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ELSEVIER

Journal of Food Engineeri ng 27 (1996) 353-375Copyright 0 1996 Elsevier Science LimitedPrinted in Great Britain. All rights reserved0260-8774/96 15.00 fO.000260-8774(95)00016-X

Modeling of Flow in a Single Screw Extruder*Y. Li & F. Hsieh?

Departments of Agricultural Engineering and Food Science and Human Nutrition,University of Missouri, Columbia, MO 65211, USA

(Received 31 August 1994; revised version received 7 March 1995;accepted 27 March 1995)

ABSTRACTA new analy t ical solut ion of an isothermal, New t onian fl ow in a singlescrew extr uder w i t h a fi ni t e channel i s devel oped w i t h t he act ualboundary condit i ons encountered. Dow n channel vel oci t y distr i buti onsare present ed i n t hree-di mensi onal pl ot s. The boundary condit i ons,velocit y di st ri but i ons, and screw characteri st i cs predi ct ed by t he newsol ut i on are t est ed usi ng t he experi ment al dat a from publ i shed l i t erat ure(Choo et al., 1980, Po l ymer Engineeri ng and Science, 20, 349-56;Grt ff i t h, 1962, I ndust ri al and Engi neeri ng Chemi st ry Fundament al s, 1,180- 7). The resul t s are found t o be more accurat e t han exi sti ngt heori es (Row e11 & Finl ayson, 1922, Engineeri ng, 126, 249-87; Tadmor& Gogos, 1979, Pri ncipl es of Pol ymer Processi ng, W i l ey; Rauw endaal ,1986, Pol ymer Ext rusi on, Hanser).

NOTATIONa,b Factors defined in eqns (7) and (13)Internal barrel diameter (m)The flight width (m)Shape factor of drag flow defined in eqn (19)Fd* Shape factor for overall drag flow defined in eqn (15)FP Shape factor of pressure flow defined in eqn (19)FP Shape factor for pressure flow defined in eqn (14)F,* Shape factor for pressure flow defined in eqn (15)FS The shape factor for drag flow caused by screw root surface*Contribution from the Missouri Agricultural Experiment Station, Journal SeriesNo. 11,988.tTo whom correspondence should be addressed.

353

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Y L i, E H siehdefined in eqn (14)The drag flow factor for the wall effect defined in eqn (14)The maximum channel depth (m)The number of leadsThe dimensionless pressurePressure gradient in the down channel directionPressure gradient in the channel depth directionPressure gradient in the cross channel directionVolumetric flow rate of leakage flow (m3/s)Down channel volumetric flow rate (m3/s)Dimensionless down channel volumetric flow rateInternal barrel radius (m)Screw root radius (m)=Rb-HVelocity of the barrel, screw flights and screw root, respectively(m/s)Component of velocity in X, y and z directions, respectively (m/s)Tangential velocity (m/s)Axial velocity (m/s)Width of channel at the internal radius of barrel(m)=2nRb sin&-eAverage width of channel in the n direction (m)Channel width coordinate (m)Channel depth coordinate (m)Down channel coordinate (m)Viscosity of the Newtonian fluid (Pa s)Helix angle of screwHelix angle of screw at radius RbRotational speed of screw (l/s)

INTRODUCTIONOver the last few decades, extrusion technology has played an increasinglyimportant role in many industries, such as polymer, food, and feedprocessing, and its future is still very promising. Among all kinds ofextruders utilized in industries, single screw extruders are certainly a typethat should not be overlooked. The theories of a single-screw extruder havebeen significantly advanced during the last 40 years. But the attempt ofapplying these theories to screw design and simulation of the extrusionprocess has not been entirely successful (Rauwendaal, 1989). First, this isbecause the existing theories are not accurate enough for the purpose ofscrew design and extrusion simulation. Secondly, these theories are oftenmisapplied to situations where their assumptions are not valid.Using the simplified flow theory as an example, it is valid only for screwswith an infinite channel. Although this theory is based on incorrectboundary conditions, the predictions of the volumetric flow rate will still bereasonably accurate if it is only applied to screws with a channel depth toradius ratio, H / Rb, less than 0.05 (Li & Hsieh, 1994). Due to its simplicity,this theory is commonly used in extrusion analysis, and is often used for

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M odel i ng of Jl ow n a si ngl e scr ew extr ud er 355screws with a finite channel width where its assumptions are violatedseverely (Squires, 1958).As the design and operations of extruders continue to evolve, commercialextruders, especially for those that are used in food and feed processingindustries, are no longer restricted to screws with a very small channel depthto width ratio which are favored by the simplified theory. The flow theoryapplied for screws with a finite channel width is urgently needed. The firstsuch flow theory was published by Rowe11 and Finlayson (1922, 1928). Manyother approaches were attempted later on, but all proved to be almostequivalent to that by Rowe11 and Finlayson (Rauwendaal, 1986).A thorough examination of the boundary conditions used by Rowe11 andFinlayson (1922) reveals that their solution is not a complete one. Twoimportant moving boundaries, the screw flights, were assumed to bestationary. Therefore, their solution does not consider the effect of thescrew flights. Besides, like the simplified theory, the moving surface whichrepresents the barrel is assumed to have the velocity of the screw tip, &a.Plenty of experimental results (Choo et a l ., 1980; McCarthy et al., 1992) haveshown that this assumption is wrong. The actual velocity of this movingsurface is the screw root velocity, R,w. A detailed examination has beengiven elsewhere (Li & Hsieh, 1994).As computer technology advances, solving a complicated analytical ornumerical problem is becoming less difficult. The effort of searching for amore accurate solution should always be encouraged. As a reward, thiseffort will provide us with more concrete foundations for extruder designand extrusion simulation. A complete solution can provide more detailedinformation of the effect of the screw geometry parameters on theperformance of the screw extruders, especially the effect of screw flights,which has not been carefully studied before.Campbell et al. (1992) designed a special screw extruder to test theexisting drag flow analysis. The barrel, screw core, and flights of thisextruder can be rotated separately or simultaneously in pairs. In theirexperiments, they tested the following situations: (i) only the barrel wasrotated, (ii) only the screw core was rotated, (iii) only the screw flights wererotated, (iv) the screw core and flights were rotated together. It wasconcluded that the flow generated by rotation of the screw does not agreewith the existing theory and that screw flights are the major contributor tothe drag flow.The objectives of this study are: (1) to develop a new analytical solutionfor an isothermal, Newtonian flow in a single screw extruder using the actualboundary conditions (i.e. rotating screw and stationary barrel); (2) to derivesimplified formulas and figures of shape factors for both drag and pressureflows; (3) to verify the new solution using the experimental data frompublished literature.

FORMULATION OF THE PROBLEMIn order to simplify the problem, the screw curvature is assumed to be smallso that the barrel surface and screw channels can be unwrapped and becomeflat plates. The following assumptions are commonly used to further simplify

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356 Y Li, E Hsiehthe problem: (1) the flow is laminar, (2) the flow is isothermal, (3) the fluidis Newtonian and incompressible, (4) gravitational forces are negligible, (5)the flow is fully developed, (6) there is no slip at the walls. Therefore, theequation of motion in rectangular coordinates are reduced to:

2 component:

y component:

I (1)+ ( %+ %)

x component: J $ ($+A%)

(2)

(3)where X, y and z are coordinates of channel width, channel depth and downchannel, respectively.Assuming that vY~0, which is a reasonable assumption for screws with asmall channel depth to width ratio, H / W, and using the continuity equation;then, C$,lax z 0, au,/& z 0, and av,/ay z 0. Thus, eqns (l)-(3) become:

$ (z k+%) (4)

ap aG ay* (6)The above simplified equation of motion can be found in Tadmor andGogos (1979) and Rauwendaal (1986). Existing solutions for eqn (4) werebased on the following boundary conditions (Rowe11 & Finlayson, 1922,1928):Barrel:Screw root:Screw flights:

u,(x, H ) =& ,o cos qb,,v&,0)=0&(O,y)=Or&(W,y)=O

This set of boundary conditions is only valid for model extruders withstationary screw and rotating barrel, not applicable for real extruders thatare always operated with a rotating screw and stationary barrel.Experimental results obtained using extruders with a rotating screw haveclearly shown the error predicted by this existing solution (Rowe11 &

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M odel i ng of l ow i n a si ngl e scr ew extr ud er 357Finlayson, 1928; McKelvey, 1953; Middleman, 1977; Griffith, 1962; Choo etal., 1980; Campbell et al ., 1992). The source of the error was elucidated byLi and Hsieh (1994) and a new solution for pure drag flow in extruders witha rotating screw and stationary barrel was provided.In extruders with rotating screw and stationary barrel, the followingboundary conditions should be used to solve eqn (4) for the down channelflow:

Barrel:Screw root:Screw flights:

z&,H)=O,z&(x,O)=R,o cos &,t:(O,y)=(Rs+y)o cos4hr(W,y)=(R,+y)(~ cos+h

Thus, different from the conventional parallel plate model, the screwchannel in this new model is kept moving with its actual velocity and thebarrel surface is stationary. This set of boundary conditions are definedbased on the absolute velocities of the rotating screw. The velocity profilesbased on this set of boundary conditions can be integrated directly to obtainthe net volumetric flow rate.Notice that the flight velocities are not zero but a function of channeldepth. In existing solutions (Rowe11 & Finlayson, 1922, 1928; Tadmor &Gogos, 1979; Rauwendaal, 1986) the flight velocities were all set to zero byassuming the screw was stationary. Since the screw is rotating, the velocitiesfrom the screw center to the tip are not constant but a function of theradius.Since eqn (4) is a nonhomogeneous, second-order partial differentialequation with nonhomogeneous boundary conditions, it needs to beseparated into two parts with two sets of boundary conditions. The finitesine transformation method can then be used to transform the problem intoordinary differential equations, and solve z;,(x,y). This is shown in theAppendix. The final result is

1 aP~z=Rstr ,cos(bhfvl +(2Rh-H )wcos~b~l +~ z.(aW2f,3+hH 2f,4) (7)where

sin $$ sinh in(H-Y)W

71 =l .3,5, . . . i i7cHsinh - Wi71y in(W-x) i7cxsin - sinh

f+2 5 H H + sinh H71 r= l .3,.5.... i i7 rWsin h - H

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358 I: Li, l Hsiehim rsinh i7c(H-y) sin - W + sinh 5 -1

inHsinh - Wi7T(W-x) inxsinh H + sinh H -1i71Wsinh - H

The constants a and b in eqn (7) are dependent on the H/W ratio. When theH/W ratio is small, a=0 and b=l; when the H/W ratio is large, a=1 andb=O.For the cross channel flow, u,, eqn (6) can be integrated twice to yield:aP y*c,=--+c,y+c*ax 2~ (8)

The constants cl and c2 are solved with the following boundary conditions:u,(H)=&uJO)= -R,w sin 4b

Then, the equation for U, becomes:aP (y2-Hy) H-Yv,=- ax 2/l - R,w sin &, - H (9)

In eqn (9) the pressure gradient aP/ax is usually not known. In order toevaluate this pressure gradient, v, can be integrated over the channel depthH: s H3 aP R,w sin&HUxdy= _- --0 12~ ax 2 Q leakage 10)or

ap lb Q R,o sin &,H-= _-ax H3 leakage+ 2 1 11)Clearly, aPli3x depends not only on the screw geometry and viscosity, butalso the clearance between the screw tip and barrel. Substituting eqn (11)

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M odel i ng of l ow i n a si ngl e scr ew extr ud erback into eqn (9) the cross channel velocity IJ,, becomes

359

Rsw sin 4bW -y>vx=H

+ 6QleakageW 91H 3

(12)If the screw geometry is provided and the leakage flow Qlrakage known, thevelocity profile v, can be determined. Unfortunately, Qleakage is difficult toevaluate. Equations for calculating Qleakage are available in the literature byRauwendaal (1986, 1988) as well as Tadmor and Gogos (1979); however,these equations have not been verified by experimental data.

FLOW RATE CHARACTERISTICSFor extruder design and extrusion simulation, the flow rate-pressurerelationship is of great interest. By integrating the down channel velocity,CX(~Y) [eqn ( 7) 1,ver the cross-sectional area of the channel, the overallflow rate can be obtained:Qz=R,cl , cosq4,W2fc,, +(2Rh-H )w cos& H 2fQ2+$ $ WHW4fc,3-12

b i3P H3W+- -P az H4fQ4- 12where

71-j-135 I-_;, ~

_ sinh - w .i71Wcash - - 1H Ii7cWsinh - H I

i nH

fQ =? f 11;72 i=l 3 .5...i7 rWcash - - 1H Ii7cWsinh - H I

(13)

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360 X Li, l? HsiehWhen H/W< 1, a common case for single-screw extruders, the coefficient

f a2, will become 0.271377272 and fQ4 converges to 0.05252075 (Rowe11 &Finlayson, 1928). Therefore, by setting a=0 and b=l and rearranging eqn(13), the flow rate can be simplified to:R; i3PQz=R;co(Fs+Fw) -- -F,IJ az (14)

whereF = (1 -H/R,) sin2&,

s nt

For better accuracy, fpl is evaluated using the average radius, orircH

whereW,= 7C(2Rb H) tan &, -e cos&,nt )

An equation suggested by Rowe11 and Finlayson (1928) can be used toestimate fnI as well:&* =0*5 (;)-0.3151(;)

Detailed analyses of F, and F, can be found in Li and Hsieh (1994). Forthe case of H/W> 1, it is convenient to set a=1 and b=O in eqn (13) andthe factors, &I, fez and&,, can be calculated in a similar manner.By defining the dimensionless flow rate Qzand pressure P, as:

Q;= 1aP H=RbW COS4bwH and P,=- -p R,,co COS 4bthe equation for dimensionless screw characteristics can be obtained:

QL=F -F;Pz (15)

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Modeling of flow in a single screw atruderwhere

361

Fd*= 1 HIR, 27ctan& e @271377272H/R, 2 -H/R,)HI&, Q -R l+b1 2ntan& e-- cos&& Rh >27ctan& e-- cos&,& Rh >

The shape factor, F 2, in eqn (15) indicates the maximum dimensionlessflow rate, or open discharge flow rate for any given screw. It is plotted inFig. 1 as a function of H/R, ratio at various helix angles &,. As the channeldepth to radius ratio, H/R,, increases, this maximum flow rate decreases ina slightly nonlinear manner. For a small helix angle, $h, a higherdimensionless flow rate will be expected at the same H/R, ratio. But thishelix angle effect will diminish quickly as the helix angle increases, especiallyfor a H/R, ratio less than 0.2. The shape factor, F,*, in eqn (15) is the slopeof the dimensionless screw curves and an indication of pressure flow effect.A similar plot for Fp* is given in Fig. 2. The relationship between Fp* andH/R, is linear; Fp* decreases with increases in H/R,. The effect of helixangle, &,, on Fp* is opposite to that on F$. For a smaller helix angle, &,F,* decreases faster with increases in H/R, than for a larger helix angle &.Using Figs 1 and 2, the screw characteristics can be constructed. If theH/R, ratio and helix angle, $h, are given, the corresponding maximum flowrate, F$ and the slope of the screw curve, Fp*, can be located from Figs 1and 2, respectively. A corresponding screw characteristic can then beplotted. The intercept of the curve with the x axis, or the ratio of

Fd

0.5100.4590.4080.3570.3060.2550.2040.1530.102

0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0H/ R,

Fig. 1. Drag flow factor, F$, as a function of H/R, for overall drag flow at varioushelix angles & (e/R,,=O.l, n,=l).

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362 2: L i , E HsiehF F,*, i s the maximum dimensionless pressure, defined in eqn (15). Fourscrew curves are shown in Fig. 3. For a very small H / Rb ratio (O-05) theeffect of the helix angle is trivial. But for a large H / R, ratio (O-.5), the effectof the helix angle &,, is significant. On the other hand, no matter what thehelix angle is, the effect of H/R, is always considerable.

DOWN CHANNEL VELOCITY PROFILESThe velocity profile, v,, as a function of the channel depth, y [i.e. u,=v, (y)],in the screw channel of single screw extruders has been discussed extensively

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0H/R,

Fig. 2. Pressure flow factor, Fp*,as a function of H/R, at various helix angles &,(e/R,=O.l, n,=l).0.5

2c 0.4tozg 0.3

i;ti6 0.2E

f; 0.1

z0.0

- &= lo , H/R,= 0.05

..----- 60. H/R,= 0.5

0 1 2 3 4 5 6 7 6Dlmenrionless Pressure. P, or Fi/Fl

Fig. 3. Dimensionless screw characteristics predicted by eqn (15).

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M odeli ng offlow in a single screw extruder 6in published literature (Tadmor & Gogos, 1979; Rauwendaal, 1986, 1988).However, little information has been given on the effect of screw flights.From this analytical solution [eqn (7)], the dependency of the down channelvelocity component v, on the cross channel position, x, and channel depthposition, y (~=v&,y)) can be analyzed. This provides information thatcannot be revealed by one-dimensional solutions.If the leakage flow is ignored, the flow in a screw channel inside a singlescrew extruder becomes a combination of drag and pressure flow. Usually,the pressure flow is caused by the restriction of the die mounted at the endof an extruder. If there is no die attached to the extruder, the back pressureflow will vanish. The drag flow is introduced by the movement of the screw,or specifically, the screw root surface and the screw flights.In the conventional studies, only the drag flow caused by the screw rootsurface is investigated. The argument for ignoring the flight effect is basedon the assumption of an infinite channel width. But in practical situations,screws with an infinite channel width do not exist. Therefore, investigationof the flight effect is imperative. The analysis given here can provide usinformation on under what conditions a screw channel can be treated asinfinite. Figure 4 is the pure drag velocity distribution introduced by thescrew root surface itself, the first term on the right-hand side of eqn (7).Figure 5 is the velocity distribution introduced by the flight surfaces only,the second term on the right-hand side of eqn (7). The overall drag flow isa combination of these two contributions, and is presented in Fig. 6. At theboundaries (screw root and screw flight surfaces), the velocities satisfy no-slip conditions. In the cross channel direction (x direction), the velocity isvery much affected by the movement of the flights. At any given crosschannel position x, a convex velocity profile is seen except at x=0 and x= W,where the velocity profiles are linear. This phenomenon was also reportedby Choo et al. (1980). This effect is caused by the drag effect of the flights.This is why it was not revealed by the simplified and Rowe11 and Finlaysonssolutions (1922, 1928) in which this effect was ignored. In Rowe11 andFinlaysons solution, the velocities at the flights are set to be zero implicitly.Therefore, their velocity profile is similar to that shown in Fig. 4.Figure 7 shows the corresponding velocity distribution of combined drag,the first two terms on the right-hand side of eqn (7) and pressure flow, thethird term on the right-hand side of eqn (7). Due to the influence ofpressure flow, the convex profile changes to concave shape in the centralpart of the screw channel. Consequently, the overall flow rate is reduced.From these three-dimensional plots, it is evident that, as long as the channelwidth is finite, the velocity distribution along the cross channel direction isnot uniform, and the maximum velocities are located at the flightboundaries. Thus, the simplified theory and the exact solution by Rowe11and Finlayson (1922, 1928) Tadmor and Gogos (1979) and Rauwendaal(1986) cannot provide sufficient accuracy when the flight effect is significant.

COMPARISON WITH PUBLISHED EXPERIMENTAL RESULTSThe predicted tangential and axial velocities, v(-, and v,, are compared withthe published data in Figs 8-11. The equations used to calculate vg and cZ3

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364 IT Li, E Hsiehare as follows:

uO=Rs~ - [uz cos & - V, sin &] (16)u, = u, sin & + v, cos q& (17)Since the clearance of the screw used by Choo et al. (1980) was over four-fold the standard clearance (0.001 I&), the leakage flow would have beenconsiderable (Rauwendaal, 1992). However, neither the cross channel

1.00.8

Fig. 4. Velocity distribution of a drag flow by screw root surface.

2. 0

0. 5

0.0

Fig. 5. Velocity distribution of a drag flow by screw flights.

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M odel i ng of l ow n a si ngle screw ext ruder

Fig. 6. Overall velocity distribution of a drag flow.

365

Fig. 7. Overall velocity distribution of a combined drag and pressure flow

pressure gradient, i3PBx, not the leakage flow, Qleakage is available (Choo etal., 1980). In order to estimate oX,Qleakagr=(6/H)R,o sin &Wr,H was used asthe first guess, which turned out to be close to the final values used. At eachgiven condition, the estimated value was adjusted to fit one of the velocityprofiles, such as vg, obtained experimentally by Choo et al . (1980). The sameakage was used to predict the axial velocity profile (v,) in eqn (17). Forthree out of four cases with dimensionless pressure, P, ~3.33 (Choo et al.,

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366 Y. Li, E HsiehV, (mm/s)

4 6 6 10 12

- 0.66 8 10 12 14 16

. V,-Choos data- Ve-Predlctrd

-.-. V#-PredIcted

6 8 10Va (mm/s>

16

Fig. 9. Comparison of the predicted vo and v, and the experimental results byChoo et al. (1980) for corn syrup (P,=1.25, H/Rb=0.505, ,=45.2, Rb=19 mm,H=9.6 mm, W=54*9 mm, q=l, e=11.9 mm).

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368 Y Li , E H sieh

L 0.60: 0.50L 0.4r:r 0.30z2 0.2Eii 0.1

. . . .. _\ . . . . ...~~~.~~ he slmpllflrd theory- - Rowell and Flnlayrons theory

.: ,.... - This analytlcal rolutlonChoos axperlmrntal data

0.0 -2 -1 0 1 2 3 4 5 6 7Dimensionless Pressure

Fig. 12. Comparison of the predicted screw characteristics by this solution, thesimplified theory, Rowe11 and Finlaysons solution and the experimental data byChoo et al . (1980) for corn syrup (H / & ,=0.505, &,=45.2, I&=19 mm, H=9.6 mm,W=54.9 mm, n,=l, e=ll.9 mm).

0.5 I I I / I_q. ....... The slmpllfled theory

2 - - Rowell and Flnlaysons theoryl.E 0.4 - Thls analytlcal solutlon: Grlfflths experlmental dataz 0.3:0$ 0.2si:E 0.1

0.0 0 1 2 3 4 5 6 7Dimensionless Pressure

Fig. 13. Comparison of the predicted characteristics by this solution, the simplifiedtheory, Rowe11 and Finlaysons solution and the experimental data by Griffith (1962)for corn syrup (H&,=O~128, q&=30, Rr,=25.4 mm, H=3.25 mm, W=69.19 mm,n,=l, e=6.35 mm).

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370 Y Li , F H si ehREFERENCES

Campbell, G. A., Sweeney, P. A. & Felton, J. N. (1992). Experimental investigationof the drag flow assumption in extruder analysis. Pol ymer Engineeri ng and Science,32. 1765-70.Chad, K. P., Neelakantan, N. R. & Pittman, J. F. T. (1980). Experimental deep-channel velocity profiles and operating characteristics for a single-screw extruder.Pol ymer Engineeri ng and Science, 20,349-56.Griffith, R. M. (1962). Fully developed flow in screw extruders. I ndust ri al andEngi neeri ng Chemi st ry Fundament al s, 1, 180-7.Li, Y. & Hsieh, F. (1994). New melt conveying models for a single screw extruder.Jour nal of Food Process Engineeri ng, 17, 299-324.McCarthy, K. L., Kauten, R. J. & Agemura, C. K. (1992). Application of NMRimaging to the study of velocity profiles during extrusion processing. Trends i nFood Science and Technol ogy , 3,21519.McKelvey, J. M. (1953). Experimental studies of melt extrusion. I ndust ri al andEngineeri ng Chemi st ry , 45,982-6.Middleman, S. (1977). Fundamenta l s of Po l ymer Processi ng. McGraw-Hill, NewYork.Rauwendaal, C. (1986). Pot )mer Ext rusion. Hanser Publishers, New York.Rauwendaal, C. (1988). Leakage flow of an isothermal power law fluid. Advances inPol ymer Techno l ogy, 8,289-316.Rauwendaal, C. (1989). The ABC of extruder screw design. Advances i n Pol ymerTechnol ogy, 9,301-8.Rowell, H. S. & Finlayson, D. (1922). Screw viscosity pumps. Engineering, 114,606-7.Rowell, H. S. & Finlayson, D. (1928). Screw viscosity pumps. Engineering, 126,249-387.Squires, P. H. (1958). Screw-extruder pumping efficiency. SPE Jour nal , 14, 24-30.Tadmor, Z. & Gogos, C. G. (1979). Pri ncipl es of Pol ymer Processi ng. John Wiley,New York.

APPENDIXa2uz a2vz 1 aP-+-- -ax2 ay -; az

Based on eqns 4) and (5), it is known that the pressure gradient aPb3.z iseither a constant or only a function of z. The unwrapped screw and barrelsystem, the coordinates defined in this study, and the absolute velocities ofthe barrel, screw root and screw flights are shown in Fig. Al. The actualboundary conditions for this problem are defined based on the absolutevelocities:

n* x, H) =fi (x) = 0, o

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M odel i ng of l ow n a si ngl e screw e& ruder 371

Fig. Al. Schematic representation of an unwrapped screw and barrel system.

This is a nonhomogeneous partial differential equation withnonhomogeneous boundary conditions. This problem can be solved byseparating it into two problems with homogeneous boundary conditions.Assuming v,(x,y)=u,,(x,y) + u&,y) and separating the boundarycondition into two sets, then the problem becomes:

a*~,, a , a aP-+-=- -ax2 ay2 P azThe first set of boundary conditions are:

1;Z,(x,H)=f,(x), o

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372 Y Li , F HsiehFirst, vZ1(x,y) is solved using the first set of boundary conditions. Thesolution of vZl(x,y) is as assumed

k(4y)= 5 My) sin(b)i=lIn order to satisfy the boundary conditions for x=0 and x=W, let k=i TC Wand define the following:

For a non-zero solution, i= 1,3,5,. . . Therefore,

Now the nonhomogeneous PDE has been transformed into anonhomogeneous ODE with the form

The boundary conditions become:

b.(H)=~suZl x,H) sin0 (. >; &CO

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M odel i ng of l ow i n a si ngl e scr ew extr ud er 373Solving the above ODE, the result is:

sinh ia-Y 1bib)= 4R,o cos & W a aP 4w2+- --i7c p dz (in )

sinh in(H-y) .W + sinh $ -1i nHsinh -WTherefore, the general solution of vzI(x,y) can be presented as:

Vzl (qJ)= i:i=1,3,5...

4R,w cos &i71

a 8P 4W2sinh wf-Y) +sinh iXY

W W+- --p CL z in ) -1i xHsinh - W

i7cxsin - W

The method used for solving u&x,y) is the same and the second set ofboundary conditions were used. The final solution for ~~~(11,~) s presentedas:i

4 Gy>= i: 4&o co@, + b CIP 4H2;=1 3 5... in p dz (i7r)

sinh (s) + sinh ( i~Ux) b ap 4H2--

i7cWsinh -( )

-1 82 (in )3H

i7tysin -( 1where

R,= & +K--zR~_~2

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374 Y Li , E HsiehThe volumetric flow rate for the overall drag flow is obtained byintegrating the overall velocity distribution over the cross channel area ofthe screw:

w H WHQz=n, ss Qx,Y) dy dX=nt ss Q,I(X ,Y) dy d.x0 0 0 0WH

n t ss & ,Y> dy h=Qs+Qw+Q,0 0Substituting the expression for R, and W and rearranging the equation,the final results can be presented as:

i sH IW

Qs=ntR,~W2cos&, inH,w,:27cRb tan&, 2=nt(Rb -H)o -e cos3 fjbnt

whereF

)2

s

where

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M odel in g of l ow i n a si ngl e scr ew extr ud eri xH

+LP fI_( a.2 r=1,3,5

i rcW16H4 cash - - 1H 8H W_p(i7c) si nh i 7cW (i7c)*

H

375