Abstract
This paper is a study on the elongational deformation induced by pins in pin-barrel cold-feed extruders. The mathematical model is established by using simple flow field, plate model, and Newton constitutive relation. It is proved that there is elongational deformation when the fluid bypasses the pin, and the elongational deformation is quantitatively calculated. The finite element method is used to simulate the flow field of the screw mixing section of the pin-barrel cold-feed extruder with different specifications. The velocity vector nephogram in the simulation results is analyzed by numerical analysis of elongational deformation and compared with the theoretical elongational deformation value. The results show that the simulated and theoretical values of elongational deformation are approximately consistent. The numerical analysis of the key factors affecting the elongational deformation shows that the elongational deformation will gradually increase with the increase of screw diameter. For the screw of the same specification, the increasing helical angle will reduce the elongational deformation, and the increasing rotational speed will linearly increase the elongational deformation in unit time. This study can be used to roughly estimate the elongational deformation induced by pins and establish the mixing theoretical model of the pin-barrel cold-feed extruder.
1. Introduction
Since its birth in 1972, the pin-barrel cold-feed extruder has been paid more and more attention by rubber industry, especially by tire manufacturers. Because of its relatively strong mixing performance, it has been widely used in rubber extrusion process. The structural characteristics of the pin-barrel cold-feed extruder are that 6~10 rows of pins are installed on the barrel and 6~12 pins are arranged on each row of pins (Figure 1) [1]. The number and rows of pins depend on the diameter and length-diameter ratio of the screw. The main role of barrel pins is to continuously conduct (i) diversion orientation, (ii) elongational deformation, and (iii) position replacement for high viscosity rubber, which greatly promotes the interface growth, enhances the distribution mixing, improves the plasticization (softening) efficiency, and achieves the requirements of deformation and temperature uniformity [2–5, 23, 24].
Many experimental and simulation studies have been carried out on the mixing behavior of pin-barrel cold-feed extruder [6–10]. Yabushita et al. used a white compound and a black compound to study the mixing performance of pin barrel extruder with two rows of pins and ordinary cold-feed extruder. Their contrastive experimental results showed that the introduction of pins could greatly improve the distribution mixing effect [6]. Shin and White used the flow analysis network method to study the influence of introducing slots in the screw flights and pins on the barrel on the screw pump characteristics. Their analysis results showed that the slots in the screw flights reduced the pumping capacity and the pumping capacity of the two extruders was very close [7]. Brzoskowski et al., through the establishment of kinematic model and the use of flow network analysis method, concluded that the reorientation and elongation of the rubber compounds induced by pin movement were the key factors to obtain good mixing effect [8]. Schöppner et al. tried to optimize the distribution and dispersion mixing effect in the screw channel by changing the geometric design and arrangement of pins. Their experimental and simulation results showed that the geometric changes of pins (cylindrical, concave, convex, and conical) had no effect on the dispersion mixing, but they could almost equally greatly enhance the distribution mixing effect [9].
However, the mixing theory of pin-barrel cold-feed extruder still lags far behind the engineering practice, because, so far, a complete mathematical model of mixing behavior of pin-barrel cold-feed extruder has not been established. The reason for this is that the flow field in the pin-barrel section is very complex and difficult to model. This condition also includes elongational deformation induced by pins. When the fluid bypasses the pin, the fluid produces elongational deformation under the convergence gap formed by the screw flight and the pin. Due to the increase of interfacial area index caused by elongational deformation, the mixing effect is greatly enhanced. Therefore, it is necessary to carry out theoretical analysis on the elongational deformation induced by pins.
The main purpose of this paper is to supplement the mixing theory of pin-barrel extruder, theoretically verify the elongational flow induced by pins, and quantitatively calculate it. However, according to the current technology, the elongational deformation induced by pins cannot be measured by experiments. Therefore, this paper attempts to compare the established simple approximate mathematical model of elongational deformation induced by pins with Polyflow simulation results. The idea of establishing such a model is without considering the precise and complex flow field in the screw channel of the pin barrel section; the existence proof and quantitative calculation of the elongational deformation of the pin are carried out only according to the simple flow field and the screw channel geometry.
2. Mathematical Model
2.1. Motion Transformation and Coordinate Establishment
For the convenience of analysis, the plate approximation and motion transformation are carried out, like the analysis of melt conveying in the extrusion section of single screw extruder [11–13, 22]. That is, the screw channel is unfolded into a flat channel, and the corresponding barrel is unfolded into a flat plate. Suppose the screw is stationary and the barrel rotates. Therefore, in this plate approximation model, the plate (barrel) slides relatively on the developed flat channel (screw channel) (Figure 2). In Figure 2, is the moving speed of the barrel (equal to the linear speed of the screw rotation). where is the screw diameter and is the screw rotational speed.
Under the drag action of the barrel, the velocity of the fluid in the screw channel is decomposed into the cross-channel velocity component and the down-channel velocity component . The coordinate system is established with the cross-channel velocity component direction as axis, the down-channel velocity component direction as axis, and the screw channel depth direction as axis, as shown in Figure 2.
2.2. Basic Equations, Assumptions, and Simplification
In the theoretical analysis, it is assumed that the rubber is incompressible Newtonian fluid and is filled with screw channel. The flow process of the rubber is an isothermal process, so the three-dimensional constitutive equation and motion equation of the fluid are the following: (1)Newton constitutive equation:(2)Mass conservation equation:(3)Momentum conservation equation:where is the stress tensor, is the Newton viscosity, is the strain rate tensor, is the differential operator, is the velocity vector, is the density, is the pressure, and is the gravity acceleration.
Since the viscosity of the rubber increases and the Reynolds value is low,
The has little effect on the elongational deformation induced by pins, so the velocity component can be ignored in the calculation of elongational deformation.
Assuming that the flow along the direction is a fully developed steady flow and ignores the inertial force, gravity, and pressure gradient along the direction, the simplified momentum equation in the direction is
2.3. Velocity Distribution
In the plate model of Figure 2, the screw channel depth is . Since the pin is fixed on the barrel and moves together with the random barrel, the velocity of the pin is , which does not change with the change of the height position of the screw channel (Figure 3(a)).
In the screw channel area without pins, the quadratic integral of Equation (6) is carried out, and the assumption of no wall slip is considered: ; ; get the velocity :
Therefore, in the screw channel area without pins, the fluid velocity was linearly related to the screw channel height (Figure 3(b)).
2.4. Generation of Elongational Deformation
As shown in Figure 4, since the typical design value of the ratio of the pin diameter to the axial width of the screw channel is about 0.2, the pin has a noticeable effect on the motion of the fluid in the screw channel. The moment when the pin occupies any position in the screw channel can be seen from Figure 4; due to the geometric constraints of the flight and the pin, the flow channel regions abgf and bcjh converge geometrically along the direction. Since the depth of the screw channel is constant, when the fluid flows through these two geometric convergent channels, the linear velocity of the fluid layer will increase at the position of the same screw channel height. Therefore, in the direction, for any screw channel height , the velocity at the cross section along the direction at the fluid layer lines fg and hj is greater than that at the lines ab and bc, respectively. Therefore, there is a velocity gradient in the direction, resulting in elongational deformation. In Sections 2.5 and 2.6, the existence of the above elongational deformation will be proved, and the elongational deformation will be quantitatively calculated.
2.5. Quantitative Proof of Elongational Deformation
To facilitate the calculation, the pin diameter is approximately equal to the axial width of the screw flight slot. In Figure 4, position “1” is the initial position of the pin movement in a screw channel, position “3” is the end position, and the screw channel region pqsl is used as the analysis area of elongational deformation. Position “2” is the position of any pin movement in the analysis area.
Set , , , and .
When the pin follows the barrel, in the direction, the interaction of the screw and the pin produces the convergent flow channel. According to the geometric characteristics, it is obvious that
Let , , , and , respectively, be the fluid velocity functions along the direction on the cross sections of hj, bc, fg, and ab. From the law of conservation of mass, we can obtain
Therefore,
In order to make Equations (11) and (12) hold, . Equations (11) and (12) show that the velocity of the fluid in the direction is inversely proportional to the width of the channel at the position of the convergent channel on the same side of the pin and the same height of the channel. Combing Equations (8) and (9), we can obtain
Therefore, at the same screw channel height , the velocity gradient of the fluid on both sides of the pin along the direction exists. This quantitatively proves the conclusion that the qualitative analysis in Section 2.4 has elongational deformation.
2.6. Quantitative Calculation of Elongational Deformation
By multiplying the sides of Equation (11) and (12), respectively, and considering the inlet velocity in the critical state, , we can obtain
As pins move in the region pqsl, has been changing, resulting in and also changing. In order to obtain a simple analytical solution of elongational deformation, the relationship between the outlet velocity and the inlet velocity of the convergent channel on both sides of the pin must be known. Therefore, it is assumed that when the pin moves to a certain position in the analysis area, , then , so
When the center of the pin moves to the width of the normal screw channel of , and . Equation (15) holds, , indicating that there is indeed such a point in the analysis area pqsl that satisfies the above hypothesis (as shown in position “2” in Figure 4).
From Equation (15),
Combining Equations (11) and (12) and considering the relationship between the outlet velocity and inlet velocity at position “2” satisfying Equation (15) can be obtained:
Since at the inlet of convergent channel on both sides of pin, so
It can be seen from Figure 4 that in the analysis region pqsl, the elongational deformation generated by the movement of the pin from position “1” to position “2” and from position “2” to position “3” is symmetrically equal to that at position “2.” Therefore, in order to simplify the calculation, the elongational deformation of the fluid induced by the pin at the position “2” of Equation (15) can be used as the average value of the elongational deformation induced by the pin at any position in the screw channel during the rotation of the screw for a round. Therefore, according to the definition of elongational strain,
From Equation (18),
Therefore,
Equation (21) shows that the elongational strain rate () of the fluid on both sides of the pin will change due to the different height of the screw channel. And because combining Equation (21), the average elongational deformation in one side (left or right side) of the pin at any pin position is where
When the screw speed is (rpm), the time of screw rotation 1r is (min). In the process of screw rotation 1r, the path length of a single pin is For a single pin, the path of elongational deformation is , and the time is so
By substituting Equations (24) and (26) into Equation (23), it is found that the elongational deformation of fluid on one side of a single pin at the height of the screw channel is
Therefore, the elongational deformation of the fluid on both sides of a single pin at the height of the screw channel is
When
By integrating Equation (28), the total elongational deformation of fluid on both sides of a single pin in the whole screw channel depth is obtained as follows:
It should be noted that in the screw flight slot section outside the analysis area pqsl ( section in Figure 4), when the pin passes, elongational deformation occurs on the left (or right), both sides or right (or left) of the pin in the direction. Since is small, the elongational deformation in section is still approximately calculated according to the above method.
When the screw rotates 1r, the number of elongational deformations produced by a single pin at distance is The total elongational deformation produced by a single pin is
Set the extruder has a total of rows and each row has pins. When the screw rotates 1r, the elongational deformation induced by all pins is
When the screw rotation numbers in unit time (s) is , the elongational deformation of all pins in unit time (s) is
Equation (33) shows that the elongational deformation of all pins per unit time (s) depends on the screw diameter, helical angle, screw channel depth, normal screw channel width, pin diameter, pin number, and screw rotation speed.
According to the general design experience, the screw of the same specification has the following typical approximate relationship:
In the above equation, is the number of screw heads, and generally in pin barrel section,
Substituting the above relation into Equation (31), we can get
Substituting Equation (34) into Equation (33), we also get
Equation (36) shows that for the pin-barrel cold-feed extruder with a given specification, and are known quantities, and the main factors affecting the elongational deformation of pins per unit time are the screw rotation speed and the number of pins.
3. Simulation
The three-dimensional model of the mixing section of pin barrels was established [20, 25, 26], which is shown in Figure 5. The finite element simulation of the three-dimensional model was carried out by Polyflow, and the velocity vector nephogram in the simulation results was numerically analyzed. The parameter settings (according to the general screw design standard, the helical angle is set at 24° to allow for the fluctuation of pressure transfer and to achieve the desired pumping capacity) of the mixing sections are shown in Table 1.
In order to simplify the calculation, the following assumptions were made in the simulation: (i) The rubber compound is full of screw channels and incompressible. (ii) The influence of gravity and inertia force is ignored because of the large rubber melt viscosity. (iii) There is no wall slip. (iv) The rubber compound is laminar flow. (v) There is isothermal flow [14–18].
The Bird-Carreau equation was used for the constitutive model to describe the rheological characteristics of rubber compound: where is the infinite shear viscosity (Pa·s), is the zero shear viscosity (Pa·s), is the relaxation time (s), is the shear rate, and is the non-Newtonian index.
The physical parameters of the rubber compound are as follows:
Table 2 shows the data satisfying Equation (37), where the standard errors of zero shear viscosity and relaxation time are minimized [19]. According to the general design standard of the screw, the screw diameter of each specification has different rotational speed ranges. In order to improve the authenticity and practicability of simulation results, the boundary conditions and rotational speed (Table 3) are selected as follows: (i) combined with the extrusion output of extruder and the typical ratio-gravity of rubber compound (1.25), the volume flow rate at the flow field inlet of the screw channel can be calculated (Table 3). (ii) Select free flow at the outlet of the flow field of the screw channel. (iii) The wall condition is set as no wall slip. (iv) Screw is the moving part.
Under the above conditions, CFD-POST was used to generate the velocity vector nephogram. For the convenience of observation, taking the screw of as an example [21], the image is generated as shown in Figure 6.
(a) Mixing section of the pin-barrel extruder
(b) Mixing section of the single screw extruder
4. Result Analysis
In the direction of the drag flow, by comparing the velocity vector of the fluid in the screw channel, it can be observed that the color of the fluid velocity vector changes obviously near the pin of the pin barrel extruder (Figure 6(a)); in the single screw extruder, the velocity vector of the fluid almost keeps the same color (Figure 6(b)). The velocity vector nephogram generated by simulation shows that in the pin-barrel extruder, the existence of pins indeed makes the fluid generate velocity gradient and elongational deformation in the direction of drag flow.
In order to clearly observe the fluid motion near the pin, the flow field near the pin in Figure 6 was enlarged. As shown in Figure 7, combined with the theoretical analysis in Section 2, the pin was selected as far as possible to move to the position that satisfies Equation (15). Then the fluid layers at the inlet and outlet when the screw channel depth are taken to calculate the velocity of the rubber compound on this fluid layer, as shown in Figure 8.
The fluid layer velocity of other specifications of the screw is shown in Figure 9.
(a)
(b)
(c)
According to Figures 8 and 9, the velocities of the region I, III, and V were ignored. The maximum velocity difference and of the fluid on both sides of the pin are taken, and then, the average velocity difference was calculated. Using Equations (19) and (22), the elongational deformation simulation values of the fluid on both sides of the pin were obtained when . The theoretical values were calculated according to Equation (29), as shown in Figure 10.
By observing Figure 10, it can be clearly seen that the simulated value and the theoretical value are approximately consistent, indicating that the theoretical analysis in Section 2 is approximately correct and can be used for practical calculation. In order to investigate the key factors affecting the elongational deformation, condition (a) was substituted into Equation (35), and condition (b) was substituted into Equation (36) for numerical simulation. The effects of screw diameter , helical angle , and rotational speed on the elongational deformation were analyzed, and the results are shown in Figure 11. (a)Screw diameter: (i)Helical angle: (b)Number of pins: (i)Screw diameter: (ii)Helical angle: (iii)Rotational speed:
(a)
(b)
It can be seen from Figure 11(a) that when the screw diameter is constant, the elongational deformation will gradually decrease with the increase of the helical angle. When the helical angle is constant, the elongational deformation increases with the increase of screw diameter. It also can be seen from Figure 11(b) that when the screw specification is given, the elongational deformation induced by pins in unit time will increase linearly with the increase of the screw rotational speed.
5. Conclusion
Through simple flow field and geometric simplification, the mathematical model was established to prove and quantitatively calculate the elongational deformation induced by pins. The mathematical model in the theoretical analysis was verified by simulating the flow field in the mixing section, and it was found that the simulated value of elongational deformation was approximately consistent with the theoretical value. According to the mathematical model and simulation results of elongational deformation, the following conclusions can be drawn that the elongational deformation will gradually increase with the increase of screw diameter. For the screw of the same specification, the increasing helical angle will reduce the elongational deformation, and the increasing rotational speed will linearly increase the elongational deformation in unit time.
Although the model is used for the rough estimation of elongational deformation, it reflects the key factors affecting the elongational deformation induced by pins. We can also approximately estimate the mixing performance of an extruder according to the calculated elongational deformation value, so as to make a more convenient and intuitive comparison. This theory can be used to establish the complete mixing theory model of pin-barrel cold-feed extruder.
Nomenclature
| : | Moving speed of the barrel |
| : | Screw diameter |
| : | Screw rotational speed |
| : | Cross-channel velocity |
| : | Down-channel velocity |
| : | Stress tensor |
| : | Newton viscosity |
| : | Strain rate tensor |
| : | Differential operator |
| : | Velocity vector |
| : | Density |
| : | Pressure |
| : | Gravity acceleration |
| : | Screw channel depth |
| : | Helical angle |
| : | Pin diameter |
| : | Normal screw channel width |
| : | Normal screw flight width |
| : | Screw flight slot length |
| : | Elongational strain |
| : | Elongational deformation |
| : | Screw heads |
| : | Infinite shear viscosity |
| : | Zero shear viscosity |
| : | Relaxation time |
| : | Shear rate |
| : | Non-Newtonian index. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Science and Technology Innovation Project of Shandong Province of China (Grant No. 2020CXGC010312). Furthermore, the authors would like to thank the Associate Professor Liu Yanchang and Dr. Pan for their discussions and guidance throughout this work.