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The main aim in this work is to investigate predicting capabilities of the Zatloukal-Vlcek model if nonisothermal conditions and non-Newtonian fluid behavior are taken into account. The studied model behavior will be compared with Tas’s PhD thesis experimental data [18] and predictions of the following two different Pearson and Petrie based models: Sarafrazi and Sharif model [16] (eXtended Pom-Pom constitutive equation is used; a variable heat transfer coefficient and stress induced crystallization is taken into account and Beaulne and Mitsoulis model [15] (integral constitutive equation of the K-BKZ type is utilized; constant heat transfer coefficient and no crystallization effects are assumed).

The Zatloukal-Vlcek model is based on the assumption that bubble during blowing can be viewed as a elastic membrane (characterized by the one constant value of compliance J where the thickness is neglected) which is bended due to the internal load, p, and take-up force, F in such a way, that bubble shape satisfies minimum energy requirements [34]. Under these assumptions, the model yields analytical expressions for bubble shape, take-up force and internal bubble pressure which are summarized in Table 1. The model is given by the following four physical parameters: freezeline height, L, bubble curvature, pJ (which is given by the bubble compliance, J, and the internal load, p), the blow-up ratio, BUR, and the die radius, R

where Q is the volume flow rate, y(x), the radius of the bubble, h(x), the thickness of the film and v(x) is the film velocity, all as functions of the distance from the die x. Secondly,

where means the extra stress tensor, D represents the deformation rate tensor and stands for the viscosity, which is not constant (as in the case of standard Newtonian law), but it is allowed to vary with the first invariant of the absolute value of deformation rate tensor

where E

Energy Equation: With the aim to take non-isothermal conditions into account, cross-sectionally averaged energy equation taken from [40], has been considered:

where Cp stands for the specific heat capacity, ρ is the polymer density, y means the local bubble radius, m is the mass flow rate, HTC represents the heat transfer coefficient, T is the bubble temperature, T

In order to reduce the problem complexity, the axial conduction, dissipation, radiation effects and crystallization are neglected. For such simplifying assumptions, the Eq. 9 is reduced in the following, the simplest version of the cross-sectionally averaged energy equation:

where the local bubble radius y is given by Eq. 1 in Table 1. The Eq. 10 applied for the whole part of the bubble takes the following form:

where T

With the aim to get equations for the temperature profile along the bubble, it is necessary to apply the Eq. 10 for any arbitrary point at the bubble i.e. in the following way:

After the integration of Eq. 13, the temperature profile takes the following analytical expression:

Velocity profile calculation: With the aim to calculate the velocity profile and the film thickness in the non-isothermal film blowing process, the force balance in vertical direction (gravity and upward force due to the airflow are neglected) proposed by Pearson and Petrie is considered in the following form:

where σ

where v and h is bubble velocity and thickness, respectively. Assuming that h << y, then

By combination of Eqs. 2, 16, 17, the σ

After substituting Eq. 18 into Eq. 15, the equation for the bubble velocity in the following form can be obtained.

where v

In Figure 1, it is clearly visible that the used generalized Newtonian model has very good capabilities to describe steady shear and steady uniaxial extensional viscosities for the Tas’s LDPE L8 sample which justifies its utilization in the film blowing modeling. The generalized Newtonian model parameters are provided in Table 4 and the parameter has been chosen to be 20 as suggested in [39].

It should be mentioned that two possible numerical schemes have been tested for the proposed model. First procedure consider that the bubble shape (i.e. pJ, BUR) is a priory known and take-up force F and internal bubble pressure Δp are unknowns parameters whereas in the second case, Δp is known and bubble shape (i.e. pJ, BUR) and F are unknown parameters.

As can be clearly seen in Figures 2-4, both numerical approaches leads to very similar predictions for all investigated variables (bubble shape, velocity and temperature) and it can be concluded that the agreement between the proposed model predictions are in very good agreement with the corresponding experimental data. Moreover, tested model predictions are comparable with the Sarafrazi/Sharif [16] model predictions (which is based on the advanced eXtended Pom-Pom constitutive equation; a variable heat transfer coefficient and stress induced crystallization).

Complete set of calculated variables in the proposed model for theoretical predictions depicted in Figures 2-4 are summarized in Table 5. It is nicely visible that predicted F and Δp for all tested polymers and processing conditions are in fairly good agreement with the corresponding Tas’s experimental data. These predictions are comparable with Sarafrazi/Sharif [16] model predictions and even better than Beaulne and Mitsoulis model [15] behavior which is based on the viscoelastic integral constitutive equation of the K-BKZ type assuming constant heat transfer coefficient and no crystallization effects. Just note that for the die volume rate calculation (from the experimentally known mass flow rate), the following definition of the LDPE density taken from [18] was used:

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TABLE 1. A Summary of the Zatloukal-Vlcek mode

TABLE 2. Relationship between A and phi functions.

TABLE. 3. Characteristics of the L8 Stamylan LDPE used in the experiments by T as [18].

TABLE 4. Film blowing model parameters for Tas’s experiments No. 29.

TABLE 5. Summarization of Tas’s experimental data [18], Zatloukal-Vlcek [39] (the calculated results for the fixed bubble shape pJ and internal bubble pressure (delta)p are provided in the brackets and without brackets, respectively), Sarafrazi/Sharif [16] and Beaulne/Mitsoulis [15] model predictions for Tas’s LDPEs and processing conditions.

- σ
_{11}, σ_{33}at the freezeline were calculated by using v_{d}, R_{0}, H_{0}, F, Δp, BUR, v_{f}provided in [15] and Pearson and

FIGURE 1. Comparison between the generalized Newtonian model fit (solid lines) [39] and PTT model predictions (symbols) characterizing L8 Stamylan LDPE material according to Tas's Ph.D. thesis [18].

FIGURE 2. Comparison of the bubble shapes between the proposed model prediction [30], experiment No. 29 taken from Tas ́s Ph.D. thesis [18] and the Beaulne/Mitsoulis model prediction [15] and the Sarafrazi/Sharif model prediction [16].

FIGURE 3. Comparison of the velocity profiles between the proposed model prediction [30], experiment No. 29 taken from Tas ́s Ph.D. thesis [18] and the Beaulne/Mitsoulis model prediction [15] and the Sarafrazi/Sharif model prediction [16].

FIGURE 4. Comparison of the temperature profiles between the proposed model prediction [30], experiment No. 29 taken from Tas ́s Ph.D. thesis [18] and the Beaulne/Mitsoulis model prediction [15] and the Sarafrazi/Sharif model prediction [16].

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