1. Introduction

Polymer plasticization is a complex process with many uncertain variables, which involves phase transfer and viscoelastic behavior. The nonlinear effect of polymer plasticization is a multidisciplinary engineering science problem that includes heat transfer, rheology, and flow dynamics among others. The thermal homogeneity and stability of polymer melts in this plasticization process is the key to determine the quality of products, especially for biodegradable nanocomposites or microcellular foam materials.

In polymer processing, plasticization screw is an added unit operation that facilitates melting and homogenization of an initially heterogeneous physical system [1, 2]. In general, the temperature distribution is not uniform in the process of plasticization; this is due to significant friction heating and the low thermal conductivity of polymers. It is very important to optimize the structural parameters and working characteristics of the screw in order to enhance plasticization of

polymers. The effect of barrel configurations and screw designs on heat and mass transfer has been investigated in the past and proved unquestionably important attributes for determining temperature uniformity and mixing effectiveness in extrusion and injection molding processes [3–8]. Different from conventional screw, the new types of screw can be roughly divided into four categories: distribution screws, barrier screws, separator screws, and channel screws with variable sections. These reconfigured screws are stated to be better than a standard one. For example, Kelly et al. [6] developed a barrier-flighted screw with Maddock mixer to achieve good melting performance and low temperature and pressure fluctuations. Spalding [9] introduced a distributive melt-mixing type screw equipped with an Eagle mixer in injection molding process and obtained better melting capacity and higher mixing than a conventional screw. Shimbo et al. [10] disclosed a mixture system by combining Pin and Dulmage type screws and also reported their beneficial effect in kneading, homogenization, and ensuring stability of gas/polymer solution. Zitzenbacher et al. [11, 12] indicated that the shearing sections like axial and spiral Maddock elements and Z-elements are often used to improve the melt homogeneity by enhancing dispersive mixing. Rydzkowski [13] developed an autothermal screw-disc extruder to induce autothermal effect. Rauwendaal [4] noted a CRD mixing screw with wedge-shaped barrier region to generate elongational flow. In this way, the mixing ability improves under the condition of lower power consumption and viscous dissipation than shear flow. Also, the melt temperatures and pressure fluctuation reduce in the flow channel. Based on the elongational flow and volume transportation, Qu et al. [14, 15] proposed a novel vane plasticization system in place of screw one to create extensional stress. Their results showed that the plasticizing capacity improved with the decrease of power consumption in the vane extruder. Diekmann [16] analyzed the direct-drive singlescrew extruders without gearing and results indicated that plasticization capacity increased in the direct-drive system. Besides, Jiang [17] introduced ultrasonic plasticizing in the extruder to achieve energy saving. And Qu et al. [18] introduced a vibrational force field in the screw plasticization system. Their results showed that mixing performance improved and extrusion pressure reduced.

2. Multi-field synergy theory

DOI: http://dx.doi.org/10.5772/intechopen.89616

In order to find out relationships among the velocity, velocity gradient, and temperature gradient fields in the plasticization process of polymer, we presented the mathematical expressions for quantitative analysis of multi-field synergy based

Multi-Field Synergy Process for Polymer Plasticization: A Novel Design Concept for Screw…

From the knowledge of polymer rheology, we have obtained the Navier-Stokes equation derived from momentum conservation equation in the form of Eq. (1).

\*

¼ ∇� τ \*

where ρ is the fluid density, v is the fluid velocity, t is the time, τ is the stress, P is

where α is the intersection angle between the velocity gradient and the velocity

0

B@

Here defines the synergy angle α to represent the synergy between the velocity and the velocity gradient. In the range 0°–90°, the dot product Eq. (3) increases with decreasing α, which leads to the increase in momentum and enhancement of mass transfer. This means the interaction of velocity and velocity gradient has an effect on the momentum of the system, that is, the increment of momentum depends not only on the magnitudes of velocity or velocity gradient, but also on the

When the synergy angle α becomes zero, the flow is a pure elongational one, and if it becomes 90°, the flow is a pure shear one. Most notably, it is generally known that the effect of elongational flow on mixing is more strong than that of shear flow [19–24], which provides evidence for the feasibility of field synergy analysis. Therefore, the synergy relationship provides a new perspective to understand the

From Eqs. (2) and (3), we can also conclude that the pressure gradient is affected by the synergy angle α when the stress and gravity terms are invariable. In other words, the synergistic effect of velocity and velocity gradient can reduce

v \* � ∇v \*

1

v \* � � � � � � ∇v � � \* � � � �

the pressure, and g is the gravitational acceleration. The left-hand term of the equation is the inertia term, reflecting the increment of fluid momentum per unit volume in unit time. In the parentheses in Eq. (2), there is a dot product of velocity

�∇P þ ρg (1)

�∇P þ ρg (2)

cosα (3)

CA (4)

on momentum conservation equation and energy conservation equation.

<sup>D</sup><sup>t</sup> <sup>¼</sup> <sup>∇</sup>� <sup>τ</sup>

2.1 Synergy between velocity and velocity gradient

Eq. (2) is obtained by further expansion.

ρ ∂ v \* ∂t þ v \* � ∇v \*

and velocity gradient, which can be expressed as

vector, and can be calculated according to Eq. (4)

overall synergy between velocity and velocity gradient.

polymer mixing.

19

ρ Dv \*

!

v \* � ∇v \* ¼ v \* � � � � � � � ∇v � � \* � � � �

α ¼ arccos

These studies may focus on mixing and rheology. However, heat transfer also plays a significant role in plasticizing. In the past, researchers paid little attention to understand heat transfer in viscous fluid. Understanding the mass and heat transfer processes in a plasticization system as a function of screw configurations is essential to further develop a more effective screw design to overcome some of the existing challenges. The properties of composites in the plasticization process also depend on the control of the velocity, temperature, shear, and pressure fields. Therefore, it is worth investigating the synergetic relationship, if any, between various physical fields in order to maximize the efficiency of the plasticization effect.

Mixing a high-viscosity or high-molecular weight polymer melt leads to shearinduced overheating due to the large torque induced, required to unleash the polymer chain entanglements. The challenge in this case is to fabricate a screw configuration that facilitates polymer chain mobility in melt phase without inducing high shear and by facilitating effective transfer of the excess local heat out of the bulk of the polymer melt. Otherwise, the local overheating effect essentially results in unwanted heat loss and poor melt quality, subsequently, the polymer chains break down and thermosensitive polymers such as biopolymers may even be degraded.

In order to overcome the challenges of an inadequate control of flow-thermal management for multicomponent melt, we explore the synergistic relationship and interaction mechanism between various physical fields for non-Newtonian viscous liquids such as polymer melts, with special emphasis on higher molecular weight thermoplastic resins, subsequently a torsion screw has been designed.

Multi-Field Synergy Process for Polymer Plasticization: A Novel Design Concept for Screw… DOI: http://dx.doi.org/10.5772/intechopen.89616

### 2. Multi-field synergy theory

polymers. The effect of barrel configurations and screw designs on heat and mass transfer has been investigated in the past and proved unquestionably important attributes for determining temperature uniformity and mixing effectiveness in extrusion and injection molding processes [3–8]. Different from conventional screw, the new types of screw can be roughly divided into four categories: distribution screws, barrier screws, separator screws, and channel screws with variable sections. These reconfigured screws are stated to be better than a standard one. For example, Kelly et al. [6] developed a barrier-flighted screw with Maddock mixer to achieve good melting performance and low temperature and pressure fluctuations. Spalding [9] introduced a distributive melt-mixing type screw equipped with an Eagle mixer in injection molding process and obtained better melting capacity and higher mixing than a conventional screw. Shimbo et al. [10] disclosed a mixture system by combining Pin and Dulmage type screws and also reported their beneficial effect in kneading, homogenization, and ensuring stability of gas/polymer solution. Zitzenbacher et al. [11, 12] indicated that the shearing sections like axial and spiral Maddock elements and Z-elements are often used to improve the melt homogeneity by enhancing dispersive mixing. Rydzkowski [13] developed an autothermal screw-disc extruder to induce autothermal effect. Rauwendaal [4] noted a CRD mixing screw with wedge-shaped barrier region to generate

Thermosoftening Plastics

elongational flow. In this way, the mixing ability improves under the condition of lower power consumption and viscous dissipation than shear flow. Also, the melt temperatures and pressure fluctuation reduce in the flow channel. Based on the elongational flow and volume transportation, Qu et al. [14, 15] proposed a novel vane plasticization system in place of screw one to create extensional stress. Their results showed that the plasticizing capacity improved with the decrease of power consumption in the vane extruder. Diekmann [16] analyzed the direct-drive singlescrew extruders without gearing and results indicated that plasticization capacity increased in the direct-drive system. Besides, Jiang [17] introduced ultrasonic plasticizing in the extruder to achieve energy saving. And Qu et al. [18] introduced a vibrational force field in the screw plasticization system. Their results showed that

These studies may focus on mixing and rheology. However, heat transfer also plays a significant role in plasticizing. In the past, researchers paid little attention to understand heat transfer in viscous fluid. Understanding the mass and heat transfer processes in a plasticization system as a function of screw configurations is essential to further develop a more effective screw design to overcome some of the existing challenges. The properties of composites in the plasticization process also depend on the control of the velocity, temperature, shear, and pressure fields. Therefore, it is worth investigating the synergetic relationship, if any, between various physical

Mixing a high-viscosity or high-molecular weight polymer melt leads to shearinduced overheating due to the large torque induced, required to unleash the polymer chain entanglements. The challenge in this case is to fabricate a screw configuration that facilitates polymer chain mobility in melt phase without inducing high shear and by facilitating effective transfer of the excess local heat out of the bulk of the polymer melt. Otherwise, the local overheating effect essentially results in unwanted heat loss and poor melt quality, subsequently, the polymer chains break down and thermosensitive polymers such as biopolymers may even be degraded. In order to overcome the challenges of an inadequate control of flow-thermal management for multicomponent melt, we explore the synergistic relationship and interaction mechanism between various physical fields for non-Newtonian viscous liquids such as polymer melts, with special emphasis on higher molecular weight

mixing performance improved and extrusion pressure reduced.

fields in order to maximize the efficiency of the plasticization effect.

thermoplastic resins, subsequently a torsion screw has been designed.

18

In order to find out relationships among the velocity, velocity gradient, and temperature gradient fields in the plasticization process of polymer, we presented the mathematical expressions for quantitative analysis of multi-field synergy based on momentum conservation equation and energy conservation equation.

#### 2.1 Synergy between velocity and velocity gradient

From the knowledge of polymer rheology, we have obtained the Navier-Stokes equation derived from momentum conservation equation in the form of Eq. (1).

$$
\rho \frac{\mathbf{D} \overline{v}}{\mathbf{D}t} = \nabla \cdot \overline{\mathbf{\tau}} - \nabla P + \rho \mathbf{g} \tag{1}
$$

Eq. (2) is obtained by further expansion.

$$
\rho \left( \frac{\partial \overline{\boldsymbol{v}}}{\partial t} + \overline{\boldsymbol{v}} \cdot \overline{\boldsymbol{\nabla}} \boldsymbol{v} \right) = \nabla \cdot \overline{\boldsymbol{\tau}} - \nabla P + \rho \mathbf{g} \tag{2}
$$

where ρ is the fluid density, v is the fluid velocity, t is the time, τ is the stress, P is the pressure, and g is the gravitational acceleration. The left-hand term of the equation is the inertia term, reflecting the increment of fluid momentum per unit volume in unit time. In the parentheses in Eq. (2), there is a dot product of velocity and velocity gradient, which can be expressed as

$$
\overrightarrow{\boldsymbol{v}} \cdot \overrightarrow{\boldsymbol{\nabla}} \boldsymbol{v} = \begin{vmatrix} \overrightarrow{\boldsymbol{v}} \\ \overrightarrow{\boldsymbol{v}} \end{vmatrix} \times \begin{vmatrix} \overrightarrow{\boldsymbol{\nabla}} \boldsymbol{v} \end{vmatrix} \cos \boldsymbol{a} \tag{3}
$$

where α is the intersection angle between the velocity gradient and the velocity vector, and can be calculated according to Eq. (4)

$$a = \arccos\left(\frac{\overrightarrow{v} \cdot \overrightarrow{\nabla v}}{\left|\overrightarrow{v}\right| \left|\overrightarrow{\nabla v}\right|}\right) \tag{4}$$

Here defines the synergy angle α to represent the synergy between the velocity and the velocity gradient. In the range 0°–90°, the dot product Eq. (3) increases with decreasing α, which leads to the increase in momentum and enhancement of mass transfer. This means the interaction of velocity and velocity gradient has an effect on the momentum of the system, that is, the increment of momentum depends not only on the magnitudes of velocity or velocity gradient, but also on the overall synergy between velocity and velocity gradient.

When the synergy angle α becomes zero, the flow is a pure elongational one, and if it becomes 90°, the flow is a pure shear one. Most notably, it is generally known that the effect of elongational flow on mixing is more strong than that of shear flow [19–24], which provides evidence for the feasibility of field synergy analysis. Therefore, the synergy relationship provides a new perspective to understand the polymer mixing.

From Eqs. (2) and (3), we can also conclude that the pressure gradient is affected by the synergy angle α when the stress and gravity terms are invariable. In other words, the synergistic effect of velocity and velocity gradient can reduce

pressure loss and energy consumption, which is of great significance in Newtonian fluid flow. However, for most polymers, which are non-Newtonian in nature, the pressure term has little effect on mechanical power and energy consumption due to its very high viscosity.

Nul ¼ Re lPrl

Re <sup>l</sup> <sup>¼</sup> <sup>ρ</sup>vδ<sup>t</sup>

3. Model design and description

DOI: http://dx.doi.org/10.5772/intechopen.89616

4. Numerical analysis examples

Figure 1.

21

ð<sup>δ</sup>,<sup>t</sup> lh 0

Multi-Field Synergy Process for Polymer Plasticization: A Novel Design Concept for Screw…

where U is the velocity vector, ∇T is the temperature gradient vector, Rel is the l-component of the Reynolds number, Prl is the l-component of the Prandtl number, and Nul is the l-component of the Nusselt number. Rel, Prl, and Nul are expressed as

<sup>λ</sup> ,Nul <sup>¼</sup> <sup>K</sup>δ<sup>t</sup>

<sup>μ</sup> ,Prl <sup>¼</sup> Cp<sup>μ</sup>

where μ is the fluid viscosity, Cp is the specific heat capacity at constant pressure, λ is the thermal conductivity, and δ<sup>t</sup> is the characteristic dimension, which refers to the thickness of the thermal boundary layer. From Eqs. (9) and (10), we can notice that an increase in interaction between the temperature gradient and the velocity fields increases the Nusselt number and the coefficient of local heat transfer, and consequently enhances the overall heat transfer. These synergy equations suggest a new approach to enhance heat transfer of the polymers with poor heat conductivity, namely by increasing the dot product in the integral (Eq. 9).

By understanding the multi-field synergy effect in the heat and mass transfer process of polymer plasticization, we can construct a specific flow field so that the directions of velocity field and temperature gradient field are no longer perpendicular, and the flow field movement is more random. The schematic diagram is shown in Figure 1. In this way, it can facilitate phase-to-phase thermal and molecular mobility, so as to significantly improve heat transfer and molecular mixing, particularly for highly viscous multicomponent polymer melts with Bio or Nano filler. Based on this method, we can design a special screw configuration to divert the fluid particles and obtain the desired flow field. Here, we propose a new type of screw element, namely, torsion element, to stimulate the spiral or torsional flow, which is the most common way of disturbing or changing flow direction in nature.

In the following sections, we develop a novel torsion element-induced torsional flow into the flow field by adapting the field synergy principle. Then, we establish a three-dimensional physical and mathematical model with finite element method

The synergy between velocity field and heat flow field: Parallel movement (a) and spiral movement (b).

<sup>U</sup> � <sup>∇</sup><sup>T</sup> � �dy (9)

<sup>λ</sup> (10)

#### 2.2 Synergy between velocity and temperature gradient

Almost all polymer processing unit operations require heat transfer processes such as energy exchange, heating, and cooling to facilitate phase-to-phase thermal and molecular mobility. Therefore, the study of energy balance and distribution has special significance in the process of melt flow. It is well known that the general energy conservation equation in the flow field can be represented in the form of Eq. (5).

$$
\rho C\_V \frac{\text{DT}}{\text{Dt}} = -\nabla \cdot \overline{q}^- - T \left( \frac{\partial P}{\partial T} \right)\_P \left( \nabla \cdot \overline{\nu}^- \right) + \left( \overline{\tau}^- : \nabla \cdot \overline{\nu}^- \right) \tag{5}
$$

By further expansion, we obtain Eq. (6) as follows.

$$\rho C\_V \left( \frac{\partial T}{\partial t} + \overline{\boldsymbol{v}} \cdot \overline{\boldsymbol{\nabla} T} \right) = -\nabla \cdot \left( -K \, \overline{\boldsymbol{\nabla} T} \right) - T \left( \frac{\partial \mathbf{P}}{\partial T} \right)\_P \left( \boldsymbol{\nabla} \cdot \overline{\boldsymbol{v}} \right) + \left( \overline{\boldsymbol{\tau}} \, : \, \boldsymbol{\nabla} \cdot \overline{\boldsymbol{v}} \right) \tag{6}$$

where ρ is the fluid density, CV is the constant-volume specific heat,T is the fluid temperature, v is the fluid velocity, t is the time, τ is the stress, P is the pressure, and K is the heat transfer coefficient. The left-hand term of the equation is the changing rate of internal energy, reflecting the change of heat caused by the temperature variation per unit time at a point in the flow field. Moreover, in the parentheses, there is also a dot product of velocity and temperature gradient. The latter term signifies the interaction between velocity and temperature gradient, and demonstrates that this interaction parameter has an effect on the thermal energy of the system, that is, the change of internal energy depends not only on the velocity field and temperature gradient field, but also on the overall synergy between the velocity field and the temperature gradient field. The dot product in the parentheses in Eq. (6) can be further expressed as

$$
\overrightarrow{\boldsymbol{\nu}} \cdot \overrightarrow{\boldsymbol{\nabla}} \mathbf{T} = \begin{vmatrix} \overrightarrow{\boldsymbol{\nu}} \\ \boldsymbol{\nu} \end{vmatrix} \times \begin{vmatrix} \overrightarrow{\boldsymbol{\nabla}} \mathbf{T} \\ \end{vmatrix} \cos \boldsymbol{\beta} \tag{7}
$$

where β is the intersection angle between the temperature gradient and the velocity vector, and can be calculated according to Eq. (8)

$$\beta = \arccos\left(\frac{\stackrel{\rightarrow}{v} \cdot \stackrel{\rightarrow}{\nabla T}}{\left|\stackrel{\rightarrow}{v}\right| \left|\stackrel{\rightarrow}{\nabla T}\right|}\right) \tag{8}$$

Here is defined as the synergy angle α β representing the synergy between the velocity and the temperature gradient. In the range 0°–90°, the dot product (Eq. (7)) increases with decreasing β. When the viscous dissipation power is constant, a small β contributes to a large heat transfer coefficient K, which leads to enhanced heat transfer and temperature uniformity.

In the case of the problem of a two-dimensional flat-plate steady-state boundary layer, Guo et al. [25, 26] have simplified the energy conservation equations into the dimensionless forms

Multi-Field Synergy Process for Polymer Plasticization: A Novel Design Concept for Screw… DOI: http://dx.doi.org/10.5772/intechopen.89616

$$Nu\_l = \operatorname{Re}\_l Pr\_l \Big|\_{0}^{\delta\_l \cdot \operatorname{lh}} (\overline{U} \cdot \overline{\nabla T}) d\overline{\mathcal{y}} \tag{9}$$

where U is the velocity vector, ∇T is the temperature gradient vector, Rel is the l-component of the Reynolds number, Prl is the l-component of the Prandtl number, and Nul is the l-component of the Nusselt number. Rel, Prl, and Nul are expressed as

$$\text{Re}\_l = \frac{\rho v \delta\_l}{\mu} \cdot Pr\_l = \frac{C\_p \mu}{\lambda} \cdot Nu\_l = \frac{K \delta\_l}{\lambda} \tag{10}$$

where μ is the fluid viscosity, Cp is the specific heat capacity at constant pressure, λ is the thermal conductivity, and δ<sup>t</sup> is the characteristic dimension, which refers to the thickness of the thermal boundary layer. From Eqs. (9) and (10), we can notice that an increase in interaction between the temperature gradient and the velocity fields increases the Nusselt number and the coefficient of local heat transfer, and consequently enhances the overall heat transfer. These synergy equations suggest a new approach to enhance heat transfer of the polymers with poor heat conductivity, namely by increasing the dot product in the integral (Eq. 9).

### 3. Model design and description

By understanding the multi-field synergy effect in the heat and mass transfer process of polymer plasticization, we can construct a specific flow field so that the directions of velocity field and temperature gradient field are no longer perpendicular, and the flow field movement is more random. The schematic diagram is shown in Figure 1. In this way, it can facilitate phase-to-phase thermal and molecular mobility, so as to significantly improve heat transfer and molecular mixing, particularly for highly viscous multicomponent polymer melts with Bio or Nano filler. Based on this method, we can design a special screw configuration to divert the fluid particles and obtain the desired flow field. Here, we propose a new type of screw element, namely, torsion element, to stimulate the spiral or torsional flow, which is the most common way of disturbing or changing flow direction in nature.

Figure 1.

pressure loss and energy consumption, which is of great significance in Newtonian fluid flow. However, for most polymers, which are non-Newtonian in nature, the pressure term has little effect on mechanical power and energy consumption due to

Almost all polymer processing unit operations require heat transfer processes such as energy exchange, heating, and cooling to facilitate phase-to-phase thermal and molecular mobility. Therefore, the study of energy balance and distribution has special significance in the process of melt flow. It is well known that the general energy conservation equation in the flow field can be represented in the form of

2.2 Synergy between velocity and temperature gradient

<sup>D</sup><sup>t</sup> ¼ �∇� <sup>q</sup>

By further expansion, we obtain Eq. (6) as follows.

\*

¼ �∇ � �K ∇T

v \* � ∇T \* ¼ v \* � � � � � � � ∇T � � \* �

velocity vector, and can be calculated according to Eq. (8)

enhanced heat transfer and temperature uniformity.

dimensionless forms

20

�<sup>T</sup> <sup>∂</sup><sup>P</sup> ∂T � �

\* � �

where ρ is the fluid density, CV is the constant-volume specific heat,T is the fluid temperature, v is the fluid velocity, t is the time, τ is the stress, P is the

pressure, and K is the heat transfer coefficient. The left-hand term of the equation is the changing rate of internal energy, reflecting the change of heat caused by the temperature variation per unit time at a point in the flow field. Moreover, in the parentheses, there is also a dot product of velocity and temperature gradient. The latter term signifies the interaction between velocity and temperature gradient, and demonstrates that this interaction parameter has an effect on the thermal energy of the system, that is, the change of internal energy depends not only on the velocity field and temperature gradient field, but also on the overall synergy between the velocity field and the temperature gradient field. The dot product in the parentheses

where β is the intersection angle between the temperature gradient and the

0

B@

Here is defined as the synergy angle α β representing the synergy between the

In the case of the problem of a two-dimensional flat-plate steady-state boundary layer, Guo et al. [25, 26] have simplified the energy conservation equations into the

v \* � ∇T \*

v \* � � � � � � ∇T � � \* �

β ¼ arccos

velocity and the temperature gradient. In the range 0°–90°, the dot product (Eq. (7)) increases with decreasing β. When the viscous dissipation power is constant, a small β contributes to a large heat transfer coefficient K, which leads to

P ∇� v \* � �

� <sup>T</sup> <sup>∂</sup><sup>P</sup> ∂T � �

> � � �

> > � � �

1

þ τ

P ∇� v \* � �

\* ∶∇� <sup>v</sup> \* � �

þ τ

cosβ (7)

CA (8)

\* ∶∇� <sup>v</sup> \* � � (5)

(6)

its very high viscosity.

Thermosoftening Plastics

ρCV DT

in Eq. (6) can be further expressed as

Eq. (5).

ρCV

∂T ∂t þ v \* � ∇T \* � �

The synergy between velocity field and heat flow field: Parallel movement (a) and spiral movement (b).

### 4. Numerical analysis examples

In the following sections, we develop a novel torsion element-induced torsional flow into the flow field by adapting the field synergy principle. Then, we establish a three-dimensional physical and mathematical model with finite element method

Figure 2.

The geometrical configuration and flow line of particles for a torsion element.

(FEM), and present results of computational fluid dynamics (CFD) simulations of the flow and heat transfer of a viscous polypropylene (PP) melt in the screw with torsion elements to confirm this field synergy method and compared them with the conventional screw in common use today.

torsion element. Screws were constructed of two kinds of polymer plasticization elements: the torsion element and the screw element. Screws A to D had six torsion elements with regular arrangement in different orders. As control subjects, screw E is a conventional screw without torsion element and screw F is a torsional screw without screw elements. Table 1 presents a summary of the geometric parameters

Parameters Dimensions (mm) Length of screw 180 Diameter of screw 30 Diameter of barrel 30.4 Length of single torsion element 10 Lead of screw element 30 Inner diameter of screw 25.8

Multi-Field Synergy Process for Polymer Plasticization: A Novel Design Concept for Screw…

In this case, we assumed that the polymer fluid had non-isothermal transient laminar flow and was incompressible. No-slip conditions were adopted at the boundary. Polypropylene (PP) was chosen to be the model polymer due to its common use in polymer processing. Compared with viscous force, the inertial force can be neglected due to the high viscosity of PP. Therefore, the governing equations representing the flow field in this situation are shown in the form of Eqs. (11)–(13).

> ∂ui ∂xi

> > ¼ ∂ ∂xj η ∂ui ∂xj

> > > ¼ λ ∂2 T ∂x<sup>2</sup> i

The apparent viscosity of PP was described by the Carreau-approximate Arrhenius model (Eq. 14), which match most polymers, to consider the factors of both

¼ 0 (11)

(12)

exp ½ � �B Tð Þ � T<sup>0</sup> (14)

þ φ (13)

of simulation models.

Geometric parameters of simulation models.

DOI: http://dx.doi.org/10.5772/intechopen.89616

Table 1.

Continuity equation:

Momentum equation:

Energy equation:

temperature and shear.

23

4.2 Governing equations and boundary conditions

ρ ∂ui ∂t þ ∂P ∂xi

ρc<sup>p</sup>

η ¼ η<sup>0</sup> 1 þ t

∂T ∂t þ u<sup>i</sup> ∂T ∂xi

2 γ\_ <sup>2</sup> ð Þ <sup>n</sup>�<sup>1</sup> <sup>=</sup><sup>2</sup>

The physical characteristics of the PP and screw material are listed in

Tables 2 and 3, respectively. The choice of the polymer affects only the constants in the constitutive equation (Eq. (14)). Moreover, the Carreau-approximate Arrhenius model has been already validated for most polymer melts (e.g., polystyrene,

The geometrical configuration of the proposed torsion element is shown in Figure 2. The torsion channels are divided into N parts along the circumferential direction by torsion flights. Between every two adjacent torsion flights, there are two surfaces twisted by 90°along the axial direction. When polymer flows over the torsion channel, it is expected to undergo a torsional rotation (tumbling) under the forces generated from viscous friction with barrel wall and with the steering between two 90° twisted surfaces. As a result, spiral-shaped or torsional-shaped flow may occur in the torsion channel. In consequence, the intersection angle between the velocity and the heat flux will decrease to less than 90° compared with that in the standard screw channel, and then the synergic effect between the velocity vector and the temperature gradient will be improved.
