Wall Slip and Its Implications in the Design of Single Screw Melt-Fed Extruders*

R. A. WORTH** and J. PARNABY

Schools of Mechanical and Manufacturing Systems Engineering University of Bradford

Bradford West Yorkshire BD7 1 DP, England

and H. A. A. HELMY***

Bone Cravens Ltd. Wembley, England

The phenomenon of wall slip during the capillary flow of polymer melt is investigated for low and high density polyethylene. It is found that wall slip occurs in both cases, and that the effect is related to melt fracture. In addition, it is shown that a silicone fluid exhibits wall slip. The performance of thc metering zone of a 38 mm diameter single-screw extruder is discussed in relation to wall slip. It is suggested that the power consumption of the extruder is reduced as a result of slip at the polymer/metal interface. Results based on experiments with the silicone fluid tend to support this hypothesis. A theoretical analysis of the effects of wall slip upon throughput rate and power consumption for a one-dimensional isothermal Newto- nian case is included.

INTRODUCTION olymer melt constitutive equations are microscopic P in that they define the stress-deformation relation-

ship at every point in the fluid. Usually it is only possible to measure the macroscopic variables. For instance, it is easier to measure the volume flow rate than to measure the velocity components at every point in the fluid. Therefore, it is necessary to relate macroscopically measurable variables to the fundamental microscopic properties of the material.

Capillary flow experiments are widely used for inves- tigating polymer melt flow characteristics. In order to derive quantitative relationships, several assumptions must be made, which are not always accurate for poly- mer melt flow, but which allow a starting point to ba obtained. The basic assumptions normally made are: (a) the flow is laminar, (t)) the flow is time independent, (c) the fluid is incompressible, (d) the flow is isothermal, (e) the flow is constant all along the tube, and (f) the velocity of the fluid at the capillary wall is zero (i.e., there is no slip at the wall).

In the discussions which follow it will be assumed that (a) to (d) are correct, particularly for small diameter, shallow channel screws which generally show isother- mal behavior. I t is accepted that the flow is not constant along the tube due to elasticity and viscosity (Carley (1)) effects which must be considered when analyzing capil-

* Based upon a paper read at the British Society of Hheology/Pla\tic\ m d H u h b e r lnstitute Conferenc? on Polymer Rheulogy arid Plastic5 Proce\aing, Univrrrtty of Loughhorough, September, 1975. ** Pre\cnt Address: hlechnnical Engineering Departmrnt, Uriivrrrity of the We\t India, Trinidad. *** Formerly a Research Student, University of Bradford

lary flow data. Such data may becorrected using the well known method due to Bagley (2), which involves the use of two or more capillaries of different LID ratios. The effect of pressure on viscosity for a given shear rate and fixed LID ratio will here be assumed constant for con- stant capillary pressure drop (l), as will the effect of compressibility for materials of high compressibility modulus such as plastic melts (circa loy N/m2). The remaining assumption, that the velocity of the fluid at the wall is zero, must be checked experimentally.

In the solution of the governing equations for flow in the extruder channel, the assumption of no slip at the screw and barrel boundaries is always made in order to sirnpl8y the anal!,sis by introducing ;i known 1,oundary condition. However, it has been found that this assnmp- tion is open to some doubt (hlaxwell and Galt ( 3 ) ) . The results of Ui (4) and Kennaway (5) together with the results presented in this paper show that the assumption is invalid and, strictly speaking, should not be used. Nevertheless, if wall slip is to he taken into consideru- tion in the screw channel governing equations, it makes the solution more difficult.

This is demonstrated by the limited one-dimensional analysis given in Appendix 1, where the effects of wall slip are taken into account in performance predictions. The effective use of such analyses requires the mea- surement of wall slip data for many important industrial polymers as described lielow.

In this paper, a procedure for investigating the no slip condition using a constant-pressure gas viscometcr will be described, and the likely effect of wall slip on screw extruder peifoririance will be discussed, Experi-

POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4 257

R. A. Worth, J . Pnrnaby, and H . A. A. Helrny

mental results, based on polymer melts and silicone fluid, are presented. Bagley plots for capillary experiments (6) gave good straight lines suggesting that viscous pressure gradient was independent of position and giving additional reasonable confidence in assumptions (c), (d), and (e) above. N o method is available to fully correct for entrylexit-diameter, pressure-viscosity and viscous heating effects when the Deborah number varies over a cross section (7) . In this work, the residue of such effects was reasonably assessed as being negligible in the light of the remarks above.

THE NO-SLIP CONDITION The Determination of Wall Velocities

The analysis by Lupton and Regester (8) is usefril in describing the capillary flow of polymer melts, where wall slip occurs. It is found that the apparent wall shear rate (4Q/n-R3) in the case of slip may be written as

which is of the form y = mr + c . A plot of apparent wall shear rate against 1lR yields a

straight line with a slope equal to four times the slip velocity, and a y-intercept equal to the apparent shear rate at the wall, corrected for slip. A horizontal line represents the no-slip condition. The form of the graph is shown in Fig. 1 .

A procedure for making use ofEq 1 to determine the slip velocity, risingaconstant pressure gas viscometcr, is as follows:

(a) Select a series of capillary dies, of different diame- ters D,, D,, . . . Di, . . . D,, but of the same LID ratio. ratio .

(b) Extrude polymer melt through each die in turn, with the same reservoir pressure, Ap, and measure the volumetric output rate through each die, (I,, Q2, . . .

(c) Calculate the steady-state shear stress at the wall, Qi, . . Pi,.

which is the same for each die, and is given by

m E e \ i) G

No s l i p

m E e \ i) G

No s l i p

l / F ? F i g . 1. Determirrcitiori of i 1 ~ 1 1 1 p l i p oelocity.

258

Note that this expression is approximate, as entrance pressure drop effects have not been taken into account. However, the correction will be approximately the same for each die due to the constancy of LID.

(d) Calculate the apparent shear rate for each die, given b y

(e) Draw a graph of .jcli vs 2I0,. The graph should be a straight line with slope equal to 4es. Hence, the slip velocity may be determined for a particular wall shear stress, T ~ ~ .

(if) Steps (11) to (e) may be repeated using different reservoir pressures in order to examine the relationship between slip velocity and shear stress.

Experimental Results for low and High Density Polyethylenes

The low density polyethylene used in the investiga- tion was Alkathene XDG 33 (I.C.I. Ltd.) and the high density material was Rigidex T9 (B. P. ). The uncorrected flow curves for these materials, obtained from a die with D = 0.508 mm and LID = 12, are shown in Figs. 2 and 3, respectively, which also illustrate the onset of melt frac- ture. As melt fracture is often associated with wall slip (Pearson and Petrie (9)), it was decided to investigate the no-slip condition at shear stresses in the region of the critical shear stress for melt fracture.

Three dies with diameters of 0.508 ( t 2 percent), 0.794, and 1.59 inrn and LID = 12 were used to obtain graphs of apparent shear rate vs UR, and these graphs are shown in Figs. 4 and 5 for low and high density polyethylenes, respectively (Worth (6)). It can be seen that the experimental points for a given shear stress fall

3 P 0 ri

M 0

r-l

5.E

5.4

5.2

5.0

4.8

4.6

4.4

4.2 0.4 1.2 2.0 2.8 3.6 4.0

l og (4Q/.rrR3 1 Melt f r ac tu re

Fig. 2. Uticorrected ,flow curve f o r Alkathene XDC33 (200C).

POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4

Wall S l i p und Its lmplicutions in the Design of Single Screw Melt-Fed Extruder.$

1.4 1.2 2.0 2.8 3.6 4.0

log1 0 ( 4Q/.rrR3 1

*Melt f rac ture Fig. 3. Uncorrectedjow curve f o r Rigidex T9 (200'C).

0 1 2 3 4 5

1/R r m - l Fig. 4 . Determination of slip velocity f o r Alkuthene XDG33 (200C).

very nearly on a straight line, which is predicted b y E q 1 . From Figs. 4 and 5, curves of slip velocity vs shear stress was deduced. These are illustrated in Figs. 6 and 7 . It can be seen froin Fig. 6 that for the low density material at low shear stresses the wall slip velocity is very small. However, at higher shear stresses (above 0.8 x 10' Nini') the wall slip velocity becomes appreciable. In the case ofthe high density polyethylene (Fig. 7 ) , the wall slip velocity is again found to be sinall at low shear stresses. At a shear stress of around 2 x 10' N/in', there is a sudden increase in the wall slip velocity, and the slip velocity becomes very considerable. Comparing Figs. 3

x 105 N/m2

0 1 2 3 4 5

1 / R rn-l Fig. 5 . Determination of slip oelocity f o r Rigidex T9 (200'C).

0.4

0.3

0.1

0.4 0.6 0.8 1.0 1.2 1.4

F i g . 6. S l i p oelocity cs sheur stre.rsfor Alkatlzene XDG33.

and 7, it is evident that the discontinuity in the uncor- rected shear stress vs shear rate curve corresponds to the discontinuity in the curve of slip velocity vs shear stress, as would be expected. For hoth polymers, the onset of melt fracture occurs at shear stresses corre- sponding to an increase in w d l slip.

It is possible that iri the case of low density polyethylene the apparent inanifestation of wall slip is not true slip but is a viscons heating effect. In the film of polymer adjacent to the capillary wall, where the shear rates are high, there may he significant viscous heating. Hence, the viscosity of the inelt would be reduced i n

POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4 259

R. A. Worth,]. Parnuby, and H . A. A. Helrny

0 0.5 1.0 1.5 2.0 2.5

T x l o 5 N/m2 w

Fig. 7 . S l i p velocity cs shear stress f o r Rigidex T9.

this region, with a corresponding increase in shear rate, which could be interpreted as wall slip. However, in the case of the high density material, the slip velocities are so great that it is unlikely that the phenomenon can be explained solely in terms of viscous heating. It appears that the shear stress at the wall exceeds the critical shear stress for breakdown of cohesion between the polymer and the boundary, although the exact nature of this mechanism is not clear (7).

The difference in behavior between low density and high density polyethylene is emphasized by the fact that at shear rates much greater than the critical shear rate, high density polyethylene extrudate becomes smooth again (Dennison (10)). Most common polymers, such as low density polyethylene, polypropylene, and polysty- rene, do not show this effect. The super extrusion region has been utilized for very high speed processing of high density polyethylene.

Silicone Fluid Experiments The silicone fluid used in the experiments described

in this paper was manufactured by Dow Corning and had a nominal viscosity of 100 poise. Three dies, with diameters of 0.69, 1.07 and 1.68 mm, and LID = 140, were used to obtain graphs of apparent shear rate vs 1IR at various shear stresses, and these graphs are shown in Fig. 8. The large LID ratio was required because of the comparatively low viscosity of the silicone fluid. Conven- tional short dies are impractical as the small pressure drop through the dies cannot be measured accurately at the shear rates of interest. The dies were manufactured from small-bore stainless steel tubes ofthe type used in hypodermic syringes. It was not possible to observe any signs of melt fracture.

It can be seen from Fig, 8 that wall slip occurs at all the shear stresses considered since the slopes of the graphs are non-zero in all cases. The magnitude of wall slip increases with increasing shear stress, from approxi- mately 0.5 cmisec at a shear stress of 4.9 x lo3 NIm2 to 5.8 cmIsec at a shear stress of 14.6 x lo3 N/m2. A graph of slip velocity versus wall shear stress is shown inFig. 9.

It should be noted that the 2 percent error in capilliary radius is equivalent to 6 percent on shear rate, which is reasonably insignificant in relation to H D P E and silicone fluid.

THE EFFECT OF WALL SLIP ON EXTRUDER PERFORMANCE

In this section of the paper, the effect of wall slip on screw power consumption is discussed. Since the power consumption is calculated from the rate of shear at the

1200

1000 4

CJ

F

a * 800 A

W

2 W

400

200

0 1.0 2.0 3.0

I R nm-1 Fig. 8 . Deterniinution of slip velocity f o r silicone fluid (20C).

0 a \ ffl4

6

0

Fig. 9. Slip celocity us shear stress f o r silicone j u i d .

260 POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4

Wall Slip and I t s Implications in the Design of Single Screw Melt-Fed Extruders

barrel/melt interface, it is expected to be directly af- fected by wall slip. Calculations will tend to predict higher power consumption than is actually the case in practice. Appendix 1 indicates for a simplified model how the effect of wall slip upon power consumption might be anticipated, and order of magnitude calcula- tions are made.

In the work reported in this paper, screw power con- sumption was both calculated and measured for an ex- perimental 38 mm single screw extruder pumping silicone fluid. Although this polymer is not directly re- lated to practical extrusion, it was chosen for the follow- ing reasons:

I t has a high chemical stability. This enabled the polymer to be recirculated several times through the extruder without any significant change in properties, so that only a few kilograms of the fluid were required for the experimental program.

0 The viscosity of the polymer (1000 poise) was cho- sen to allow adequate gravity feeding through the feed hopper.

0 The fluid was sufficiently non-Newtonian (n = 0.85) to enable the theoretical solutions for the non- Newtonian flow solution to be investigated.

Due to the relatively low viscosity of the polymer, viscous heating effects were negligible, allowing the isothermal model to be applied.

The dimensions of the screw used to pump the silicone fluid and of the circular orifice die are as follows:

Screw Diameter = Root diameter = Flight width, e Helix angle, 0 = Pitch - Channel Depth, H - Length, 1 = Flight clearance, 6 -

Diameter = Die length = Entry length = Entry angle =

- -

-

-

-

Die

37.90 mm 33.65 mm

2.54 mm 17.65 degrees 37.90 mm

2.125 mm 365.00 mm

0.0632 mm

4.76 mm 62.50 mm 25.00 mm 30.00 degrees

The experimental extruder was equipped with a strain gauge torque transducer to measure the screw torque. An electronic device for measuring the screw speed was also available. The screw power consumption was ob- tained as the product of the screw speed multiplied by the torque.

A variable-speed drive facilitated infinitely variable screw speed. The extruder output was adjustable by means of a restrictor at the die on the end of the ex- truder. Flow rates were measured by weighing the ex- trudate over five min time intervals.

Theoretical Power Calculations

The mechanical power consumption in an extruder may be calcnlated as the rate of working at the melt barrel interface in a fixed screw moving barrel analogue.

e + 7 s y I y = ~ tan0)) i- ~fVb2 s (4)

Now, considering the power consumption in the screw channel only, i.e., the first term in E 9 4 (see Appendix 11,

Dimensionless power per unit channel length is defined as :

The above equation describes the ratio of the actual power to that consumed in uniform simple shear flow with the same shear rateV,,JH, and average shear stress, 7 = fV,,/H.

It is apparent from E 9 5, in order to compute values for the screw power consumption, that values for the shear stresses at the barrel melt interface are required. These wall shear stresses are easily calculated for a New- tonian fluid where the two velocity profiles in the yz and xy planes are independent. For the more realistic case of a non-Newtonian fluid, however, the velocity profiles in the yz andry planes are interdependent, hence calculat- ing the shear stress at the barrel wall involves solving the two-dimensional momentum equation coupled with the fluid constitutive model (e.g., power law) in two dimen- sions. This renders the solution very complicated.

The one-dimensional isothermal solution was used here to calculate the screw channel power consumption in the yz plane. The dimensionless screw power coil- sumption for the isothermal non-Newtonian, one- dimensional power law case is written a s

( 7 )

where tk is a dimensionless position of the plane of zero shear stress in the velocity profile and the z sign repre- sents the two cases of predominant drag or pressure flows (Helmy (ll), Helmy and Parnaby (12)). Appendix 1 examines fiirther the effects ofwall slip in these cases.

The contribution of the flow in the xy plane to the screw channel power consumption was included here by using a simple semi-empirical method. This method was suggested by Fenner's results (13), where the plots of the dimensionless power against dimensionless output G from the one- and two-dimensional isothermal solu- tion produce two remarkably parallel curves, as shown in Fig. 10. This feature stimulated the idea that the addition of a constant factor to vertically shift the isothermal one-dimensional curve to coincide with the two-dimensional curve would be reasonable for design

-

E = (Tp,)" (1 f 5s)

POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4 26 1

R . A. Worth,]. P n m t i b y , und H . A. A. Helmy

purposes. The constant factor suggested is based on the Newtonian isothermal transverse-flow power compo- nent and is expressed by the equation

E~ = 4n tan% (8 )

where q i s the dimensionless power contribution due to the channel transverse flow. Figure 10 shows adjust- ments made using E q 8 to derive superimposed curves C and D simply from the one-dimensional theory of curves A and B.

Hence, a semi-analytical expression for the screw channel power consumption in the two dimensions would be (curves C and D , Fig. 10)

E = (TJ (1 f ch.) + 4n tan% (9) When the dimensionless power for a given flow rate was compared with that calculated by the accurate numeri- cal solution, the superimposition effect was undetecta- ble in Fig. 10.

Experimental Results

Figure 11 shows the power consumption plotted against dimensionless flow rates for the silicone polymer at different screw speeds. For each experiment, two points for the dimensionless flow rate G corresponding to leakage ( L ) and no leakage ( N L ) are plotted. The theoretical points calculated from Eys 9 and 6 for the no leakage case and from Eqs 9 and 6 plus the term qfV2 el6 for the leakage case are also plotted. Here the theoreti- cal points in the figures are joined by straight lines so as not to imply continuous relationships. The degree of isothermality is also indicated on the figures by the viscous heating factor *rH.

The results in Fig. 12 show that at slow speeds (30 r.p. in. ) the experimental points lie between the leakage

I l i r n e n s i o n l P s s floi.: r a t e G

Fig. 10. Relationships between dimensionless flow rute G rind isothermd one ~72d two diinen.vionml I)o"erconsumption E. {A,n = 0.5, B,tt = 1 .0 ) 1 dimensional; ( C , n = 0.5, D,n=1.0} 2 dimc,nsional. { C , n = 0.5, D,n = 1 .O} 1 dimensional w i th empiri- cril correction to giue scii)"rinip[).sition, i .e., udjusted curce,s A cind B.

262

and no-leakage theoretical solutions. At higher speed (50 r.p.m.), however, the experimental results lie below the theoretical no leakage predictions. At still higher speed (80 r.p. m.), greater deviation from the theoretical

90

eo

70

b l

Y Y

3

53

Y

4 g 40 ci

3 L

30

iC

10

C

'. ,

0.1 0.2 0.3 G

Fig. 1 1 . Power consumption P (wat t s ) against dimensionless f l o w ra te , C, f o r si l icone p u m p i n g .screw a t d i f f e ren t r.p.m.--- N o leakage, - - - - Leukage: 1 dimens ional isothermal empirically adjusted model. N L +--x L n, = 0.0841, 80 r.p.m.; N L 0- 0 L x,, = 0.052, 5 0 r.p.m.; LVLfl- 0 L rII = 0.032, 30 r.p.m.: experimental points.

H

r i t

V I V )

Fig. 12ja). Velocity profiles und wall slip. Drag pow.

POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4

Wall Slip and I t s Implications in the Design of Single Screw Melt-Fed Extruders

y I Vb _IcI

, . / / / / / / / / A , - / / /

vh

t

v ( y 1

* Y Y c r i t

I v I y 1

V 1 v2 Fig. 12(b). Velocity profiles und wu11 ,slip. Drug and pressure $OW, apiaz 1 u. prediction is observed and a discrepancy of approxi- mately 30 percent occurs between the experiments and the no-leakage theoretical results.

It is significant in the above results that no measured power consumptions were greater than those predicted for the leakage conditions in spite of the fact that additional power losses due to the barrel feed pocket made the measured power somewhat higher than ex- pected. Clearly, viscous heating in the clearance was not very significant at the speeds reported, since the theoretical curves for isothermal and adiabatic flow in the clearance were close together and norinally both predicted total power consumption greater than those measured.

The experimental results suggest that in addition to any inadequacies in the the ore t ical model :

(1) Significant leakage only occurred at low spceds when wall slip was less likely. At high speeds, ifwall slip occurs, then the flights possibly scrape the 1)arrel clean and the measured power consumption lies at or below the theoretical no leakage solution. The gradual increase in the cliscrepancy with increased speed might be attri- buted to the departure from full leakage flow over the flights to partial and finally no leakage at high speeds.

The elastic properties of the polymer are likely to be significant in affecting the onset of wall slip here in relation to short transient slip as the flight scrapes the barrel wall clean.

The results from further experiments conducted on the same screw and polymer, but with a screw clearance 6 of 0.126 mm, showed a slight improvement in agree- ment between the theory and experiments. The latter were approximately 27 percent lower than those calcu- lated based on the no-leakage solution, compared with 30 percent discrepancy obtained with a smaller screw clearance (Fig. 11 ). The results of the experiments with the large clearance add more evidence to the hypothesis of only partial leakage flow due to slip at and near the flight tips once a critical wall shear stress is exceeded.

(2) The gradual increase in the discrepancy between the experimental points arid the no leakage flow with increased screw speed suggest that slip occurs at the barrel wall and increases with the increase in wall shear stresses which correspond to the increase in screw speed. Unfortunately, the results described in Fig. 1 1 do not give sufficient information to explain the mechanism of slip at the barrel wall. One can reasonahly suppose, however, that slip at the wall occurs when the boundary shear rate exceeds a limiting value which re- sults in large stresses being induced in the polymer at and near the top of the screw flights, such forces being in excess of those sustainable by wall friction (7). This would limit the tnaxiinum level of wall shear stresses, hence reducing the power usage.

In relation to the experimental data quoted above, it was e s t i m a t e d t h a t the follow in g ex peri in en t al ac- curacies were achieved: measurement of G z 5%, mea- surement of power I 2.5%, measurement of speed x 0.5%, and measurement of torque f 2.0%.

CONCLUSIONS Both of the polymer melts under consideration exhih-

ited wall slip at shear stresses above the critical shear stress for melt fracture. The effect was more severe in the case of high density polyethylene, which showed a sharp increase in wall slip velocity at the onset of melt fracture. It is possible that different inechanisins are responsihle for the two different manifestations of slip.

The silicone fluid also exhibited wall slip and it was found that the slip velocity increased with increasing shear stress for the range of shear stresses considered

We might hazard a tentative comment on the phenomenological nature of wall slip in relation to the Huids discussed in this paper, as follows:

Wall slip appears to commence when the wall shear stress reaches acritical value. Wall shear stress increases with flow rate and for a given rate is higher for high viscosity fluids such as polymer melts than for low viscos- ie fluids. The critical shear stress at which slip occurs will presumably depend iipon the particular interfacial situation and the elastico-viscoiis natiire of the H i d , thus leading to vagueness in the utility of terms such a s low and high viscosity. However, with low viscosity fluids, the flow rate may exceed that giving t u r h l e n t flow before the critical shear stress for wall slip is

POLYMER ENGINEERlNG AND SCIENCE, APRIL, 1977, VoI. 17, No. 4 263

R. A. worth,J. Parnuby, und H . A. A. Helmy

reached and so with some fluids wall slip would not be observable. In principle, however, wall slip may be potentially likely for all fluids. Silicone fluids, although relatively simple low viscosity fluids compared to poly- mer melts, do not appear to adhere strongly to metal surfaces (14). (It is of interest here to observe the effect of rapidly withdrawing a smooth metal rod dipped into a bath of silicone fluid.)

From the results obtained during the extruder exper- iments using silicone fluid, it seems reasonable to pos- tulate the slip at the barrel wall is responsible for a large part of the discrepancy between the calculated and mea- sured screw power consumptions. The gradual increase in error with increasing screw speed may be attributed to the increase in slip velocity due to the rising shear stresses. This agrees qualitatively with the viscometer results shown in Fig. 9. It should be noted that wall slip reduces throughput rate Q to a greater degree than it reduces power consumption, thus accounting for the manner in which efficiency varies. Significant slip seems likeliest in the flight clearance.

APPENDLX 1

Wall Slip in a Newtonian, One-Dimensional, Isothermal Parallel-Plate Screw-Barrel Analogy

By simple parallel plate analogy, often used as a basis for approximate screw design (Tadmor and Klein (15)), it is relatively easy to show how once an approximately asymptotically limiting shear stress is reached after the onset of' wall slip, output rate and power consumption decrease relative to the zero wall slip case.

The velocity profiles in Fig. 12 illustrate the way in which increasing the velocity V, of the moving plate (barrel) relative to the fixed plate (screw) in the conven- tional representation (12, 15) leads to relative slip be- tween polymer and plate at one or both boundaries when the shear rate there reaches jcrit , the critical value. In the case of simple drag flow illustrated in Fig. 12(u), it is assumed for simplicity that effectively no slip occurs for + < yCrit and that for higher moving plate velocity V, there is a relative slip velocity V, at each boundary and shear rate ycrit is maintained in the fluid.

Figure 12 (b) illustrates the case where, due to a sig- nificant back pressure exerted by the extrusion die, a reverse pressure flow is superimposed on the drag flow (12). This superimposition is a non-linear interaction with non-Newtonian fluids but for a Newtonian fluid can be included by direct linear addition. It is assumed that slip starts first at the boundary where = yCrif and then eventually occurs at the other boundary, asV, is in- creased, only if ycrit can be reached here in the presence of slip occurring at the moving boiindary. It seems likely that slip will only occur at the moving plate for the case of Fig. 12(b), but much depends in general cases upon the way in which boundary shear stress varies with relative slip.

If in Fig. 12(b), the relative slip velocities, fluid to boundary, areV,, andV,, at the fixed and moving boundaries, respectively, let

at y = 0 and y = H , respectively, be the velocity bound- ary conditions for the momentum equation

.(p is the Newtonian viscosity), which, after integrating and substituting E 9 10, gives the velocity profile

Flow rate per unit width of plate,

i.e.,

and wall shear rates corresponding to wall stress 7(0), 7 ( H ) are

1 H aP +(0) = - (V, - V,) - -- H 2p az

If, for Fig. 12 (b), f(H) > j ( 0 ) , there is only slip at the moving boundary thenV, = 0, and E q 12 shows that flow rate reduces due to the replacement of V, by V,.

For a sinall element of length Az and unit width, the power consumed

AP = [7(0) . V, + 7 ( H ) . V,] AZ (14) and again forV, = 0 this is reduced by slip occurring at the moving boundary.

To calculate E 9 s 12 and 14 we need: (a) A relation between T(H) andVs, fitted from experi-

mental data, snch as given in Figs. 6 and 7 . Here a simple approximation such as

V,s, = 0 for f < +,,.it T ( H ) = constant for + > qcril

may be a sufficiently accurate description for practical design purposes.

(b) An interactive stepwise trial and error procedure for computation of a design to achieve a particular spec- ified value of Q as follows: 1. AssumeV, = 0, V, = V, initiallv and use Ey 12 tocalculate $I/&. 2. Use E q 13 to see if j ( O ) , y ( H ) < jcr i t . 3 . Ifwall shear rate exceeds f c r i t , then use empirical data such as (a) above to choose a slip velocity and then repeat steps 1 to 3 usingVtJ =V2 until the calculations converge.

I n the case of a non-Newtonian fluid, it seems reason- able that the procedure outlined above in deriving Eqs 12-14 by replacingv,, byV, could be followed. For the two and three dimensional cases, however, the situation becomes much more complex and empirically adjusted equations (Helmy and Parnaby (12), Parnaby (16)) may offer the only satisfactory procedure in practice.

Slip can also occur i n the extrusion die and may have a somewhat self-compensating eflect on throughput rate 9.

264 POLYMER ENGINEERING AND SCIENCE, APRIL, 1977, Vol. 17, No. 4

Wall Slip and 1t.y Implications i n the Design of Single Screw Melt-Fed Extruders

ACKNOWLEDGMENT The authors are very grateful to Mr. J . Whitworth and

his staff in the workshop of the Schools of Mechanical and Manufacturing Systems Engineering at the Univer- sity of Bradford for their help with equipment manufac- ture and operation. Some use of equipment provided by S.R.C. funds is also acknowledged.

NOMENCLATURE

D =diameter H = channel depth L = d i e length P =power Q = volumetric flow rate R =radius V = screw peripheral velocity V,, = screw velocity component in the direction of the

channel axis W = screw channel width e = flight width 1 = screw axial length n = power law index Ap = pressure drop us = slip velocity .j = shear rate 6 = screw clearance E

E~

,$,, 7 = viscosity qf

= dimensionless screw power consumption = dimensionless power consumption due to cross

= dimensionless co-ordinate of zero shear stress

= viscosity in the clearance

channel flow

8 = helix angle 7ij = shear stress 7u. = wall shear stress rH = viscous heat factor (10) rp, = dimensionless pressure gradient

REFERENCES

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