Procedure for optimal infrared heating of PET preform via a simplified 3D Modelling with ventilation

Abstract

The thermal condition plays an important role in the final thickness distribution and in the mechanical behavior of the Polyethylene Terephthalate (PET) bottle obtained from the stretch blow molding (SBM) process. A complete 3D modelling of the heating stage during the SBM process under industrial condition is very time-consuming. Based on a simplified approach to quickly achieve the numerical simulation of the preform heating, an optimization procedure is proposed to adjust the settings of the infrared lamps by comparing our simulation results to the target temperature profile. In this numerical approach, the radiation source is simulated by using a model for intensity of the incident radiation and the Beer-Lambert law. On the other hand, the ventilation effect under industrial conditions is taken into account by modelling the forced convection around a cylinder. The infrared (IR) flux and ventilation effects are implemented as thermal boundary conditions in COMSOL software for a 3D computation of the thermal problem for the preform only. Since the simulation has a very reasonable computational time, an optimization procedure can be generated to adjust the setting of IR lamps. This optimization tool provides quickly a first set of parameters to help industry to obtain the desired temperature profile.

Appendix: IR radiation and ventilation modelling

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 13a 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. 13b).

Fig. 13

(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:

$${\displaystyle \begin{array}{c}{M}^{\prime }\Big\{\begin{array}{c}{x}^{\prime }=r\cos \varphi +{x}_L\\ {}{y}^{\prime }\\ {}{z}^{\prime }=d-r\sin \varphi \end{array}\\ {}\overrightarrow{e_{r\prime }}=\left(\cos \varphi, 0,-\sin \varphi \right)\end{array}},{\displaystyle \begin{array}{c}M\Big\{\begin{array}{c}x\\ {}y=R\cos \alpha \\ {}z=R\sin \varphi \end{array}\\ {}\overrightarrow{e_R}=\left(0,\cos \alpha, \sin \alpha \right)\end{array}}$$

(5)

where x_L 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:

$$\overrightarrow{w}=\frac{\overrightarrow{M^{\prime }M}}{MM^{\prime }}=\left\{\begin{array}{c}\frac{x-{x}^{\prime }}{MM^{\prime }}=\frac{x-r\cos \varphi -{x}_L}{MM^{\prime }}\approx \frac{x-{x}_L}{MM^{\prime }}\\ {}\frac{y-{y}^{\prime }}{MM^{\prime }}=\frac{R\cos \alpha -{y}^{\prime }}{MM^{\prime }}\\ {}\frac{z-{z}^{\prime }}{MM^{\prime }}=\frac{z-d+r\sin \varphi }{MM^{\prime }}\approx \frac{z-d}{MM^{\prime }}\end{array}\right.$$

(6)

where

$${MM}^{\prime }=\sqrt{{\left(x-{x}_L\right)}^2+{\left(y-y\prime \right)}^2+{\left(z-d\right)}^2}.$$

The two angles θ^' and θ represent respectively, the angle between the direction normal to the lamp surface $\overrightarrow{n\hbox{'}}$ 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$\overrightarrow{w}$:

$$\left\{\begin{array}{l}\cos \theta =-\overrightarrow{w}.\overrightarrow{e_R}=-\frac{\left(y-{y}^{\hbox{'}}\right)\cos \alpha +\left(z-d\right)\sin \alpha }{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^2}=-\frac{\left(y-{y}^{\hbox{'}}\right)\frac{y}{R}+\left(z-d\right)\frac{z}{R}}{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^2}=-\frac{\left(y-{y}^{\hbox{'}}\right)y+\left(z-d\right)z}{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^2R}\\ {}\cos {\theta}^{\hbox{'}}=\overrightarrow{w}.\overrightarrow{e_{r\hbox{'}}}=\frac{\left(x-{x}_L\right)\cos \varphi -\left(z-d\right)\sin \varphi }{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^2}\end{array}\right.$$

(7)

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

$$\begin{array}{ll}{dQ}_{lamp\to dA}={\int}_{\lambda_1}^{\lambda_2}{i}_{\lambda b}{\varepsilon}_{\lambda }{d}_{\lambda }.{\int}_{\varphi =0}^{\pi }{\int}_{y^\prime=-l/2}^{l/2}\left(\frac{\left[\left({y}^{\prime }-y\right)y+\left(d-z\right)z\right]}{R}.\left[\left(x-{x}_L\right)\cos \varphi +\left(d-z\right)\sin \varphi \right]\right)\frac{dA}{{\left\Vert {MM}^{\prime}\right\Vert}^4}{rd\varphi dy}^{\prime}\\{dQ}_{lamp\to dA}={\int}_{\lambda_1}^{\lambda_2}{i}_{\lambda b}{\varepsilon}_{\lambda }{d}_{\lambda }.r.{\int}_{\varphi =0}^{\pi}\frac{\left[\left(x-{x}_L\right)\cos \varphi +\left(d-z\right)\sin \varphi \right]}{R} d\varphi .{\int}_{y\prime=-l/2}^{l/2}\frac{\left({y}^{\prime}-y\right)y+\left(d-z\right)z}{{\left\Vert {MM}^{\prime}\right\Vert}^4}{dy}^{\prime}. dA\\{dQ}_{lamp\to dA}={\int}_{\lambda_1}^{\lambda_2}{i}_{\lambda b}{\varepsilon}_{\lambda }{d}_{\lambda}\frac{r.2\left(d-z\right)}{R}{\int}_{y\hbox{'}=-l/2}^{l/2}\frac{\left({y}^{\hbox{'}}-y\right)y+\left(d-z\right)z}{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^4}{dy}^{\hbox{'}}. dA\end{array}$$

(8)

whereλ is a given wavelength between 0.2 and 10 μm and ε_λ is the average tungsten emissivity equal to 0.26 for a wavelength between 0.2 and 10 μm [9]. The emissive power for a blackbody ${i}_{\lambda}^b$ is given by Planck’s law:

$${i}_{\lambda}^b=\frac{2{C}_1}{\lambda^5\left({e}^{{~}^{{C}_2}\!\left/ \!{~}_{\lambda {T}_{fil}}\right.}-1\right)}$$

(9)

where C₁ ≈ 1.19 ⋅ 10⁸W.m⁻².μm⁴ and C₂ ≈ 14388μm.K. We assume that the temperature of the filament is uniform and equals to T_fil = 1700 K [24].

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

$${\varphi}_S(M)=\frac{dQ_{lamp\to dA}}{dA}={\int}_{\lambda_1}^{\lambda_2}\frac{2{C}_1{\varepsilon}_{\lambda }}{\lambda^5\left({e}^{{~}^{{C}_2}\!\left/ \!{~}_{\lambda {T}_{fil}}\right.}-1\right)}{d}_{\lambda}\frac{r.2\left(d-z\right)}{R}{\int}_{y\hbox{'}=-l/2}^{l/2}\frac{\left({y}^{\hbox{'}}-y\right)y+\left(d-z\right)z}{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^4}{dy}^{\hbox{'}}$$

(10)

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 ϕ_S(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(x_M, y_M)and the line of the lamp can be written:

$$\left(z-{z}_M\right){z}_M+\left(y-{y}_M\right){y}_M=0$$

(11)

$$\left\{\begin{array}{l}z=d\\ {}x={x}_L\\ {}-l/2\le y\le l/2\end{array}\right.$$

(12)

Therefore, the coordinates of D can be calculated as:

$$\left(z-{z}_M\right){z}_M+\left(y-{y}_M\right){y}_M=0$$

$$\left\{\begin{array}{l}\left(z-{z}_M\right){z}_M+\left(y-{y}_M\right){y}_M=0\\ {}z=d\\ {}x={x}_L\end{array}\right.\Rightarrow \left\{\begin{array}{l}{z}_D=d\\ {}{x}_D={x}_L\\ {}{y}_D=\frac{\left({z}_M-d\right){z}_M}{y_M}+{y}_M\end{array}\right.$$

(13)

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:

$${\phi}_S(M)=\frac{dQ_{lamp\to dA}}{dA}={\int}_{\lambda_1}^{\lambda_2}\frac{2{C}_1{\varepsilon}_{\lambda }}{\lambda^5\left({e}^{{~}^{{C}_2}\!\left/ \!{~}_{\lambda {T}_{fil}}\right.}-1\right)}{d}_{\lambda}\frac{r.2\left(d-z\right)}{R}{\int}_{y\hbox{'}=-l/2}^{l/2}\frac{\left(x-{x}_L\right)\left({x}_{O_1}-x\right)+\left({y}^{\hbox{'}}-y\right)y+\left(d-z\right)z}{{\left\Vert {MM}^{\hbox{'}}\right\Vert}^4}{dy}^{\hbox{'}}$$

(14)

There are also three cases as in the previous case. The coordinates of the intersection D are given in Eq. 15:

$$\left\{\begin{array}{l}\left(z-{z}_M\right){z}_M+\left(y-{y}_M\right){y}_M=0\\ {}z=d\\ {}x={h}_{lamp}\end{array}\right.\Rightarrow \left\{\begin{array}{l}{z}_D=d\\ {}{x}_D={h}_{lamp}\\ {}{y}_D=\frac{\left({h}_{lamp}-{x}_M\right)\left({x}_M-{x}_{O_1}\right){z}_M+\left({z}_M-d\right){z}_M}{y_M}+{y}_M\end{array}\right.$$

(15)

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.

Convection model for ventilation

Considering the convective exchange with the air coming from ventilation, we consider a simplified modelling of ventilation effect inspired from the work of Eckert and Drake [35]. In each elementary surface of the boundary, one must define the local convective flux:

$$d\phi =h\left(T-{T}_{\infty}\right) dA$$

(16)

where dA is the area around the current point of the surface, T is the temperature at this point and T_∞ is the ambient temperature. h is the exchange convection parameter that is related to the Nusselt number N_u by:

$$Nu=\frac{hL_c}{k}$$

(17)

with k the thermal conductivity of the air and L_c a characteristic length of the preform

In Eckert and Drake [35], the authors measured flow around a cylinder for different situation characterized by their Reynolds number Re and proposed a representation of the Nusselt number as a function of angular position that is related to the normal of outside surface (Fig. 14)

Fig. 14

Azimuthal distribution of exchange convection coefficient around a cylinder under air flow when Reynolds number Re> > 1 (a) left for Re from 20 to 600(b) for Re from 4000 to 50,000 [35]

In our case, measures managed on the SBO ventilation system lead to the air velocity that is near 8 m/s when blowing on the neck, and only 1.3 m/s in the cylindrical region of the preform. Consequently we can estimate the Reynolds number:

$$\operatorname{Re}=\frac{\rho VD}{\eta }$$

(18)

In our case, the ventilation system blows the air at the velocity V ≈ 2m/s around the preform with 20 mm diameter, consequently Re = 2800. The extrapolation of the curve of the Nusselt number for Re = 2800 is presented on Fig. 15 and interpolated as a function of θ that is the angle between the air flow and the normal of outer surface of preform.

$$Nu\left(\theta \right)=A\exp \left(-{\left(\frac{\theta }{\theta_A}\right)}^2\right)+B\exp \left(-{\left(\frac{\pi -\theta }{\theta_B}\right)}^2\right)$$

(19)

In the zone [0, π]: A = 88.5, θ_A = 4.7, θ_B = 2.4

In the zone [π, 2π]: A = 64.9, θ_A = 1.64, θ_B = 2.55

This Nusselt number enables to evaluate the convective exchange coefficient h as a function of the position on the preform surface and this is how we take the air ventilation into account

Fig. 15

Nusselt number as a function of angle θ