# Dynamics Analysis of the Melt Conveying Process in a Novel Extruder

Authors: Zan Huang
Journal: International Journal of Polymer Science
 (2015)

---

## Abstract

This study investigates dynamics of the melt conveying process in a novel extruder in which a polymer experiences a primarily elongational deformation field. The working principle of the novel extruder is completely different from that of conventional screw extruders. At the last stage of polymer processing in the proposed novel extruder, melt is conveyed through feeding and discharging processes. Here we present a mathematical model to analyze dynamics in the melt conveying process. Melt extrusion is primarily influenced by the position of vane chamber and the structural parameters of the novel extruder such as eccentricity.


---

## Body

## 1. Introduction

Polymer manufacturing involves melting and plasticizing material from glass to viscous states. During this process, the solid bed disappears completely and the polymer is fully melted. The role of a metering section is to further plasticize and homogenize the viscous polymer melt. At the end of the metering section, the melt is dispensed from the extruder headpiece at constant pressure and temperature.Polymer melt flow in the metering section must abide by the laws of conservation of mass, momentum, and energy. To solve the corresponding equations of continuity, motion, and energy, the material parameters of the polymer must be defined by an appropriate constitutive equation. Because polymer melt generally behaves as a non-Newtonian fluid, a complex constitutive equation is required to accurately model the melt behavior. Because of this complexity, it is difficult to obtain an analytical solution for the complex flow field and geometric parameters of the extruder [1, 2]. To simplify the system, approximations are made which reduce the problem into less complex terms. A linear constitutive model or the equivalent or average Newtonian viscosity has been used to simplify the constitutive equation of the melt [3, 4]. Alternatively, the melt flow in the metering section of a screw extruder can be considered as the linear superposition of drag flow and pressure flow by simplifying the flow field [5]. More recently, numerical simulations have been used to understand the melt conveying process [6–12]. In a dynamic polymer extruder, changes in the melt elasticity are more apparent when the flow is disturbed by the action of a vibrational field. Zhang and Qu [13] corrected the Tanner constitutive equation and established a dynamic model of the melt conveying process when exposed to a vibration force field.In recent years, many scholars have proposed methods to reduce the power consumption of an extruder. For example, Qu et al. [14–16] introduced the pulsating electromagnetic force field in the extrusion process, which reduced overall power consumption. The novel vane extruder introduced in this study provides a reduction in extruder power consumption, which can be verified by experiments [17]. Qu et al. [18] have also studied the power consumption of the extrusion process in a vane extruder.In this paper, an analytical model of polymer melt extrusion is established, including solving the velocity distribution, pressure gradient, extrusion flow rate, and dynamics of the melt flow field.


## 2. Description of the Melt Conveying Process in the Extensional Extruder

Figure1 shows the internal structure of a vane extruder. As introduced by Qu et al. [19], a vane extruder includes several Vane Conveying and Plasticizing Units (VCPUs) throughout the chamber. Figure 2 shows the VCPUs in the cylindrical coordinate system (r,θ,z), where the origin O is defined as the center of the rotor, r gives the radial position along the rotor shaft, θ is defined along the circumferential direction of the VCPUs where the rotation direction of the rotor is defined as positive, and z is the axial direction of the rotor where the positive direction is defined along the rotor shaft from the feed material baffle to the discharge material baffle. The main geometric parameters of the VCPU system are (1) the outer radius of the rotor r0; (2) the inner radius of the stator r1; (3) the axial width of each VCPU W; (4) the radial coordinate value of the point of the inner arc of the stator r2; (5) the eccentricity e between the center of the rotor O and the center of the stator O1; (6) the length of the vane Δrv; (7) the angular coordinate of the front vane θf; (8) the angular coordinate of the rear vane θr.Figure 1
The schematic diagram of the novel extruder.Figure 2
The cylindrical coordinate system in VCPU.According to Figure2 and the cosine theorem, the inner arc of the stator and the length of the vane are related to the VCPU system geometry by(1)r2θ=r12-e2sin2⁡θ+ecos⁡θ,Δrvθ=e+ecos⁡θ-e2sin2⁡θ2r1.A cross section of the melt conveying zone in the vane extruder is shown in Figure 3. To simplify the melt conveying process and allow for an analytical solution, the following assumptions are made:(i)
The chamber is always filled with polymer melt, even when the volume changes with the rotation of the rotor.(ii)
The melt in the vane chamber is always in isothermal laminar flow.(iii)
The melt in vane chamber is incompressible.(iv)
Leakage and the influence of the thickness of each vane and its geometry are neglected.(v)
The influence of gravity is neglected.(vi)
A Newtonian constitutive equation is used for the melt.Using the above assumptions, the melt conveying model of the vane extruder can be analytically obtained.Figure 3
The melt conveying model in a chamber of VCPU.


## 3. Mathematical Model

### 3.1. Governing Equations

Melt conveyance is examined in the melt-mixing zone of the fully filled chamber of the vane extruder. The Reynolds number was assumed to be much less than unity so that inertial effects are neglected. The governing equations for this situation are shown below.Continuity Equation. Consider(2)1r∂vθ∂θ+∂vz∂z=0,where r, θ, and z are cylindrical coordinate components, respectively; vr, vθ, and vz are velocity components of r-direction, θ-direction, and z-direction.Constitutive Equation. Consider(3)τ=μγ˙,where τ is stress tensor; μ is viscosity coefficient dependent deformation rate; γ˙ is rate of deformation tensor.Motion Equations. Consider(4)∂p∂r=-2μr2∂vθ∂θ,(5)1r∂p∂θ=μ∂∂r1r∂∂rrvθ+1r2∂2vθ∂θ2,(6)∂p∂z=μ1r∂∂rr∂vz∂r,where p is pressure; r, θ, and z are cylindrical coordinate components, respectively; vr, vθ, and vz are velocity components of r-direction, θ-direction, and z-direction; μ is viscosity coefficient dependent deformation rate. Furthermore, the boundary conditions are vz(r0,π)=0 and vz(r2,π/4)=0.Equation (2) can be simplified to(7)∂vz∂z=-1r∂vθ∂θ.Taking the partial derivative of (7) with respect to z and assuming ∂vθ/∂z=0 gives(8)∂2vz∂z2=-1r∂2vθ∂θ∂z=0.Melt velocity is assumed to vary linearly along the length of the extruder, ∂vz/∂z=0. Thus, vz can be defined as(9)vz=f1r,θz+f2r,θ,where f1 and f2 are only functions of r and θ.Material enters the vane chamber at the origin of thez axis, z=0; thus, the maximum axial velocity of melt occurs at the origin. In contrast, the velocity of the melt at the exit, z=W, in the most extreme case can be zero, vz=W=0. Substituting this boundary condition into (9) gives f2(r,θ)=-f1(r,θ)W. At the origin z=0, the axial velocity of melt feeding into the vane chamber is vz0=f2(r,θ)=-f1(r,θ)W. Thus, (9) can be simplified to(10)vz=vz01-zW,where W is the axial width of VCPU.In contrast, the terminal end of the chamber discharges melt, so the axial velocity of melt at the terminal side is maximum, while at the origin,z=0, the melt velocity is zero, vz=0=0. Substituting this boundary condition into (9), the following expression is obtained: f2(r,θ)=0. The axial velocity of the melt at discharge from the vane chamber is vz0=f1(r,θ)W. Thus, (9) can be further simplified to(11)vz=vz0zW.Assuming both sides of (4) and (5) are zero, and ∂vθ/∂z=0, the following expressions are obtained:(12)∂2p∂z∂r=∂∂z-2μr2∂vθ∂θ=0,1r∂2p∂z∂θ=∂∂zμ∂∂r1r∂∂rrvθ+1r2∂2vθ∂θ2=0.Examining (12), ∂p/∂z is independent of r and θ.Substituting (10) into (6) and rearranging give(13)1-zW1r∂∂rr∂vz0∂r+1r2∂2vz0∂θ2=1μ∂p∂z.In the geometry of a vane extruder, the circumferential distance is far greater than radial distance, and vz is independent of θ; that is, ∂vz0/∂θ=0. To solve for vz0, we first examine the inlet, z=0. Equation (13) can be simplified to(14)1r∂∂rr∂vz0∂r=1μ∂p∂zz=0.The boundary conditions of (14) are (1) when r=r0, vz0=0 and (2) when r=r2, vz0=0.Solving (14) gives(15)vz0=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0∂p∂zz=0,where vz0 is the axial velocity of the melt at discharge from the vane chamber; p is pressure; μ is viscosity coefficient dependent deformation rate; r2 is the radial coordinate value of the point of the inner arc of the stator; r0 is the outer radius of the rotor; r and z are cylindrical coordinate component, respectively.Substituting (15) into (10), the axial velocity vz of melt flow in the vane chamber at the origin (feed location) is given by(16)vz=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r01-zW∂p∂zz=0,where W is the axial width of VCPU and ∂p/∂zz=0 can be obtained using the expression for volumetric flow rate:(17)Q=∬vz0rdθdr,where Q is the rate of volume flow, Q=dV/dt; V is the volume of the vane chamber; and vz0 is the velocity of the melt entering the vane chamber.Substituting (15) into (17) and rearranging give(18)14μ∂p∂zz=0∫θrθf∫r0r2r2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0rdrdθ=Q,where μ is viscosity coefficient dependent deformation rate; p is pressure; r and z are cylindrical coordinate components, respectively; θf is the angular coordinate of front vane; θr is the angular coordinate of rear vane; r2 is the radial coordinate value of the point of the inner arc of the stator; r0 is the outer radius of the rotor; Q is the rate of volume flow.To simplify, the distancer2 is defined as r2≈r1+1/2ecos⁡θf+cos⁡θr, which can be substituted into (18):(19)∂p∂zz=0=4μQkθr,where(20)kθr=14θf-θr-r1+12ecos⁡θf+cos⁡θr4+r1+1/2ecos⁡θf+cos⁡θr2-r022ln⁡r1+1/2ecos⁡θf+cos⁡θr-ln⁡r0,where kθr is the function for simplification; θf is the angular coordinate of front vane; θr is the angular coordinate of rear vane; r1 is the inner radius of the stator; r0 is the outer radius of the rotor; e is the eccentricity between the center of the rotor and the center of the stator.The same steps can be performed on the discharge process in the vane chamber, where the volumetric flow rate of melt discharge can be written as(21)Q=∬vz0rdθdr,where Q is the rate of volume flow; vz0 is the axial velocity at the discharge point from the vane chamber; r and θ are cylindrical coordinate components, respectively.At discharge, the volume of the chamber continues to shrink, so the volumetric flow rate is negative and the absolute value ofQ must be used in (21).Substituting (11) into (6) and setting z=W, the axial velocity of the melt at discharge is given by(22)vz0=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0∂p∂zz=W.Substituting (22) into (21) gives the flow rate Q, which can then be substituted into (19). The pressure gradient of the melt flow along the z-axis at the discharge location is given by(23)∂p∂zz=W=-4μQkθr,where the value of k(θr) is given by (20).Because the volume of the vane chamber varies periodically with the rotation of the rotor, there is a large change in chamber volume during the feeding process.


### 3.2. Axial Velocity

#### 3.2.1. Feeding Process (π/2<θr≤7π/4)

Substituting (19) into (16), the axial velocity vz of the melt in the vane chamber at the inlet can be given by(24)vz=Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r01-zW,where r2 is the radial coordinate of the point on the inner surface of the stator in the cylindrical coordinate system in which the origin is the center of the rotor and r2=r1+ecos⁡θ. Here, r1 is the inner radius of the stator, e is the eccentricity between the rotor and the stator, r0 is the outer radius of the rotor, W is the axial width of the vane, and θr is the angular coordinate of the rear vane of the VCPU in the cylindrical coordinate system. In addition, Q is the volumetric flow rate of the vane chamber at the position θr. The value of Q is directly proportional to the speed of the rotor. The coefficient function k(θr) relates to the geometric parameters of the vane chamber and the position of vane.


#### 3.2.2. Discharge Process (-π/4≤θr<π)

Substituting (22) and (23) into (11), the axial velocity vz of melt in the vane chamber at the discharge point is given by(25)vz=-Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0zW.According to (24) and (25), the axial velocity vz of melt in the VCPU is directly proportional to the speed of the rotor and is related to the geometric parameters of the vane chamber and the position of vane.The elongational strain rate along the axis is defined in the cylindrical coordinate system in Figure2 and can be calculated as (26)γ˙zz=2∂vz∂z,γ˙zz=-2QW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0,where γ˙ is rate of deformation tensor; vz is the velocity components of z-direction; r and z are the cylindrical coordinate components, respectively; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator; W is the axial width of VCPU; kθr is the function for simplification; Q is the rate of volume flow.Note. The values of all parameters are identical to those given in (24) for the feeding process and to those given in (25) for the discharge process.For example,r0=20 mm, r1=23 mm, e=3 mm, W=35 mm, and the rotation speed of the rotor N=60 rpm.The relationship of the axial elongational strain rate of melt moving in the chamberγ˙zz and the circumferential position of the rear vane θr is shown in Figures 4 and 5.Figure 4
The axial tensile strain rate of melt moving in a chamber of VCPU: feed process.Figure 5
The axial tensile strain rate of melt moving in a chamber of VCPU: discharge process.In Figures4 and 5, the axial rate of strain of the melt γ˙zz<0 at the inlet, but γ˙zz>0. In brief, the axial strain rate is at a maximum for a given circumferential position, namely, r=r0+2δ, and others in inner arc surface of stator or in outer arc surface of rotor are smaller; that is, r=r0+δ or r=r0+3δ, where δ is the dimensionless gap.


### 3.3. Circumferential Velocity

#### 3.3.1. Feeding Process

Substituting (24) into continuity (2), the expression can be simplified:(27)∂vθ∂θ=Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0.The boundary conditions of melt circumferential velocity are as follows: (1) when θ=θr, vθ=ωr; (2) when r=r0, vθ=ωr0. Substituting these boundary conditions into (27), the circumferential velocity vθ can be obtained:(28)vθ=ωr+Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0θ-θr.


#### 3.3.2. Discharge Process

Similarly, substituting (29) into continuity (2), the circumferential velocity vθ matches the expression given by (28).


## 3.1. Governing Equations

Melt conveyance is examined in the melt-mixing zone of the fully filled chamber of the vane extruder. The Reynolds number was assumed to be much less than unity so that inertial effects are neglected. The governing equations for this situation are shown below.Continuity Equation. Consider(2)1r∂vθ∂θ+∂vz∂z=0,where r, θ, and z are cylindrical coordinate components, respectively; vr, vθ, and vz are velocity components of r-direction, θ-direction, and z-direction.Constitutive Equation. Consider(3)τ=μγ˙,where τ is stress tensor; μ is viscosity coefficient dependent deformation rate; γ˙ is rate of deformation tensor.Motion Equations. Consider(4)∂p∂r=-2μr2∂vθ∂θ,(5)1r∂p∂θ=μ∂∂r1r∂∂rrvθ+1r2∂2vθ∂θ2,(6)∂p∂z=μ1r∂∂rr∂vz∂r,where p is pressure; r, θ, and z are cylindrical coordinate components, respectively; vr, vθ, and vz are velocity components of r-direction, θ-direction, and z-direction; μ is viscosity coefficient dependent deformation rate. Furthermore, the boundary conditions are vz(r0,π)=0 and vz(r2,π/4)=0.Equation (2) can be simplified to(7)∂vz∂z=-1r∂vθ∂θ.Taking the partial derivative of (7) with respect to z and assuming ∂vθ/∂z=0 gives(8)∂2vz∂z2=-1r∂2vθ∂θ∂z=0.Melt velocity is assumed to vary linearly along the length of the extruder, ∂vz/∂z=0. Thus, vz can be defined as(9)vz=f1r,θz+f2r,θ,where f1 and f2 are only functions of r and θ.Material enters the vane chamber at the origin of thez axis, z=0; thus, the maximum axial velocity of melt occurs at the origin. In contrast, the velocity of the melt at the exit, z=W, in the most extreme case can be zero, vz=W=0. Substituting this boundary condition into (9) gives f2(r,θ)=-f1(r,θ)W. At the origin z=0, the axial velocity of melt feeding into the vane chamber is vz0=f2(r,θ)=-f1(r,θ)W. Thus, (9) can be simplified to(10)vz=vz01-zW,where W is the axial width of VCPU.In contrast, the terminal end of the chamber discharges melt, so the axial velocity of melt at the terminal side is maximum, while at the origin,z=0, the melt velocity is zero, vz=0=0. Substituting this boundary condition into (9), the following expression is obtained: f2(r,θ)=0. The axial velocity of the melt at discharge from the vane chamber is vz0=f1(r,θ)W. Thus, (9) can be further simplified to(11)vz=vz0zW.Assuming both sides of (4) and (5) are zero, and ∂vθ/∂z=0, the following expressions are obtained:(12)∂2p∂z∂r=∂∂z-2μr2∂vθ∂θ=0,1r∂2p∂z∂θ=∂∂zμ∂∂r1r∂∂rrvθ+1r2∂2vθ∂θ2=0.Examining (12), ∂p/∂z is independent of r and θ.Substituting (10) into (6) and rearranging give(13)1-zW1r∂∂rr∂vz0∂r+1r2∂2vz0∂θ2=1μ∂p∂z.In the geometry of a vane extruder, the circumferential distance is far greater than radial distance, and vz is independent of θ; that is, ∂vz0/∂θ=0. To solve for vz0, we first examine the inlet, z=0. Equation (13) can be simplified to(14)1r∂∂rr∂vz0∂r=1μ∂p∂zz=0.The boundary conditions of (14) are (1) when r=r0, vz0=0 and (2) when r=r2, vz0=0.Solving (14) gives(15)vz0=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0∂p∂zz=0,where vz0 is the axial velocity of the melt at discharge from the vane chamber; p is pressure; μ is viscosity coefficient dependent deformation rate; r2 is the radial coordinate value of the point of the inner arc of the stator; r0 is the outer radius of the rotor; r and z are cylindrical coordinate component, respectively.Substituting (15) into (10), the axial velocity vz of melt flow in the vane chamber at the origin (feed location) is given by(16)vz=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r01-zW∂p∂zz=0,where W is the axial width of VCPU and ∂p/∂zz=0 can be obtained using the expression for volumetric flow rate:(17)Q=∬vz0rdθdr,where Q is the rate of volume flow, Q=dV/dt; V is the volume of the vane chamber; and vz0 is the velocity of the melt entering the vane chamber.Substituting (15) into (17) and rearranging give(18)14μ∂p∂zz=0∫θrθf∫r0r2r2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0rdrdθ=Q,where μ is viscosity coefficient dependent deformation rate; p is pressure; r and z are cylindrical coordinate components, respectively; θf is the angular coordinate of front vane; θr is the angular coordinate of rear vane; r2 is the radial coordinate value of the point of the inner arc of the stator; r0 is the outer radius of the rotor; Q is the rate of volume flow.To simplify, the distancer2 is defined as r2≈r1+1/2ecos⁡θf+cos⁡θr, which can be substituted into (18):(19)∂p∂zz=0=4μQkθr,where(20)kθr=14θf-θr-r1+12ecos⁡θf+cos⁡θr4+r1+1/2ecos⁡θf+cos⁡θr2-r022ln⁡r1+1/2ecos⁡θf+cos⁡θr-ln⁡r0,where kθr is the function for simplification; θf is the angular coordinate of front vane; θr is the angular coordinate of rear vane; r1 is the inner radius of the stator; r0 is the outer radius of the rotor; e is the eccentricity between the center of the rotor and the center of the stator.The same steps can be performed on the discharge process in the vane chamber, where the volumetric flow rate of melt discharge can be written as(21)Q=∬vz0rdθdr,where Q is the rate of volume flow; vz0 is the axial velocity at the discharge point from the vane chamber; r and θ are cylindrical coordinate components, respectively.At discharge, the volume of the chamber continues to shrink, so the volumetric flow rate is negative and the absolute value ofQ must be used in (21).Substituting (11) into (6) and setting z=W, the axial velocity of the melt at discharge is given by(22)vz0=14μr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0∂p∂zz=W.Substituting (22) into (21) gives the flow rate Q, which can then be substituted into (19). The pressure gradient of the melt flow along the z-axis at the discharge location is given by(23)∂p∂zz=W=-4μQkθr,where the value of k(θr) is given by (20).Because the volume of the vane chamber varies periodically with the rotation of the rotor, there is a large change in chamber volume during the feeding process.


## 3.2. Axial Velocity

### 3.2.1. Feeding Process (π/2<θr≤7π/4)

Substituting (19) into (16), the axial velocity vz of the melt in the vane chamber at the inlet can be given by(24)vz=Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r01-zW,where r2 is the radial coordinate of the point on the inner surface of the stator in the cylindrical coordinate system in which the origin is the center of the rotor and r2=r1+ecos⁡θ. Here, r1 is the inner radius of the stator, e is the eccentricity between the rotor and the stator, r0 is the outer radius of the rotor, W is the axial width of the vane, and θr is the angular coordinate of the rear vane of the VCPU in the cylindrical coordinate system. In addition, Q is the volumetric flow rate of the vane chamber at the position θr. The value of Q is directly proportional to the speed of the rotor. The coefficient function k(θr) relates to the geometric parameters of the vane chamber and the position of vane.


### 3.2.2. Discharge Process (-π/4≤θr<π)

Substituting (22) and (23) into (11), the axial velocity vz of melt in the vane chamber at the discharge point is given by(25)vz=-Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0zW.According to (24) and (25), the axial velocity vz of melt in the VCPU is directly proportional to the speed of the rotor and is related to the geometric parameters of the vane chamber and the position of vane.The elongational strain rate along the axis is defined in the cylindrical coordinate system in Figure2 and can be calculated as (26)γ˙zz=2∂vz∂z,γ˙zz=-2QW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0,where γ˙ is rate of deformation tensor; vz is the velocity components of z-direction; r and z are the cylindrical coordinate components, respectively; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator; W is the axial width of VCPU; kθr is the function for simplification; Q is the rate of volume flow.Note. The values of all parameters are identical to those given in (24) for the feeding process and to those given in (25) for the discharge process.For example,r0=20 mm, r1=23 mm, e=3 mm, W=35 mm, and the rotation speed of the rotor N=60 rpm.The relationship of the axial elongational strain rate of melt moving in the chamberγ˙zz and the circumferential position of the rear vane θr is shown in Figures 4 and 5.Figure 4
The axial tensile strain rate of melt moving in a chamber of VCPU: feed process.Figure 5
The axial tensile strain rate of melt moving in a chamber of VCPU: discharge process.In Figures4 and 5, the axial rate of strain of the melt γ˙zz<0 at the inlet, but γ˙zz>0. In brief, the axial strain rate is at a maximum for a given circumferential position, namely, r=r0+2δ, and others in inner arc surface of stator or in outer arc surface of rotor are smaller; that is, r=r0+δ or r=r0+3δ, where δ is the dimensionless gap.


## 3.2.1. Feeding Process (π/2<θr≤7π/4)

Substituting (19) into (16), the axial velocity vz of the melt in the vane chamber at the inlet can be given by(24)vz=Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r01-zW,where r2 is the radial coordinate of the point on the inner surface of the stator in the cylindrical coordinate system in which the origin is the center of the rotor and r2=r1+ecos⁡θ. Here, r1 is the inner radius of the stator, e is the eccentricity between the rotor and the stator, r0 is the outer radius of the rotor, W is the axial width of the vane, and θr is the angular coordinate of the rear vane of the VCPU in the cylindrical coordinate system. In addition, Q is the volumetric flow rate of the vane chamber at the position θr. The value of Q is directly proportional to the speed of the rotor. The coefficient function k(θr) relates to the geometric parameters of the vane chamber and the position of vane.


## 3.2.2. Discharge Process (-π/4≤θr<π)

Substituting (22) and (23) into (11), the axial velocity vz of melt in the vane chamber at the discharge point is given by(25)vz=-Qkθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0zW.According to (24) and (25), the axial velocity vz of melt in the VCPU is directly proportional to the speed of the rotor and is related to the geometric parameters of the vane chamber and the position of vane.The elongational strain rate along the axis is defined in the cylindrical coordinate system in Figure2 and can be calculated as (26)γ˙zz=2∂vz∂z,γ˙zz=-2QW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0,where γ˙ is rate of deformation tensor; vz is the velocity components of z-direction; r and z are the cylindrical coordinate components, respectively; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator; W is the axial width of VCPU; kθr is the function for simplification; Q is the rate of volume flow.Note. The values of all parameters are identical to those given in (24) for the feeding process and to those given in (25) for the discharge process.For example,r0=20 mm, r1=23 mm, e=3 mm, W=35 mm, and the rotation speed of the rotor N=60 rpm.The relationship of the axial elongational strain rate of melt moving in the chamberγ˙zz and the circumferential position of the rear vane θr is shown in Figures 4 and 5.Figure 4
The axial tensile strain rate of melt moving in a chamber of VCPU: feed process.Figure 5
The axial tensile strain rate of melt moving in a chamber of VCPU: discharge process.In Figures4 and 5, the axial rate of strain of the melt γ˙zz<0 at the inlet, but γ˙zz>0. In brief, the axial strain rate is at a maximum for a given circumferential position, namely, r=r0+2δ, and others in inner arc surface of stator or in outer arc surface of rotor are smaller; that is, r=r0+δ or r=r0+3δ, where δ is the dimensionless gap.


## 3.3. Circumferential Velocity

### 3.3.1. Feeding Process

Substituting (24) into continuity (2), the expression can be simplified:(27)∂vθ∂θ=Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0.The boundary conditions of melt circumferential velocity are as follows: (1) when θ=θr, vθ=ωr; (2) when r=r0, vθ=ωr0. Substituting these boundary conditions into (27), the circumferential velocity vθ can be obtained:(28)vθ=ωr+Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0θ-θr.


### 3.3.2. Discharge Process

Similarly, substituting (29) into continuity (2), the circumferential velocity vθ matches the expression given by (28).


## 3.3.1. Feeding Process

Substituting (24) into continuity (2), the expression can be simplified:(27)∂vθ∂θ=Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0.The boundary conditions of melt circumferential velocity are as follows: (1) when θ=θr, vθ=ωr; (2) when r=r0, vθ=ωr0. Substituting these boundary conditions into (27), the circumferential velocity vθ can be obtained:(28)vθ=ωr+Q·rW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0θ-θr.


## 3.3.2. Discharge Process

Similarly, substituting (29) into continuity (2), the circumferential velocity vθ matches the expression given by (28).


## 4. Results and Discussion

### 4.1. Elongational Strain Rate in the Circumferential Direction

The elongational strain rate in the circumferential direction is defined in the cylindrical coordinate system in Figure2 and given below:(29)γ˙θθ=2r∂vθ∂θ,(30)γ˙θθ=2QW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0.Note. The values of all parameters are identical to those given in (28) in both the feeding and the discharge processes.For example,r0=20 mm, r1=23 mm, e=3 mm, W=35 mm, and the rotational speed of the rotor N=60 rpm.The relationship between the axial elongational strain rate of melt moving in a chamber of the VCPUγ˙θθ and the circumferential position of rear vane θr is shown in Figures 6 and 7.Figure 6
The circumferential tensile strain rate of melt moving in a chamber of VCPU: feed process.Figure 7
The circumferential tensile strain rate of melt moving in a chamber of VCPU: discharge process.In Figures6 and 7, the circumferential strain rate of the melt γ˙θθ>0 at the inlet, but γ˙θθ<0. In brief, the value of the circumferential strain rate is maximum at any given circumferential position in the vane chamber, namely, r=r0+2δ, and others in inner arc surface of stator or in outer arc surface of rotor are smaller, namely, r=r0+δ or r=r0+3δ.


### 4.2. Pressure Distribution in the Vane Extruder

In the melt conveying section, the polymer is driven circumferentially through the VCPU by a rear vane and motion of the rotor. In addition, the VCPU produces positive displacement of the melt, conveying it in the axial direction as the volume of the chamber changes periodically. The axial and circumferential velocity of the melt is different in every position in the vane extruder. Similarly, the axial and circumferential pressure in the melt vary continuously throughout the extruder.


#### 4.2.1. Pressure Distribution in the Axial Direction at the Inlet

Substituting (24) into the motion equation (6) and simplifying give(31)∂p∂z=4μQkθr1-zW,where the values of all variables including Q and k(θr) are identical to those given in (6).Because the volumetric rate along the axisQ>0 and the coefficient function k(θr)<0, ∂p/∂z≤0 at the inlet to the vane chamber. This means that the pressure within the vane chamber decreases gradually from the inlet to the outlet of the chamber. As a normal stress field in the axial direction is produced in the vane chamber, melt is conveyed forcibly forward.The axial pressure difference from the inlet to a point in the vane chamber can be given by(32)Δpz=-∫0z∂p∂zdz,where Δpz is the pressure difference for z-direction.Substituting (31) into (32), the following expression can be obtained:(33)Δpz=2μQkθrz2W-2z.With z=W, the axial pressure difference from the inlet (z=0) to the outlet is given by(34)Δpz=-2μQWkθr.


#### 4.2.2. Pressure Distribution in the Circumferential Direction at the Inlet

Substituting (28) into the motion equation (6) and simplifying give(35)∂p∂θ=2μQW·kθr4r2-r22-r02ln⁡r2-ln⁡r0θ-θr,where the values of all variables, including Q and k(θr), are identical to those given in (6).Because the volumetric axial flow rateQ>0 and the coefficient function k(θr)<0, θ-θr≥0, 4r2-r22-r02/ln⁡r2-ln⁡r0>0, and ∂p/∂θ≤0 in the entire range θr≤θ≤θf of the vane chamber. Thus, the circumferential pressure decreases gradually within the vane chamber from rear vane to front vane. The inner arc of the stator gradually moves away from the rotor so that negative pressure is produced in the front of the vane chamber along with the periodic volume expansion of the melt. The circumferential pressure difference from the edge of the chamber (θ=θr) to a point within the vane chamber is given by(36)Δpθ=-∫θrθ∂p∂θdθ,where Δpθ is the pressure difference for θ-direction.Substituting (35) into (36) and setting r=r0+r2/2 give(37)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θ-θr2.At θ=θf, the circumferential pressure difference from the extreme side (θ=θr) to the front side is given by(38)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θf-θr2,where θf and θr are the most front and the most rear circumferential coordinates, respectively. When π≤θr≤7π/4, θf-θr=π/2; when π/2<θr<π, θf-θr=θr-π/2.Synthesizing (30) and (37), the characteristic pressure function at the inlet is given by (39)p=pi+2μQkθr2z-z2W+μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θr2,where pi is the pressure at the edge of the vane chamber (z=0, θ=θr).A gap is produced due to the volume enlargement at the front vane in the vane chamber, and the final side of the chamber can be filled with melt. The pressure at the front of the vane in the vane chamber is therefore roughly equal to atmospheric pressure; that is,pθf,W=p0. Substituting into (36) gives(40)pi=p0-2μQWkθr-μQπ24W·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0,where pi is the pressure at the final side of the vane chamber; p0 is the atmospheric pressure; μ is viscosity coefficient dependent deformation rate; Q is the rate of volume flow; W is the axial width of VCPU; kθr is the function for simplification; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator.


#### 4.2.3. Pressure Distribution in the Axial Direction at the Outlet

Substituting (25) into the motion equation (6) and simplifying give(41)∂p∂z=-4zμQW·kθr.Substituting (41) into (32) gives(42)Δpz=2μQz2W·kθr.At z=W, the axial pressure difference from the inlet to the outlet is given by(43)Δpz=2μQWkθr.


#### 4.2.4. Pressure Distribution in the Circumferential Direction at the Outlet

The circumferential pressure differential expression is identical to (37).Combining (43) and (37), the characteristic pressure function at the outlet of the vane chamber is given by(44)p=p0+2μQz-W2W·kθr-μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θf2,where p is the pressure; p0 is the atmospheric pressure.


## 4.1. Elongational Strain Rate in the Circumferential Direction

The elongational strain rate in the circumferential direction is defined in the cylindrical coordinate system in Figure2 and given below:(29)γ˙θθ=2r∂vθ∂θ,(30)γ˙θθ=2QW·kθrr2-r02-r22-r02ln⁡r-ln⁡r0ln⁡r2-ln⁡r0.Note. The values of all parameters are identical to those given in (28) in both the feeding and the discharge processes.For example,r0=20 mm, r1=23 mm, e=3 mm, W=35 mm, and the rotational speed of the rotor N=60 rpm.The relationship between the axial elongational strain rate of melt moving in a chamber of the VCPUγ˙θθ and the circumferential position of rear vane θr is shown in Figures 6 and 7.Figure 6
The circumferential tensile strain rate of melt moving in a chamber of VCPU: feed process.Figure 7
The circumferential tensile strain rate of melt moving in a chamber of VCPU: discharge process.In Figures6 and 7, the circumferential strain rate of the melt γ˙θθ>0 at the inlet, but γ˙θθ<0. In brief, the value of the circumferential strain rate is maximum at any given circumferential position in the vane chamber, namely, r=r0+2δ, and others in inner arc surface of stator or in outer arc surface of rotor are smaller, namely, r=r0+δ or r=r0+3δ.


## 4.2. Pressure Distribution in the Vane Extruder

In the melt conveying section, the polymer is driven circumferentially through the VCPU by a rear vane and motion of the rotor. In addition, the VCPU produces positive displacement of the melt, conveying it in the axial direction as the volume of the chamber changes periodically. The axial and circumferential velocity of the melt is different in every position in the vane extruder. Similarly, the axial and circumferential pressure in the melt vary continuously throughout the extruder.


### 4.2.1. Pressure Distribution in the Axial Direction at the Inlet

Substituting (24) into the motion equation (6) and simplifying give(31)∂p∂z=4μQkθr1-zW,where the values of all variables including Q and k(θr) are identical to those given in (6).Because the volumetric rate along the axisQ>0 and the coefficient function k(θr)<0, ∂p/∂z≤0 at the inlet to the vane chamber. This means that the pressure within the vane chamber decreases gradually from the inlet to the outlet of the chamber. As a normal stress field in the axial direction is produced in the vane chamber, melt is conveyed forcibly forward.The axial pressure difference from the inlet to a point in the vane chamber can be given by(32)Δpz=-∫0z∂p∂zdz,where Δpz is the pressure difference for z-direction.Substituting (31) into (32), the following expression can be obtained:(33)Δpz=2μQkθrz2W-2z.With z=W, the axial pressure difference from the inlet (z=0) to the outlet is given by(34)Δpz=-2μQWkθr.


### 4.2.2. Pressure Distribution in the Circumferential Direction at the Inlet

Substituting (28) into the motion equation (6) and simplifying give(35)∂p∂θ=2μQW·kθr4r2-r22-r02ln⁡r2-ln⁡r0θ-θr,where the values of all variables, including Q and k(θr), are identical to those given in (6).Because the volumetric axial flow rateQ>0 and the coefficient function k(θr)<0, θ-θr≥0, 4r2-r22-r02/ln⁡r2-ln⁡r0>0, and ∂p/∂θ≤0 in the entire range θr≤θ≤θf of the vane chamber. Thus, the circumferential pressure decreases gradually within the vane chamber from rear vane to front vane. The inner arc of the stator gradually moves away from the rotor so that negative pressure is produced in the front of the vane chamber along with the periodic volume expansion of the melt. The circumferential pressure difference from the edge of the chamber (θ=θr) to a point within the vane chamber is given by(36)Δpθ=-∫θrθ∂p∂θdθ,where Δpθ is the pressure difference for θ-direction.Substituting (35) into (36) and setting r=r0+r2/2 give(37)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θ-θr2.At θ=θf, the circumferential pressure difference from the extreme side (θ=θr) to the front side is given by(38)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θf-θr2,where θf and θr are the most front and the most rear circumferential coordinates, respectively. When π≤θr≤7π/4, θf-θr=π/2; when π/2<θr<π, θf-θr=θr-π/2.Synthesizing (30) and (37), the characteristic pressure function at the inlet is given by (39)p=pi+2μQkθr2z-z2W+μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θr2,where pi is the pressure at the edge of the vane chamber (z=0, θ=θr).A gap is produced due to the volume enlargement at the front vane in the vane chamber, and the final side of the chamber can be filled with melt. The pressure at the front of the vane in the vane chamber is therefore roughly equal to atmospheric pressure; that is,pθf,W=p0. Substituting into (36) gives(40)pi=p0-2μQWkθr-μQπ24W·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0,where pi is the pressure at the final side of the vane chamber; p0 is the atmospheric pressure; μ is viscosity coefficient dependent deformation rate; Q is the rate of volume flow; W is the axial width of VCPU; kθr is the function for simplification; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator.


### 4.2.3. Pressure Distribution in the Axial Direction at the Outlet

Substituting (25) into the motion equation (6) and simplifying give(41)∂p∂z=-4zμQW·kθr.Substituting (41) into (32) gives(42)Δpz=2μQz2W·kθr.At z=W, the axial pressure difference from the inlet to the outlet is given by(43)Δpz=2μQWkθr.


### 4.2.4. Pressure Distribution in the Circumferential Direction at the Outlet

The circumferential pressure differential expression is identical to (37).Combining (43) and (37), the characteristic pressure function at the outlet of the vane chamber is given by(44)p=p0+2μQz-W2W·kθr-μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θf2,where p is the pressure; p0 is the atmospheric pressure.


## 4.2.1. Pressure Distribution in the Axial Direction at the Inlet

Substituting (24) into the motion equation (6) and simplifying give(31)∂p∂z=4μQkθr1-zW,where the values of all variables including Q and k(θr) are identical to those given in (6).Because the volumetric rate along the axisQ>0 and the coefficient function k(θr)<0, ∂p/∂z≤0 at the inlet to the vane chamber. This means that the pressure within the vane chamber decreases gradually from the inlet to the outlet of the chamber. As a normal stress field in the axial direction is produced in the vane chamber, melt is conveyed forcibly forward.The axial pressure difference from the inlet to a point in the vane chamber can be given by(32)Δpz=-∫0z∂p∂zdz,where Δpz is the pressure difference for z-direction.Substituting (31) into (32), the following expression can be obtained:(33)Δpz=2μQkθrz2W-2z.With z=W, the axial pressure difference from the inlet (z=0) to the outlet is given by(34)Δpz=-2μQWkθr.


## 4.2.2. Pressure Distribution in the Circumferential Direction at the Inlet

Substituting (28) into the motion equation (6) and simplifying give(35)∂p∂θ=2μQW·kθr4r2-r22-r02ln⁡r2-ln⁡r0θ-θr,where the values of all variables, including Q and k(θr), are identical to those given in (6).Because the volumetric axial flow rateQ>0 and the coefficient function k(θr)<0, θ-θr≥0, 4r2-r22-r02/ln⁡r2-ln⁡r0>0, and ∂p/∂θ≤0 in the entire range θr≤θ≤θf of the vane chamber. Thus, the circumferential pressure decreases gradually within the vane chamber from rear vane to front vane. The inner arc of the stator gradually moves away from the rotor so that negative pressure is produced in the front of the vane chamber along with the periodic volume expansion of the melt. The circumferential pressure difference from the edge of the chamber (θ=θr) to a point within the vane chamber is given by(36)Δpθ=-∫θrθ∂p∂θdθ,where Δpθ is the pressure difference for θ-direction.Substituting (35) into (36) and setting r=r0+r2/2 give(37)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θ-θr2.At θ=θf, the circumferential pressure difference from the extreme side (θ=θr) to the front side is given by(38)Δpθ=μQW·kθrr22-r02ln⁡r2-ln⁡r0-r0+r22θf-θr2,where θf and θr are the most front and the most rear circumferential coordinates, respectively. When π≤θr≤7π/4, θf-θr=π/2; when π/2<θr<π, θf-θr=θr-π/2.Synthesizing (30) and (37), the characteristic pressure function at the inlet is given by (39)p=pi+2μQkθr2z-z2W+μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θr2,where pi is the pressure at the edge of the vane chamber (z=0, θ=θr).A gap is produced due to the volume enlargement at the front vane in the vane chamber, and the final side of the chamber can be filled with melt. The pressure at the front of the vane in the vane chamber is therefore roughly equal to atmospheric pressure; that is,pθf,W=p0. Substituting into (36) gives(40)pi=p0-2μQWkθr-μQπ24W·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0,where pi is the pressure at the final side of the vane chamber; p0 is the atmospheric pressure; μ is viscosity coefficient dependent deformation rate; Q is the rate of volume flow; W is the axial width of VCPU; kθr is the function for simplification; r0 is the outer radius of the rotor; r2 is the radial coordinate value of the point of the inner arc of the stator.


## 4.2.3. Pressure Distribution in the Axial Direction at the Outlet

Substituting (25) into the motion equation (6) and simplifying give(41)∂p∂z=-4zμQW·kθr.Substituting (41) into (32) gives(42)Δpz=2μQz2W·kθr.At z=W, the axial pressure difference from the inlet to the outlet is given by(43)Δpz=2μQWkθr.


## 4.2.4. Pressure Distribution in the Circumferential Direction at the Outlet

The circumferential pressure differential expression is identical to (37).Combining (43) and (37), the characteristic pressure function at the outlet of the vane chamber is given by(44)p=p0+2μQz-W2W·kθr-μQW·kθrr0+r22-r22-r02ln⁡r2-ln⁡r0θ-θf2,where p is the pressure; p0 is the atmospheric pressure.


## 5. Conclusions

The above analysis supports a number of conclusions:(i)
The polymer melt conveying process in the vane extruder is completely different from the process in conventional screw extruders.(ii)
The pressure within the chamber is influenced by the location and volume within the chamber.(iii)
The dynamics of melt conveyance in a vane chamber depend mostly on pressure to overcome viscous forces. Therefore, the power consumption of a vane extruder increases with pressure and viscosity within the chamber and is directly proportional to the speed of the rotor.



---

*Source: 714356-2015-12-01.xml*