estimated x(t) of system ( 2 From Figs. 1-3, we can see that trajectories of the proposed observer converge exactly at the prescribed time σ = 0.3s to those of system (2). From Figs. 4 -6, the same conclusion can be made for the case σ = 0.5s.

Conclusion

In this paper, an impulsive observer with prescribed fixed-time convergence for linear systems with commensurate delay has been presented, which can involve the delays both in the state and in the output. It has been shown that the proposed observer converges in fixed time and the convergence time σ can be chosen almost arbitrary provided that σ > h × d max . Numerical simulations are provided for an academic example for different choices of σ in order to highlight the feasibility of the proposed method.

  ) is depicted in Figs. 1 -3 when σ = 0.3s and in Figs.4 -6when σ = 0.5s.

Fig. 1 :

 1 Fig.1:x 1 and its estimation x1 when σ = 0.3s. Fig.2:x 2 and its estimation x2 when σ = 0.3s.

Fig. 3 :Fig. 5 :

 35 Fig. 3: x 3 and its estimation x3 when σ = 0.3s. Fig. 4: x 1 and its estimation x1 when σ = 0.5s.


Serge Monteix

F Schmidt

Y Le Maoult

G Denis

M Vigny

COMPARISON BETWEEN A NUMERICAL MODEL AND AN EXPERIMENTAL APPROACH OF PREFORM INFRARED RADIATIVE HEATING -RECENT RESULTS

The injection stretch-blow moulding process of thermoplastic bottles requires an heating step before forming. An amorphous preform is heated above the glass transition temperature (80°C for the P.E.T.) using an infrared oven. This step is fundamental in order to determine the thickness distribution along the preform height and then insure high quality bottles. Thus, the optimisation of the infrared oven is necessary. Various experiments have been conducted to characterise the heat source and the semi-transparent properties of the P.E.T. Measurements of air temperature inside the infrared oven and air cooling speed have been processed. These parameters have been implemented in a control volume software that simulates the heating step. The surface temperature distribution of the preform has been measured using an infrared camera. Comparisons between experimental and numerical results for a rectangular sheet and a rotating preform are presented.

Introduction

The injection stretch-blow moulding process of thermoplastic bottles requires a heating step before forming. An amorphous tube-shaped preform is heated above the glass transition temperature (80°C for the P.E.T.) using an infrared oven. As the P.E.T. conductivity is low (0.29 W/m.K), halogen lamps permit a rapid heating with a high flux intensity. In addition, a cooling fan is necessary in order to prevent P.E.T from thermal crystallisation that will induce nonamorphous preform. However, there are plenty of parameters such as number of lamps, distance between halogen lamps and preform, electrical power of the lamps, air cooling speed, reflectors, rotation speed, Therefore optimising infrared oven in order to design new plastic bottles is still a long and costly work. The main objective of this work was to develop a numerical software using control volume method that simulates the heating step of the preform. For that, we have to take into account influent parameters of the heating step such as halogen lamps, characterisation and P.E.T, spectral properties. In addition, we need to measure the preform temperature. We have used an 880 LW AGEMA infrared camera (connected to real time software) in order to measure external and internal preform surface temperature distribution, heat transfer coefficient along preform height. The air temperature surrounding the preform and inside the preform is measured using thermocouples.

Modelling of infrared PET preform heating

A 3D control-volume software, called PLASTIRAD, has been developed in order to compute heat transfer during the infrared-heating step. The preform (sheet or cylinder) is meshed using cubic or hexahedral elements so-called control volumes [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF]. The temperature balance equation including radiative transfer (thermal power absorbed by the semitransparent P.E.T. preform and radiated from halogen lamps) is integrated over each control volume and over the time from t to t+t (1):

                      t r t t c p e e e dt d n q dt d n q dt d t T c ) . ( ) . (    (1)
Where  e is the control-volume, q c and q r the Fourier's conductive and the radiative fluxes.The unknown temperatures are computed at the cell centres of the elements. The specific lamp geometry is also considered, taken into account from a view factor computation. The particular lamps used in the infrared oven are composed of a coiled tungsten filament (length : 300 mm), contained in a quartz tubular enclosure (diameter : 10 mm) filled with a neutral gas (Argon) and coated on its back with a ceramic reflector in order to increase the heat flux received by the product. In our model, we assume a constant filament temperature and then an uniform source temperature. Thus the amount of radiation, reaching each front elements of the irradiated preform, is performed with the contour method (2) using Stoke's theorem [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF] :

       p r H p r H H p H dr dr r S F . . ln 2 1  (2)
where the subscripts H refers to the heater and p to the preform front element under consideration, r is the distance between two elemental linear elements, issued from the discretisation of the entire heater boundary  H and r p for each surface of control element's boundary reaching the irradiated preform's face. This contour method integration was chosen between different numerical view factor methods (Monte Carlo or geometrical method). This method allows a good accuracy with a very low computational time. An efficient numerical method based on Gaussian quadrature has been applied in order to perform the contour integration. The computation of the amount of incident radiation, for each surface element, is given in equation (3):

      d T L T S S F k r Fil Fil r H r H r p p p ) ( ) ( ) ( 0      (3)
where   is the spectral tungsten emissivity, T fil the filament temperature, 0  L the blackbody intensity, a given wavelength between [0.2,10] m and k r the efficiency factor of the ceramic reflector. In a first approximation, internal radiative transfer is assumed to be mono-dimensional across the preform thickness. The PET bulk temperature (  400K) is very low in comparison to the source temperature (T fil > 2000 K). So the assumption of cold material is convenient. This leads to express the transmitted flux across the thickness of the material, for an homogenous and amorphous PET, using the Beer-Lambert's law, if the coordinate y describe the thickness, we have, in a simple way, for the flux (4):

y k e y y         ) 0 ( ) ( (4)
where k  is the spectral absorption coefficient of our material.

Further details on this specific numerical model are given in [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF].

2. Heat source characterisation : Filament and Quartz temperatures, efficiency of the reflector.

Spectral properties of these elements have already been measured in a previous work [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF]. Using these data, the tungsten filament and quartz tube temperatures are calculated using a net radiation method [START_REF] Monteix | Modélisation du chauffage infrarouge de plaques en P.E.T", IV e Colloque Interuniversitaire Franco-Québecois[END_REF] with (5):

                      0 ) ( ) ln( *) ( 2 ) ( ) ( 0 ) ln( *) ( 2 ) ( 4 4 4 T T hS T S d d T T T k L T S T T d d T T T k L T S T P Q Q Q Q Q Fil Q Q Fil Argon Q Fil Fil Fil Fil Fil Q Fil Q Q Fil Argon Q Fil Fil Fil Fil          (5)
where P is the electrical power of the lamps; the indices Fil and Q refer respectively to the filament and the quartz tube, T is the temperature, L is the length of the lamp, d the filament or the quartz tube diameter, ,  the emissivity and absorptivity. T* is assumed to be equal to

2 Q Fil T T 
. We have simplified this model by computing directly total parameters instead of spectral ones (as emissivity and absorptivity) with polynomial functions depending on temperatures which is easier and saves time. The temperatures of the filament and the quartz tube are computed at steady state using a Newton-Raphson method. The results of this computation are shown on figure 1. Additional curves are plotted on the same graph, they are related to experimental mesurements : Colour temperature of the filament and temperature computed with a resistivity method. At the nominal power of 1 kW, we have found the following values :

table 1 Filament temperature (K) Quartz temperature (K) 2360 730
As shown on figure 1, the discrepancy between the extremal curves is important : greater than 100 K around 2350 K. This fact is due to the lack of knowledge on the properties of this particular kind of tungsten (coiled filament). On the other hand, "colour" temperature remains difficult to measure at such level of power.To conclude, we have chosen the computed temperature as a "mean" temperature, lower and upper limits of the temperature, at 1 kW (see graph on figure 1), will be considered as the global uncertainty. Using experimental setup described in [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF], the efficiency factor of the ceramic reflector has been measured :

k r ( r ) = ) , ( ) , ( r w r r T T     (6)
The indices r and w refer respectively to the lamp with a ceramic reflector and without reflector. the angle describe by a detector around the heater and T r the temperature of the reflector. This value is obtained using a thermopile detector with a large spectral bandwidth [0.25,26m] .

The output signal delivered by the thermopile is related to the flux emitted by the lamp, so the ratio of the two fluxes measured by the detector and introduced in (6) gives the efficiency of the coating. We have checked that this efficiency is unaffected by the variation of the temperature of the filament and that the emission of the lamp is quasi isotropic. Therefore, we have obtained : k r =1.36 02 . 0  3.Characterisation of P.E.T. spectral properties P.E.T. spectral properties measurements were performed using a 3-mm thick sheet. These sheets were processed by injection moulding in the same conditions than tubular preforms. Thermal analysis experiments were carried out in order to measure the sheets crystallinity. Samples of 10-15 mg weight were cut away from the thick sheet. The DSC experiments were performed on a Mettler calorimeter at a heating rate of 10°C/min. The average crystallinity value of these measurements was about 5 %, corresponding to an amorphous P.E.T, which confirms that diffusion of the radiation can be neglected. Then we have proceeded to P.E.T spectral properties measurements (transmitivity, reflectivity) using a Perkin-Elmer FT-IR spectrometer. The previous thick sheets were polished in order to obtain samples of 0.3-mm thickness, spectra are used as a database in the numerical simulation [START_REF] Monteix | Experimental study and Numerical Simulation of Sheet and Tubular preform Infrared Heating[END_REF]. In addition, we have found that the polymer is opaque for wavelengths in the range of 8 to 12 m with an integrated emissivity of 0.93. This bandwidth corresponds to the one of the infrared camera.

Experimental data

Figure 2 shows the experimental set-up developed on the basis of industrial infrared oven using in injection blow moulding machines. This device consists of six halogen lamps (Philips 1000 W-235 V SK15), monitored by power device. The preform is fixed on a rotating support to provide homogeneous hoop temperature. Behind the preform, there is a cooling fan controlled by a DC supply. A pneumatic system allows the preform displacement inside and outside the oven.

Measurement of the air temperature inside the oven

For the measurement of the air temperature inside the oven, J type thermocouples were used. They are connected to a Keithley recorder instrument. All thermocouples are shielded with aluminium adhesive tape. In order to place a thermocouple inside the preform, the top of this one is drilled (Figure 3).

Measurement of surface temperature distribution

The internal surface temperature measurement of a tubular preform is more tedious and necessitates to develop a specific process. First, we have to cut the preform in its longitudinalsection. Then, the two half-parts of the preform are sticking together using a conductive grease to insure joining during the heating time. All the lamps are set to the same temperature, coupled with the cooling fan, during 20 seconds. Then the rebuild preform exits from the oven in front of the camera. In order to make possible the internal surface temperature measurement, one halfparts is removed (immediately after preform recovering or after a characteristic inversion time). This particular method and the results associated (Infrared thermograms) have been presented in reference [START_REF] Monteix | Experimental study and Numerical Simulation of Sheet and Tubular preform Infrared Heating[END_REF].

Measurement of air forced convection heat transfer coefficient

In this section, we want to determine the air forced convection heat transfer coefficient, generated by the cooling fan along the preform height. Two different methods are developed and compared. The first one consists in measuring air-cooling speed using a hot wire anemometer (Ans Snelco). The calculated Reynolds numbers deduced from these data are in the range between 4000 and 10000 that is to say a turbulent flow. Then, we identify from the literature [START_REF] Monteix | Modélisation du chauffage infrarouge de plaques en P.E.T", IV e Colloque Interuniversitaire Franco-Québecois[END_REF] different classical relationships that are suitable for a cylinder in cross flow, i.e. Nu  f Re, Pr  . The second method is based upon the preform surface temperature measured by the infrared camera in presence of cooling flow. The following specific experimental procedure has to be observed in order to compute the heat transfer coefficient: first, a 20 seconds heating time coupled with cooling; then turn off heating and cooling in order to insure an uniform temperature throughout the preform thickness; then restart the cooling fan. Thus, the last step of the procedure allows solving analytically the energy balance integrated over the thickness if we assume that the main part of the preform is a hollow tube. For a given height, the relationship for the heat transfer coefficient h yields:

  Re Ri Re T T dt T d Cp h o t 2 2 0                (7)

Where

T is the thickness-average temperature, C p the P.E.T heat capacity,  the P.E.T specific mass, T  the thickness-average ambient temperature, R e and R i are respectively the external and internal preform radii. R i is computed using the depth penetration throughout the preform over a characteristic time t, for which we have sufficient temperature data to calculate the initial slope

dT dt         t 0
of the temperature-time plot (figures 4).

  t Re e Re Ri      2 (8)
Where  is the P.E.T diffusivity and  the argument of the erf function, computing assuming the surface temperature has decreased of 90% at the depth thickness. In figures 4, we present a comparison between the heat coefficient measured using both methods. The agreement between the two different methods is fair. It permits to calculate an average air cooling heat transfer coefficient of 45 ± 5 W/m 2 .K. The same experiment has been conducted on different industrial ovens and the values of h are in the range [35,60] W/m 2 .K which is very close to our laboratory results.

Numerical simulation

We performed simulation of the heating step for a P.E.T sheet and a tubular preform, taking into account processes parameters introduced previously. The values of the main parameters are summarised in the tables 2 and 3 below, results are presented on figures 5,6 and 7: Discussion : The comparison on figure 5 shows a relative error of 5 % on the peak temperature and 10 % on the boundary on the sheet which appears as a 3D effect related to the transmission of the radiative flux inside the material. Figures 6 shows simulated surface temperatures distribution on a tubular preform. More important is the Figure 7 that shows a discrepancy of 17 °C between numerical computation and experimental data (versus time). As we have investigated this problem, we can say that, the model, doesn't take into account the reflectivity (in the oven) of the cooling fan nozzle, composed of a metallic plate with several slits, nor does it take into account the flux reflected by the preform from the lamps. This first effect has been evaluated thru measurements. Its influence is close to 3°C in the final apparent temperature indicated by the camera. In the same way, the reflectivity coefficient deduced from spectrometry measurements : 7 % for the 8-12 m spectral band, leads to a 4°C error on the measurement (computed from a thermal image difference). At last, the total error is equal to 7°C and the true measured temperature should be equal to 110°C. Therefore, the real relative error between experimental results and computations as shown on figure 7 will be close to 10%.

Conclusion

Numerical simulations of P.E.T preform heating stage have been performed by using Characterisations of halogen lamps and P.E.T spectral properties. All the measured parameters have been included in the numerical model. The agreement between numerical simulations and experimental data is fair considering the simple model used for absorption (Beer-Lambert law).

The infrared camera appears as a powerful device to make quantitative temperature measurements on the preform and to assess the numerical simulations. Future developments will include new techniques in measurements as temperatures in the preform thickness and improvements in the radiative transfer computations used in the model (rotation of the preform).

5- 1 )

 1 PET sheet -dimensions : 180*100*3 mm / lateral faces are insulated.

Figures figure 1 : 2 :

 12 Figures

table 2

 2

	Heating time (s)	Cooling time (s)	Number of lamps	Dist.lamps-preform /mm	h (W/m 2. K)	Lamps Temp. (K)	Rotation speed (r s-1 )
	65	-	1	40	8 (natural)	2360	-
	5-2) Tubular Preform (diameter : 25 mm,height : 100 mm, thickness : 3 mm)

Acknowledgements : Perrier-Vittel Company supports this work. Special thanks go to J-P. Arcens for his help on the experiments.


Serge Monteix

F Schmidt

Y Le Maoult

G Denis

M Vigny

Recent issues in preform radiative heating modelling

The injection stretch-blow moulding process of thermoplastic bottles requires an heating step before forming. During this step, an amorphous preform is heated above the glass transition temperature using an infrared oven. This step is fundamental in order to determine the thickness distribution along the preform height and then insure high quality bottles. Basically, the infrared oven is constituted of a row of halogen lamps, a cooling device, different aluminium reflectors, a rotating support where the preform is fixed. These facts lead to manage various parameters such as lamps temperature, the shape of the lamps, the position of the lamps, the heat transfer coefficient between cooling air and the preform. Thus, the optimisation of the infrared oven based on trial and errors even becomes tedious and yields to develop numerical modelling. In this work, various experiments have been conducted in order to characterise the heat source and the semi-transparent properties of the P.E.T. Measurements of air temperature inside the infrared oven and air cooling speed have been processed. The surface temperature distribution of the preform has been measured using an infrared camera. The relevant parameters have been implemented in a software called PLASTIRAD based on the control volume method and developed to simulate the infrared heating of polymers. Different comparisons between experimental measurements and numerical simulations for rectangular sheets, fixed as well as rotating preforms, have been successfully achieved.

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Introduction

The injection stretch-blow moulding process of PET bottles requires a heating step prior forming. During this step, the amorphous preform is heated from the ambient temperature to the forming temperature (above the glass transition temperature ≅ 80°C). As the polymer conductivity is low ( ≅ 0.29 W/m/K for the PET), the heating step is performed using an infrared oven. This device is constituted of a row of halogen lamps, which permit a rapid heating with high flux intensities; and a cooling fan to heat-loss the external surface of the preform. However, the infrared heating causes high temperature gradients inside the preform. Thus, the optimisation of the heating step is critical in order to control the final thickness distribution of the bottle and the polymer microstructure, which in turn govern the mechanical properties of the product.

However, there are plenty of parameters such as number of lamps, distance between halogen lamps and preform, electrical power of the lamps, aircooling speed, reflectors, rotation speed. Therefore, optimising infrared oven in order to design new plastic bottles is still a long and costly work. The main objective of this work was to develop numerical software using control volume method that simulates the heating step of the preform. For that, we 1 HOME have to take into account influent parameters of the heating step such as halogen lamps, characterisation and P.E.T, spectral properties. In addition, we need to measure the preform temperature. We have used an 880 LW AGEMA infrared camera (connected to real time software) in order to measure external and internal preform surface temperature distribution, heat transfer coefficient along preform height. The air temperature surrounding the preform and inside the preform is measured using thermocouples.

Modelling of infrared PET preform heating

3D control-volume software, called PLASTIRAD, has been developed in order to compute heat transfer during the infrared-heating step. The preform (sheet or cylinder) is meshed using cubic or hexahedral elements so-called control volumes [START_REF] Patankar | Numerical Heat Transfer and Fluid Flow[END_REF]. The temperature balance equation including radiative transfer (thermal power absorbed by the semitransparent P.E.T. preform and radiated from halogen lamps) is integrated over each control volume and over the time from t to t+∆t (1):

ρ c p ∂T ∂t dΩ dt = -(q c → .n → Γ e ∫ ∆t ∫ Ω e ∫ ∆t ∫ ) dΓdt + (q r → .n → ) dΓdt Γ e ∫ ∆t ∫ (1)
Where Ω is the control-volume, q e c and q r the Fourier's conductive and the radiative fluxes. The unknown temperatures are computed at the cell centres of the elements. The specific lamp geometry is also considered, taken into account from a view factor computation. The particular lamps used in the infrared oven are composed of a coiled tungsten filament (length: 300 mm), contained in a quartz tubular enclosure (diameter: 10 mm) filled with a neutral gas (Argon) and coated on its back with a ceramic reflector in order to increase the heat flux received by the product. In our model, we assume a constant filament temperature and then a uniform source temperature. Thus the amount of radiation, reaching each front elements of the irradiated preform, is performed with the contour method (2) using Stokes theorem [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF]:

F H→ p = 1 2πS H ln r.dr H → .dr r p → ΓH Γr p ∫∫ (2)
Where the subscripts H refers to the heater and p to the preform front element under consideration, r is the distance between two elemental linear elements, issued from the discretisation of the entire heater boundary Γ H and Γr p for each surface of control element's boundary reaching the irradiated preform's face. This contour method integration was chosen between different numerical view factor methods (Monte Carlo or geometrical method). This method allows a good accuracy with a very low computational time. An efficient numerical method based on Gaussian quadrature has been applied in order to perform the contour integration. The computation of the amount of incident radiation, for each surface element, is given in equation (3):

Φ(r p ) = k r F H→ r p S H S r p ε λ (T Fil )π ∆λ ∫ L λ 0 (T Fil )dλ (3)
Where ε λ is the spectral tungsten emissivity, T fil the filament temperature, the blackbody intensity, λ a given wavelength between [0.2,10] µm and k 0 λ L r the efficiency factor of the ceramic reflector. In a first approximation, internal radiative transfer is assumed mono-dimensional across the preform thickness. The PET bulk temperature ( ≤ 400K) is very low in comparison to the source temperature (T fil > 2000 K). Therefore, the assumption of cold material is convenient. This leads to express the transmitted flux across the thickness of the material, for an homogenous and amorphous PET, using the Beer-Lambert's law, if the coordinate y describe the thickness, we have, in a simple way, for the flux (4):

Φ λ (y) = Φ λ (y = 0)e -k λ y (4)
Where k r is the spectral absorption coefficient of our material. Further details on this specific numerical model are given in [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF].

Heat source characterisation: Filament and Quartz temperatures, efficiency of the reflector.

Spectral properties of these elements have already been measured in a previous work [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF]. Using these data, the tungsten filament and quartz tube temperatures are calculated using a net radiation method [START_REF] Monteix | Modélisation du chauffage infrarouge de plaques en P.E.T", IV e Colloque Interuniversitaire Franco-Québecois[END_REF] with (5):

         = - - - - + = - - - ∞ 0 ) ( ) ln( *) ( 2 ) ( ) ( 0 ) ln( *) ( 2 ) ( 4 4 4 T T hS T S d d T T T k L T S T T d d T T T k L T S T P Q Q Q Q Q Fil Q Q Fil Argon Q Fil Fil Fil Fil Fil Q Fil Q Q Fil Argon Q Fil Fil Fil Fil σ ε π σ ε α π σ ε (5)
Where P is the electrical power of the lamps; the indices Fil and Q refer respectively to the filament and the quartz tube, T is the temperature, L is the length of the lamp, d the filament or the quartz tube diameter, ε, α the emissivity and absorptivity. T* is assumed to be equal to 2

Q Fil T T +
. We have simplified this model by computing directly total parameters instead of spectral ones (as emissivity and absorptivity) with polynomial functions depending on temperatures which is easier and saves time. The temperatures of the filament and the quartz tube are computed at steady state using a Newton-Raphson method. The results of this computation are shown on figure 1. Additional curves are plotted on the same graph, they are related to experimental measurements: Colour temperature of the filament and temperature computed with a resistivity method. At the nominal power of one kW, we have found the following values:

Filament temperature (K) Quartz temperature (K) 2360 730
Table 1 As shown on figure 1, the discrepancy between the extremal curves is important: greater than 100 K around 2350 K. This fact is due to the lack of knowledge on the properties of this particular kind of tungsten (coiled filament). On the other hand, "colour" temperature remains difficult to measure at such level of power. In order to conclude, we have chosen the computed temperature as a "mean" temperature, lower and upper limits of the temperature, at 1 kW (see graph on figure 1), will be considered as the global uncertainty. Using experimental set-up described in [START_REF] Rammohan | Efficient Evaluation of Diffuse View Factors For Radiation[END_REF], the efficiency factor of the ceramic reflector has been measured:

k r (θ,Τ r ) = ) , ( ) , ( r w r r T T θ θ Φ Φ (6)
The index r and w refer respectively to the lamp with a ceramic reflector and without reflector. The angle θ describes by a detector around the heater and T r the temperature of the reflector. This value is obtained using a thermopile detector with a large spectral bandwidth [0.25,26µm]. The output signal delivered by the thermopile is related to the flux emitted by the lamp, so the ratio of the two fluxes measured by the detector and introduced in (6) gives the efficiency of the coating. We have checked that this efficiency is unaffected by the variation of the temperature of the filament and that the emission of the lamp is quasi isotropic. Therefore, we have obtained: k r =1.36 ± 02 . 0

Characterisation of P.E.T. spectral properties

P.E.T. spectral properties measurements were performed using a 3-mm thick sheet. These sheets were processed by injection moulding in the same conditions than tubular preforms. Thermal analysis experiments were carried out in order to measure the sheet crystallinity. Samples of 10-15 mg weight were cut away from the thick sheet. The DSC experiments were performed on a Mettler calorimeter at a heating rate of 10°C/min. The average crystallinity value of these measurements was about 5 %, corresponding to an amorphous P.E.T, which confirms that diffusion of the radiation can be neglected. Then we have proceeded to P.E.T spectral properties measurements (transmitivity, reflectivity) using a Perkin-Elmer FT-IR spectrometer. The previous thick sheets were polished in order to obtain samples of 0.3-mm thickness, spectra are used as a database in the numerical simulation [START_REF] Monteix | Experimental study and Numerical Simulation of Sheet and Tubular preform Infrared Heating[END_REF]. In addition, we have found that the polymer is opaque for wavelengths in the range of 8 to 12 µm with an integrated emissivity of 0.93. This bandwidth corresponds to the one of the infrared camera.

Experimental data

Figure 2 shows the experimental set-up developed based on industrial infrared oven using in injection blow moulding machines. This device consists of six halogen lamps (Philips 1000 W-235 V SK15), monitored by power device. The preform is fixed on a rotating support to provide homogeneous hoop temperature. Behind the preform, there is a cooling fan controlled by a DC supply. A pneumatic system allows the preform displacement inside and outside the oven.

Measurement of the air temperature inside the oven

For the measurement of the air temperature inside the oven, J type thermocouples were used. They are connected to a Keithley recorder instrument. All thermocouples are shielded with aluminium adhesive tape. In order to place a thermocouple inside the preform, the top of this one is drilled (Figure 3).

Measurement of surface temperature distribution

The internal surface temperature measurement of a tubular preform is more tedious and necessitates developing a specific process. First, we have to cut the preform in its longitudinalsection. Then, the two half-parts of the preform are sticking together using a conductive grease to insure joining during the heating time. All the lamps are set to the same temperature, coupled with the cooling fan, during 20 seconds. Then the rebuild preform exits from the oven in front of the camera. In order to make possible the internal surface temperature measurement, one halfparts is removed (immediately after preform recovering or after a characteristic inversion time). This particular method and the results associated (Infrared thermograms) have been presented in reference [START_REF] Monteix | Experimental study and Numerical Simulation of Sheet and Tubular preform Infrared Heating[END_REF].

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Measurement of air forced convection heat transfer coefficient

In this section, we want to determine the air forced convection heat transfer coefficient, generated by the cooling fan along the preform height. Two different methods are developed and compared. The first one consists in measuring air-cooling speed using a hot wire anemometer (Ans Snelco). The calculated Reynolds numbers deduced from these data are in the range between 4000 and 10000 that is to say a turbulent flow. Then, we identify from the literature [START_REF] Monteix | Modélisation du chauffage infrarouge de plaques en P.E.T", IV e Colloque Interuniversitaire Franco-Québecois[END_REF] different classical relationships that are suitable for a cylinder in cross flow, i.e.

. The second method is based upon the preform surface temperature measured by the infrared camera in presence of cooling flow. The following specific experimental procedure has to be observed in order to compute the heat transfer coefficient: first, a 20 seconds heating time coupled with cooling; then turn off heating and cooling in order to insure an uniform temperature throughout the preform thickness; then restart the cooling fan. Thus, the last step of the procedure allows solving analytically the energy balance integrated over the thickness if we assume that the main part of the preform is a hollow tube. For a given height, the relationship for the heat transfer coefficient h yields:

( Pr , Re f Nu = ) ( ) Re Ri Re T T dt T d Cp h o t 2 2 0 - -         - = ∞ = ρ (7)
Where T is the thickness-average temperature, C p the P.E.T heat capacity, ρ the P.E.T specific mass, ∞ T the thickness-average ambient temperature, R e and R i are respectively the external and internal preform radii. R i is computed using the depth penetration throughout the preform over a characteristic time t, for which we have sufficient temperature data to calculate the initial slope

0 =         t dt T d
of the temperature-time plot (see figures 4).

β α t Re e Re Ri ∆ - = - = 2 ( 8
)
Where α is the P.E.T diffusivity and β the argument of the erf function, computing assuming the surface temperature has decreased of 90% at the depth thickness. In figures 4, we present a comparison between the heat coefficient measured using both methods. The agreement between the two different methods is fair. It permits to calculate an average air cooling heat transfer coefficient of 45 ± 5 W/m 2 .K. The same experiment has been conducted on different industrial ovens and the values of h are in the range [35,60] W/m 2 .K that is very close to our laboratory results.

Numerical simulation

We performed simulations of typical heating for a P.E.T tubular preform, taking into account processes parameters introduced previously. The values of the main parameters are summarised in the tables 2 below, results are presented on figures 5,6 and 7:

Tubular PET Preform (diameter : 25 mm, height : 100 mm, variational thickness : 2-3 mm)

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Table 2 Two different configuration have been examined; considering a fixed or a rotating preform in front of halogen lamps with a consequent time of 15 and 20 seconds respectively.

Discussion

Figure 5 shows the characteristics thermograms obtained by numerical computation. They prove the geometrical distribution of irradiated flux in agreement with crystallisation profile of PET preform after the corresponding heating. In the same way temperature across the preform thickness versus the time describes irradiated flux absorption and the influence of conductive heat transfer after the heating stop. The modelling of radiative heating of PET preform take into account the transmitted heat flux through the front part of the preform to the second one shaded of the direct heat flux of halogen lamps [START_REF] Monteix | Modelisation du chauffage convecto-radiatif de preformes en PET pour la realisation de corps creux[END_REF].

Comparison of experimental and numerical temperature results (Figure 6) from internal and external preform surfaces lead to conclude that the model is convenient. Even on the back part, where numerical temperature results are close to measurements with a computed relative error about 10%.

Finally simulation of a rotating preform has been realised, and compared to measurements, taking into account all the process parameters as the electrical power of halogen lamps and the cooling fan flow. To do this we processed of comparison between experimental and numerical temperature in order to determine the participation of the cooling fan boxes. This equipment in fact appears as a front reflector for the preform from the halogen lamp irradiance. Making iterations considering at this step natural convection, we determined this reflector efficiency equal to 1,55. Taking into account this participation, we added to the modelling parameters the cooling fan influence from the measurement of the heat transfer coefficient (Figure 4).

Temperature distribution versus the time (Figure 7) from temporal marking inside and outside preform lead to observe the typical temperature inversion through the preform thickness after the heating stop. Comparison with experiments are plotted (Figure 8)

Conclusion

Numerical simulations of P.E.T preform heating stage have been performed, by using characterisations of halogen lamps and P.E.T spectral properties. All the measured parameters have been included in the numerical model. The agreement between numerical simulations and experimental data is fair considering the simple model used for absorption (Beer-Lambert law).

The infrared camera appears as a powerful device to make quantitative temperature measurements on the preform and to assess the numerical simulations. This study has revealed the main influence of the transparence of the PET, showing a consequent increase of the temperature of the shaded back part of preforms. It confirmed the temperature inversion through the preform thickness due to the conductive heat transfer after heating stop. Also making comparison between experimental and numerical temperature distribution of preforms the relative error is less than 10%. These results have been obtained over a laboratory oven. In further developments this study will be conducted over industrial oven of blow moulding machines.

Figures

Figure 1 :

 1 Figure 1 : Filament temperature versus electrical power

Filament temperature versus electrical powerFigure 2 :Figure 3 :Figures 4 :Figure 5 Figure 6 Figure 7 Figure 8

 2345678 Figure 1 : Filament temperature versus electrical power

Acknowledgements

Perrier-Vittel Company supports this work. Special thanks go to J-P. Arcens for his help on the experiments.