*2.1.1 Characteristics of material:*

The material under investigation is a rigid PVC compound from the company ALFAVINIL RE97/1-773, which is currently used widely in the manufacture of all types of profiles for construction, in the refrigeration industry, among other applications, due to its excellent properties of resistance to corrosive media, good thermal, electrical and acoustic insulation, low weight, and mechanical properties suitable for each use. This is possible due to the possibility of designing its formulation with different types of materials (resin, plasticizer, additives, mineral fillers, pigments, thermal stabilizers, and others). It is also an ecological material, since it has the lowest carbon footprint and can be recycled.

On the other hand, due to its chemical composition, it is a thermally sensitive material, degrading from 140°C; therefore, for its processing, in addition to containing a thermal stabilizer in its composition, the use of special extruders is required, with screws with a low compression ratio (2:1), which perform the plasticization of the molten material with a low shear deformation or also known as shear rate, so that the heat provided for this process comes mainly from the heating bands of the cylinder and that the movement of the screw is mainly to improve the mixing and homogenization of the different components of the compound without generating heat by friction between the material, screw, and cylinder. Due mentoned reason, twin screw extruders, which have the best of these mentioned characteristics are recommended for use for this material, especially for medium and high productions.

#### **2.2 Rheological characterization of the material**

Most of the molten polymers behave as non-Newtonian fluids; that is to say, their shear viscosity (*η*), at a constant temperature, depends on the shear deformation or shear rate (*ɣ*), being able to define it (Equation 1) is the quotient between shear stress (*τ*) and shear rate (*ɣ*) Vlachopoulos [6].

$$
\eta = \mathfrak{r}/\mathfrak{y} \tag{1}
$$

On the other hand, thermoplastics, especially PVC, is classified as pseudoplastics; that is, the viscosity decreases with shear deformation (e.g., that is exerted by the extruder screw). To carry out computer simulation, precise information on the behavior of the material under the processing conditions must be possessed: temperature, flow, density, and the rheological characteristics of the material, especially these are very important because it is a composite. Regarding the rheological characteristics, it is necessary to know the *K* and *n* values of the model to be used and that in the case of this work, the value of the shear deformations that occur in the last zone of the extruder and inside the head would apply the Power or Ostwald's Law (Eq. (2)). To obtain the values of the mentioned variables, a special capillary rheometer connected to an extruder equipment, Limper and Fattmann [7], is used, and from this test, the curve deformations of cuts versus viscosities for six conditions are obtained, as shown in **Table 1** and it is carried out by the company ALFAVINIL S.A., manufacturers of PVC composites, with a RheoDrive 7 capillary rheometer, model: Rheomex 19/25 OS, in line with an


**Table 1.**

*Rheological values of the rigid PVC compound of ALFAVINIL code: RE97/1-773.*

extruder, whose screw has a compression ratio of 2:1, L/D: 25 and a slit capillary matrix H: 2.0 mm W: 20 mm and with a material temperature profile of 150, 170, and 190°C in the extruder and 200°C in the head.

$$
\eta = K.(\mathfrak{y}n - \mathfrak{1})\tag{2}
$$

A practical way to identify a PVC compound is through its *K* value, which is the relative viscosity index [4] and which in our case has a value of 65; it is determined by testing a solution with a concentration of 5 g/L. PVC-U in cyclohexanone measures the time of passage through a capillary containing the Ostwald viscometer. This test was also carried out by the company ALFAVINIL S.A. Most of the molten polymers behave as non-Newtonian fluids, that is, their shear viscosity (*η*), at a constant temperature, depending on the shear deformation (*γ*), being able to define it (Equation 1) is the quotient between shear stress (*τ*) and shear rate (*ɣ*). Thermoplastics, especially PVC, is classified as pseudoplastics; that is, their shear viscosity decreases with increasing shear deformation (*γ*). To carry out a simulation, you must have precise information on the behavior of the material under the following processing conditions: temperature, flow, density, and rheological characteristics. Due to the fact that it is desired to model the extrusion process in the final part of the screw and in the head, the flow of the molten material has low speeds and shear deformations (*γ*), the latter are of the order of 50–100 1/s, For this reason, the Ostwald model or Power Law (Eq. (2)) is applied for which the k and n values need to be known, which are obtained from the rheological test. *η* = *k*.*ɣ* (*n* � 1) (2). The value *n* is the exponent of the power law and is defined as the relationship between the stress and the rate of deformation or shear strain (*γ*), and the value *k* is the viscosity at shear rate or speed of deformation = 0 [3]. The rheological test is carried out with a special capillary rheometer connected to an extruder [8] and the values of the shear deformations vs. viscosities are obtained for six points or measurement conditions (**Table 2**).

#### **2.3 Methods**

#### *2.3.1 Drawing of the flow channel in the head*

The existing head flow channel is drawn using 3D solid Computer Aided Design software (**Figure 1**).

#### *2.3.2 Simulation*

The flow channel geometry drawing is imported into the head using the ANSYS Polyflow simulation software.


*Design, Simulation, and Analysis of the Extrusion Process of a PVC Thermoplastic Profile… DOI: http://dx.doi.org/10.5772/intechopen.100909*

**Table 2.**

*Rheological values obtained from the rigid PVC test—ALFAVINIL code: RE97/1-773.*

**Figure 1.** *Geometry flow interior head and outlet distance 60 mm (green).*

Geometry meshing (**Figure 2**) is performed using the Polyflow default system, which performs the automatic subdivision into anisotropic tetrahedral elements for three-dimensional flows that have free surfaces such as the one presented in our case (flow of molten material at the head outlet).

In the same module, the names of each subdomain are incorporated, which are the areas of the melt inside the head: SD1 (**Figure 3**) and outside of it: SD2 (**Figure 4**) so that the simulation software recognizes which is each zone for the later calculations.

The names of the contours (surfaces) of each subdomain SD1\_BS2 (**Figure 5**) and SD2\_BS2 (**Figure 6**) are defined; the flow input surfaces to the head: SD1\_BS1 (**Figure 7**) and the flow output surfaces of the same: SD2\_BS1 (**Figure 8**) are also defined.

Subsequently, the setup module is entered and the input variables are entered using the Polydata program: flow (8.5 kg/h), material density (1.42 g/cm3 ), and the rheological values *k* = 7645 and *n* = 0.69 of the Power Law, also defining the boundary conditions, that is, the values of speed, voltages, input, and output flows among other parameters of each subdomain and contours (surfaces). In addition, in this step, the zone in which the meshing must be performed again is defined in the programming, which is the exit zone of the molten flow from the head (see green region in **Figure 1**), so that the geometry of the profile in the molten state with the deformations that are the product of the swelling effect of the melt [3]. This effect occurs when a non-Newtonian fluid (polymer in viscoelastic molten state) passes through a restricted area, such as the internal geometry of the flow channel in the head, tensions are generated in its interior and when leaving said matrix, the material relaxes, decreasing its length and increasing its cross section. To determine the magnitude of this geometric variation of the flow at the outlet of the head, ANSYS Polyflow uses a technique called overlock optimization, which uses a

**Figure 2.** *Mesh of the flow channel inside the head.*

**Figure 3.** *Subdomain SD1—molten zone inside the head.*

**Figure 4.** *Subdomain SD2—fade zone at the head output.*

*Design, Simulation, and Analysis of the Extrusion Process of a PVC Thermoplastic Profile… DOI: http://dx.doi.org/10.5772/intechopen.100909*

**Figure 7.** *SD1\_BS1: head flow inlet surface.*

kinematic equation f (s) that introduces nonlinear terms in the problem that lead to the convergence of the model. As in our case, we have a problem of a numerically complex flow because it is a non-Newtonian fluid with low Power Law indices (the rheological values were determined at deformation speeds of 50–100 1/s), which are those that could be measured with the special capillary rheometer used in line with an extruder [7], an incremental numerical scheme is used to facilitate convergence. To solve these highly nonlinear problems, a low flow solution is first calculated and then projected for a higher flow until the required value is reached. This method is called "Evolution" that is to say that the flow evolves from an initial value (see **Figure 9**), and then, it increases this parameter (*s*) and finds a second solution

**Figure 9.** *Evolution of the* S *parameter.*

*Design, Simulation, and Analysis of the Extrusion Process of a PVC Thermoplastic Profile… DOI: http://dx.doi.org/10.5772/intechopen.100909*

and so on until the model is optimized, and there may be several intermediate steps and in each of them, the solution of the previous step is obtained and so on repeatedly until convergence is achieved. The concept can be applied to different boundary conditions (flow, temperature, drag force, amount of mass slip, etc.) and material properties (shear thinning index, relaxation time, specific heat, etc.). This process is carried out automatically and the increments are adapted in the same way, until the best solution is found. *Q* = *Q*nom. *f* (*s*), *Q*nom is the value of nominal, flow *f* (*s*) is the evolution function and "*S*" is the evolution variable.

Next, in solution, the computational run is made to calculate the results for the model defined in the previous steps.

Finally, the post-processing is carried out with which the results of the calculations performed are obtained, such as the values of the distribution of speeds, shear deformations, pressures, contours of the profile at the exit of the head, and the graphs of each of them for analysis.

### *2.3.3 Experimental design and construction of the calibrator*

The caliper is experimentally designed and drawn with 3D solid computer-aided design software (**Figure 10**).

The different parts that make up the calibrator are built and assembled (**Figure 11**), placing the assembly in the chamber or calibration-cooling pan and leaving the fixing screws of the upper part unadjusted, just positioning them, leaving the necessary play to enter the profile for calibration.

The gauge defines the final external dimensions of the profile [1], so that the profile in the molten state at the outlet of the head, at temperatures close to 200°C,

**Figure 10.** *CAD-3D caliper drawing.*

**Figure 11.** *Assembled calibrator.*

enters the gauge, where vacuum is applied and the material copies the internal dimension calibrator.

On the other hand, when the calibrator is immersed in a pan with water at 20°C, the profile cools, solidifying.

In order to define the internal dimensions of the caliper, it was taken into account that the molten material, when it cools, contracts and in the case of rigid PVC compounds it can reach high values of up to 4%.

In addition, another decrease in the cross-sectional area must be taken into account due to the stretching caused to the material by the dragging carried out by

**Figure 12.** *Collins extruder.*

*Design, Simulation, and Analysis of the Extrusion Process of a PVC Thermoplastic Profile… DOI: http://dx.doi.org/10.5772/intechopen.100909*

the draft equipment, which can be between 5 and 10% for low-thickness rigid PVC profiles [8].
