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Numerical Simulation of Infrared Heating and Ventilation before Stretch Blow Molding of PET Bottles

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Abstract

The initial temperature of the preform has an important influence on the stretch and blowing step of the process to produce PET bottles. A complete 3D modelling of the heat part of the stretch blow molding machine including meshing is a long and complex task. Solving Navier Stokes equation coupled with the thermal problem takes more than one week using ANSYS/Fluent software. The numerical simulation of infrared (IR) heating taking into account the ventilation effect is very time-consuming. This work proposes a simplified approach to achieve quickly the numerical simulation in order to have an estimation of the temperature distribution in the preform. In this approach, the IR heating flux coming from IR lamps and the ventilation model are calculated in a semi analytical way and are applied as the boundary conditions of the simulation in COMSOL where only the preform is meshed. This approach is validated by comparing our numerical results with the experimental temperature distribution of PET preform.

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Appendix: Radiative model for infrared heating

Appendix: Radiative model for infrared heating

In the case of the PET cylindrical preforms, we have modelled the radiative heat transfer in a simplified way, as presented in the following. The infrared radiation received by the preform can be estimated by integrating the spectral energy and by taking into account the view (or form) factors between the lamps and the preform. Figure 12a presents the geometry of the lamps and the preform. N identical IR lamps are modeled as horizontal cylinders in a vertical plane separated by a distance d of the preform axis (Fig. 12b).

Fig. 12
figure 12

A) Geometrical configuration of the lamps and the preform; (b) Position of the lamps

The amount of the radiation energy that comes from the surface element dA’ at the collocation point M’ and reaches the surface element dA at the collocation point M is first calculated. Firstly, one focuses on the cylindrical part of the preform. The coordinates of the points M and M ' are:

$$\begin{array}{cc}M'\left\{\begin{array}{l}x'=r\cos\varphi+x_L\\y'\\z'=d-r\sin\varphi\end{array}\right.,&M\left\{\begin{array}{l}x\\y=R\cos\alpha\\z=R\sin\varphi\end{array}\right.\\\overrightarrow{e_{r'}}=\left(\cos\varphi,0,-\sin\varphi\right)&\overrightarrow{e_r}=\left(0,\cos\alpha,\sin\varphi\right)\end{array}$$
(9)

where xL is the lamp coordinate along the axis x.

The path vector from M’ to M is denoted by \(\overrightarrow{w}\). Assuming that the radius r is negligible compared to the distance d (r <  < d), this vector can be written as follow:

$$\overrightarrow w=\frac{\overrightarrow{M'M}}{MM'}=\left\{\begin{array}{l}\frac{\overrightarrow{x-x'}}{MM'}=\frac{x-r\cos\varphi-x_L}{MM'}\approx\frac{x-x_L}{MM'}\\\frac{y-y'}{MM'}=\frac{R\cos\alpha-y'}{MM'}\\\frac{z-z'}{MM'}=\frac{z-d+r\sin\varphi}{MM'}\approx\frac{z-d}{MM'}\end{array}\right.$$
(10)

where \(MM'=\sqrt{\left(x-x_L\right)^2+\left(y-y'\right)^2+\left(z-d\right)^2}\).

The two angles θ′ and θ represent respectively, the angle between the direction normal to the lamp surface \(\overrightarrow{n'}\) at point M' and the path direction \(\overrightarrow{w}\); the angle between the direction normal to the PET sheet \(\overrightarrow{n}\) at point M and the path direction \(\overrightarrow{w}\):

$$\left\{\begin{array}{l}\cos\theta=-\overrightarrow w.\overrightarrow{e_R}=-\frac{\left(y-y'\right)\cos\alpha+\left(z-d\right)\sin\alpha}{{\Arrowvert MM'\Arrowvert}^2}=-\frac{\left(y-y'\right)\frac yR+\left(z-d\right)\frac ZR}{{\Arrowvert MM'\Arrowvert}^2}=-\\\cos\theta'=\overrightarrow w.\overrightarrow{e_{R'}}=\frac{\left(x-x_L\right)\cos\varphi-\left(z-d\right)\sin\varphi}{{\Arrowvert MM'\Arrowvert}^2}\end{array}\frac{\left(y-y'\right)y+\left(z-d\right)z}{{\Arrowvert MM'\Arrowvert}^2R}\right.$$
(11)

The amount of the radiation heat energy can be written in the following way:

$$\begin{array}{l}dQ_{lamp\rightarrow dA}=\int_{\lambda_1}^{\lambda_2}i_{\lambda b}\varepsilon_\lambda d_\lambda.\int_{\varphi=0}^\pi\int_{y'=-l/2}^{l/2}\left(\frac{\left[\left(y'-y\right)y+\left(d-z\right)z\right]}R.\left[\left(x-x_L\right)\cos\varphi+\left(d-z\right)\sin\varphi\right]\right)\frac{dA}{{\Arrowvert MM'\Arrowvert}^4}rd\varphi dy'\\dQ_{lamp\rightarrow dA}=\int_{\lambda_1}^{\lambda_2}i_{\lambda b}\varepsilon_\lambda d_\lambda.r.\int_{\varphi=0}^\pi\frac{\left[\left(x-x_L\right)\cos\varphi+\left(d-z\right)\sin\varphi\right]}Rd\varphi.\int_{y'=l/2}^{l/2}\frac{\left(y'-y\right)y+\left(d-z\right)z}{{\Arrowvert MM'\Arrowvert}^4}dy'.dA\\dQ_{lamp\rightarrow dA}=\int_{\lambda_1}^{\lambda_2}i_{\lambda b}\varepsilon_\lambda d_\lambda\frac{r.2\left(d-z\right)}R\int_{y'=-l/2}^{l/2}\frac{\left(y'-y\right)y+\left(d-z\right)z}{{\Arrowvert MM'\Arrowvert}^4}dy'.dA\end{array}$$
(12)

where λ is a given wavelength between 0.2 and 10 μm and ελ is the average tungsten emissivity equal to 0.26 for a wavelength between 0.2 and 10 µm [14]. The emissive power for a blackbody iλb is given by Planck’s law:

$${i}_{\lambda }^{b}=\frac{2{C}_{1}}{{\lambda }^{5}\left({e}^{{~}^{{C}_{2}}\!\left/ \!{~}_{{\lambda T}_{fil}}\right.}-1\right)}$$
(13)

where C1 ≈ 1.19·108 W.m−2.μm4 and C2 ≈ 14,388·μm.K. We assume that the temperature of the filament is uniform and equals to Tfil = 1700 K [26].

Finally, the intensity per unit area of the incident radiation can be written as follow:

$${\phi }_{S}\left(M\right)=\frac{d{Q}_{lamp\to dA}}{dA}={\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\frac{2{C}_{1}{\varepsilon }_{\lambda }}{{\lambda }^{5}\left({e}^{{~}^{{C}_{2}}\!\left/ \!{~}_{{\lambda T}_{fil}}\right.}-1\right)}{d}_{\lambda }\frac{r.2\left(d-z\right)}{R}{\int }_{{y}^{'}=-l/2}^{l/2}\frac{\left({y}^{'}-y\right)y+\left(d-z\right)z}{{\Vert MM'\Vert }^{4}}d{y}^{'}$$
(14)

The incident radiation on the point M depends on its position with respect to the points A and B of the lamp (y′ =  ± l / 2). There are three cases:

  • Point M can receive the radiation from point A to point B. The interval of integration for ϕS(M) is [–l / 2, l / 2].

  • Point M receives no radiation from the lamp (M is located in the rear part of the preform). In this case, the incident radiation on point M is zero.

  • Point M can get only a part of the total radiation from point A to point B. That means that point M can receive the radiation from the interval AD or BD, where D is an intermediate point of the lamp. In this case, points A and B are on the opposite sides of the tangent plane of the tube at point M.

In the third case, it is necessary to find the intersection D between the tangent plane and the lamp. The equation of the tangent plane at the point M(xM, yM) and the line of the lamp can be written:

$$\left(z-{z}_{M}\right){z}_{M}+\left(y-{y}_{M}\right){y}_{M}=0$$
(15)
$$\left\{\begin{array}{l}z=d\\ x={x}_{L}\\ -l/2\le y\le l/2\end{array}\right.$$
(16)

Therefore, the coordinates of D can be calculated as:

$$\begin{array}{l}\left(z-z_M\right)z_M+\left(y-y_M\right)y_M=0\\\left\{\begin{array}{l}\left(z-z_M\right)z_M+\left(y-y_M\right)y_M=0\\z=d\\x=x_L\end{array}\Rightarrow\left\{\begin{array}{l}z_D=d\\x_D=x_L\\y_D=\frac{\left(z_M-d\right)z_M}{y_M}+y_M\end{array}\right.\right.\end{array}$$
(17)

Then we consider the semi-spherical part of the preform. After a calculation similar to that of the cylindrical part, the intensity per unit area of the incident radiation at the point M is:

$${\phi }_{S}\left(M\right)=\frac{d{Q}_{lamp\to dA}}{dA}={\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\frac{2{C}_{1}{\varepsilon }_{\lambda }}{{\lambda }^{5}\left({e}^{{~}^{{C}_{2}}\!\left/ \!{~}_{{\lambda T}_{fil}}\right.}-1\right)}{d}_{\lambda }\frac{r.2\left(d-z\right)}{R}{\int }_{{y}^{'}=-l/2}^{l/2}\frac{\left(x-{x}_{L}\right)\left({x}_{{O}_{1}}-x\right)+\left(y'-y\right)y+\left(d-z\right)z}{{\Vert MM'\Vert }^{4}}d{y}^{'}$$
(18)

There are also three cases as in the previous case. The coordinates of the intersection D are given in Eq. 11:

$$\left\{\begin{array}{l}\left(z-z_M\right)z_M+\left(y-y_M\right)y_M=0\\z=d\\x=h_{lamp}\end{array}\Rightarrow\left\{\begin{array}{l}z_D=d\\x_D=h_{lamp}\\y_D=\frac{\left(h_{lamp}-x_M\right)\left(x_M-x_{O_1}\right)z_M+\left(z_M-d\right)z_M}{y_M}+y_M\end{array}\right.\right.$$
(19)

From the calculation of the incident radiation intensity on the cylindrical or semi-spherical part, one obtains the incident heat flux on the outer surface of the preform.

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Nguyen, T.T., Luo, YM., Chevalier, L. et al. Numerical Simulation of Infrared Heating and Ventilation before Stretch Blow Molding of PET Bottles. Int J Mater Form 16, 37 (2023). https://doi.org/10.1007/s12289-023-01763-2

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