Theoretical modelling of torque requirements for single screw feeders

Powder Technology 93 (1997) 151–162

0032-5910/97/$17.00 q 1997 Elsevier Science S.A. All rights reservedPII S0032- 5910 (97)03265 -8

Journal: PTEC (Powder Technology) Article: 3318

Theoretical modelling of torque requirements for single screw feeders

Y. Yu, P.C. Arnold U

Centre for Bulk Solids and Particulate Technologies, Department of Mechanical Engineering, University of Wollongong, Wollongong, NSW 2522, Australia

Received 13 December 1996; revised 20 March 1997; accepted 21 April 1997

Abstract

In the design or selection of a screw feeder the torque requirement is a principal parameter which is related to the feeder loads, propertiesof the bulk solid and constructional features of the screw. In this paper the load which is imposed on a screw feeder by the bulk solid in thehopper is assumed to be the flow load determined on the basis of the major consolidation stress. Five boundaries around the bulk materialwithin a pitch are considered and forces acting on these surfaces are analysed. Particular attention is paid to the pressure distribution on thelower region of the screw. An analytical solution for the calculation of torque is determined, which allows the torque characteristics of screwfeeders to be predicted. Experimental studies on the required torque for screw feeders are also reported. Two types of material, three troughswith different inside diameter and two screws with different configurations have been investigated. The results from the experiments arepresented and compared with the theoretical predictions.

Keywords: Torque characteristics; Screw feeders

1. Introduction

The screw feeder is one of the most useful feeding deviceswhich not only has good metering characteristics, but alsouses relatively simple components and can be designed tofeed many kinds of bulk solids reliably in a variety of appli-cations. In the design of a screw feeder the torque requirementis an important parameter which is related to the feeder load,properties of the bulk solid and configuration of the screw.

Metcalf [1] considered the mechanics of a screw feeder,concentrating on the rate of delivery and the torque requiredto feed different types of coal using mining drill rods asscrews. The model chosen was that of a rigid plug of materialmoving in a helix at an angle to the screw axis. A detailedexperimental investigation was conducted by Burkhardt [2].The tests included the effect, on the performance of a screwfeeder, of the pitch, the radial clearance between the screwflight and trough, the hopper exposure and the head of thebulk solid contained in a hopper. Carleton et al. [3] discussedboth screw conveyors and feeders, but from the experimentalapparatus and results described in their paper there was moreemphasis on screw conveyors rather than on screw feeders.Rautenbach and Schumacher [4] carried out scale-up exper-iments with two geometrically similar screws. By dimen-sional analysis the relevant set of dimensional numbers was

U Corresponding author. Tel.: q61 42 214 566; fax: q61 42 214 577.

derived for the calculation of power consumption and capac-ity. More recently, Roberts and Manjunath [5] analysed themechanics of screw feeder performance in relation to the bulksolid draw-down characteristics in the feed hopper. In theirstudy the force exerted on the screw flights is assumed to beuniformly distributed along the whole feed length and threeempirical pressure ratios are used in the determination of therequired torque.

In this paper, five boundaries around the bulk materialwithin a pitch are considered and the forces acting on theseboundaries are analysed. Particular attention is paid to thepressure distribution on the lower region of the screw. Ananalytical solution for calculation of torque is determinedwhich allows the torque characteristics of screw feeders to bepredicted. Experimental studies on the required torque arereported.

2. Feeder loads

A typical form of hopper with a screw feeder is shown inFig. 1. The load which is exerted on a feeder by the bulksolids in a hopper was discussed in Refs. [6–9]. There aretwo main loading conditions: the initial filling condition whenthe bin is filled from the empty state and the flow conditionwhen discharge has occurred. The experimental evidence ofthis study suggests that the feeder load on a screw feeder can

Y. Yu, P.C. Arnold / Powder Technology 93 (1997) 151–162152


Fig. 1. Typical form of a hopper fitted with a screw feeder.

Fig. 2. Bulk material boundary within a pitch: (a) five boundary surfaces;(b) two basic regions.

be considered to be that arising under flow conditions. Carson[10] also noted that the starting torque is close to the runningtorque for many bulk materials and situations, but warnedthat there would be exceptions, for example: bulk materialswhich adhere to surfaces with storage at rest (effectivelyincreasing wall friction angles); bulk materials which gainstrength with storage and require increased additional shearforces to commence flow; bins or hoppers which are vibratedduring storage at rest (greatly increasing the vertical stress,i.e. feeder load). The recommended flow loads can beobtained by using methods proposed by Reisner andEisenhart Rothe [11].

According to McLean and Arnold [6], the feeder load Qacting at the outlet of the hopper is given by

1ym mq2QsqgL B (1)

q is a non-dimensional surcharge factor; m is a hopper shapefactor: ms1 for axisymmetric flow or a conical hopper, ms0for plane flow or a wedge-shaped hopper.

The flow loads on screw feeders can be determined on thebasis of the major consolidation stress s1. The non-dimen-sional flow surcharge factor may be expressed as

mp Y(1qsin d)

q s (2)fs1 ž /4 2(Xy1) sin a

where

1 sin fhy1bs f qsin (3)h≥ ž / ¥2 sin d

m2 sin d sin(2bqa)Xs q1 (4)≥ ¥1ysin d sin a

m 1ymYs[{2[1ycos(bqa)]} (bqa) sin a

1qmqsin b sin (bqa)]2qm y1=[(1ysin d) sin (bqa)] (5)

In Eq. (5) both b and a must be in radians for the term(bqa)1ym.

For a hopper fitted with a screw feeder, in general, ms0.Based on the flow case, the feeder load can be written as

2Qsq gLB (6)fs1

It is reasonable to assume that the feeder load is uniformlydistributed over the hopper outlet. The resulting stress so canbe obtained by

Qs s sq gB (7)o fs1LB

3. Pressure on surfaces of bulk material

3.1. The five boundary surfaces

Considering the bulk material boundary in a pitch, ‘pres-sures’ are imposed on five surfaces, as indicated in Fig. 2(a).Taking account of the boundary conditions applying to thebulk material moving within screw flights, two basic regionscan be specified: an upper region in which a ‘shear surface’exists between the bulk solid surrounding the screw and thebulk solid propelled by the screw and a lower region in whichthe bulk solid is moving within a limited space which com-prises rigid surfaces, as shown in Fig. 2(b). The surfaces towhich pressure is applied are:

Y. Yu, P.C. Arnold / Powder Technology 93 (1997) 151–162 153


Fig. 3. Stress on an element at the lower region of the screw.

c the ‘shear surface’ on the upper region of the screw;c the trailing side of the screw flight;c the driving side of the screw flight;c the outside surface of the core shaft;c the inside surface of the trough.

3.2. Pressure distribution on bulk material in the lowerregion

Consider the bulk material axial cross-section in the lowerregion in a pitch, as depicted in Fig. 3. The bulk materialboundary in this region is composed of four sides: the trailingside and the driving side of the screw flight, the inside surfaceof the trough and the outside surface of the shaft. This bound-ary can be assumed to be rectangular in shape and of unitthickness. Considering the forces acting on the bulk materiallimited by this boundary, the force due to gravity is neglected.Because the speed of rotation is relatively low, the centrifugaleffects are also regarded as negligible.

Stress sw is the normal wall pressure acting perpendicu-larly to the wall of the trough and the core shaft. s x is theaxial compression stress. The ratio

l ss /s (8)s w x

is known as the stress ratio of the bulk material sliding on theconfining surfaces (i.e. trough and core shaft surfaces). Ageneral expression can be obtained:

s 1wl s s (9)s 2 2 2 2 1/2s 1q2m q2[(1qm )(m ym )]x d d d w

mdstan d and d is the effective angle of internal friction ofthe bulk solid. mw is the friction coefficient between the bulksolid and a confining surface. The derivation of Eq. (9) ispresented in Appendix A. For free flowing materials, mdsm,where m is the friction coefficient of the bulk solid. Then, Eq.(9) has the same form as that given by Hong and Ling [12].

When a moving bulk solid reaches steady state, there isequilibrium between the driving force and the resisting force.Assuming the axial stress and the radial stress are functionsof x only, as shown in Fig. 3, the balance of forces acting onthe element of length dx results in

2m l dsw s xs y s0 (10)xR yR dxt c

A solution to Eq. (10) is

2m lw ss sc exp x (11)x 1 ž /R yRt c

where c1 is an integration constant. For the purposes of thisanalysis, c1 is determined by making the following simpli-fying assumption concerning the boundary condition:

s ss at xs0x o

so is the stress exerted on the screw feeder by the bulk solidin a hopper. The solution to Eq. (10) can then be written as

2m lw ss ss exp x (12)x o ž /R yRt c

To simplify the calculation an average radial stress along apitch is introduced as

P

lss s s dx (13)wa x|P

0

Substituting s x from Eq. (12), the average radial stress maybe derived by

R yR 2m l Pt c w ss ss exp y1 (14)wa o ≥ ž / ¥2m P R yRw t c

For convenience, non-dimensional parameters areintroduced:

c sd/DsR /Rd c o

c sP/Dp

c s(Dq2c)/Ds2R /Dt t

Eq. (14) can be expressed as

c yc 4m l ct d w s ps ss exp y1 (15)wa o ≥ ž / ¥4m c c ycw p t d

In an actual application the wall friction between the bulksolid and the core shaft and the trough surface may be thesame. If they are not identical, then the wall friction coeffi-cient between the bulk solid and the trough surface can bechosen to replace mw, as the trough surface is dominant.

3.3. Forces acting on individual surfaces

The analysis of the forces acting upon the bulk solid ele-ment on the individual surfaces where non-cohesive bulkmaterial is transported in a vertical screw conveyor was madeby Nilsson [13]. For screw feeders the surfaces upon whichforces are exerted and the status of the acting forces shouldbe distinguished from those produced in vertical screwconveyors.



Fig. 4. A material sector in a pitch.

Fig. 6. Forces on the shaft surface.Fig. 5. Forces on the shear surface.

A bulk material sector in a pitch is used for the calculationof the axial forces acting on an individual surface, as depictedin Fig. 4.

It is assumed that within the length of a pitch the forcesacting on individual surfaces are uniformly distributed (onthe upper shear surface, trailing side and driving side of aflight) or these forces can be represented by average forces(on the outside surface of the core shaft and the inside surfaceof the trough).

4. Axial resisting forces

4.1. Axial resisting force on the shear surface

The axial resisting force acting on the element of the bulksolid on the shear surface, as shown in Fig. 5, is given by

dF sm s R P du cos(a qf ) (16)ua e o o o f

ao is the helical angle of the flight at the outside radius. me isthe equivalent friction coefficient. According to Roberts etal. [7–9], mes(0.8–1.0) sin d.

The total axial force acting over a pitch length of the screwis

p

F sm s R P cos(a qf ) du (17)ua e o o o f |0

After integration and by means of non-dimensional parame-ters, Eq. (17) can be written as

p 2 2F s m c cos(a qf ) s D sk s D (18)ua e p o f o u o2

and

pk s m c cos(a qf ) (19)u e p o f2

4.2. Axial resisting force on the core shaft

As shown in Fig. 6, the axial resisting force acting on theelement of the bulk solid is

dF sm s R P du sin a (20)ca w wa c c

mc is the wall friction coefficient between the bulk solid andthe core shaft. ac is the helical angle of the flight at the coreshaft.

The total axial force acting over a pitch length of the screwis

2p

F sm s R P sin a du (21)ca w wa c c |0

swa can be obtained from Eq. (15). Integrating Eq. (21)leads to

pc (c yc ) 4m l cd t d w s p 2F s sin a exp y1 s Dca c o≥ ž / ¥4 c yct d

2sk s D (22)c o

and



Fig. 7. Forces on the trailing side of a flight.

Fig. 8. Forces on the driving side of a flight.pc (c yc ) 4m l cd t d w s pk s sin a exp y1 (23)c c ≥ ž / ¥4 c yct d

4.3. Axial resisting force on the trailing side of a flight

The axial resisting force acting on the trailing side of aflight as presented in Fig. 7 is

r dr dudF sl s cos(f ya )la s o f rcos a cos fr f

sl s r dr du (1qtan a tan f ) (24)s o r f

Substituting tan arsP/2pr and tan ffsmf, and integratingfor r from Rc to Ro and for u from 0 to 2p:

p m cf p2 2 2F sl (1yc )q (1yc ) s D sk s D (25)la s d d o l o≥ ¥4 2

and

p m cf p2k sl (1yc )q (1yc ) (26)l s d d≥ ¥4 2

4.4. Axial resisting force on the trough surface

From Fig. 5, the axial resisting force on the trough surfaceis given by

dF sm s R P du cos(a qf ) (27)ta w wa t o f

After integrating for r from Rc to Ro and for u from 0 to p,the total axial force over a pitch is

pF s c (c yc ) cos(a qf )ta t t d o f8

4m l cw s p 2 2= exp y1 s D sk s D (28)o t o≥ ž / ¥c yct d

and

p 4m l cw s pk s c (c yc ) cos(a qf ) exp y1 (29)t t t d o f ≥ ž / ¥8 c yct d

4.5. Axial force and stress on the driving side of a flight

The axial force acting on the driving side of a flight isshown in Fig. 8. This force should be equal to the total resist-ing axial forces

F sF qF qF qF (30)da ua ca la ta

It is assumed that the total force is uniformly exerted on thesurface of the driving side. The axial stress can be determinedby

F 4(k qk qk qk )da u c l ts s s s sK s (31)a o s o2 2 2p(R yR ) p(1yc )o c d

where

4(k qk qk qk )u c l tK s (32)s 2p(1yc )d

5. Torque requirement in the feed section

Most screw conveyors can be designed with little thoughtgiven to thrust as the thrust force or axial force in an ordinaryscrew conveyor is moderate and commonly used screw con-veyor drives will accommodate thrust in either direction.However, in a screw feeder, especially with long inlet open-ings, axial force can be very severe. Thus, determination ofthe axial force is necessary for the design of a screw feeder.

Roberts and Manjunath [5] proposed a method by whichthe torque requirement can be calculated based on the axialforces. An obvious advantage of this method is that bothtorque and axial force can be obtained from one calculationprocess.

Referring to Fig. 8, the tangential force on the bulk materialelement is

dF ss du r dr tan(a qf ) (33)dt a r f

The torque required for turning the screw isRo

2Ts2ps r tan(a qf ) dr (34)a r f|Rc



which becomes3TsK s D (35)s a

andRo

2p 2K s r tan(a qf ) dr (36)s r f|3DRc

An analytical solution to Eq. (36), by means of non-dimen-sional parameters, is

m 1qmf f3 3 2K spc (1yc )q (1yc )s p d d3 2≥ 12c 8pcp p

2 2 2m (1qm ) 3m (1qm )f f f fq (1yc )y

(37)

d2 34p c 8pp

2 2m (1qm ) pym cf f f pq ln3 ž / ¥4p pc ym cd f p

The derivation of Eq. (37) is presented in Appendix B.Substituting s a from Eq. (31), the torque requirement can

be expressed by3TsK K s D (38)s s o

Eq. (38) is a general expression for the torque requiredfor one pitch in the feed section. For pitch i, Eq. (38) can bewritten as

3T sK K s D (39)fi si s i o

The torque required for all pitches in the feed section isnf

T s T (40)ft fi8is1

where nf is the number of pitches in the feed section.It can be seen from Eq. (38) that the required torque is

proportional to the stress exerted on the feeder by the bulksolid in the hopper and to the third power of the screw diam-eter. An increase of 50% in screw diameter will result in a50% increase in opening width of the hopper and a 50%increase in the stress exerted by the bulk solid in the hopper(from Eq. (7)). According to Eq. (38) the total increase intorque will be 500%. This conclusion agrees with the dimen-sional analysis and experimental results reported in Ref. [4].

6. Torque calculation in the choke section

A choke section is adjacent to the hopper, as shown inFig. 1. This section is cylindrical and has the same radialclearance as the lower part of the trough. For effective flowcontrol the choke section should extend for at least two stan-dard pitch lengths [14].

It is assumed that in the choke section the screw isoperating100% ‘full’. The shear surface in the feed section does notexist, but is replaced by a cylindrical sliding surface. Thus,in the choke section Eq. (32) can be replaced by

4(k qk q2k )c l tK s (41)sc 2p(1yc )d

A general expression for the torque required for a pitch in thechoke section is

3T sK K s D (42)c s sc o

Normally, pitches in the choke section have the same geom-etry. The total torque required for pitches in the choke sectioncan be expressed as

T sn T (43)ct c c

where nc is the number of pitches in the choke section.

7. Torque characteristics of screw feeders

7.1. Torque components

From the analyses presented in Section 5, the torquerequirement for a screw feeder is dependent on the resistingforces on the four surfaces. When the stress exerted by thebulk material in the hopper and the screw diameter are deter-mined, the required torque can be calculated from the resist-ing forces acting on the individual surfaces. For the sake ofconvenience, the equation for the calculation of the torquerequirement is rewritten as

3TsK K s Ds s o

The factor Ks is related to the axial resisting forces actingon the individual surfaces and is represented by the factorsin Eqs. (19), (23), (26), (29) and (37). The factor Ks isderived from the tangential force acting on the driving sideof a flight. To simplify the expressions, let

4KsKsK K s (k qk qk qk ) (44)s s u c l t2p(1yc )d

Let

4k Ku sK s (45)u 2p(1yc )d

Ku reflects the torque contribution from the shear surface.

4k Kc sK s (46)c 2p(1yc )d

Kc reflects the torque contribution from the core shaft surface.

4k Kl sK s (47)l 2p(1yc )d

Kl reflects the torque contribution from the trailing side of theflight.

4k Kt sK s (48)t 2p(1yc )d

Kt reflects the torque contribution from the trough surface.



Fig. 9. Variation of factors with ratio P/D (mds0.8, mfsmws0.5, cds0.3).Fig. 10. Influence of c/D on K (mds0.8, mfsmws0.5, cds0.3).

Fig. 11. Influence of mw on K (mds0.8, mfs0.5, cds0.3).

Fig. 9 shows the contributions of these factors to the totaltorque. It can be seen that the major contribution is the resist-ing torque acting on the upper shear surface, varying from43% for P/Ds0.3 to 50% for P/Ds1. When P/D)0.5,factor Ku contributes 50% of the whole torque. The resistingtorques on the trough surface and on the shaft surface increasewith an increase of the P/D ratio, varying from 13% to 19%and from 3% to 11%, respectively. Compared with thetorques on the other three surfaces, the resisting torque on theshaft surface is low, even when the value of the ratio of thecore shaft diameter to screw diameter increases up to 0.5. Thecontribution of the resisting torque on the trailing side of theflight decreases with an increase of the P/D ratio, from 42%for P/Ds0.3 to 20% for P/Ds1.

7.2. Influence of clearance on the torque

The clearance between the trough and the tips of the flightsis necessary to prevent metallic contact from taking placeduring rotation, due to various adverse factors such as deflec-tion, minor manufacturing eccentricities, and tolerance on thescrew and the trough. It is also essential to avoid nipping orwedging of particles to prevent damage and the generationof extreme contact pressure and, hence, high torques resistingrotation.

Fig. 10 gives the influence of the values of the c/D andP/D ratios on the torque requirements. The values of c/Dvary from 1/64 to 1/8. Results from the theoretical calcula-tion show that there is no obvious increase in factor K with achange in the ratio c/D from 1/64 to 1/8 for the differentvalues of the P/D ratio. Although an increase in the clearancewill result in some increase in the resisting torque (due to anincrease in Rt), it also results in a decrease in radial pressure(from Eq. (12)), which feeds back to decrease the resistingtorque on the trough surface and shaft surface. However, fromEq. (7), the stress exerted by the bulk solid in the hopper islinearly proportional to the opening width of the hopperoutlet. Normally, the opening width of the hopper outletBs2(Dq2c)s2Rt, which means that the stress so increases

with an increase of the clearance. This influence of the clear-ance cannot be neglected in the calculation of the torquerequirement.

7.3. Influence of the trough wall friction coefficient

Fig. 11 shows the effect of the trough wall friction coeffi-cient on factor K. It can be clearly seen that K increases withan increase of the trough wall friction coefficient mw. Suchan increase is due to two influences. First, the radial pressurealong the pitch length increases with an increase in mw. Sec-ond, the resisting torque increases with both the radial pres-sure and the friction coefficient. Furthermore, the effect isstronger with a larger pitch length.

7.4. Influence of the effective angle of internal friction

The influence of the effective angle of internal friction isrelated to the stress ratio, shear force on the upper part of thescrew and the radial pressure on the lower part of the screw.The stress ratio and the radial pressure decrease with anincrease of the effective angle of internal friction, althoughthe increase of the shear force caused by the effective angle



Fig. 12. Influence of d on K (mfsmws0.5, cds0.3).

Fig. 13. Influence of mf on K (mds0.8, mws0.5, cds0.3).

Fig. 14. Influence of cd on K (mds0.8, mfsmws0.5).

Fig. 15. Test rig for screw feeders: 1, hopper; 2, trough; 3, test screw; 4,driving unit; 5, receiving and weighing silo.

of internal friction coefficient is not sufficient to compensatefor the decrease of the stress ratio and the radial pressure.Fig. 12 shows the effect of the effective angle of internalfriction on the factor K.

7.5. Influence of the flight friction coefficient

Fig. 13 gives the results for K with variation of mf from 0.3to 0.7. It can be seen that K becomes larger with increasingflight friction coefficient. The forces acting on both the trail-ing side and the driving side of the flight increase withincrease of the friction coefficient between the bulk solid andthe flight, resulting in an increase in the torque requirement.

7.6. Influence of the ratio d/D

The increase of cd, the ratio of the core shaft diameter d tothe screw diameter D, will result in an increase of the stressratio, which affects the radial pressure, especially for longerpitch lengths. The resisting force and the torque on the shaftsurface also increase with an increase of the ratio cd. Fig. 14shows the influence of the ratio cd on K.

8. Test rig and test materials

Experimental investigations have been undertaken tounderstand the influence of geometric parameters, propertiesof the bulk solid conveyed and operating conditions on thetorque required for screw feeders. A test rig, as shown inFig. 15, was designed to monitor the performance of differenttypes of screw (such as stepped pitch, tapered shaft andstepped shaft) in conjunction with different bulk solids,troughs and rotational speeds. This test rig consisted of fiveparts: hopper, trough, test screw, driving unit, receiving andweighing silo.



Fig. 16. Configuration of test screws.

Table 1Physical properties of test materials

Bulk solid dm (mm) rb (kg my3) d (deg) fw (deg)

Semolina 0.39 736 31.0 27.5Cement 0.028 791 51.5 27.0

Fig. 17. Comparison of torques for a screw feeder using the screw with atapered shaft and stepped pitches.

Fig. 18. Comparison of torques for a screw feeder using the screw with astepped shaft and stepped pitches.

Screws employed for the test program had the same outsidediameter (Ds150 mm); Fig. 16 shows the configurations ofthe two screws. Three troughs were used with inside diame-ters of 160, 170 and 190 mm, giving the radial clearances of5, 10 and 20 mm. Two bulk solids were chosen to examinethe influence of material properties on the performance ofscrew feeders. The main physical properties of the bulk solidshave been measured and are listed in Table 1. The effectiveangle of internal friction and the wall frictional angle on mildsteel were determined by measurements with a Jenike-typedirect shear tester.

Each material was poured into the hopper to a desired level.Three different states of material in the hopper have beenexamined: hopper filled from an empty state to a level of 600mm (high initial); hopper filled from an empty state to a levelof 300 mm (low initial); after some discharge the level ofbulk material was reduced to 300 mm (low flow).

9. Experimental results

Detailed experiments under three different filling states inthe hopper and for a screw speed range of 10–80 rpm werecarried out using a trough with clearance cs5 mm. Theexperimental results did not show any obvious difference dueto the different filling states and screw speeds for the testmaterials.

Figs. 17 and 18 give the comparison between the theoret-ical predictions and the experimental results for two screwswith three troughs. In the keys, C is short for cement and Sfor semolina. The three radial clearances are 5, 10 and 20.For the tapered shaft screw (No. 1) an average value of theshaft diameter within an individual pitch has been assumedin the calculations. For screw No. 2 with a stepped diametershaft the corresponding shaft diameter for an individual pitchhas been used. It can be seen that the theoretical predictionsare reasonably consistent with the experimental data, except

for the screw operating in the largest trough (Rts95 mm)with semolina, where the calculated values are lower than theobserved results.

Semolina is a very free flowing material. The output of thescrew feeder with semolina increases with increasing radialclearance. However, the torque requirement does not showthe same pattern. It can be observed from the experimentalresults that for both screw feeders the torque required for themiddle trough (Rts85 mm, cs10 mm) is less than that forthe other two troughs. There are two reasons for this situation.For bigger shaft diameters of the screws, the small clearance(cs5 mm) would result in an increase in the pressure in thelower region, thereby increasing the torque requirement. Onthe other hand, the large radial clearance (cs20 mm) mayresult in an additional resisting force due to the difficulty ofthe larger mass of bulk solids (assumed to be sliding on the



trough surface in the lower region) trying to ‘enter’ the regionof the shear surface above the screw.

10. Conclusions

1. A theoretical model for the torque requirement of a screwfeeder is developed by applying principles of powdermechanics to a moving material element within a pitch. Themodel indicates that the torque requirement is proportionalto the stress exerted on the feeder by the bulk solid in thehopper and to the third power of the screw diameter.

2. Consideration of the forces acting on the five confiningsurfaces surrounding the bulk solid contained within a pitch,and the pressure distribution in the lower region of the screw,leads to a reasonable prediction of the torque requirement fora screw feeder.

3. The analytical solution for the calculation of the torquerequirement can reveal the relationships among the screwgeometry parameters, properties of the bulk solid and feederload arising from the hopper. Thus, the torque characteristicscan be better understood.

4. Based on the parameters used for this study, the torquerequired for a screw feeder is determined mainly by the resist-ing torques acting on the shear surface; the proportion of thetotal torque requirement is about 50%.

5. Experimental results indicate that the starting torque isclose to the running torque for the test materials and situa-tions. The feeder load exerted by bulk solid in the hopper canbe determined by the flow load based on the major consoli-dation stress.

11. List of symbols

B opening width of hopper outlet (m)c clearance between trough and tip of flight (m)c1 integration constant in Eq. (11)cd ratio of shaft diameter to screw diametercp ratio of pitch to screw diameterct ratio of trough inside radius to screw outside

radiusd core shaft diameter (m)dm mean equivalent particle diameter (m)D screw diameter (m)Fca axial resisting force on core shaft surface (N)Fda axial resisting force on driving side of flight (N)Fdt tangential force on driving side of flight (N)Fla axial resisting force on trailing side of flight (N)Fta axial resisting force on trough surface (N)Fua axial resisting shear force on upper region of screw

(N)kc factor defined in Eq. (23)kl factor defined in Eq. (26)kt factor defined in Eq. (29)ku factor defined in Eq. (19)

K factor defined in Eq. (44)Kc factor defined in Eq. (46)Kl factor defined in Eq. (47)Ks factor defined in Eq. (37)Kt factor defined in Eq. (48)Ku factor defined in Eq. (45)Ks factor defined in Eq. (32)Ksc factor defined in Eq. (41)L length of feed section (m)m hopper shape factor in Eq. (1)nc number of pitches in choke sectionnf number of pitches in feed sectionP pitch length (m)q non-dimensional surcharge factorqfs1 surcharge factor for flow condition based on s1

Q feeder load exerted by bulk solids in hopper (N)r radius of flight (m)Rc radius of core shaft (m)Ro outside radius of flight (m)Rt inside radius of trough (m)T torque required for driving screw (N m)Tc torque required in choke section (N m)Tf torque required in feed section (N m)x coordinateX, Y factors in feeder load equations, flow condition

Greek letters

a hopper half-angle (deg)ac flight helix angle at core shaft (deg)ao flight helix angle at outside diameter (deg)ar flight helix angle at radius r (deg)b angle in Eq. (3) (deg)g specific weight of bulk solid (N/m3)d effective angle of internal friction of bulk solid

(deg)u polar coordinatels stress ratio of bulk solid sliding on surfacem friction coefficient of bulk solid, mstan f

md tangent of effective angle of internal friction,mdstan d

me equivalent friction coefficient of bulk solid,mes(0.8–1) sin d

mf wall friction coefficient between bulk solid andflight

mw wall friction coefficient between bulk solid andconfining surface

rb loose poured bulk density (kg/m3)s1 major consolidation stress (Pa)s a axial stress on driving side of flight (Pa)sm stress in Fig. 19 (Pa)so stress exerted by bulk solid in hopper (Pa)sw wall stress in Fig. 3 (Pa)swa average wall stress defined in Eq. (13) (Pa)s x axial stress in Fig. 3 (Pa)tw shear stress on confining surface (Pa)



Fig. A1. Mohr circle representation of the stress in an element on a confiningsurface.

f kinematic angle of internal friction of bulk solid(deg)

ff wall friction angle of bulk solid on flight surface(deg)

fh wall friction angle of bulk solid on hopper wall inEq. (3) (deg)

fw wall friction angle between bulk solid andconfining surface, e.g. trough or core shaft surface(deg)

Appendix A

The Mohr circle representing the stress of a bulk solid elementon a confining surface is shown in Fig. A1.

From the right angle triangle ACD in Fig. A1,2 2 2(s ys ) qt sr (A1)m w w m

Substituting tw with mwsw ,2 2 2 2(s ys ) qm s sr (A2)m w w w m

The solution to Eq. (A2) is2 2 2 2 1/2s y[r (1qm )ys t ]m m w m w

s s (A3)w 21qmw

Since mdstan d

2 1/2r r (1qm )m m ds s sm sin d md

and

s qs s2sw x m

The stress ratio of the bulk solid sliding on a confining surfacecan be obtained by

s sw wl s ss

s 2s ysx m w

1s (A4)2 2 2 2 1/21q2m q2[(1qm )(m ym )]d d d w

Appendix B

Eq. (31) is rewritten as follows:

Ro

2p 2K s r tan(a qf ) dr (B1)s r f|3DRc

Substituting tan arsP/2pr and ffsmf, Eq. (B1) becomes

Ro

2p 1q2pm r/Pf2K s r dr (B2)s |3 ž /D 2pr/PymfRc

Let xsr/P, then

drsP dx

xsR /P at rsRc c

xsR /P at rsRo o

Eq. (B2) becomes

R /P R /Po o3 2 32pP x 2pm xfK s dxq dx (B3)s | |3 ž /D 2pxym 2pxymf f

R /P R /Pc c

The solution to the first integral is

R /Po 2 22x 1 R Ro cdxs y| ≥ ž / ž / ¥2pxym 4p P PfR /Pc

2 2m R R 3m m 2pR /Pymf o c f f o fq y y q ln (B4)2 3 3ž / ž /4p P P 16p 8p 2pR /Pymc f

The solution to the second integral is

R /Po 3 332pm x m R Rf f o cdxs y| ≥ ž / ž / ¥2pxym 3 P PfR /Pc

R /Po2m xfq dx (B5)|2p x/m y1/(2p)f

R /Pc

For the integral in Eq. (B5), the solution is

R /Po 2 22x m R Rf o cdxs y| ≥ ž / ž / ¥x/m y1/(2p) 2 P PfR /Pc

2 3 3m R R 3m m 2pR /Pymf o c f f o fq y y q ln (B6)2 2 2ž / ž /2p P P 8p 4p 2pR /Pymc f

Combining Eqs. (B4), (B5) and (B6), Ks can be expressedas



3 33P m R Rf o cK s2p ys 3 µ ≥ ž / ž / ¥D 3 P P2 2

1qm R Rf o cq y≥ ž / ž / ¥4p P P

(B7)

2 2 2m (1qm ) R R 3m (1qm )f f o c f fq y y2 3ž /4p P P 16p

2 2m (1qm ) 2pR /Pymf f o fq ln3 ž / ∂8p 2pR /Pymc f

Employing non-dimensional parameters allows Ks to beexpressed as

m 1qmf f3 3 2K spc (1yc )q (1yc )s p d d3 2≥ 12c 8pcp p

2 2 2m (1qm ) 3m (1qm )f f f fq (1yc )y

(B8)

d2 34p c 8pp

2 2m (1qm ) pym cf f f pq ln3 ž / ¥4p pc ym cd f p

which is Eq. (37) of the main text.

References

[1] J.R. Metcalf, The mechanics of the screw feeder, Proc. Inst. Mech.Eng., 180 (6) (1965–1966) 131–146.

[2] G.J. Burkhardt, Effect of pitch, radial clearance, hopper exposure andhead on performance of screw feeders, Trans. ASAE, 10 (1967) 685–690.

[3] A. Carleton, J. Miles and F. Valentin, A study of factors affecting theperformance of screw conveyers and feeders, Trans. ASME, J. Eng.Ind., 91 (2) (1969) 329–334.

[4] R. Rautenbach and W. Schumacher, Theoretical and experimentalanalysis of screw feeders, Bulk Solids Handling, 7 (5) (1987) 675–680.

[5] A.W. Roberts and K.S. Manjunath, Volumetric and torque character-istics of screw feeders, Proc. Powder and Bulk Solids Conf., Chicago,IL, USA, 1994, pp. 189–208.

[6] A.G. McLean and P.C. Arnold, A simplified approach for theevaluation of feeder loads for mass flow bins, Powder Bulk SolidsTechnol., 3 (3) (1979) 25–28.

[7] A.W. Roberts, M. Ooms and K.S. Manjunath, Feeder load and powerrequirements in the controlled gravity flow of bulk solids from mass-flow bins, Mech. Trans. IEAust., ME9 (1) (1984) 49–61.

[8] K.S. Manjunath and A.W. Roberts, Wall pressure–feeder loadinteractions in mass flow hopper/feeder combinations, Part I, BulkSolids Handling, 6 (4) (1986) 769–775.

[9] K.S. Manjunath and A.W. Roberts, Wall pressure–feeder loadinteractions in mass flow hopper/feeder combinations, Part II, BulkSolids Handling, 6 (5) (1986) 903–911.

[10] J.W. Carson, Designing effective screw feeders, Powder Bulk Eng.,(Dec.) (1987) 32–36, 41–42.

[11] A. Reisner and M.V. Eisenhart Rothe, Bins and Bunkers for HandlingBulk Materials, Trans. Tech. Publications, Clausthal-Zellerfeld,Germany, 1971.

[12] Z.Y. Hong and L.M. Ling, Continuous Conveying Mechanisms,Mechanical Industry Press, Beijing, 1982 (in Chinese).

[13] L.G. Nilsson, On the vertical screw conveyor for non-cohesive bulkmaterials, Acta Polytech. Scand. Mech. Eng. Ser. 64, (1971).

[14] Screw Conveyors, CEMA Book No. 350, Conveyor EquipmentManufacturers’ Association, Washington, DC, 1980.

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Theoretical modelling of torque requirements for single screw feeders

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