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Abstract: This paper reports on a simplified approach for analysing the temperature evolution in a semi-transparent thin membrane of the amorphous polyethylene terephtalate type (PET) exposed to a radiative source. It is based on a 3D finite elements method. The thermophysical properties of the PET are assumed independent of temperature while the internal radiative intensity absorption is taken as one-dimensional and is governed by the Beer–Lambert law. To avoid the difficult problem of computing the shape factors, a semi-analytical approach is used. Finally, the numerical simulations have allowed to validate the analytical and experimental results. In the first place, we have written the energy conservation equation in absence of convection (based on the first law of thermodynamics) (1.a), the radiative source term (1.b) and the boundary conditions (2). As for the finite elements method, the Galerkin approach is used for the formulation of the 3D heat transfer equation (3) and the diagonalisation of the heat capacity matrix, “lumped matrix”, is adopted [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. A single step implicit time integration scheme is used for the computation [M.A. Dokainish, K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics, Comput. Struct. 32 (6) (1989) 1371–1386]. After recalling the classical expressions for the radiative flux divergence (5) and the flux itself (6), we have rewritten the radiative source term for an homogeneous medium as a function of spectral intensity (7) and given the integro-differential equation of radiative transfer [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992], Eq. (8). The spectral relations linking coefficients of extinction, absorption, scattering and scattering albedo are given by Eqs. (9) and (10), while Eq. (11) expresses the boundary conditions. In the second place, on the basis of the non-scattering behaviour of amorphous PET [K. Esser, E. Haberstroh, U. Hüsgen, D. Weinand, Infrared radiation in the processing of plastics: Precise adjustment-the key to productivity, Adv. Polymer Technol. 7 (2) (1987) 89–128; M.D. Shelby, Effects of infrared lamp temperature and other variables on the reheat rate of PET, in: Proceedings of ANTEC''91 Conference, 1991, pp. 1420–1424; G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249], we have assumed that extinction and absorption coefficients are equal, the corresponding albedo being zero. The radiative transfer equation can thus be rewritten in a reduced form (12). Moreover, for the temperature range prevailing in the processes of PET thermoforming and preforms blowing, temperatures of radiation sources are generally much higher than those used for the forming of these thermoplastic media. Under these conditions, the cold medium hypothesis is used [Y. Le Maoult, F.M. Schmidt, V. Laborde, M. El Hafi, P. Lebaudy, Measurement and calculation of perform infrared heating: a first approach, in: Proceeedings of the Fourth International Workshop on Advanced Infrared, 1997, pp. 321–331], which allows to express transmission of spectral intensity across the material as a function of position and direction (13), and to write the radiative flux divergence in a simplified form as well (14). For one-dimensional radiation, solution of the radiative transfer equation [R. Siegel, J.R. Howel, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 1992] is given by Eq. (15), which combines with Eq. (14) to yield the spectral flux generated inside the semi-transparent medium, Eq. (16). In case of radiation propagation in the same direction as the normal to the polymeric membrane surface, radiation intensity can then be expressed by the Beer–Lambert law (17). This permits to deduce the expression of spectral flux transmitted across the PET membrane thickness (18). Infrared radiation intercepted by the membrane For determining the infrared radiation intercepted by the membrane surface, the medium between the radiation source and the thermoplastic material is assumed to be completely transparent to radiation. It is also assumed that the surface of the source is diffuse. This allows to infer, from spectral expressions of the energy emitted by the source (Eqs. (19) and (20)) and of the energy received by the semi-transparent membrane (Eq. (21)), a relation for radiation intercepted by the surface at all wavelengths (Eqs. (22) and (23)). For an isotropically radiating source (Lambertian source) when considering an average emissivity of the heating source, Eq. (24), total radiation intercepted by the surface is given by Eq. (25). Infrared radiation absorbed by the membrane For the determination of infrared radiation absorbed by the polymeric membrane, first, based on the principle of energy conservation, spectral relation (27) which links reflectivity, absorptivity and transmissivity is recalled, as well as relation (28) associating spectral absorption and transmission coefficients following the Beer–Lambert law. Consequently, this allows to find the spectral energy absorbed by the semi-transparent medium (29) with the aid of Eq. (25). Concentrating on wavelength bands that constitute the transmission spectrum of the medium, Eq. (29) is then replaced by Eq. (30) where average values of reflectivity (31) and transmissivity (32) are considered. As a result, the volumetric total energy absorbed by the medium, the flux divergence, is then given by Eqs. (33) and (34). Furthermore, considering that reflectivity is low for polymeric materials (generally lower than 5%), the latter is neglected leading to Eq. (35) instead of Eq. (34). Fig. 1 illustrates a typical transmissivity curve for PET [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249]. Shape factor In order to take into account arbitrary shapes of sources and preforms, a semi-analytical approach is used for the computation of shape factors [G. Venkateswaran, M.R. Cameron, S.A. Jabarin, Effect of temperature profiles trough preform thickness on the properties of reheat-blown PET containers, Adv. Polymer Technol. 17 (3) (1997) 237–249]. The definition of shape factor is recalled in Eq. (36) and Fig. 2. An equivalent form based on the contour principle [R. Rammohan, Efficient evaluation of diffuse view factors for radiation, Int. J. Heat Mass Transfer. 39 (1996) 1281–1286] is given by Eq. (37), while the semi-analytical formula [F. Erchiqui, N.G. Dituba, Analyse comparative des méthodes de calcul des facteurs de formes pour des surfaces à contours rectilignes, Internat. J. Thermal Sci. 46 (2007) 284–293] is expressed by Eq. (39). Validation of this semi-analytical approach is obtained by comparison with the analytical solution [H.C. Hottel, Radiant heat transmission between surfaces separated by non-absorbing media, Trans. ASME 53 (1931) 265–273, FSP-53-196; A. Feingold, Radiant-interchange configuration factors between various selected plane surfaces, Proc. Roy. Soc. London Ser. A 292 (1996) 51–60; J.R. Ehlert, T.F. Smith, View factors for perpendicular and parallel rectangular plates, J. Thermophys. Heat Trans. 7 (1993) 173–174] as well as with results yielded by three different techniques, i.e. the area-integration method, the Gauss quadratic technique and the contour method. Table 1 summarizes the numerical results obtained in each case and gives the relative error. Analytical validation of the reheating Regarding the numerical validation, we have considered a PET semi infinite medium subjected to a uniform incident radiative flux density. The boundaries of the semi transparent medium are taken as adiabatic. Thermophysical properties and geometrics are given in Table 2. The one-dimensional Laplace equation which governs the radiation heat transfer is solved analytically with the aid of Laplace transforms. The temperature evolution within the depth of the semi transparent medium is obtained by analytical solution [A.B. De Vriendt, in: G. Morin, (Ed.), La transmission de la chaleur, vol. 1, Chicoutimi, Québec, 1984], Eq. (40). Fig. 3 shows a comparison between the numerical solution obtained by the 3D finite elements method and the analytical one. It is seen that, for the three cases, the relative error between numerical and analytical results stays lower than 0.1%. Numerical modeling of infrared heating An amorphous PET sheet is considered. The face of the membrane is a square of side 20 cm with a thickness of 1.5 mm. The lateral walls are assumed adiabatic. For modeling with the 3D finite elements method, the sheet is meshed with identical hexahedra comprising eight nodes. The thermophysical properties used for this study are given in [S. Monteix, F. Schmidt, Y. Le Maoult, R Ben Yedder, R.W. Diraddo, D. Laroche, Experimental study and numerical simulation of perform or sheet exposed to infrared radiative heating, Journal of Materials Processing Technology 119 (2001) 90–97] and the reheating time is 35 seconds. The heat transfer coefficient h on front and rear faces is 10 W m−2 K−1. Heat flux received by the polymer during heating To estimate the incident heat flux distribution on the surface of the polymer, we first calculate the shape factors between each pair of emitting/receiving surfaces. Then the flux distribution is obtained with Eq. (25). Fig. 4 illustrates this distribution. Validation against experimental results Figs. 5 and 6 compare the numerical results obtained by the simulation (via MEF 3D) with those obtained experimentally. Fig. 5 displays the temperature distribution on the front and rear surfaces. Results Figs. 7–10 illustrate the numerical results obtained for the temporal temperature evolution, during the first five seconds of heating, of the upper receiving face (surface directly exposed to the infrared source), the central plane and the lower face of the PET membrane, respectively. Figs. 11 and 12 give the temperature evolution at four positions on upper and lower faces of the membrane: 0 cm (edge), 2.5 cm, 5 cm and 10 cm (middle). It is observed that the trend is nearly linear during the period of infrared heating. On Figs. 13 and 14 is displayed the temperature evolution in the center of the membrane along its thickness during periods 0–4 and 5–35 seconds. [Copyright &y& Elsevier]

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