Studies on positive conveying in helically channeled single ...1. Introduction Single screw extruders were widely used as one of the basic and convenient elements to melt and con-vey

1. IntroductionSingle screw extruders were widely used as one ofthe basic and convenient elements to melt and con-vey polymer materials. It was usual to consider sin-gle-screw extruders being composed of solids con-veying zone, melting zone, and melt conveyingzone. Attention was here paid to the solids convey-ing zone closely related to the performance of theextruders.The classical solids conveying theory, called Dar-nell-Mol theory, demonstrated that the frictionforce on the barrel internal surface must be largerthan that on the screw surface for effective solidsconveying and steady extrusion. An effective solu-tion to improve feeding efficiency is to groove bar-rel surface along the axial direction as introducedby Menges at the Institute of Plastics Processing

(IKV) in Aachen in 1960’s because the groovesgreatly increases the apparent friction coefficient atthe barrel-solid plug interface. Later, the effects ofstructural parameters on the apparent friction coef-ficient of the barrel were studied by Rautenbach [1],Grünschloß [2] and Potente [3]. Due to the improvedbarrel friction coefficient, higher pressure results inhigher throughput with better stability than conven-tional systems, which facilitates processing highmolecular weight polymer and highly filled materi-als [4–6]. However, such screw consumes moreenergy as well as greater wearing than commonscrews [5, 7–9].In recent years, some advances were made on thedesigns of the groove structural parameters toimprove the performance of the solids conveyingzone. Helical grooves were first invented in the

543

Studies on positive conveying in helically channeled singlescrew extrudersL. Pan, M. Y. Jia, P. Xue*, K. J. Wang, Z. M. Jin

Institute of Plastic Machinery and Engineering, Beijing University of Chemical Technology, 100029 Beijing, China

Received 5 November 2011; accepted in revised form 20 January 2012

Abstract. A solids conveying theory called double-flight driving theory was proposed for helically channeled single screwextruders. In the extruder, screw channel rotates against static barrel channel, which behaves as cooperative embeddedtwin-screws for the positive conveying. They turn as two parallel arc plates, between which an arc-plate solid-plug wasassumed. By analyzing the forces on the solid-plug in the barrel channel and screw channel, the boundary conditions whenthe solid-plug is waived of being cut off on barrel wall, were found to have the capacity of the positive conveying. Experi-mental data were obtained using a specially designed extruder with a helically channeled barrel in the feeding zone and apressure-adjustable die. The effects of the barrel channel geometry and friction coefficients on the conveying mechanismwere presented and compared with the experimental results. The simulations showed that the positive conveying could beachieved after optimizing extruder designs. Compared with the traditional design with the friction-drag conveying, thethroughput is higher while screw torque and energy consumption are decreased. Besides, the design criteria of the barrelchannel were also discussed.

Keywords: processing technologies, double-flight driving theory, positive conveying, helical barrel channels, solids con-veying

eXPRESS Polymer Letters Vol.6, No.7 (2012) 543–560Available online at www.expresspolymlett.comDOI: 10.3144/expresspolymlett.2012.58

*Corresponding author, e-mail: [email protected]© BME-PT

1970’s for decreasing the deflection of the grooveflow in the peripheral direction. Later, the superior-ity of helical grooves was testified by Kramer’ exper-iment [10] and analyzed mathematically by Grün-schloß [11, 12] and Miethlinger [13]. Their researchesindicated that helical grooves improved the through-put much higher than axial grooves. Later, thegroove adjustable continuously in geometry wasused in extruders [14] and showed great advantagesin controlling the solids conveying efficiency and inwidening the range of materials, which was con-firmed by Rauwendaal and Sikora [14], Kowalska[15] and Sikora [16]. However, the adjustableextruder has not been widely applied due to its com-plexity and high cost of the adjustable mechanism.The theory on solids conveying of grooved barrelextruders were also studied and modified. Rauten-bach and Peiffer [17] proposed an in-depth modelto determine the conveying performance of thegrooved section in single-screw extruders in 1982.Potente [18] established a new throughput model inwhich the solids were divided into two parts,including pellets flow in the screw channel and thoseflow in the barrel grooves. The effects of the pres-sure gradient and particle size on the solids convey-ing angle were also analyzed by Potente [19].Besides, optimal design of the groove helical anglewas conducted by considering the effects of the pres-sure gradient and friction coefficient on the convey-ing efficiency by Rauwendaal [20]. In 2001, Potenteand Phol [21] attempted to model the two-dimen-sional approach flow of pellets using the discreteelement method. Moysey and coworkers [22–24]utilized the discrete element method to further ana-lyze the flow characteristics of pellets with a three-dimensional model. Michelangelli et al. [25]extended Potente and Monysey’ efforts and investi-gated the effect of the average pellet size on thedynamics of the granular flow.However, in the previous models based on the fric-tion-drag conveying mechanism, the effects of bothaxial and helical grooves were only assumed toincrease the mean friction coefficient of the barrelon solid pellets, by which the friction force betweenthe barrel and the solids is the active force and thefriction force between the screw and the solids isthe resistant force. Shear fracture occurs on theinterface between the groove and screw channel inabove friction-drag conveying mechanism, so there

is no positive conveying in single screw extruders.Thus, these designs based on the friction-drag con-veying models have disadvantages such as largermotor load, energy consumption and rapid wear.For solving these questions, large barrel channels,not grooves, were developed in this study to achievethe positive conveying in the solids conveying zonebased on a new solids conveying theory called dou-ble-flight driving theory. In the theory, static helicalbarrel channels and rotating screw were regarded asa cooperative system similar to the counter-rotatingtwin-screws. The solids in the barrel channel andscrew channel form one arc-plate solid-plug model,which is different from the traditional parallel-platemodel as shown in Figure 1. From the mathematicalmodel, the boundary condition equations for thepositive conveying in single-screw extruders andthe pressure equation in the solids conveying zonewere established. Based on these, the design criteriafor the helically channeled single screw extruderswith the positive conveying were discussed, simu-lations were conducted, and experimental studieswere carried out to verify this model.

2. Physical model2.1. Arc-plate modelThe physical model called arc-plate model in thedouble-flight driving theory is shown in Figure 1.The feeding barrel sleeve is helically channeledsuch that pellets flow into the screw channel and fillin the barrel channels. In the modeling, one solid-plug element is assumed to be composed of thesolids filled in the screw channel and the connectedones limited in one helical barrel channel as Figure 1shows. The element volume is formed by surround-ing the plug element forward side and backwardside along the helical screw channel, active flight

Pan et al. – eXPRESS Polymer Letters Vol.6, No.7 (2012) 543–560

544

Figure 1. The arc-plate model for the screw extruder withthe helically channeld barrel. 1. the active flightof the barrel channel, 2. the trailing flight of thebarrel channel, 3. the trailing flight of the screw,4. the active flight of the screw.

and trailing flight of the screw, and active flight andtrailing flight of the barrel channel. The bottom sur-face and the top surface of the solid-plug elementare in arcs parallel to the barrel channel bottom. Theother four sides of the element are in planes. Thus,the element is called one arc-plate in modeling.If no shear fracture occurs inside the arc-plate ele-ment with the screw rotating, the element will bepushed forward along the barrel channel directionby the barrel channel and screw flight. Such solidsconveying process is called the positive conveying,similar to the counter-rotating twin-screw extruder.In the positive conveying, active forces from theactive flights of the barrel channel and screw chan-nel are driving forces, and friction forces from thebarrel and screw surface are resistant forces. Thus,the barrel and screw can be manufactured smoothlyto have low friction heat. More importantly, thematerial with low friction coefficient can besteadily conveyed along the barrel channel direc-tion by the active forces. In order to achieve thepositive conveying, we carried out the theoreticalanalysis in Section 3 and experimental study inSections 4 and 5.

2.2. Basic assumptions(1) The granules or powder added in the hopper is

compressed to form the solid-plug withoutinternal slip in the solids conveying zone.

(2)!The solid-plug closely contacts with the six sur-faces: the screw channel bottom, the active andtrailing flight of the screw, the barrel channelbottom, and the active and trailing flight of thebarrel channel.

(3)!The solid-plug is compressible and the bulkdensity is the function of the pressure.

(4)!The pressure of the solid-plug is the function ofthe distance along the screw channel direction.

(5)!In the barrel channel located in each screwpitch distance, the pressure gradient is ignored.

(6)!The friction coefficient, in terms of theCoulomb’s Law, is a constant.

(7)!The barrel channels are approximated to be rec-tangular.

(8)!The clearance between the barrel and the screwis negligible.

(9)!The effects of the material gravity and the vari-ation of the internal temperature inside thesolid-plug are negligible.

3. Mathematical model3.1. Velocities and accelerations of the

solid-plugIn modeling, the barrel is assumed as the movingreference system relative to the static screw. Thevelocity of the solid-plug element is decomposed asshown in Figure 2. Axis Z is axial direction. ! is thescrew channel helical angle at barrel wall, !N is thebarrel channel helical angle in the barrel. Vb is thecircumferential velocity of the barrel. The solidsconveying mechanism is either the positive convey-ing or the friction-drag conveying. In the positiveconveying, the solids move at a speed of VS alongthe barrel channel direction while moving at Vr inthe screw channel. In the friction-drag conveying,the solids are forwarded at VSf composited with Vrf.It can be seen that the conveying angle of the solid-plug in the positive conveying is much greater thanthat in the friction-drag conveying.The inertia forces are not negligible in the solidsmoving, which make the solid pellets or powderinto the barrel channels and further compact them.Figure 3 presents the acceleration decompositionsof the solid-plug embedded in the barrel channeland screw channel.Axis X is radial direction and axis Y is tangentialdirection. Due to the position variation of the mov-


545

Figure 2. Velocity decompositions of the solid-plug in thepositive conveying and the friction-drag conveying

Figure 3. Acceleration decompositions of the solid-plugembedded in the barrel channel and screw channel

ing solid-plug against rotating screw, Coriolisacceleration is generated when the solid-plugmoves at Vr along the screw channel direction asshown by Equation (1):

(1)

The relative acceleration ar resulting from the vari-ation in Vr is given by Equation (2):

(2)

The velocities Vb and Vr can be decomposed intothe tangential direction (Axis Y). That is, normalacceleration an in the radial direction (Axis X)includes aexn and arxn which can be calculated byEquation (3):

(3)

where D is the outer diameter of the screw and t thetime.

3.2. Force analysisWhen pushing the solid-plug forward, the two activeflights can be viewed as scissors resulting in a shearstress on the interface between the part of the solid-plug in the barrel channel and the rest in the screwchannel. In order to determine the shear stress, theforce analysis on the arc-plate model must be car-ried out. Figure 4 shows the forces on the solid-plug. Figure 5 displays the forces on the partembedded in the barrel channel. All of the forcesare included in the following:(1) Main Inertia forces include In, Ik and Ir caused

by the centripetal acceleration, Coriolis acceler-ation and the relative acceleration, respectively.

(2) Normal forces, some of them resulting frompressure P, P1, P2, P3, P4, P51, P52, P61 and P62per unit area on the surfaces of the solid-plug,others from friction FFP1,FFP2, FFP3, FFP4, FFP51and FFP61.

(3) Active forces pushing solids forward are theforce F3 from the active flight of the screw andthe force F5 from the active flight of the barrelchannel.

(4) The screw rotates against the barrel such thatthere is an interface shear stress ô between thepart of the solid-plug in the barrel channel andthe rest in the screw channel.

In all of these forces, the active force of the screwflight F3, the active force of the barrel flight F5 andthe interface shear stress ô are unknown. Activeforces F3 and F5 in Figure 4 can be determined fromthe torque balance around the screw axis (Axis Y)and the force balance along the screw axis (Axis Z).The following Equation (4) is obtained through thetorque balance around the screw axis (Axis Y):

an 52 2Vb2 sin2w cos2wN

D sin21wN 1 w 2

ar 5dVrdt

ak 54Vb2 cosw

D sin1wN 1 w 2ak 54Vb2 cosw

D sin1wN 1 w 2

ar 5dVrdt

an 52 2Vb2 sin2w cos2wN

D sin21wN 1 w 2


546

Figure 4. Forces on the solid-plug embedded in the barrelchannel and screw channel

Figure 5. Forces on the solid-plug part embedded in thebarrel channel

(4)1FP61sinwNDNm

22FFP61coswN

DNm2

1FP62sinwNDm2

5 0

1FFP4coswmDm2

2F5sinwNDNm

22FP51sinwN

DNm2

2FFP51coswNDNm

22 Ircoswm

Dm2

2FP52sinwNDm2

2FFP1coswNDNS

21FFP2coswS

DS2

1F3sinwmDm2

1FP3sinwmDm2

1FFP3coswmDm2

2FP4sinwmDm2

2FFP1coswNDNS

21FFP2coswS

DS2

1F3sinwmDm2

1FP3sinwmDm2

1FFP3coswmDm2

2FP4sinwmDm2

1FFP4coswmDm2

2F5sinwNDNm

22FP51sinwN

DNm2

2FFP51coswNDNm

22 Ircoswm

Dm2

2FP52sinwNDm2

1FP61sinwNDNm

22FFP61coswN

DNm2

1FP62sinwNDm2

5 0

The force balance along the screw axis (Axis Z) gives Equation (5):

(5)

Combining Equations (4) and (5), it leads to Equation (6):

(6)

where

A1 = DNm(cos!m – fLsin!m)(sin!N + fTcos!N) + Dm(sin!m + fLcos!m)(cos!N – fTsin!N)

A2 = fT[cos!NDNS(fLsin!m – cos!m) + sin!NDm(sin!m + fLcos!m)]

A3 = fL[cos!SDS(cos!m – fLsin!m) + sin!SDm(sin!m + fLcos!m)]

A4 = 2DmfL

A5 = Dm[sin!N(cos!m – fLsin!m) + cos!N(sin!m + fLcos!m)]

A6 = DNm(cos!m – fLsin!m)(sin!N – fTcos!N) + Dm(sin!m + fLcos!m)(cos!N + fTsin!N)

A7 = Dm

FP1 = P1bNb

FP2 = P2bNbS

FP4 = P4bNh

FP51 = P51bhN

FP52 = P52bmh

FP61 = P61bhN

FP62 = P62bmh

F55 1FP11In 2A2A1

1 1FP21Ik 2A3A1

1 FP4A4A1

2FP511 1FP62 2FP52 2A5A1

1FP61A6A1

2IrA7A1

1FP51coswN 2FFP51sinwN 1FP52coswN 1 Itsinwm 2FP61coswN 2FFP61sinwN 2FP62coswN 5 0

2FFP1sinwN 2FFP2sinwS 1F3coswm 1FP3coswm 2FFP3sinwm 2FP4coswm 2FFP4sinwmN 1F5cosw2FFP1sinwN 2FFP2sinwS 1F3coswm 1FP3coswm 2FFP3sinwm 2FP4coswm 2FFP4sinwmN 1F5cosw

1FP51coswN 2FFP51sinwN 1FP52coswN 1 Itsinwm 2FP61coswN 2FFP61sinwN 2FP62coswN 5 0

F55 1FP11In 2A2A1

1 1FP21Ik 2A3A1

1 FP4A4A1

2FP511 1FP62 2FP52 2A5A1

1FP61A6A1

2IrA7A1


547

where Ai (i = 1, 2, 3, 4, 5, 6, 7) are the parametersthat are constant for a given barrel and screw as wellas material, fT the friction coefficient of the solidson the barrel, fL the friction coefficient of the solidson the screw, DNS the barrel channel root diameter,DNm the mean barrel channel diameter, DS the screwroot diameter, Dm the mean screw diameter, bN thebarrel channel width, b the screw channel width atbarrel wall, bS the screw channel width at screwroot, bm the mean screw channel width, hN the bar-rel channel depth, h the screw channel depth, !S the

screw channel helical angle at screw root, !m themean screw channel helical angle, FP1, FP2, FP4,FP51, FP52, FP61 and FP62 the pressure forces actingon the surfaces of the solid-plug.Once the active force F5 is known, the interfaceshear stress ô in Figure 5 can similarly be obtained bythe torque balance around the screw axis (Axis Y)and the force balance along the screw axis (Axis Z).In Figure 5, the torque balance around the screwaxis (Y direction) is thus written as shown by Equa-tion (7):

(7)2FFP61coswNDNm

25 0

2FFP1coswNDNS

21 ôbNbsina

DN2

2F5sinwNDNm

22FP51sinwN

DNm2

2 FFP51coswNDNm

21FP61sinwN

DNm2

2FFP1coswNDNS

21 ôbNbsina

DN2

2F5sinwNDNm

22FP51sinwN

DNm2

2 FFP51coswNDNm

21FP61sinwN

DNm2

2FFP61coswNDNm

25 0

where " is the angle between the direction of theinterface shear stress ô and the axial direction asshown in Figure 6 and the barrel channel diameterat barrel wall.The equilibrium of forces on the solid-plug partembedded in the barrel channel along the screwaxis (Axis Z) is:

– FFP1sin!N + ôbNbcos" + F5cos!N – FP51cos!N –FFP51sin!N – FP61cos!N – FFP61sin!N = 0 (8)

By substituting F5 in Equation (8) into Equation (7),the interface shear stress ô amounts to:

3.3. Boundary conditions for positiveconveying

In order to have the capacity of the positive convey-ing, the shear stress ô must be less than the internalfriction force per unit area at the interface betweenthe part of the solid-plug embedded in the barrelchannel and the rest in the screw channel so that the

plug could move as a whole. Therefore, the firstboundary condition equation is:

ô ! Pfi (10)

Substituting Equation (9) into Equation (10), itbecomes Equation (11):


548

Figure 6. Motion and forces on the differential element inthe screw channel

(9)ô 5!FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2

bNbô 5

!FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2bNb

(11)

or

(12)

When Equation (12) is divided by FP12 fT2, the first boundary condition equation can approximately be sim-

plified as follows:

(13)a F5FP1b 2 c1 1 1

fT2d 1 F5

FP1c 2 1 4FP51

FP1d 1 c 2 FP51

FP11 1 d 2 # a fi

fTb 2

FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2 # 1PbNbfi 2 2

!FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2bNb

# Pfi!FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2

bNb# Pfi

FP12 fT2 1 F5211 1 fT22 1 4FP512 fT2 1 2FP1F5 fT2 1 4FP51F5 fT2 1 4FP1FP51 fT2 # 1PbNbfi 2 2

a F5FP1b 2 c1 1 1

fT2d 1 F5

FP1c 2 1 4FP51

FP1d 1 c 2 FP51

FP11 1 d 2 # a fi

fTb 2

where fi is internal friction coefficient in solids, and

From Equation (13), it can be seen that the bound-ary condition is the function of the friction coeffi-cients and structural parameters. hN/bN is defined asthe barrel channel aspect ratio. It is known from

Equation (13) that the lower the friction coefficientof the barrel surface is, the easier the positive con-veying can be achieved. More importantly, whenthe solid-plug is positively conveyed by the twoactive forces of the barrel and screw channel flights,the extruder has an operating mode where the solidsconveying is independent of the friction coefficienton the barrel surface. Therefore, different frictioncoefficients on the barrel surface only induce differ-ent energy consumption and pressure peak at theend of the solids conveying zone and don’t affect

FP51FP1

5hNbN

F5FP1

5A2A1

1A3A1

1hb

A4A1

2hNbN

1hNbN

A6A1

F5FP1

5A2A1

1A3A1

1hb

A4A1

2hNbN

1hNbN

A6A1

FP51FP1

5hNbN

the solids conveying process. If the friction coeffi-cient on the barrel surface becomes nil, the solid-plug is also conveyed along the barrel channel heli-cal angle. However, in the case of the friction-dragconveying, an effective solids conveying is depend-ent on a sufficient high friction coefficient on thebarrel surface. The greater the friction coefficient ofthe barrel surface is, the steadier the friction-dragconveying is. As a result, higher energy consump-tion is required for the friction-drag conveying,compared to that in terms of the positive conveying.

3.4. Pressure equationContinuity equation and kinematic equation areused to analyze the motion and forces of the differ-ential element in the screw channel. The pressureequation in the solids conveying zone can beobtained after resolving the two equations.A down-channel differential element is displayed inFigure 6, where z is the distance along the screwchannel direction. The continuity equation is givenby Equation (14):

(14)

Where the material density is varied along thescrew channel, given by Equation (15) [26]:

# = #m – (#m – #0)e–C0P (15)

Inserting Equation (15) into Equation (14), the con-tinuity equation becomes Equation (16):

(16)

where # is the density under pressure P at time t, #mthe density under utmost pressure, #0 the bulk den-sity under the atmospheric pressure and C0 a coeffi-cient.The kinematics equation can be determined by theapplication of the force balance to the differentialelement in Figure 6 in both the down-channel direc-tion and the direction perpendicular to the screwchannel.

The equilibrium of forces in the direction perpendi-cular to the screw channel is given by Equation (17):

F"3 = ôcos(" – !)bdzb (17)

The equilibrium of forces in the down-channeldirection gives Equation (18):

F"FP2 + F"FP3 + F"FP4 + (F"P6 – F"P5) – ôsin(" – !)bdzb = m"|ar| (18)

where m" is the mass of the differential element inthe screw channel, F"P3, F"P4, F"P5, F"P6, F"FP2, F"FP3,F"FP4 the normal forces on the differential elementin the screw channel and F"3 the active force on thedifferential element from the active flight of thescrew.Substituting Equation (17) into Equation (18) leadsthe kinematic equation to be expressed as shown byEquation (19):

(19)

where Di (i = 1, 2, 3) are the parameters that areconstant for a given barrel and screw as well asmaterial.The boundary conditions for the movement ofsolids are:

Vr(z,t)|z#=#0 = V0 = Vbcos!, Vr(z,t)|t#=#0 = V0P(z,t)|z#=#0 = P0, P(z,t)|t#=#0 = P

–

where V0 is the inlet velocity of solids along thescrew channel direction, P0 the inlet pressure and P

–

the average pressure.Equations (16) and (19) are resolved using Laplacetransformation, Laplace ultimate theory and dimen-sional transformation [27]. The pressure on thesolid-plug along the screw channel direction can bewritten as shown by Equation (20):

(20)

where $ is the modification factor, L the length ofthe barrel channel along the axial direction (Axis Z)and

P 5 P0~e2bz

L 1g2rmV02D2

D1~a e2bzL 2 1 b

0P0z

1 PD1 1 r cVr2D2 1 D3 a dVrdt b d 5 0

0P0t

1 Vr0P0z

11C0c rmeC0Prm 2 r0

2 1 d 0Vr0z

5 0

0r

0t1 Vr

0r

0z1 r

0Vr0z

5 00r

0t1 Vr

0r

0z1 r

0Vr0z

5 0

0P0t

1 Vr0P0z

11C0c rmeC0Prm 2 r0

2 1 d 0Vr0z

5 0

0P0z

1 PD1 1 r cVr2D2 1 D3 a dVrdt b d 5 0

P 5 P0~e2bz

L 1g2rmV02D2

D1~a e2bzL 2 1 b


549

b5r0LD1

r02g2C0rmV02D31rm2r0 2 D15

sinw abS fLsinwmsinwS 12fLhb1sinwm a bC11bSC21hC3 1bhNC5

bNb

bmh 3sinw 1 sinwmC4 4b5r0LD1

r02g2C0rmV02D31rm2r0 2 D15

sinw abS fLsinwmsinwS 12fLhb1sinwm a bC11bSC21hC3 1bhNC5

bNb

bmh 3sinw 1 sinwmC4 4

Equation (20) indicates that the pressure on thesolids conveying zone is not only the function ofthe friction coefficients, material density and struc-tural parameters, but also the function of the cir-cumferential velocity. More importantly, it also dis-closes that the pressure can be well establishedalong the screw channel direction when the inletpressure is equal to zero, which can not be effec-tively explained by the Darnell-Mol theory.In Equation (20), when the parameter ! is largerthan zero, the pressure is minus along the screwchannel direction, which is no practical signifi-cance. If the parameter ! is negative, the pressure ispositive and can be built along the screw channel

direction. Therefore, the second boundary equationcan be obtained and showed as Equation (21):

(21)

Equations (13) and (21) are used to determinewhether the given parameters of the barrel channeland screw are good or not so that the optimal barrelchannel and screw channel can be designed forlarger positive conveying.To summarize, the important components of thedouble-flight driving theory were established: twopositive conveying boundary equations and thepressure equation in the solids conveying zone.

b 5r0LD1

r0 2 g2C0rmV02D31rm 2 r0 2 6 0b 5

r0LD1r0 2 g

2C0rmV02D31rm 2 r0 2 6 0


550

Ct 5 1fLcosw 1 sinw 2A7A1 1coswN 2 fTsinwN 2 1A7A1

DNm1sinwN 1 fTcoswN 2 fLsinw 2 coswDN

1 DNm c 1fTcoswN 2 sinwN 2 1 A6A1 1sinwN 1 fTcoswN 2 dfLsinw 2 cosw

DN

C5 5 1fLcosw 1 sinw 2 c 1coswN 1 fTsinwN 2 1 A6A1 1fTsinwN 2 coswN 2 d

C4 5 1fLcosw 1 sinw 2A5A1 1fTsinwN 2 coswN 2 1A5A1




C2 5 Ck 5 1fLcosw 1 sinw 2A3A1 1fTsinwN 2 coswN 2 1A3A1


1 c fTcoswNDNS 1 A2A1DNm1sinwN 1 fTcoswN 2 dfLsinw 2 cosw

DN

C1 5 Cn 5 1fLcosw 1 sinw 2 c fTsinwN 1 A2A1 1fTsinwN 2 coswN 2 d

D3 5sinw

sinw 1 sinwmC4£1 2

sinwm a1 1 bhNbmh bCtsinw

§

D25sinw

sinw1sinwmC4£

2sinwm a11 bhNbmhbCn1sinwcotwN 22

Dsinw1 °

sinwm a11bhNbmhbCksinw

1 fL¢ 4coswsin1wN 1w 2DsinwN §D25sinw

sinw1sinwmC4£

2sinwm a11 bhNbmhbCn1sinwcotwN 22

Dsinw1 °

sinwm a11bhNbmhbCksinw

1 fL¢ 4coswsin1wN 1w 2DsinwN §

D3 5sinw

sinw 1 sinwmC4£1 2

sinwm a1 1 bhNbmh bCtsinw

§

C1 5 Cn 5 1fLcosw 1 sinw 2 c fTsinwN 1 A2A1 1fTsinwN 2 coswN 2 d

1 c fTcoswNDNS 1 A2A1DNm1sinwN 1 fTcoswN 2 dfLsinw 2 cosw

DN

C2 5 Ck 5 1fLcosw 1 sinw 2A3A1 1fTsinwN 2 coswN 2 1A3A1






C5 5 1fLcosw 1 sinw 2 c 1coswN 1 fTsinwN 2 1 A6A1 1fTsinwN 2 coswN 2 d

1 DNm c 1fTcoswN 2 sinwN 2 1 A6A1 1sinwN 1 fTcoswN 2 dfLsinw 2 cosw

DN

Ct 5 1fLcosw 1 sinw 2A7A1 1coswN 2 fTsinwN 2 1A7A1


Sections 4 and 5 will present the experiments usedto verify the double-flight driving theory with ourextruder.

4. Experimental4.1. ApparatusOne helically channeled extruder was speciallydesigned and manufactured to study the positiveconveying mechanism as shown in Figure 7. Theextruder can be used to measure the pressure valueon-line at the end of the solids conveying zone andthe throughput that was only composed of the solidsconveying zone and equipped with one detachablefeeding barrel sleeve, one screw and one pressure-adjustable die. One taper barrel sleeve with tworows of small circular holes shown in Figure 8 isinstalled between the screw and diversion cone.Two rows of circular holes with their axis leaning toscrew axis are evenly and alternately arrangedalong the circumferential direction. Their gross areais equal to the total area of the barrel channels andscrew channel. While the extruder working, smallcircular holes are used as the outlets of the die forextruding solids so that solids pellets are extrudedevenly in the peripheral direction. Therefore, themotion trace of the solids is approaching in the realcase. The outlet area of the die is constant to makesure the steady extrusion.

Besides, in order to discuss the effects of the geo-metrical parameters on the positive conveying mech-anism and the performance of the solids conveyingzone, two helically channeled sleeves with differentbarrel channel widths and two screws with differentpitches were made, as shown in Figure 9. The meanbarrel channel helical angles are both 50° with eightbarrel channels in the two helically channeledsleeves. The screws are of diameter 45 mm and


551

Figure 7. A schematic diagram of the extruder speciallydesigned for the experiments. 1. spring, 2. springbumper, 3. weighing sensor, 4. flange, 5. tapersleeve, 6. outlets, 7. diversion cone, 8. screwchannel, 9. barrel channel, 10. feeding sleeve,11. barrel, 12. screw.

Figure 8. A taper sleeve with two rows of small circularholes

Figure 9. A schematic diagram of the feeding sleeves andscrews. (a) two helically channeled sleeves withdifferent barrel channel widths and one axiallygrooved sleeve, (b) two screws with differentpitches.

Table 1. Geometrical parameters of the solids conveying zone of the experimental extruder

Experimentalcomponents Thread number

Lead

[mm]

Length

[mm]

Barrel channeldepth[mm]

Barrel channelwidth[mm]

Flight width

[mm]Sleeve a 8 From 180 to 156 261 2.0 From 8.5 to 7.7 5.5Sleeve b 8 From 180 to 156 261 2.0 From 7.0 to 6.2 7.0Sleeve IKV 8 – 261 1.5 8 9.6Screw c 1 45 261 3.2 – 4.5Screw d 1 From 45 to 39 261 3.2 – 4.5

have a length to diameter ratio of 5.8. In addition, thefeeding sleeve IKV with axial grooves was alsodesigned to compare the effects of the geometricalparameters on the solids conveying mechanism.The basic geometrical parameters of the experimen-tal extruder are given in Table 1.

4.2. MaterialsLow-density Polyethylene (LDPE, LD607 type)from Beijing Yanshan Plant was used with the phys-ical properties listed in Table 2. In the experiments,

the friction coefficients were assumed to be con-stant.

5. Results and discussions5.1. Boundary conditions for positive

conveyingTwo critical condition equations (Equation (22) and(23)) for the positive conveying can be obtainedfrom the two boundary condition Equations (13)and (21).When ô = Pfi, the first critical condition Equationcan be determined as follows:


552

Table 2. Physical properties of LDPE used in the experiments

Material Melt flow index[g/10 min]Bulk density

[kg/m3]

Density underutmost pressure

[kg/m3]

External frictioncoefficient

(fL = fT)

Internal frictioncoefficient

(fi)

Mean diameterof particles

[mm]LDPE 7.5 485 920 0.13 0.45 1.6–2.0

Figure 10. Predicted critical curves for positive conveying and experimental data. (a) the predicted critical curves andexperimental values 1, 2 and 3 for the case with screw c, (b) the predicted critical curves and experimental values4, 5 and 6 for the case with screw d. 1, 2, 3, 4, 5 and 6 are the experimental values for the combination c-a of thescrew c and the feeding sleeve a, c-b, c-IKV, d-a, d-b and d-IKV.

(22)X1 5 aF5FP1b2 a11 1

fT2b 1 F5

FP1a214FP51

FP1b 1 a2FP51

FP111b 2 2 a fi

fTb 2 5 0X1 5 aF5FP1b

2 a11 1fT2b 1 F5

FP1a214FP51

FP1b 1 a2FP51

FP111b 2 2 a fi

fTb 2 5 0

If ! = 0, the second critical condition equation canbe written as Equation (23):

(23)

When the geometrical parameters of the screw, thephysical parameters of material and the screw rota-tion are known, the unknown barrel channel aspectratio hN/bN is resolved from Equations (22) and (23)as a function of the barrel channel helical angle.Thus, two critical curves can be drawn.

In order to compare with the experimental datadirectly, theoretical simulations are carried outusing the same parameters in Table 1. The two criti-cal curves for the given screw c and d and materialcan be drawn at the screw rotation speed of 40 rpm,shown in Figure 10. It can be seen that only whenthe values of the barrel channel aspect ratio and bar-rel channel helical angle are both simultaneouslyless than their corresponding critical points in thecritical curves, the boundary condition Equa-tions (13) and (21) can be both maintained. There-

X2 5r0LD1

r0 2 g2C0rmV02D31rm 2 r0 2 5 0X2 5

r0LD1r0 2 g

2C0rmV02D31rm 2 r0 2 5 0

fore, it can be concluded that, to have the capacityof the positive conveying, both the barrel channelaspect ratio and barrel channel helical angle mustbe beneath the two critical curves so that they aresimultaneously less than their corresponding criticalpoints. Otherwise, it is the friction-drag conveying.On the one hand, it can be seen from Figure 10a and10b that the four experimental points 1 with thecombination c-a of the screw c and the feedingsleeve a, 2 with c-b, 4 with d-a and 5 with d-b areall in the area beneath the two critical curves, whichmeans that the positive conveying works in the fourexperiments. As for the experimental point 3 with thecombination c-IKV of the screw c and the feedingsleeve IKV, and point 6 with d-IKV, they both fallout of the area beneath the two critical curves, so itis the friction-drag conveying in the two experi-ments. Positive conveying can be achieved in axialgrooves only in the case of the minimum aspectratio (zero), which has no practical significance.Thus, the feeding sleeve IKV with axial groovescan not achieve positive conveying in practice.On the other hand, it can be calculated by Equa-tions (22) and (23) that the critical values of thebarrel channel aspect ratio for the two experimentalpoints 1 and 2 are 1.27 and 0.44 corresponding to thefirst and second critical curves respectively in Fig-ure 10a while those for the two experimental points4 and 5 are 1.23 and 0.4 respectively in Figure 10bwith the given barrel channel helical angle of 50° infeeding sleeve a and b. Based on the theoreticalanalysis of two critical curves, the positive convey-ing can be achieved if the maximum values of thebarrel channel aspect ratio are less than 0.44 in Fig-ure 10a and 0.4 in Figure 10b with the given barrelchannel helical angle of 50°. Otherwise, the friction-drag conveying works. Similarly, The maximumbarrel channel helical angle in Figure 10a must beless than 73° for the point 1 to get positive convey-ing with the given value of the barrel channel aspectratio of 0.25 in the feeding sleeve a and that is 67°for the point 2 with the given value of the barrelchannel aspect ratio of 0.31 in the feeding sleeve b.Besides, the maximum barrel channel helical anglein Figure 10b must be less than 69°for the point 4with the given value of the barrel channel aspectratio of 0.25 in the feeding sleeve a and that is 63°for the point 5 with the given value of the barrelchannel aspect ratio of 0.31 in the feeding sleeve b.

Therefore, the critical curves define such two char-acteristics in the positive conveying mechanism.One characteristic is that the maximum barrel chan-nel helical angle can be determined when the barrelchannel aspect ratio is known. It can be explainedby the fact that the interface shear stress is raisedremarkably with the increasing barrel channel heli-cal angle. When the interface shear stress is biggerthan the internal friction force, the solid-plug can becut off at the interface as shown in Figure 11, andthen the conveying is not the positive conveying.The other characteristic is that the maximum barrelchannel aspect ratio also can be obtained when thebarrel channel helical angle is given. It can beknown from above theoretical studies that the fric-tion forces from the barrel channel are the resistantforces and the active forces of the screw and barrelchannel flight are the driving forces when the solid-plug in the barrel channel and screw channel is pos-itively conveyed. However, it is worthwhile to men-tion that the active forces of the screw and barrelchannel flight are both derived from the interfaceshear stress. Therefore, the deeper the barrel chan-nel depth is, the greater the resistant force is. Thewider the barrel channel width is, the greater theactive force is. If the maximum barrel channelaspect ratio is exceeded, the active forces will notbe enough so that the solid-plug will be cut off atthe interface and there is no barrel channel con-veyance. This is also the reason why the friction-drag conveying prevails in the experimental points3 and 6. Thus, single screw extruders with the posi-tive conveying can be designed by the two criticalcurves.Figure 12 shows the theoretical simulations aboutthe effects of different external friction coefficientson the critical curves using the same parameters as


553

Figure 11. A schematic diagram of fracture interfacesinside the solid-plug

screw c at the screw rotation speed of 40 rpm at aconstant internal friction coefficient of 0.6.It can be found from Figure 12 that different exter-nal friction coefficients lead to the variation of thetwo critical curves corresponding to different criti-cal barrel channel aspect ratios and barrel channelhelical angles. When the external friction coeffi-cient increases, the location of the first criticalcurve descends while the location of the secondcritical curve ascends. Besides, it can also be seenthat the greater the external friction coefficient is,the smaller the available area for positive convey-ing beneath the two critical curves is. It can be con-cluded that big external friction coefficients canresult in the transformation from the positive con-

veying to the friction-drag conveying, which is dueto the fact that great friction coefficient of the barrelsurface induces high resistance force for barrelchannel conveyance in the positive conveying sothat the solid-plug is cut off at barrel wall.Figure 13 shows a whole solid-plug sample in asteady solids conveying process using the experi-mental extruder with the combination c-a at a screwrotation speed of 40 rpm. When the extruder runssteadily, we stop the screw suddenly to disassemblethe pressure-adjustable die and to subsequently peeloff the polymer sample located in the screw head. Itcan be seen from the sample in Figure 13 that thesolids in the barrel channel and screw channel wereextruded as a whole solid-plug indicating no inter-nal circumferential shear fracture at the shear inter-face and the positive conveying prevailing. Thisexperiment showed the typical positive conveyingusing the combination c-a, which is close to the pre-dicted results from the point 1 in Figure 10a.

5.2. Pressure distributionIn order to verify the positive conveying mecha-nism further, some theoretical simulations about theeffects of the geometrical parameters on the pres-sure distribution in the solids conveying zone werecarried out at a screw rotation speed of 40 rpm.Moreover, the on-line measured results of experi-mental pressure were compared with theoreticalanalysis.The pressure distribution curves in the solids con-veying zone with different solids conveying mecha-nisms and external friction coefficients are pre-sented in Figure 14. The first experiment was madeusing the combination c-IKV for the friction-dragconveying. The second experiment was carried outusing the combination c-a for the positive convey-ing. It can be found from Figure 14 that the simu-lated pressure distribution in the friction-drag con-veying by the Darnell-Mol theory increases sharplyand is seriously deflected from the experimentaldata. The reason may be that the density of thesolid-plug is assumed to be invariable and theacceleration of the solid-plug is not considered. Incontrast, the simulated pressure distribution in thepositive conveying by the present model increaseslinearly and is well consistent with the experimentaldata at the end of the solids conveying zone. Thisconfirms that the double-flight driving theory can


554

Figure 12. Effect of different external friction coefficientson the critical curves

Figure 13. A whole solid-plug sample in a steady solidsconveying process in the experimental extruder

better describe the pressure distribution in the solidsconveying zone than the Darnell-Mol theory.Besides, it is worthwhile to notice that the two exper-imental values in different solids conveying mecha-nisms are not obviously different indicating thatboth positive and friction-drag conveying mecha-nisms also have the good building pressure capabil-ity. In addition, it also can be seen that bigger exter-nal friction coefficients induce higher pressureincrease in the solids conveying zone, which is con-sistent with researches of Fang et al. [28].Effect of the barrel channel helical angle on thepressure value at the end of the solids conveyingzone is shown in Figure 15. The experiment wasperformed with the combination c-a. The pressurevalue at the end of the solids conveying zone was

found to decrease with the increasing barrel channelhelical angle. In the case of the positive conveying,the conveying angle of the solid-plug in the screwchannel is equal to the barrel channel helical angle.Thus, high barrel channel helical angle induces bigconveying angle of the solid-plug in the screwchannel, which improves the conveying rate of thesolids in the screw channel resulting in low pressurevalue. In addition, it also can be known from thefigure that the experimental value approaches to thetheoretical data with the given external frictioncoefficient of 0.13, which confirms the accuracy ofthe theory.Figure 16 displays the effect of the screw pitch onthe pressure value at the end of the solids conveyingzone, including the experimental data for the com-binations c-a and d-a , respectively. It can be foundthat the simulated pressure value at the end of thesolids conveying zone is raised with the increasingscrew pitch size because the throughput in thescrew channel is higher at a constant barrel channelconveying angle, which is helpful to compact solid-plug. The predicted values are consistent with theexperimental data. However, such tendency is notthat obvious at high barrel channel helical angle,which can be due to the fact that increasing barrelchannel helical angle results in the decrease of thepressure value. Figure 17 presents the predicted pressure value andexperimental data at the end of the solids conveyingzone for the combinations c-a and c-b. The simu-lated pressure values at the end of the solids con-veying zone vary insignificantly with different barrelchannel widths, as is consistent with the experimen-


555

Figure 15. Effect of the barrel channel helical angle on thepressure value at the end of the solids conveyingzone

Figure 16. Effect of the screw pitch size on the pressurevalue at the end of the solids conveying zone

Figure 14. Pressure distribution in the solids conveyingzone with different mechanisms and differentfriction coefficients

tal data. This can well be understood by comparingthe difference between the double-flight drivingtheory and the Darnell-Mol theory. In the case ofthe friction-drag conveying, the barrel friction forceis the active force and the mean friction coefficientof the barrel is raised greatly with the increasingbarrel channel width by some previous reports [1–3, 13], which results in high pressure value at theend of the solids conveying zone. In contrast, all thefriction forces from the barrel and screw surface are

resistance forces in the double-flight driving theoryfor the positive conveying and the solid-plug is onlypushed forward along the barrel channel helicalangle by the two active flights of the barrel channeland screw. Therefore, the solids conveyance in thescrew channel in the case of the positive conveyingis hardly affected by the variation of the barrelchannel width with a steady pressure value at theend of the solids conveying zone.

5.3. Throughput and energy consumptionAccording to above theoretical analysis on the pos-itive conveying mechanism, the advantages of highthroughput and low energy consumption can berevealed when one extruder works with the positiveconveying. Experimental verifications for the posi-tive conveying mechanism were also implementedfrom the view of throughput and energy consump-tion.In the case of the positive conveying, the solidsembedded in the barrel channel and screw channelas a whole plug are conveyed. Thus, taking thematerial velocity along the axial direction and thebarrel channel and screw channel area around thescrew axis into consideration, the total throughputm is given by Equation (24):


556

Figure 17. Effect of the barrel channel width on the pres-sure value at the end of the solids conveyingzone

(24)m 5 npDm— cp

41DNS2 2 DS2 2 2 M e1hsinwm 2 N

e2hNsinwN

d sinwsinwNsin1w 1 wN 2r0~60m 5 npDm

— cp41DNS2 2 DS2 2 2 M e1hsinwm 2 N

e2hNsinwN

d sinwsinwNsin1w 1 wN 2r0~60

where n is the screw rotation, Dm— the average diam-

eter of the solid-plug, e1 and e2 the flight land widthof the screw and barrel channel respectively, and Mand N the thread numbers of the screw channel andbarrel channel respectively.Figure 18 describes the throughput measured andsimulated for the combinations of c-a, c-b andc-IKV. It can be seen that the measured and simu-lated throughput data of the extrusion system are allimproved greatly with the increasing screw rotationfor the three combinations. The simulated through-put by the positive conveying mechanism for thecombinations c-a and c-b are both close to theexperimental values, which shows that the positiveconveying mechanism is achieved during extrusion.Based on the positive conveying mechanism, greaterbarrel channel width induces larger barrel channelconveying square, which is consistent with the factthat the experimental throughput using the feeding

sleeve a with great barrel channel width is largerthan that of the feeding sleeve b with small barrelchannel width. Thus, the positive conveying mecha-nism is confirmed by the above experiments. Inaddition, it also can be seen that the measured

Figure 18. The measured and simulated throughput withthe combinations c-a, c-b and c-IKV

throughput with the combination c-IKV is less thanthose of the combinations c-a and c-b. This isbecause the solid-plug experiences fracture at barrelwall and the friction-drag conveying prevails dur-ing extrusion, which results in the decrease of thebarrel channel conveyance. Besides, it is worthnoticing that the simulated results for the combina-tion c-IKV are seriously deflected from the meas-ured results because only the solids in the screwchannel are considered and those in the barrel chan-nel are neglected when the friction-drag conveyingprevailing in the Darnell-Mol theory.The experimental electric currents, total power andtheoretically useful power at different screw rota-tions for the three combinations c-a, c-b and c-IKVare listed in Table 3.I1, I2 and I3 are the experimental electric currentsfor the three combinations c-a, c-b and c-IKV respec-tively and PW1, PW2 and PW3 are the total powercorresponding to I1, I2 and I3. PW11 and PW22 are thetheoretically useful power for the combinations c-aand c-b by the positive conveying theory and PW33the theoretically useful power for the combinationof c-IKV by the Darnell-Mol theory.The variation of the electric current reflects thechange of the total power in the motor because thetotal power of a three-phase induction motor is pro-portional to the electric current. In the present modelfor the combinations c-a and c-b, only the usefulpower of PW11 and PW22 for solids conveyance arecalculated. That results in the theoretical data lessthan the experimental data of PW1 and PW2, respec-tively. It can be noticed that because the deviationbetween the experimental and theoretical data stillkeeps in a small range with the increase of thescrew rotation, the experimental data shows theactual energy consumption indicating that the posi-tive conveying mechanism works during extrusion.

In addition, the comparison of PW11 and PW22 alsoshows that the simulated useful power in the posi-tive conveying mechanism is enhanced with theincreasing barrel channel width for the combina-tions c-a and c-b, which can also be explained bythe fact that the active forces of the barrel channeland screw flight are improved with the increase ofthe barrel channel width resulting in higher frictionheat on the active flights of the barrel channel andscrew. It also can be seen that the experimentalvalue of the total power for the combination c-IKVwith friction-drag conveying is remarkably higherthan those for the combinations c-a and c-b withpositive conveying. This may be due to that thesolid-plug in the barrel channel and screw channelexperiences fracture at barrel wall for the combina-tion c-IKV when the friction-drag conveying mech-anism prevails during extrusion resulting in highfriction heat. From Table 3, it also can be found thatthe theoretical value of PW33 for the combinationc-IKV is much greater than the experimental valuedue to the high pressure value predicted by the Dar-nell-Mol theory.

6. ConclusionsA solids conveying theory called double-flight driv-ing theory was proposed for helically channeledsingle screw extruders where the solids embeddedin the barrel channel and screw channel behaving asa whole solid-plug were pushed forward along thebarrel channel helical angle direction by activeflights of the barrel channel and screw. In order toachieve positive conveying in the double-flightdriving theory, two boundary conditions for thepositive conveying were determined by analyzingforce equations and pressure equation. On the basisof the theoretical and experimental studies on thehelically channeled feed zone, the positive convey-


557

Table 3. Experimental electric currents, total power and theoretically useful power at different screw rotations

n[r/min]

Combination c-a Combination c-b Combination c-IKV

Experimental dataTheoreticaldata by the

present modelExperimental data

Theoreticaldata by the

present modelExperimental data

Theoretical databy Darnel-Mol

theoryI1

[A]PW1

[kW]PW11[kW]

I2[A]

PW2[kW]

PW22[kW]

I3[A]

PW3[kW]

PW33[kW]

10 3.7 1.41 1.13 3.3 1.24 0.94 5.2 1.95 56.9620 6.1 2.31 1.96 5.2 1.96 1.68 8.8 3.35 113.9230 8.6 3.25 2.93 7.6 2.89 2.52 11.3 4.29 170.8840 11.6 4.41 3.80 10.4 3.95 3.45 14.5 5.51 227.82

ing mechanism was confirmed by the results meas-ured and simulated by our theory.From the boundary condition equations, the maxi-mum barrel channel helical angle could be deter-mined with a given barrel channel aspect ratio. ViceVersa, the maximum barrel channel aspect ratioalso could be obtained with a given barrel channelhelical angle for positive conveying. The externalfriction coefficients determine if the conveying isthe positive conveying or the friction-drag convey-ing. More importantly, an extruder with the positiveconveying can be designed from the analysis ofboundary condition equations.Compared with the sharp increase of the pressurevalue in the friction-drag conveying theory of theDarnel-Mol Model, the pressure distribution in thepositive conveying increases linearly. In addition,the pressure value at the end of the solids conveyingzone in the positive conveying is higher at lowerbarrel channel helical angle, higher external frictioncoefficients and larger screw pitch size. However,the effect of the barrel channel width is not signifi-cant. Therefore, the ability to build pressure in thesolids conveying zone can be effectively controlled.The extruder designed on the positive conveyingmechanism showed higher throughput and lowerenergy consumption than that on the friction-dragconveying mechanism. Besides, bigger barrel chan-nel width is helpful to obtain a steady solids con-veying and high throughput in the case of the posi-tive conveying.

AcknowledgementsThe authors would like to acknowledge the support of theNational Natural Science Foundation of China (No.50873014).

NomenclatureX radial directionY tangential directionZ axial directionz distance along screw channel directionVb circumferential velocity of barrelVS velocity along barrel channel direction in

positive conveyingVr velocity along screw channel direction in

positive conveyingVSf velocity along barrel channel direction in

friction-drag conveying

Vrf velocity along screw channel direction infriction-drag conveying

V0 inlet velocity of solids along screw channeldirection

n screw rotationan normal accelerationak coriolis accelerationar relative accelerationIn, Ik, Ir inertia forcesP, P1, P2, P3, P4, P51, P52, P61, P62

pressureP0 inlet pressure P– average pressureFP1, FP2, FP3, FP4, FP51, FP52, FP61, FP62

normal forces on the solid-plug resultingfrom pressure

FFP1, FFP2, FFP3, FFP4, FFP51, FFP61normal forces on the solid-plug resultingfrom friction

F3 active force on solid-plug from active flightof screw

F5 active force on solid-plug from active flightof barrel channel

F"P3, F"P4, F"P5, F"P6normal forces on differential element inscrew channel resulting from pressure

F"FP2, F"FP3, F"FP4normal forces on differential element inscrew channel resulting from friction

F"3 active force on differential element fromactive flight of screw

ô interface shear stress between part of solid-plug in barrel channel and the rest in screwchannel

Ai (i = 1, 2, 3, 4, 5, 6, 7)parameters that are constant for given bar-rel and screw as well as material

Di (i =1, 2, 3)parameters that are constant for given bar-rel and screw as well as material

fT friction coefficient of solids on barrelfL friction coefficient of solids on screwfi internal friction coefficient in solidst time# density at pressure P at time t#m density at utmost pressure#0 bulk density at atmospheric pressureC0 coefficient of material


558

" angle between direction of interface shearstress and axial direction

$ modification factorL length of barrel channel along axial direc-

tionDN barrel channel diameter at barrel wallDNS barrel channel root diameterDNm mean barrel channel diameterD outer diameter of screwDS screw root diameterDm mean screw diameterDm— average diameter of the solid-plugbN barrel channel widthb screw channel width at barrel wallbS screw channel width at screw rootbm mean screw channel widthhN barrel channel depthh screw channel depth!N barrel channel helical angle! screw channel helical angle at barrel wall!S screw channel helical angle at screw root!m mean screw channel helical anglee1 flight land width of screwe2 flight land width of barrel channelM thread numbers of screw channelN thread numbers of barrel channelm total throughput in solids conveying zonem" mass of differential element in screw chan-

nel in Figure 6Ii (i = 1, 2, 3)

experimental electric currentPWi (i = 1, 2, 3)

total powerPWii (i = 1, 2, 3)

theoretically useful power

References [1] Rautenbach R.: Model tests on the transport of plastics

powders in the feed section of single-screw extruders.Kunststoffe-German Plastics, 69, 377–380 (1979).

[2] Grünschloß E.: Calculation of the mean coefficient ofbarrel friction in grooved feed sections. Kunststoffe-German Plastics, 74, 405–409 (1984).

[3] Potente H.: Methods of calculating grooved extruderfeed sections. Kunststoffe-German Plastics, 75, 439–441 (1985).

[4] Franzkoch B., Menges G.: Grooved forced-feedingzones can improve extruder performance. PlasticsEngineering, 34, 51–54 (1978).

[5] Rautenbach R., Peiffer H.: Throughput and torquecharacteristics of grooved feed sections in single-screw extruders. Kunststoffe-German Plastics, 72,262–266 (1982).

[6] Davis B. A., Gramann P. J., Del P. Norieqa E. M., Oss-wald T. A.: Grooved feed single screw extruders –Improving productivity and reducing viscous heatingeffects. Polymer Engineering and Science, 38, 1199–1204 (1998).DOI: 10.1002/pen.10288

[7] Menges G., Feistkorn W., Fischbach G.: Hot-operatedgrooved bushes increase the output and reduce theenergy consumption of single screw extruders. Kunst-stoffe-German Plastics, 74, 695–699 (1984).

[8] Grünschloß E.: Process improvements in single-screwextruders with grooved feed sections. Kunststoffe-German Plastics, 75, 850–854 (1985).

[9] Schuele H., Fritz H. G.: Minimizing the abrasive wearin the feed zone of grooved barrel extruders. Kunst-stoffe-German Plastics, 77, 387–393 (1987).

[10] Kramer A.: Experience in using extruders with groovedfeed zones. Kunststoffe-German Plastics, 78, 21–26(1988).

[11] Grünschloß E.: A new style single screw extruder withimproved plastification and output power. Interna-tional Polymer Processing, 17, 291–300 (2002).

[12] Grünschloß E.: A powerful universal plasticating sys-tem for single-screw-extruders and injection-mouldingmachines. International Polymer Processing, 18, 226–234 (2003).

[13] Miethlinger J.: Modelling the solids feed section ingrooved-feed extruders. Kunststoffe Plast Europe, 93,49–53 (2003).

[14] Rauwendaal C., Sikora J.: The adjustable grooved feedextruder. Plastics Additives and Compounding, 2, 26–30 (2000).DOI: 10.1016/S1464-391X(00)88882-0

[15] Kowalska B.: Grooved feed sections: Design variantsfor single-screw extruders. Kunststoffe Plast Europe,90, 10–11 (2000).

[16] Sikora J. W.: The effect of the feed section groovetaper angle on the performance of a single-screwextruder. Polymer Engineering and Science, 41, 1636–1643 (2001).DOI: 10.1002/pen.10861

[17] Rautenbach R., Peiffer H.: Model calculation for thedesign of the grooved feed section of single-screwextruders. Kunststoffe-German Plastics, 72, 137–143(1982).

[18] Potente H.: The forced feed extruder must be reconsid-ered. Kunststoffe-German Plastics, 78, 355–363 (1988).

[19] Potente H., Schöppner V.: A throughput model forgrooved bush extruders. International Polymer Pro-cessing, 10, 289–295 (1995).

[20] Rauwendaal C.: Polymer extrusion. Hanser, Munich(2010).


559

http://dx.doi.org/10.1002/pen.10288http://dx.doi.org/10.1016/S1464-391X(00)88882-0http://dx.doi.org/10.1002/pen.10861

[21] Potente H., Phol T. C.: Polymer pellet flow out of thehopper into the first section of a single screw. Interna-tional Polymer Processing, 17, 11–21 (2001).

[22] Moysey P. A., Thompson M. R.: Investigation of solidstransport in a single-screw extruder using a 3-D dis-crete particle simulation. Polymer Engineering andScience, 44, 2203–2215 (2004).DOI: 10.1002/pen.20248

[23] Moysey P. A., Thompson M. R.: Modelling the solidsinflow and solids conveying of single-screw extrudersusing the discrete element method. Powder Technol-ogy, 153, 95–107 (2005).DOI: 10.1016/j.powtec.2005.03.001

[24] Moysey P. A., Thompson M. R.: Determining the col-lision properties of semi-crystalline and amorphousthermoplastics for DEM simulations of solids trans-port in an extruder. Chemical Engineering Science, 62,3699–3709 (2007).DOI: 10.1016/j.ces.2007.03.033

[25] Michelangelli O. P., Yamanoi M., Gaspar-Cunha A.,Covas J. A.: Modelling pellet flow in single extrusionwith DEM. Journal of Process Mechanical Engineer-ing, 225, 255–268 (2011).DOI: 10.1177/0954408911418159

[26] Chung C. I.: Plasticating single-screw extrusion the-ory. Polymer Engineering and Science, 11, 93–98(1971).DOI: 10.1002/pen.760110204

[27] Qu J., Shi B., Feng Y., He H.: Dependence of solidsconveying on screw axial vibration in single screwextruders. Journal of Applied Polymer Science, 102,2998–3007 (2006).DOI: 10.1002/app.24658

[28] Fang S., Chen L., Zhu F.: Studies on the theory of sin-gle screw plasticating extrusion. Part II: Non-plug flowsolid conveying. Polymer Engineering and Science,31, 1117–1122 (1991).DOI: 10.1002/pen.760311508


560
http://dx.doi.org/10.1002/pen.20248http://dx.doi.org/10.1016/j.powtec.2005.03.001http://dx.doi.org/10.1016/j.ces.2007.03.033http://dx.doi.org/10.1177/0954408911418159http://dx.doi.org/10.1002/pen.760110204http://dx.doi.org/10.1002/app.24658http://dx.doi.org/10.1002/pen.760311508

Documents

Studies on positive conveying in helically channeled single ...1. Introduction Single screw extruders were widely used as one of the basic and convenient elements to melt and con-vey