Jinnan Chen，Pan Dai，Hui Yao and Tung Chan
1 Introduction
Conventional full-flight single-screw extruders have been widely used because they have many advantages，such as low cost. casy manufacture and operation，parts resistant to abuse，and worn parts easily replaced. But the mixing capability of the conventional full-flight single-screw extruder is low. As revealed in an experimental study by Bigio et al. on the mixing performance of a full-flight single-screw extruder，laminar flow ex ists in the mixing section of the extruder. Other screw configurations have been designed to improve the mix capability of single-screw extruder. The pinscrew singlescrew extruder，which has pins installed on the screw，has been designed to enhance the mixing performance through the reorientation of polymer flow. However，comparing various mixers in the single-screw extruder. Rawendaal pointed out that dead spots were found behind the pins and at a 90° angle between the pins and the root of the screw. As a result，Yao et al. designed holed-pins to improve the mixing capability of pin-screw single-screw extruder. The holed-pins were channeled from top-front to bottom-rear. The experimental investigation by Yao et al. indicated that holedpins can eliminate two shortcomings of the normal pin-screw singlescrew extruder，namely，inefficient mixing and the formation of dead spots.
The pin-barrel single-screw extruder，which has fixed pins on the barrel，has also been studied. Many patents exist on pin-barrel screw extruders and singlescrew extruder. Yabushita et al. used a white rubber compound and a black rubber compound as flow markers to study the mixing performance of a pin-barrel singlescrew extruder with two rows of pins and a regular singlescrew extruder. Their experimental results indicated that the introduction of pins can greatly increase the mixing performance. Shin and White used the flow analysis network technique to study the effects of the introduction of slices in screw flights and the introduction of pins fixed on the barrel on the screw pumping characteristics. Their results showed that slices in screw flights reduced pumping characteristics，and the pumping capacity of a pinbarrel extruder closely resembles that of a screw extruder with slices in its screw flights. The reciprocating pin-barrel single-screw extruder is one type of pin-barrel single-screw extruder. The full-flight screw interrupted by pins rotates and oscillates simultaneously in the barrel. This special structure results in excellent distributive and dispersive mixing and self-cleaning perform-ance. Bi and Jiang studied the effects of operating conditions on the residence time distribution (RTD) of material in a reciprocating pinbarrel single-screw extruder.
The polymer mixing state directly influences the properties of the extruded product. The extrusion process is complex，and mixing capability is affected by the polymer properties，the operating conditions，and the geometry configurations of the screw extruder. To obtain a polymer sample for mixing state analysis，some researchers took the screw out after cooling the barrel. The polymer sample was cut into several pieces，and sections of the pieces were analyzed by a photoelectric microscope and other instruments. This experimental method was inconvenient and sometimes limited the study in-depth.
The moving trajectories of particles (MTPs) have been experimentally studied to understand material transport and mixing mechanisms in the extrusion process. Generally，tracer particles are added to the polymers，and the flows of polymers are observed through a transparent window that can withstand high temperature and pressure. MTPs are monitored in realtime to study the transport and mixing properties of polymers. Because the polymer mixing state is not the same at different groove depths. only the superficial polymer mixing state can be observed. It is sometimes difficult to clearly observe the mixing state of the polymer through the window because some polymers are not transparent or are strongly adhesive to the window.
The particle tracking analysis (PTA) method is a useful tool for the numerical analysis of polymer melt mixing state in screw extruders. In this study，we performed numerical simulations of the flow field of a rubber melt and used the PTA method to statistically analyze the polymer melt mixing state in the mixing sections of both a pin-barrel and the corresponding conventional fullflight single-screw extruders. In particular，the effects of the number of pins on the mixing capability of pinbarrel single-screw extruders were investigated.
2 Mathematical and physical models
2.1 Geometric model
The screw is a double-flighted screw.The origin of the geometric model is established at the center of the inlet plane.The positivez-direction is the direction of extrusion Fig.1A shows the geometric configuration of the pin-barrel extruder.The main geometric parameters are as follows: the diameter of the pin-barrel extruder is 90mm，the diameter of the pins is 10mm，the height of the pins is 13.8mm，the axial gap of the pins is 72mm，the radius of the barrel is 45.3mm，the root radius of the screw is 30mm，the tip radius of the screw is 45mm，the width of the screw flight is 6mm，the width of the screw groove is 54mm，the depth of the screw groove is 15mm，and the width of the cutting slot on the screw flight is 14mm.Both the screw length and the barrel length are 216mm.There are three rows of pins.The first row of pins is atz＝36mm.the second row of pins is atz＝108mm，and the third row of pins is atz＝180mm.Each row has six pins.Hexahedral and tetrahedral elements were used to mesh the barrel mixing section and the screws，respectively.The utility of mesh superposition technique provided by Polyflow combined the screw and the barrel grids into the grid of the flow channel.Different meshes were tested.When the values of velocity and pressure at the same points were less than 10－3，minimum finer meshes were used.Fig.1B shows the finite-element grid of the pin-barrel extruder mixing section needed for convergence，with 161829 elements and degree of freedom of 34818.The corre sponding number of elements for the conventional screw extruder was 18781 and degree of freedom was 39079.
Fig.1 Geometric configuration and finite element mesh of mixing section of pin-barrel extruder.
2.2 Mathematical model
The Carreau model was used to describe the shear-rate dependence of viscosity of the CIS60 rubber melt investigated in this study:
whereη is the shear viscosity，η0is the zero shear vis cosity，
is the shear rate，λ is the relaxation time，and n is the non-Newtonian index.
The module Polymat of Polyflow was used to fit the constitutive equation Eq.(1)，to the viscosity data for CIS60 given in Fig.2.Curve 1 was measured by a parallel-plate Haake rheometer in the shear rate range of 10－3to 5×10－1s－1.Curve 2 was obtained by Monsanto capillary rheometer measurements in the shear rate range of 10 to 103s－1.Curve 3 is the best fit of Eq.(1) to the experimental data given by Curves 1 and 2，and the standard error is 4%.The following Carreau model parameters were obtained:η0＝5.5×106Pa· s，λ＝130s andn＝0.2.
Fig.2 Best fit of constitutive equation to rheological data. Curve 1 was measured by a parallel-plate Haake rheometer in the shear rate range of 10－3to 5×10－1s－1. Curve 2 was obtained by Monsanto capillary rheometer measurements in the shear rate range of 10 to 103s－1. Curve 3 is the best fir of Eq. (1) to the experimental data given by Curves 1 and 2.
The following assumptions were made in the numberical simulations: (i) the rubber melt in the mixing section of the screw extruder is considered as an incom-pressible fluid；(ii) inertial and gravitational forces are negligible；(iii) the flow in the screw channel is fully developed，steady state，laminar，and isothermal；(iv) there is no slip at the walls；and (v) the screw channel is always full of rubber melt.
Based on these assumptions，the equations of continuity and motion can be reduced to:
where u is the velocity vector p is the pressure， τis the viscous stress tensor，and D is the rate of deformation tensor.
2.3 Mixing model
The PTA method was used to analyze the mixing process on the basis of the following assumptions: (i) the particles have no volume；(ii) the particles have no impact on the flow field；(iii) there are no interactions among the particles；and (iv) the velocity field totally determines the movement of particles.
The following processing conditions were used in the computations.the screw speed is 30rpm.the volumetric flow rate at the inlet isq＝1.5×10－5m3/s.and the outlet is a free boundary condition.
The RTD function reflects the time range of all materials being extruded under the given processing conditions RTD can be divided into internal and external functions. The internal RTD functiong(t) dt is defined as the fluid volume fraction during the residence timet→t+dt. The cumulative internal RTD function is
The external RTD functionf(t) dt is the flow rate fraction at the outlet of the mixing section. The cumulative external RTD function is
The average residence time is
It is equivalent to the system volume divided by the volumetric flow rate.
Dispersive mixing. which is associated with the extension and break up of droplets，is considered to be strongly related to the shear stress and stretch stress level. If the stresses are too low，no break up will occur.The higher the stresses and the more frequent the particles undergoing high stresses. the better the dispersive mixing. To estimate the fraction of material undergoing extension，the mixing indexλ is defined as follows:
whereω is the vorticity tensor.The mixing index can range from 0 to 1.At 0，the flow is locally a plug flow；at 0.5，the flow is locally a pure shear flow；at 1，the flow is locally a pure elongational flow.The plug flow is not beneficial for the melt mixing.The shear flow and elongational flow is beneficial for the melt mixing.
For three-dimensional flow，the local stretching of the material infinitesimal surface can be calculated from the average value of the stretchingζ per calculated unit area. In the initial state. dA is an infinitesimal surface with normal direction
in flow channel. The surface deforms with time. The surface deformation at time dt is da with normal direction
. It follows that
whereX is the spatial coordinates. Because the fluid is incompressible，Eq.(10) can be expressedas
C and F are the right Cauchy Green strain tensor and the deformation gradient，respectively. The larger the value ofζthe better the distributive mixing. The instantaneous stretching efficiency (a local evaluation of the mixing efficiency) eζis defined as
where
is the rate of stretching and D is the rate of deformation tensor at time t. The values of eζare always within the interval. The mixing efficiency is best，When eζ＝1，and the mixing efficiency is worst when eζ＝0.(www.xing528.com)
The time-averaged stretching efficiency <eζ> is defined as
The segregation scale of the material particles，a measure of dispersive mixing，is calculated using the PTA method. Fluid particles of two different colors(fluid A and Fluid B) of the same material are to be mixed. Assume that there is no diffusion between the fluid particles of different colors. Letc(r，t) denote the concentration of particles of one color. At timet，consider a set of M pairs of material points separated by a distancer. For thej-th pair，let
and
denote the concentrations at the two material points. Letcidenote the concentration(0 or 1) of thei-th material point；the average concentration of all material points is given
. The correlation coefficientR(r，t)gives the probability of finding a pair of random points with a relative distancer and with the same concentrationR(r，t) is defined as follows.
whereσcis the standard deviation of the concentration given by
The segregation scaleS(t) is the integral ofR(r，t) defined as
Initially，1000 particles (500 of each color) are arbitrarily dispersed at the entrance of the mixing section of each of the conventional single-screw extruder and the pin-barrel extruder. For a double-flighted screw element. the flow field at an angle of rotationαis identical to that at an angle of (α+180°). According to this cyclical characteristic，it is sufficient to calculate only the flow field of the melt between the angles 0° and 180°. The velocity fields in the mixing sections of the pin-barrel extruder and the conventional single-screw extruder have been simulated by using Eqs. (1) to (5). The trajectories of the 1000 particles in the flow domain are obtained by integrating the calculated velocity. and the Runge-Kutta method is used to integrate the velocity. The mixing performance of both types of mixing section is quantitatively evaluated by using Eqs. (6) to (18). A relative convergence criterion of 10－3was used for all solutions. All the computations were carried out on a Hewlett-Pakard HPXW9300 workstation. The longest computation time for a single run was approximately 90h.
3 Results and discussion
3.1 Flow field
Fig.3 shows the velocity fields of melt on thex-z plane in the mixing section of different extruders. Differences can be seen between the velocity fields given in Fig.3A and B. As shown in Fig.3A，there is no significant change in the velocity along the screw channel in the mixing section with no pin. In Fig.3B，the velocity changes along the screw channel in the mixing section with pins. The pins offer resistance to the flow and change its direction. The pins can scatter stagnant melt and increase the velocity gradient.
Fig.4 shows the flow field of melt on placez＝108mm in the mixing section of different extruders.As shown in Fig.4A，there are two flow domains in the mixing section with no pin.The six pins divide the cross-section into six domains in the mixing section with pins.As shown in Fig.4A，the maximal of velocity in no pin mixing section and in pin mixing section is 0.121 and 0.094m/s，respectively；theminimum of velocity in no pin mixing section and in pin mixing section is 0.015and 0.016m/s，respectively.In the whole，the velocity of the melt in the mixing section with pins is lower than that in the mixing section without pins.thus increasing the residence time of the melt in the screw channel.The velocity gradient is increased and stagnant melt is scattered.As shown in Fig.4B，the regions closest to the screw edge and the barrel wall have high shear stress.The shear stress of the melt in the whole screw channel also increases according to Eq.(4).The highest shear stress around the pins is 27MPa.which is 30 times higher than the maximum shear stress in the mixing section with no pin.The pressure fields on this cross-section in the mixing sections with pins and with no pin are also calculated，and the highest and lowest pressures in the mixing section with pins are approximately 50% lower than the corresponding values in the mixing section with no pin.The decreased melt pressure in the mixing section with pins adversely affects the plasticizing capacity.However，the effect of shear stress on plasticizing capacity is larger than that of pressure.As shown in Fig.4C，the mixing index is between 0.087 and 0.784in the mixing section with no pin and between 0.413 and 0.661 in the mixing section with pins.When the value of the mixing index is 0，the flow is locally a plug flow，and the plug flow is not beneficial for the melt mixing.Although the maximum mixing index in the mixing section with no pin is higher than that in the mixing section with pins，the mixing index is in the mixing section with pins around 0.5.The shear flow is beneficial for the melt mixing.Therefore，the mixing effect is better in the mixing section with pins than in the mixing section without pin.
Fig.3 Velocity field of melt onx-z plane in mixing section of different extruders. Velocity in m/s.
Fig.4 Flow field of melt on planez＝108mm in mixing section of different extruders.
3.2 Trajectories and mixing process of particles
The trajectories of the 1000 particles were numerically calculated in the mixing section of each of the conventional single-screw extruder and the pin-barrel extruder. Fig.5A and B show the trajectories of particles in the no pin mixing section and pin mixing section for a time period of 24s，respectively. The trajectories of some par-ticles in the no pin mixing section are screw spirals，whereas those in the pin mixing section have been partially disorganized.
On the basis of the computed trajectories of the 1000 particles，the dynamic mixing process of two incompatible fluid particles of two different colors (fluid A and fluid B of the same material) is statistically analyzed at different times using the statistical analysis module Polystat of Polyflow.Fluid A and fluid B，respec-tively，occupy the left and right sides of the entrance to the mixing section.Fig.6 show the dynamic mixing process of the material particles in the flow channel of the no pin mixing section.The material particles are evenly distributed at the channel entrance in the beginning.As time passes the material particles gradually concentrate at the bottom of the groove as they move forward along the screw.At 14.7s，some particles have arrived at the exit of the mixing section.Fig.7 shows the dynamic mixing process of the material particles in the flow channel of the pin mixing section.Here the material particles fill the flow channel as they move forward along the screw.The particle velocity in the pin mixing section is almost the same as that in the no pin mixing section before 6.3s，but the velocity in the pin mixing section becomes lower than that in the mixing section with no pin after 8.4s.It takes the particles 21.0s to reach the exit of the pin mixing section.This means that the particles encounter the pins between 6.3s and 8.4s in the pin mixing section.The particle movement is disrupted and the particles are obstructed by the pins.With the decrease in particle velocity，the residence time of particles in the pin mixing section is longer than that in the no pin mixing section.As a result，the mixing is better and more efficient in the pin-barrel extruder than in the conventional extruder.This result is in agreement with the experimental result of Yabushita et al..Pins passing through the grooves changed the flow direction of material in the flow channel of the extruder and periodically reoriented and stretched the rubber compound.
3.3 Residence time distribution (RTD)
The RTD of polymer melt in the screw extruder is also an important indicator of mixing performance.A statistical analysis of the residence time of particles in both the pin and no pin mixing sections was carried out.The cumulative RTDs of the material particles in the two mixing sections were calculated by Eqs.(6) to (8).Fig.8 shows the cumulative RTDs of particles in the no pin mixing section and pin mixing section.As shown in Fig.8A，the times for 90% of the particles to pass through the no pin mixing section and pin mixing section are 200s and 450s，respectively.The residence time of the particles in the no pin mixing section is much shorter than that in the pin mixing section.Fig.8B is a partial amplification of Fig.8A in the time range 0≤t≤70 (0≤RTD≤0.5).This figure shows that the time of the particles first arriving at the exit of the mixing section is 14.8s for the no pin mixing section and 20.2s for the pin mixing section，consistent with the results given in Section 3.1.The times taken for half of the particles to pass through the no pin and pin mixing section are 23s and 69s，respectively.
3.4 Distributive mixing
The material concentration distributionsc(r，t) were numerically computed for the case with no pin and the case with pins. To discuss the distributive mixing，the length of screw extruder 216mm was divided into 12 parts where each part is 18-mm log. Therefore，there are 13 cross-sections between the entrance and exit of screw extruder，the results were shown in 13 cross-sections. The logarithm of stretching lgζ，the instantaneous efficiency of stretchingeζand the time averaged efficiency of stretching<eζ>were calculated by Eqs.(10) to(15). The term percentage of particles is used to analyze the mixing state of different proportions of particles in the mixing section.
Fig.5 Particle trajectories in mixing section of different extruders：(A) no pin，(B) with pins.
Fig.6 Dynamic mixing process of two types of material particles in no pin mixing section.
Fig.7 Dynamic mixing process of two types of material particles in pin mixing section.
Fig.8 (A) Cumulative residence time distribu-tion of particles in mixing section of different extrud-ers，(B) partial amplification of (A) at small times.
Fig.9A and B show the logarithm of stretching lgζalong the axial direction for 10%，50%，and 90% of particles in the no pin and pin mixing sections，respectively. It can be seen from Fig.9A and B，the maximal lgζ that 90% of particles undergoing in the pin mixing section and no pin mixing section are very different. It means that the pin mixing. section and no pin mixing section both can produce the same maximal stretching. However，the maximal lgζthat 10% and 50% of particles undergoing in the pin mixing section is higher than that in the no pin mixing section. This means that most particles undergoing stretching effect in pin mixing section are bigger than that in the no pin mixing section. It can be seen from Fig.9A that the two curves for 10%and 50% of particles are almost horizontal in the no pin mixing section. The stretching effect of the no pin mixing section does not change along the axial distance. The corresponding curves in Fig.9B for the pin mixing section show an upward trend. The stretching effect of the pin mixing section increases along the axial distance. As shown in Fig.9A and B，the logarithm of stretching lgζ is generally higher in the pin mixing section than in the no pin mixing section. Therefore，the pin mixing section has better mixing performance than the no pin mixing section.
Fig.10A and B show the time-averaged efficiency of stretching <eζ>along the axial direction for 10%，50%，and 90% of particles in the no pin and pin mixing sections，respectively.It can be seen form Fig.10A that the time-averaged efficiency of stretching<eζ> declines after the second cross-section for each of 10%，50%，and 90% of particles in the no pin mixing section.As shown in Fig.10B，the time averaged efficiency of stretching<eζ>gradually increases along the axial direction for each of 10%，50%，and 90% of particles in the pin mixing section.The time-averaged efficiency of stretching<eζ>increases significantly for all percentages of particles with the addition of pins.Particularly for 90% of particles，the average stretching efficiency is 0.08 at the exit of the no pin mixing section，but increases to 0.24 with the addition of pins，a three-fold increase.Furthermore，a local maximum of the stretching efficiency is reached at each pin.
Fig.9 Logarithm of stretching lgζ along axial direction for 10%，50%，and 90%of particles in mixing section of different extruders：(A) no pin，(B) with pins.
3.5 Influence of number of pins on mixing per-formance
Three mixing section configurations of a pin-barrel extruder were investigated:(i) three pins in each row，(ii) six pins in each row，and(iii) 12 pins in each row.The mixing performance of the rubber melt was statistically analyzed in each of the above three extruders.The pins are located at the third，seventh，and 11th cross-sections as defined in Section 3.4.The segregation scale of the trajectories of 1000 particles was computed using Eqs.(16) to(18).Fig.11 shows the segregation scale of particles on each cross-section of the above three extruders along the axial direction.It can be seen that the segregation scale of particles on the second，sixth，and 10th cross-sections are lower than that on the respective adjacent cross-sections.This is true for all three extruders，indicating that the addition of pins to the mixing section enhances mixing performance.Because particles are not detained before the third cross-section，the higher the number of pins at a cross-section，the lower the segregation scale of particles and the higher the mixing efficiency.Beyond the third cross-section，the segregation scale of particles is the lowest for the six-pin model and the highest for the 12-pin model.The average segregation scale of particles at the 13 cross-sections except the first and 13th is 14.12×10－4for the three-pin model.10.95×10－4for the six-pin model and 13.35×10－4for the 12-pin model.The mixing efficiency increases by 22.5% when the number of pins goes from three to six.However，the mixing efficiency decreases by 21.9% when the number of pins increases further to 12.When there are too many pins，the material particles are detained in the screw extruder flow channel，thus increasing the distancer between the remaining particles.Asr increases，the segregation scale of particles also increases.The distancer has a larger effect on the mixing performance than that of the number of pins per cross-section.
Fig.10 Time-averaged efficiency of stretching <eζ>along axial direction for 10%，50%，and 90%of par-ticles in mixing section of different extruders：(A) no pin，(B) with pins.
Fig.11 Segregation scale of particles at each crosssection of three extruders along axial direction.
4 Conclusions
Numerical results show that pins disrupt the particle trajectories in the mixing section，split the material flow，change the direction of movement of the particles，and increase the mixing performance. With an increase of the material RTD in the pin mixing section over a no pin mixing section，the mixing efficiency increases. From a statistical analysis of the distributive mixing of material particles in both the no pin and pin mixing sections，it was found that pins increase the efficiency of stretching lgζ，the time-averaged efficiency of stretching<eζ>，and hence the mixing ability of the extruder. Increasing the number of pins does not ensure better mixing. With too many pins，material particles are detained in the screw extruder flow channel and the distance between the remaining particles increases. Pins in the mixing section can effectively enhance the mixing ability of a single-screw extruder. The numerical results reported here provide a theoretical basis for the optimal design of pin-barrel extruder.
免责声明：以上内容源自网络，版权归原作者所有，如有侵犯您的原创版权请告知，我们将尽快删除相关内容。
