9.

PREDICTING THE VOLUMETRIC AND TORQUE

CHARACTERISTICS OF SCREW FEEDERS

Alan W. Roberts

Emeritus Professor, Centre for Bulk Solids and Particulate Technologies,

The University of Newcastle, N.S.W., Australia.

SUMMARY
The mechanics of screw feeder performance in relation to the bulk solid draw-down
characteristics in the feed hopper are examined. Taking account of the changing screw
geometry and volumetric efficiency along the screw at the hopper interface, a design
methodology is presented which permits the determination of the required parameters of
pitch, screw diameter and core or shaft diameter to meet specified draw-down objectives.
Particular attention is given to the determination of volumetric efficiency, for which a new
theory is presented. A theory is also presented for the determination of screw feeder torque;
for design convenience, non-dimensional torque parameters are introduced. Both the
volumetric efficiency and torque parameters take into account screw geometry and the
frictional characteristics of the bulk material on the screw surface. In addition to the
foregoing, the limiting condition for axial conveying and feeding to occur is investigated.

1. INTRODUCTION
Screw feeders are widely used for bulk solids which are generally of low or cohesion and free
flowing such as fine and granular materials which have to be dispensed under controlled
conditions at low flow rates. Because of their positive displacement mode of operation, they
have a particular advantage in providing good volumetric feed control. In the case of the
feeding of fine powders, they also permit 'automatic' sealing of the hopper against flooding
and uncontrollable discharge. This is achieved through the extended conveying section of the
screw that projects beyond the hopper at the discharge end. This is illustrated in Figure 1.

Desirability for
Uniform Draw-Down

Extended Section
for Sealing

Figure 1. Screw Feeder

While screw feeders are sometimes fitted to conical hoppers, they are most commonly used in
conjunction with plane-flow hoppers where it is necessary to feed along slotted hopper
openings. Primarily, the objective in the design of a screw feeder and its interfacing with the

©AWR CBSPT 9.1

hopper is to achieve uniform draw-down of the powder or bulk solid as illustrated in Figure 1.
This is to ensure that segregation and, hence, degradation problems do not occur. For
uniform draw-down to be achieved, the capacity of the screw needs to increase, in a
controlled way, in the direction of feed. The correct design of screw feeders involves the
selection of combinations of screw diameter, shaft diameter and pitch to give the necessary
volume increase along the screw.

A selection of references on the subject of screw feeder design is given at the end of this
paper [***]. The material presented herein is based on the research of Roberts et al (Refs.[2-
6]). The influence of volumetric efficiency on feeder throughput is investigated and the
interrelation between the geometric variables of the screw feeder and hopper draw-down
patterns is examined. Procedures for determining the loads acting on screw feeders and the
corresponding torques and powers are presented.

2. SCREW FEEDER DESIGN – GENERAL CONSIDERATIONS

As indicated above, the overriding aim is to select a screw geometry to achieve uniform
draw-down in the hopper. As is obvious, this cannot be achieved in the case of a constant
pitch, constant diameter screw since material is drawn from the back of the hopper by funnel-
flow as illustrated in Figure 2. This is because the screw space over the first pitch is
completely filled in one rotation of the screw. As this material is transported to the next and
subsequent pitches, no more material can be drawn down from the hopper creating a large
dead zone as indicated in Figure1. In this case the screw is acting as a conveyor and not as a
feeder. A conveyor has constant capacity along its length. A feeder must have variable,
increasing capacity in the direction of feed.

Flow
Channel Dead
Material

Figure 2. Constant Pitch, Constant Diameter Screw Showing Feed at Rear of Hopper

Several arrangements are advocated for providing an increasing screw capacity in the
direction of feed such as:
• Stepped pitch
• Variable pitch
• Variable pitch and diameter
• Variable shaft diameter

Not all of these produce an ideal arrangement. For instance Figure 3 shows a screw feeder in
which the screw and shaft diameters are each constant, while the pitch progressively
increases from the rear to the front as illustrated. This is not a satisfactory arrangement,

©AWR CBSPT 9.2

mainly due to the fact that the volumetric efficiency of the feeder decreases with the
expanding pitch in the direction of feed. Although an improvement over the arrangement
shown in Figure 2, the feeder will draw preferentially from the rear as shown. To overcome
this problem, the screw requires both a tapered shaft in addition to the expanding pitch as
illustrated in Figure 4.
Time 0

Time t

Figure 3. Screw Feeder with Constant Screw Diameter, Constant Shaft Diameter
and Expanding Pitch. Feed Occurs Preferentially from Rear of Hopper.

Constant
Increasing Pitch Pitch

Tapered Shaft

Figure 4. Screw Feeder with Constant Screw Diameter, Tapered Shaft Diameter
and Expanding Pitch. Results in Uniform Draw-Down in Hopper.

The screw pitch variation is generally limited to a range between 0.3 screw diameters
minimum to 1.5 diameters as an absolute maximum, although a screw pitch to screw diameter
ratio of 1.0 is normally regarded as the maximum from a practical point of view. Also, the
length to diameter ratio for a screw feeder is limited to about 6:1, making them unsuitable for
hoppers with long slotted openings. An important factor in selecting the minimum pitch,
particularly in the case of cohesive bulk solids, is the need to ensure that the pitch is large
enough to prevent the bulk solid arching of jamming in the pitch space. The flow properties
of the bulk solid will provide the necessary information to determine whether jamming will
occur. In addition, it needs to be noted that screw feeders generally incur high frictional
losses and are prone to high abrasive wear, particularly when abrasive bulk solids are

©AWR CBSPT 9.3

handled. It is necessary to select an appropriate, low friction, wear resistant material for the
screw and shaft.

The section of the screw leading from the hopper to the feeder outlet is fundamental in
determining the quantity of material discharged per revolution of the screw. At the point
when the screw leaves the hopper, it is essential for control purposes to cover the screw,
normally by a ‘choke’ section having the same radial clearance as the trough. This choke
section should extend for at least 1.5 pitch lengths to prevent material cascading over the
flights.

For practical reasons, the outside diameter of the screw is usually fixed and variable capacity
is achieved through combinations of increasing pitch and tapered shafts as depicted in Figure
4. Occasionally, screws with increasing tapered outside screw diameters are employed but
these required a corresponding tapered casing which makes the interfacing with the hopper
more difficult. Tapered screws and casings can only be used in the case of free flowing
powders or bulk solids where arching across the narrow section of the casing would not be a
problem.

3. VOLUMETRIC CAPACITY OF SCREW FEEDER

3.1 Screw Geometry

In the analysis that follows, the general form of screw geometry as depicted in Figure 5, is
considered. Actual screws used in practice can then be treated as special cases of the general
form.
HOPPER
v(x)
Rear Front
p(x)
bi b(x)
FEED
Q(x) Q
D(x)
Di b D

x
dx

Figure 5. Screw Feeder Geometry.

(a) Screw Geometrical Variables

At any location x, the relevant screw geometrical variables are,

p(x) = pitch, D(x) = screw diameter and b(x) = core or shaft diameter

When x = 0, the initial values apply. That is,

©AWR CBSPT 9.4

 pi = initial or starting pitch , Di = initial screw diameter and bi = initial shaft diameter

When x = L, the geometry at the outlet is defined. That is,

D = screw diameter at the outlet, po = pitch at outlet and b = shaft diameter at outlet

3.2 Volumetric Flow Rate

At location x, the volumetric feed rate is given by

Q(x) = QT(x) ηv(x) (1)

where QT(x) = maximum theoretical volumetric throughput with the feeder running
100% full and the bulk solid moving axially without rotation
ηV(x) = volumetric efficiency

QT(x) is given by

QT(x) = A(x) ηg(x) v(x) (2)

A(x) = cross-sectional area of screw ηg(x) = feeder geometry factor

v(x) = axial conveying velocity

The feeder geometry factor adjusts the throughput to allow for the influence of clearance
between the feeder and casing and the blade thickness.

The axial conveying velocity of the screw is related to the pitch p(x) by
ω
v(x) = p(x) (3)
2π
dθ
Where ω = = angular velocity of the screw, rad/sec, and θ = angular displacement (rad)
dt
The cross-sectional area A(x) is given by

π
A(x) = [ D(x)2 - b(x)2 ] (4)
4

Hence (1) becomes

ω
Q(x) = [ D(x)2 - b(x)2 ] p(x) ηg(x) ηv(x) (5)
8

3.3 Feeder Capacity

At the feeder outlet, when x = L, the capacity of the feeder is given by

ω
Q = 8 [ D2 - b2] po ηg ηv (6)

©AWR CBSPT 9.5

where D and b are, respectively, the screw and shaft diameters at the outlet of the feeder, ηg is
the feeder geometry factor at the outlet and ηv is the volumetric efficiency at the outlet

3.4 Feeder Geometry Factor ηg

The throughput of a screw feeder is influenced, to a small extent, by the thickness of the
helical blade, a loss in throughput, and the effective clearance between the screw and the
feeder casing, a gain in throughput. The effective clearance is the space between the tip of
the screw and the casing which is influenced by the conveying motion of the screw. The
effective clearance is approximately equal to the actual clearance for the lower half of the
screw in the region of the casing. For the upper half where internal shear occurs along the tip
of the screw, the effective clearance is zero. The feeder geometry factor ηg(x), previously
defined, is introduced to account for the variation in throughput resulting from the clearance
and blade thickness. For convenience, the various screw geometrical variables are expressed
as ratios of the screw diameter D(x).

[(1 + n κ ) 2 − χ 2 ][β − ε]
ηg(x) = (7)
β[1 − χ 2 ]

c( x ) b( x ) ts p( x )
where κ= , χ= , ε= and β =
D( x ) D( x ) D( x ) D( x )

c(x) = screw tip clearance ts = blade thickness

Other variables as previously defined.

The constant ‘n’ is introduced to account for the effective clearance. Normally, n ≈ 1.0

By way of example, for the case when κ = 0.02, χ = 0.5, ε = 0.02 and β = 0.8, then from
equation (7), ηg = 1.03. In this case the combined effect of the clearance and blade thickness
has negligible influence on the throughput.

3.5 Volumetric Efficiency - General Discussion

The volumetric efficiency at location x is given by

Q( x )
ηv(x) = (8)
QT ( x )

where QT (x) = Maximum theoretical throughput at location ‘x’ as previously defined

The actual throughput Q(x) will be less than QT(x) due to the following reasons:

(i) The axial velocity of bulk solid will be less than ideal or optimum velocity owing to
rotary or vortex motion imparted by the screw.

(iii) The screw may be operating less than 100% full. This is more likely to be a problem
in screw conveyors operating at higher speeds when the screw has the capacity to

©AWR CBSPT 9.6

 convey the bulk solid faster than the bulk solid can flow by gravity in the hopper to
fill the screw space.

In general terms, the volumetric efficiency is expressed by

ηV = f(ηVR . ηVF) (9)

where ηVR = rotational or vortex component representing the conveying efficiency -

ηVF = fill ratio or 'fullness'

Screw feeders operate at low speeds normally less than 1 rev/sec, and, in almost all cases, the
screw runs completely full, that is ηVF = 100%. Hence the volumetric efficiency is directly
related to the vortex or conveying efficiency.

4. MECHANICS OF SCREW FEEDER OPERATION

4.1 Velocity and Force Diagrams

The mechanics of screw feeder operation are depicted in Figure 6. At an arbitrary coordinate
distance along the feeder “x” and an arbitrary radius “r”, the velocity and force diagram are as
illustrated in Figure 6(a). Figure 6(b) shows the velocity diagram in more detail. As the
screw rotates, a particle of bulk solid moves in a helical path of opposite hand to that of the
screw. VS is the tangential velocity of the screw at the radius considered, VR is the relative
velocity of the particle with respect to the screw surface and VA is the absolute velocity of the
particle. VA is the vector addition of VS and VR. That is,

~ ~ ~
VA = VS + VR (10)

The angle λ defines the direction of the absolute velocity and, hence, the helix angle of the
path followed by the particles at the radius considered.
Vs
VS
α ΔF α
T
λ V α+φ VL
A s VR
VT
VA
λ ΔF
A
ΔF λ
β
VR R
φs
Velocity Diagram Force Diagram Axis of
α
Screw
VLt

(a) Velocity and Force Diagrams (b) Velocity Diagram Showing

Components
Figure 6. Force and Velocity Diagrams for Screw Feeder

©AWR CBSPT 9.7

Referring to the force diagram of Figure 6(a), FA is the axial force due to the internal and
casing friction, FT is the tangential driving force which contributes to the drive torque and FR
is the resultant force due to blade and casing friction. FR acts at the angle λ in the opposite
direction to the absolute velocity VA. The angle φs shown in Figure 6 is the friction angle for
the bulk solid in contact with the screw surface.

In the case of screw conveyors operating in the vertical or steeply inclined position, the helix
angle λ of the path is not constant but increases with increase in conveyor speed approaching
the limit of [90o – (α + φs)], for which β = 0, under high speed operation. In the case of the
horizontal screw feeder, it has been established that β ≈ 0 under all operating conditions.
That is,

λ = 90o – (α + φs) (11)

The screw velocity at the radius ‘r’ is defined by

VS = r ω (12)

πN
where ω = 30 = angular velocity of screw and N = rotational speed in rev/min
The helix angle ‘α ‘ of the screw at the radius ‘r’, expressed in terms of the pitch 'p' is given
by
p
α = tan-1 (13)
2πr
The helix angle of the screw increases from αo at the outer periphery of the screw to αi at the
core or shaft. This means that the helix angle λ of the path of particles of bulk material
decreases from λo at the outer periphery to λi at the core.

The motion of bulk particles relative to the screw surface is defined by the relative velocity
VR where
VS sin λ
VR = (14)
sin (α + λ)

The magnitude of the absolute velocity is given by

VS sin α
VA = (15)
sin (α + λ)

The absolute velocity VA has two components, the useful lifting or conveying velocity
component VL and the undesirable rotating or turning component VT. VL and VT,
respectively, are given by

VL = VA sin λ (16)

and VT = VA cos λ (17)

©AWR CBSPT 9.8

VLt is the theoretical maximum conveying velocity which would occur if the bulk material
could move axially without rotation. This is impossible to achieve in practice, but it does
provide a basis against which to measure the actual performance.

4.2 Vortex Motion in Screw Feeder

Vortex motion in vertical screw conveyors and screw feeders has been studied in some detail
by Roberts [7-10]. Vortex motion is described by the variation of rotational velocity VT with
radius r. The general relationship is

VT rn = const. (18)

where n = vortex motion constant

In general, it has been shown that for screw conveyors and feeders, the index n ≈ 0. That is,
the rotational velocity component VT ≈ constant. The validity of this result may be examined
by investigating the variation of the ratio of the rotational velocity component VT to screw
velocity VS. Referring to Figure 6(b) it follows that

VT sin α cos λ
= (19)
VS sin(α + λ)

Noting that from equation (11), that for a screw feeder, the helix angle of the particle path is
given by λ = 90o - (α + φs), it may be shown that equation (20) can be expressed in terms of
the radius to pitch ratio as

r
VT 1+ 2 π μs p
VS = r (20)
4 π2 (p)2 + 1

where μs = tan φs = coefficient of friction for the screw surface and φs = friction angle

VT
For a given pitch p, this shows that V varies with the radius. It is convenient to compare
S
the rotational velocity component VT with the screw velocity VSo at the outer periphery.
Since VSo = Ro ω, where Ro is the screw tip radius at the location considered and VS = r ω,
then
VT r VT
VSo = (Ro ) ( VS ) (21)

Hence
r
VT 1+ 2 π μs p
r
VSo = (R )[ r ] (22)
o
4 π2 ( p )2 + 1

©AWR CBSPT 9.9

Figure 7 illustrates the predicted vortex motion in a horizontal screw feeder. The graphs
VT VT VT
show and from equations (20) and (22) respectively. As indicated, is
VS VSo VSo
substantially constant. Since, for a given speed of rotation, VSo is constant, then the predicted
vortex motion shows that VT as a function of radius is approximately constant. This is
consistent with a constant tangential velocity vortex in which the vortex index n = 0.

0.7
VELOCITY RATIOS VT/VS & VT/VSo

VT/VS
0.6
VT/VSo

0.5

0.4

0.3

0.2

0.1
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
RATIO r/p

Figure 7. Predicted Vortex Motion in Horizontal Screw Feeder

p
Pitch to Diameter Ratio = 1.0; φs = 20o
D

4.3 Conveying Efficiency

Referring to Figure 6(b), the actual conveying velocity VL, when expressed as a ratio of the
maximum theoretical conveying velocity VLt, provides a measure of the conveying efficiency
allowing for losses resulting from the rotational or vortex motion. It may be shown that

VL tan λ
VLt = tan α + tan λ (23)

Noting from equation (11) that λ = 90o - (α + φs), it may be shown that equation (23)
becomes

VL 1
VLt = tan α tan (α + φs) + 1 (24)

where α is the helix angle at the radius ‘r’ as defined by equation (13) and φs is the friction
angle for the bulk solid on the screw surface.

©AWR CBSPT 9.10

 VL
An alternative expression for the ratio is
V Lt
VL 1 −μs tan α
= (25)
V Lt tan 2 α + 1

where μs = tan φs

r
Equation (25) may be expressed in terms of the radius to pitch ratio, as
p

VL 1+ 2 π μs ζ
=1 - (26)
V Lt 4 π2 ζ2 + 1

r
where ζ=
p

VL
Figure 8 shows the variation of as given by equation (26) for a range of friction angles
V Lt
r
φs. The loss in conveying efficiency with decrease in is quite significant. It is also evident
p
from Figure 8 that good conveying efficiency is dependent on low screw friction.

0.9

0.8
VELOCITY RATIO VL/VLt

0.7

0.6

0.5
Øs = 15 deg
Øs = 20 deg
0.4 Øs = 25 deg
Øs = 30 deg
0.3

0.2
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
RATIO r/p

VL r p
Figure 8. versus ζ = for various values of φs. = 1.0
V Lt p D

©AWR CBSPT 9.11

5. VOLUMETRIC EFFICIENCY

5.1 “Exact” Expression for Volumetric Efficiency

It is assumed that the feeder runs 100% full and that the feeder geometry factor ηg = 1.0.
Referring to Figure 5, the total volume throughput at any location 'x' along the screw is

Ro
Q = 2π ⌠
⌡ VL r dr (27)
Ri

Where Ro = screw tip radius, Ri = core or shaft radius and r = arbitrary radius.

Since, at each location ‘x’, the pitch does not vary with the radius, VLt is constant. The
maximum theoretical throughput is

Qt = π (Ro2 - Ri2) VLt (28)

Hence the volumetric efficiency is

Ro
2 ⌠ VL
ηV = r dr (29)
(Ro2 - Ri2) ⎮ ⌡ VLt
Ri
VL
Substitute for from equation (26), it may be shown that
V Lt
ζo
2 ⌠ 1+ 2 π μs ζ
ηV = ⎮ ζ {1 - }dζ (30)
ζo2 - ζi2 ⌡ 4 π2 ζ2 + 1
ζi

r Ro Ri
where, ζ = p , ζo = p and ζi = p .

The closed form solution of equation (30) is

1 4π2ζ o2 + 1 μs μ
ηV = 1 − ln [ ]− + 2 2s 2 [tan −1 (2πζ o ) − tan −1 (2πζi )]
4 π (ζ o − ζ i )
2 2 2
4 π ζ i + 1 π( ζ o + ζ i ) 2 π ( ζ o − ζ i )
2 2

(31)

Alternatively, equation (30) may be solved by numerical integration.

5.2 Approximate Expression for Volumetric Efficiency

A simpler form of an expression for the volumetric efficiency is obtained by using the
V
average value for the conveying efficiency ( L )av calculated at the mean radius
V Lt

©AWR CBSPT 9.12

 Ro
2 V VL
ηVav = ( L )av
(R o −R i ) V Lt
2 2 ∫ rdr = ( V
Ri Lt
)av (32)

VL
( )av is obtained from equation (26). Hence ηVav is given by
V Lt

1+ 2 π μs ζav
ηVav = 1 - (33)
4 π2 ζav2 + 1

ζo + ζi
where ζav = 2

From equation (24) an alternative form of equation for ηVav may be obtained. That is,

1
ηVav = (34)
tan α av tan(α av + φs ) + 1

where

1
tan αav = (35)
2πζ av

φs = tan-1(μs) = friction angle for bulk solid on screw surface

Values computed using either equations (32) or (34) are compared with the exact solution
using equation (31) in Table 1. As shown, the approximate and exact values are almost
identical.

TABLE 1 - COMPARISON OF EXACT & APPROXIMATE

VOLUMETRIC EFFICIENCIES –
Core or Shaft Diameter One Third Screw Diameter (b/D =1/3)

Ratio φs = 15 deg φs = 20 deg φs = 25 deg φs = 30 deg

p/D Exact Approx. Exact Approx. Exact Approx. Exact Approx.

0.1 0.98474 0.98493 0.98018 0.98037 0.97532 0.9755 0.96999 0.97016

0.2 0.96482 0.96554 0.95578 0.95647 0.94616 0.94681 0.93559 0.9362
0.3 0.94076 0.94217 0.9274 0.92872 0.91318 0.91439 0.89756 0.89865
0.4 0.9132 0.91527 0.89575 0.89761 0.87715 0.87879 0.85674 0.85814
0.5 0.88286 0.88534 0.86157 0.86369 0.83889 0.84063 0.81399 0.81531
0.6 0.85043 0.85293 0.82561 0.82756 0.79916 0.80053 0.77013 0.77086
0.7 0.81658 0.81862 0.78854 0.78982 0.75867 0.75912 0.72587 0.72542
0.8 0.78191 0.78296 0.75097 0.75102 0.71801 0.71699 0.68182 0.67963
0.9 0.74692 0.74648 0.7134 0.71173 0.67768 0.6747 0.63847 0.63405
1.0 0.71203 0.70967 0.67623 0.67242 0.63809 0.63273 0.59622 0.58916

©AWR CBSPT 9.13

5.3 Some Plotted Results

(a) Exact Form of ηV

By way of illustration, equation (31) has been used to determine the volumetric efficiencies
ηV for screws in which the core or shaft diameter is one third that of the screw diameter. The
variation of volumetric efficiency with pitch to screw diameter ratio for a range of screw
friction angles is shown in Figure 9. It is quite evident that the volumetric efficiency
decreases quite substantially with both in crease in pitch and increase in screw surface friction
angle.
1

0.9
VOLUMETRIC EFFICIENCY

0.8

Øs = 15 deg.
0.7 Øs = 20 deg.
Øs = 25 deg.
Øs = 30 deg.

0.6

0.5
0 0.2 0.4 0.6 0.8 1 1.2
RATIO p/D

Figure 9. Variation of Volumetric Efficiency with p/D

Core or Shaft Diameter One Third Screw Diameter (b/D =1/3)

(b) Linear Approximation of ηV

Inspection of the graphs plotted in Figure 9 show that ηV decreases approximately linearly
with increase in p/D. Fitting straight line relationships to each of the curves in Figure 9,
equations of the form given below were obtained
p
ηV = CO - CE D (34)

The values of CO and CE, together with correlation coefficients RC, are tabulated below:

TABLE 2.

φs CO CE RC
15 1.0293 0.30886 0.997
20 1.0266 0.34376 0.998
25 1.0237 0.38094 0.999
30 1.0206 0.42178 0.9995

Equation (34) confirms the earlier research of Roberts et al [3,4] in which the linear
relationship of equation (34) was assumed with the values CO = 1.0 and CE ≤ 0.5

©AWR CBSPT 9.14

(c) Influence on Core or Shaft Diameter on Volumetric Efficiency

The influence of core or shaft diameter on the volumetric efficiency is shown in Figures 10
and 11. These two sets of graphs are based on the screw friction angle of φs = 20o. Figure
10 shows the volumetric efficiency plotted against b ratio for a range of p ratios ranging
D D
from 0.25 to 1.0.

0.95
VOLUMETRIC EFFICIENCY

0.9

0.85

0.8

0.75
p/D = 0.25
p/D = 0.5
0.7 p/D = 0.75
p/D = 1.0

0.65
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
RATIO d/D
Figure 10. Volumetric Efficiency versus b/D Ratio - φs = 20o

0.95 Di/D = 0.25

VOLUMETRIC EFFICIENCY

Di/D = 0.5
0.9 Di/D = 0.75

0.85

0.8

0.75

0.7

0.65
0 0.2 0.4 0.6 0.8 1 1.2

RATIO p/D

Figure 11. Volumetric Efficiency versus p/D Ratio - φs = 20o

©AWR CBSPT 9.15

As would be expected, the volumetric efficiency increases with b ratio, the increase being
D
most pronounced for the p = 1.0. The volumetric efficiency approaches limiting values as
D
b → 1.0, that is when the helical blade ceases to exist. This is of theoretical interest only,
D
the fact that limiting values of the volumetric efficiency are less than 1.0, is explained by the
limiting rotational driving effect of the blade. In practical terms, as the core or shaft increases
in diameter, the rotational driving influence of the shaft becomes more significant.

Figure 11 shows, for the screw friction angle φs = 20o, the volumetric efficiency plotted
against p ratio for three values of b .
D D

6. DISTRIBUTION OF THROUGHPUT ALONG SCREW

The geometry of the screw needs to be examined in relation to the draw-down characteristics
of the hopper. Following the procedures described in Refs.[2-4}, the draw-down
characteristics for the plane-flow, mass-flow hopper depicted in Figure 12 is now
investigated.

Vo(x) Datum Time t = 0

Volume Time t
Distribution

HOPPER
Vt(x) dQy(x)

Q(x) Feed Rate Q

Vt(x)

x dx Q(x)+dQ(x)
FEEDER
L

Figure 12. Operation of Hopper and Screw Feeder System

6.1 Variation of Throughput along Screw

Since the screw capacity increases in the direction of feed, the throughput or feed rate
increases by dQ(x) over distance dx. Hence the gradient of the feed rate or screw capacity at
location x is

dQ(x)
Q(x)' = dx (35)

©AWR CBSPT 9.16

The total throughput at location x is

x
⌠Q(x)' dx
Q(x) =⌡ (36)
0

The units are Q(x) = m3/s and Q(x)’= m2/s

6.2 Draw-Down Pattern in Hopper

Referring to Figure 14, at location x, the continuity of feed from the hopper to the screw is
such that

dQy(x) = dQ(x) = Q(x)' dx (37)

or Qy(x)' = Q(x)' (38)

where dQy(x) = incremental volume feed rate in hopper due to gravity flow

Qy(x)' = volume feed rate from hopper per unit length (m2/s)

At location x, the total volumetric feed V(x) per unit length (m3/m = m2) from the hopper
after time t has elapsed is

t
V(x) = ⌠
⌡Qy(x)' dt (39)
0

If the screw feeder operates at constant speed, then Q(x)' is constant and equation (39)
becomes

V(x) at time t = Q(x)' t (40)

The draw-down pattern may be plotted for increasing values of time t as follows:

Vo(x) = Initial volume per unit length at time t = 0. (m2)

Volume distribution in hopper after time t has elapsed during constant feed

Vt(x) = Vo(x) - Q(x)' t (41)

Expressing "t" in terms of screw revolutions. The time T for one revolution is
2π
T = (42)
ω
Then (51) can be written in terms of the number of screw revolutions 'n' as
2πn
VnT(x) = Vo(x) - Q(x)' (43)
ω
n = 0, 1, 2, 3.....

©AWR CBSPT 9.17

6.3 Uniform Draw-Down

Ideally, the feeder geometry may be selected to achieve a desired draw-down pattern in the
hopper. In most cases, the aim is to obtain uniform draw-down, for which the feeder should
be designed so that
dQ(x)
Q(x)' = dx = Constant (44)

While this is the ideal performance requirement, as discussed in more detail in Section ##,
practical constraints make this virtually impossible to achieve. The aim then becomes to
achieve, as close as possible, the condition described by equation (44).

7. NON-DIMENSIONAL REPRESENTATION

It is convenient to present the foregoing performance parameters in non-dimensional form.

The hopper feeder representation is as shown in Figure 13.

NV (n = 0) Datum Time t = 0

Time t = 2 π n
ωT

NV NQy ' d(x/L) HOPPER

NQ(x/L) Feed Rate NQ

x NQ+NQ'd(x/L)
L

0 1.0 FEEDER

Figure 13. Non-Dimensional Representation of Hopper and Feeder

The screw diameter D at the outlet and feeder length L are selected as the characteristic
dimensions; all other screw geometrical variables are expressed as functions of D and the
coordinate distance x is expressed as a function of L.

7.1 Non-Dimensional Coordinate Nx

x
Nx = (45)
L

7.2 Non-Dimensional Throughput NQ

Q( x )
NQ = (46)
ω D3

©AWR CBSPT 9.18

 πs
where ω = = angular velocity of screw; s = rev/min
30

7.3 Non-Dimensional Gradient of Throughput NQ'

Q( x ) '
NQ' = (47)
ω D2

7.4 Draw-Down Pattern NV

VnT ( x )
NV = (48)
D2

NV is the non-dimensional volume per unit length which is computed at discrete time values
to indicate the draw-down patterns.

7.5 Non-Dimensional Speed Ns

For corresponding speeds

g
Ns = 2
(49)
ω D

8. VARIATIONS IN SCREW GEOMETRY - GENERAL CASE

As demonstrated in Refs.[2-5], the foregoing may be examined in relation to specific screw

geometrical configurations. For practical purposes, linear variations in screw geometries are
considered as follows:

8.1 Screw Diameter

x
D(x) = D [ C11 + (1 - C11) L ] (50)
Di
where C11 = D (51)

8.2 Shaft or Core Diameter

x
b(x) = D [ C21 + (C21 - C22) L (52)
bi b
where C21 = D and C22 = D (53)

Note that C21 ≤ C11

8.3 Pitch
x
p(x) = D [ C31 + (C32 - C31) L (54)

©AWR CBSPT 9.19

 pi p
where C31 = D and C32 = D (55)

It needs to be noted that manufacturing constraints normally require the pitch to increase in
discrete steps of one pitch length as illustrated in Figure 14.

8.4 Area of Cross-Section

In this case, equation (4) becomes

π D2 2
A(x) =
4
[K 1
+ K (x )
2 L
+ K
3
( xL ) ]
(56)
where
K = C2 – C2
1 11 21

K = 2
2
{C 11
(1 – C11) + C
21
(C21 – C
22
}
)
2 2
K =
3
(1 – C11 ) – (C
21
– C
22
)
(57)

p8

p7
p6
p5

p4
p3

p2
p1 p2 p3 p4 p5 p6 p7 p8
Pitch p1

Figure 14. Stepwise Increase in Pitch

9. VERIFICATION OF ANALYTICAL MODEL

Tests were conducted on an experimental screw feeder, depicted in Figure 15, handling
plastic powder [4]. The hopper is constructed of perspex and different screws may be fitted
to allow various geometries to be examined. The results reported here are for a screw with a
constant screw diameter and constant shaft diameter but with an increasing pitch in the
direction of feed. The screw pitch is represented by equation (49) with C31= 0.32 and C32 =
0.78. In order to overcome the end effect at the rear of the feeder, a 'buffer' zone is
introduced as illustrated in Figure 15. Layers of coloured powder were introduced into the

©AWR CBSPT 9.20

hopper to permit the draw-down patterns to be observed. The initial surface of the powder in
the hopper was horizontal.

W
Time 0

Time t

Buffer Zone
L b (Const.)

D (Const.)

x p(x)
Lh

Figure 15. Schematic Arrangement of Experimental Screw Feeder Test Rig

Lh = 700mm; L = 650mm; W = 600mm; D = 150mm; b = 40mm;
α = 19o; C31 = 0.32; C32 = 0.78

Figure 16. Test Hopper and Screw Feeder Looking from Rear

As would be expected for the test screw used in the experiments, draw-down was more
pronounced at the rear end of the hopper. This is verified in the photograph of Figure 16. The

©AWR CBSPT 9.21

measured and predicted draw-down pattern at a particular stage of discharge are compared in
Figure 17. The predicted draw-down assumes the approximated volumetric efficiency as
given by equation (34). The value CE = 0.35 gave the best fit.
250

300

350 Measured Draw-Down

400

450

500

550

Computed Draw-Down C = 0.35

E
600

650
0 0.2 0.4 0.6 0.8 1.0

x/L

Figure 17. Comparison of Measured and Predicted Draw-Down for Experimental

Screw. Initial Surface Horizontal

1 0.04
VOLUMETRIC EFFICIENCY

0.9 0.035
0.8 0.03
AND Q(x)

0.7 0.025
Q(x)'

0.6 0.02
0.5 Vol. Eff. 0.015
Q(x)
0.4 0.01
0.3 Q(x)' 0.005
0.2 0
0 0.2 0.4 0.6 0.8 1 1.2
RATIO x/L

Figure 18. Predicted Performance of Screw Feeder of Figure 15

The predicted performance of the feeder using the more rigorous volumetric efficiency has
been examined, the predicted results being shown in Figure 18. The decreasing volumetric

©AWR CBSPT 9.22

efficiency with increase in x/L is due to the increasing pitch. Even though the throughput
Q(x) increases along the screw as is necessary, the gradient Q(x)' decreases, which confirms
that the feeder will draw-down preferentially from the rear of the hopper.

10. PREDICTING HOPPER DRAW-DOWN PERFORMANCE

The foregoing procedures may be used to predict the interrelation between screw geometry
and hopper draw-down performance.

10.1 Example of Tapered Screw with Constant Pitch

As an example, the case of the tapered screw with constant shaft diameter and constant pitch
as shown in Figure 19 is considered.

(Const.)
0.3D
0.5D

p = D (Const.)

D
L = 5D

Figure 19. Tapered Screw with Constant Shaft Diameter and Constant Pitch
C11 = 0.5; C21 = 0.3; C22 = 0.3; C31 = 1.0; C32 = 1.0; CE = 0.2

The performance characteristics are presented in Figures 20 and 21. Figure 20 shows the non-
dimensional parameters NQ and NQ' as functions of x/L while Figure 21 shows the volumetric
feed per unit length NV at non-dimensional time increments n = 0, 10, 20 and 50. The NV
graphs indicate the draw-down patterns. As can be observed, the feed is biased towards the
front of the hopper. This is also indicated by the NQ' graph which shows an increase with
x/L.
SCREW FEEDER CASE 1 - VARIABLE DIAMETER,
CONSTANT PITCH AND SHAFT DIAMETER
0.10
NQ
0.08 NQ'
NQ and NQ'

0.06

0.04

0.02

0.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2
RATIO x/L
Figure 20. Screw Feeder Case 1. Non-Dimensional Parameters

©AWR CBSPT 9.23

 SCREW FEEDER CASE 1 - VARIABLE DIAMETER,
CONSTANT PITCH AND SHAFT DIAMETER
2

DRAW-DOWN NV
1

0 NV(0)
NV(10)
NV(20)
NV(50)

-1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
RATIO x/L
Figure 21. Screw Feeder Case 1. Draw-Down Patterns

10.2 Optimum Performance

For practical reasons, a screw with a constant diameter is preferred. To achieve the desired
draw-down performance, a tapered shaft in conjunction with a variable pitch is required as
illustrated in Figure 22[2-4].
p
pi Q(x)
p(x) do

Q
di D

x dx
Lt Lp
L

Figure 22. Screw with Constant Diameter, Tapered Shaft and Expanding Pitch

The draw-down performance of the screw feeder is evaluated by considering the gradient of
the throughput along the feed zone. From equation (2) it follows that

dQ(x) ω dA(x) dp(x) d ηV(x)

Q(x)' = = [ dx p(x) ηV(x) + A(x) dx ηV(x) + A(x) p(x) dx ] (58)
dx 2π

For uniform draw-down, Q(x)' should be constant. Since the equations for both Q(x) and
Q’(x) are both non linear with x, the ideal draw-down condition, theoretically, cannot be
achieved. However, it may be approximated by setting Q’(x) to be a maximum at x = L/2.
dQ ' ( x ) L
This requires = 0 at x = as illustrated in Figure 23.
dx 2

©AWR CBSPT 9.24

 dQ(x) = 0
dx
Draw-Down

Constant
Increasing Pitch Pitch

Tapered Shaft

Figure 23. Condition for Optimum Draw-Down

As an illustration, the following example of a screw with tapered shaft and expanding pitch as
illustrated in Figure 22 is considered:
D = 0.15 m; po = 0.05 m ; p = 0.15 m; di = 0.13 m; do = 0.052 m; LT = 2.0 m; φs= 20o; N =
50 rev/min. The predicted performance of the screw feeder is shown in Figure 23
5 0.95

Q(x)
Q'(x)
Q(x) (m^3/h) and Q'(x) (m^2/h)

VOLUMETRIC EFFICIENCY

4 0.9

Vol. Effcy.
3 0.85

2 0.8

1 0.75

0 0.7
0 0.5 1 1.5 2 2.5

COORDINATE DISTANCE x (m)

Figure 24. Predicted Performance of “Optimum” Screw Feeder

dQ ' ( x ) L
Figure 24 shows that = 0 at x = . The throughput curve Q(x) is slightly ‘s-shaped’
dx 2
indicating that the draw-down will slightly more pronounced in the centre that at the ends as
indicated in Figure 23. In practice, there will be some ‘cross flow’ of material so that the
draw-down will tend to be more uniform than indicated in Figure 23.

©AWR CBSPT 9.25

11. FEEDER LOADS

11.1 Vertical Load due to Bulk Solid in Hopper

The loads acting on the feeder and pressure distributions around the screw are illustrated,
respectively, in Figures 25 and 26. FV is the resultant vertical load arising from the vertical
pressure acting over the hopper outlet. As demonstrated in Refs.[11-13], the load FV is
dependent on the stress field generated in the hopper.

p
v
FV Shear
Surface

K p K2 p
1 v v

FA

x dx K3 p
v
L

Figure 25. Forces Acting in Screw Feeder Figure 26. Assumed Pressure
Distribution Around Boundary

During initial filling of the hopper, an active state of stress exists with the stress field peaked
such that the major consolidation stress almost vertical. For this condition, FV will have its
maximum value. Once flow has been initiated, a passive state of stress or arched stress field
is established in the hopper with the major principal consolidation stress acting tangential to
the arch. As a result, the hopper walls provide increased support for the bulk material in the
hopper and the force F reduces to its minimum value. The arched stress field is quite stable
and is retained throughout discharge. Even if flow is commenced and then stopped while the
hopper is still substantially full, the arched or flow stress field is retained. The peaked stress
field is only re-established by refilling the hopper from the empty condition.

11.2 Axial Force

The pressure distribution around the boundary is assumed to be as indicated in Figure 24.
The axial force FA comprises two components:

(i) The force FAS to shear the bulk solid along the shear surface

(ii) The force FAC to slide the bulk material along the casing

The component FAS is given by

FAS = μE1 FV (59)

Where the equivalent friction coefficient μE1 is

©AWR CBSPT 9.26

 μE1 = Kf sin δ (60)

δ = Effective angle of internal friction Kf = Constant such that 0.8 ≤ Kf ≤ 1.0

The component FAC may be estimated by

FAC = μc FV (K1 + K2 + K3 ) (61)

where K1, K2 and K3 = Pressure ratios μc = Casing friction coefficient

The pressure ratios K1, K2 and K3 are assumed to have the following values

0.6 ≤ K3 ≤ 1.0 ]
] (62)
0.4 ≤ (K1 = K2) ≤ 1.0 ]

For the pressure ratios K1 and K2, the lower limit 0.4 may be approached for the static case
and the upper limit 1.0 for steady flow. In the case of low feed velocities, the values of K1
and K2 will normally be in the middle of the range, say 0.6 to 0.7.

The total axial force FA is

FA = FAS + FAC (63)

The force is assumed to be uniformly distributed. The average force acting over one screw
pitch is
p
ΔFA = FA L (64)

12. SCREW TORQUE

Consider a section of the screw as depicted in Figure 27. Since the screw is driven, then the
forces acting on the blade are as illustrated. ΔFT is the tangential force required to drive the
screw and move the particle, the other force components being as previously described.

p
ΔFR
Ro
α+φs
ΔFA
r r
Ri

ΔFT dr
dr
Figure 27. Forces Acting on Screw Blade

©AWR CBSPT 9.27

12.1 Screw Torque per Pitch

Referring to Figure 25, the gradient of the torque with respect to radius is

dT
dr = 2 π σa tan (φs + α) r2 (65)

σa = axial pressure. The total torque acting over one pitch length of the screw is

Ro
T = 2π ⌠
⌡ σa tan (φs + α) r dr
2 (66)
Ri
which becomes
Ro
r
⌠ 1 + 2 π μs p
T =
2 ΔFA ⎮ r2 [ ] dr (67)
Ro2 - Ri2 ⎮ r
2 π p - μs
⌡
Ri

Equation (67) may be solved by numerical integration

12.2 Approximate Solution

The screw torque may be approximated by lumping the mass at the effective radius Re for
which the helix angle is αe such that

p
tan αe = (68)
2πRe
In this case equation (66) becomes
Ro
Tapprox = 2 π σa tan (φs + αe) ⌠
⌡ r2 dr (69)
Ri
Also Tapprox = σ a π(R o2 - R i2 ) R e tan (φs + αe) (70)

From equations (69) and (70),

3 3
2 Ro - Ri
Re = [ 2 ] (71)
3 R o - R i2

Hence Tapprox = ΔFA Re tan (φs + αe) (72)

Where ΔFA = σ a π(R o2 - R i2 ) (73)

Alternatively, Tapprox may be expressed as

Tapprox = ΔFA p NTa (74)

©AWR CBSPT 9.28

 2 ζo3 - ζi3 1 + 2 π μs ζe
where NTa = 3 [ 2 ][ ] (75)
ζo - ζi2 2 π ζe - μs

NTa = non-dimensional torque

r Ro Ri Re
ζ = p , ζo = p , ζi = p and ζe = p

As an example, plots of the non-dimensional torque for a range of screw friction angle for a
shaft one third of the screw diameter and a p/D ratio of 1,0 is given in Figure 28.

2.5

Øs = 15 deg

2 Øs = 20 deg
NON-DIMENSIONAL TORQUE NT

Øs = 25 deg
Øs = 30 deg
1.5

0.5

0
0 0.2 0.4 0.6 0.8 1 1.2
RATIO p/D

Figure 28. Non-Dimensional Torque versus Pitch to Diameter Ratio.

p/D = 1.0; Core or Shaft Diameter = 1/3 Screw Diameter

12.3 Total Torque due to Screw

Noting that the screw geometry varies, as illustrated in Figure 3, the total torque is obtained
by summing the torque along the length of the feeder. That is,

L
T = ⌠
⌡ T(x) dx (76)
0
From equation (63)
L
T = FA' ⌠
⌡ p(x) NT(x/L) dx (77)
0
FA
Where FA' = L = Distributed axial load (assumed constant)

p(x) = Pitch at location x

NT(x/L) = NT at location x

L = Length of feeder

©AWR CBSPT 9.29

Where the pitch increases in discrete steps as illustrated in Figure 14, the total torque is given
by
T = ∑ Tp (78)

where Tp = torque per pitch

12.4 Torque due to Shaft

For the shaft, the torque per pitch 'p' is

Tsh = σn 2 π Ri2 p μsh (79)

FV
where σn = Ks L(D + 2C) (80)

FN = Normal Force
L = Length of screw in feed hopper zone
D = Screw diameter
C = Clearance around screw
μsh = Friction coefficient between bulk material and shaft
σn = Average pressure acting normal to shaft

The factor Ks in equation (70) accounts for the pressure distribution around the shaft. It is
assumed that

Ks = 0.75 K + 0.25 (81)

Where 0.4 ≤ K ≤ 1.0 depending on degree of agitation

The total torque due to the screw shaft is obtained by summing the torque obtained by
equation (79) for each pitch location.

12.5 Example

Consider the experimental screw feeder of Figure 15 which handles plastic powder of bulk
density ρ = 550 kg/m3. The effective angle of internal friction is δ = 50o, the hopper wall
friction angle is φ = 20o, the friction angle for the screw surface is φs = 20o, the friction angle
for the powder on the screw casing surface is φc = 25o. The feeder loads are calculated in
accordance with the method described in Ref.[12]. For the flow condition in the hopper, the
non-dimensional surcharge factor is qf = 1.15. The vertical load acting on the feeder is

FV = qf ρ g L B2 = 1.15 x 550 x 9.81 x .65 x .162 = 103 N

From equations (59) and (60),

FAS = 0.8 sin 50o x 103 = 63.1 N

©AWR CBSPT 9.30

From equation (53) with K1 = K2 = 0.4 and K3 = 0.6; μc = tan 25o = 0.466,

FAC = 0.466 x (0.4 + 0.4 + 0.6) x 103 = 67.2 N

Hence FA = 130.3 N and FA’ = 103/.65 = 158.5 N/m

The non-dimensional torque NT, distributed torque and cumulative torque for the screw blade
have been calculated and are shown in Figure 18. The total screw blade torque for the feeder
is 34.3 Nm. To this must be added the torque computed for the screw shaft, which is Tsh = 5.5

Nm. The total screw torque is 39.8 Nm. To this must be added the torque computed for the
extension of the screw beyond the discharge end of the casing.

40 0.6

35 0.55

NON-DIMENSIONAL TORUE
30 0.5
NT
TORQUE N m

25 0.45

20 0.4

15 0.35
Cumulative Torque
10 0.3
Distributed Torque
5 0.25

0 0.2
0 0.2 0.4 0.6 0.8 1 1.2
x/L

Figure 29. Computed Torque Values for the Screw Feeder of Figure 15.

13. CONDITION FOR AXIAL CONVEYING

It has been assumed throughout this paper that axial conveying occurs. For axial conveying,
the maximum retarding torque resistance due to the combined effect of the feeder trough or
casing and the bulk material in the hopper immediately above the screw is equal to or greater
than the fully developed torque due to the screw and shaft. This is illustrated in Figure 30.
That is, the friction torque developed by the screw surface and the screw shaft, Tscrew, should
be less than the external friction braking torque, Tbraking, developed by the casing and
hopper.

Attention to the axial conveying problem has been drawn by Rademacher [14] who proposed
an analysis which simulated the screw behaviour by rotating discs spaced at a distance equal
to the screw pitch. This allowed a critical pitch to be determined which, in turn, enabled the
screw helix angle to be calculated. An alternative analysis has been undertaken in which the
actual screw geometry is taken into account. The condition for forward motion is examined
under low rotational speed of the screw. This will give the worst case. Once the bulk
material rotates under higher speed of operation, then the centrifugal pressure will generate

©AWR CBSPT 9.31

additional braking torque, thus enhancing the feeding action. The analysis summarised
below ignores any centrifugal pressure effects.
T screw

Shear Surface

T braking

Figure 30. Diagram to Illustrate Torques Generated in Screw Feeder

13.1 Torque Ratio for Axial Transport

Tscrew
Defining the torque ratio as ΤR = T , it may be shown that
braking

Cp Kj 1 - χ 3
[ 3 ] Nta + χ 2 μsh
2β
TR = (82)
μav

2 β + (χ +1) π μs
where Nta = tan (αav + φs) = (83)
(χ + 1) π - 2 β μs

Ri b
χ = R =D
o
p
β = D
p
Cp = p = Number of screw starts
i
p = lead or pitch of single blade
pi = pitch of adjacent blades
μs = coefficient of friction for bulk solid in contact with screw surface
μsh = coefficient of friction for bulk solid in contact with the shaft
μav = average resistive friction coefficient

μc + μi
μav = 2 for hopper section (84)

μav = C μc for extended section beyond hopper (85)

μi = coefficient of internal friction of bulk solid
μc = coefficient of friction for bulk solid in contact with casing

©AWR CBSPT 9.32

If extended section is 100% full, C = 1.0. If extended section is greater than 50% full but less
than 100% full, 0 < C < 1.0
σa
Kj =
σn
σa = axial stress or pressure σn = normal stress or pressure

For axial transport, the resistive torque Tc must be greater than the total screw torque (Ts +
Tsh). This means that ΤR < 1.0, otherwise the bulk material will rotate without moving
axially. The limiting condition is for TR = 1. The average friction coefficient for the trough
or casing and hopper combination should be such that

Cp Kj 1 - χ 3
μav ≥ [ 3 ] Nta + χ 2 μsh (86)
2β

It is reasonable to assume that the pressures σn and σa are approximately the same, that is Kj
= 1.0. With this assumption, for illustration purposes, the limiting or lower bound values for
μav have been computed and are plotted in Figures 31 and 32. The two graphs of Figure 31
compare a three start screw with a single start screw. Significantly higher values of μav are
required in the case of the three start screw in view of the greater surface area per pitch/lead.
In the case for the single start screws of Figure 21, conveying or feeding is more readily
achieved. The graphs show that a wider variation in the μav values occur at the larger values
b
of χ = D .

1.2
AVERAGE FRICTION COEFF. µav

Single Start Screw

1 3 Start Screw

0.8

0.6

0.4

0.2
.

0 0.2 0.4 0.6 0.8 1 1.2

RATIO p/D

p
Figure 31. Limiting or Lower Bound Values of μav for Transport as a Function of
D
b
for Single and Three Start Screws - k = = 0.45; μs= tan 20ο = 0.364.
D

©AWR CBSPT 9.33

 AVER AGE FR IC TIO N C OEFF . µ
0.6 .

b/D = 0.2
0.5 b/D = 0.4
b/D = .6
0.4 b/D = 0.8

0.3

0.2

0.1
0 0.2 0.4 0.6 0.8 1 1.2
RATIO p/D
p
Figure 32. Limiting or Lower Bound Values of μav for Transport as a Function
D
for Single Start Screws - μs= tan 20ο = 0.364

14. CONCLUDING REMARKS

This paper has examined the influence of screw geometry on the volumetric performance and
torque requirements of screw feeders. The validity of the theory and methodology presented
has been confirmed by results obtained from tests on an experimental screw feeder. The
volumetric efficiency has been examined in some detail and equations have been developed
to permit the volumetric efficiency to be determined. Procedures for obtaining a desired
feeder performance such as uniform draw-down in the hopper are given, uniform draw-down
being a desired goal if segregation of feed is to be avoided. Equations for the torque required
to drive screw feeders have been derived, the methodology involving the application of non-
dimensional torque parameters. The conditions for axial conveying or feeding have been
investigated.

13. REFERENCES

1. Bates, L. "Entrainment Patterns of Screw Hopper Dischargers". Trans. A.S.M.E., Jnl.

of Engng for Industry, Vol. 91, Series B, No.2, May 1969. (pp.295-302).

2. Roberts, A.W. "Volumetric Capacity of Screw Feeders". Interim internal Report to

POSTEC Research, Porsgrunn, Norway, May 1990.

3. Roberts, A.W. "Determining Screw Geometry for Specified Hopper Draw-Down

Performance". Proc. Bulk 2000 Conference, Instn. of Mech. Engrs., London Oct.
1991.(pp.111-116).

4. Roberts, A.W., Manjunath, K.S. and McBride, W. “The Mechanics of Screw Feeder
Performance for Bulk Solids Flow Control" Transactions of Mechanical Engineering,
The Institution. of Engineers., Australia, Vol. ME 18, No. 1, 1993. (pp.67-73).

©AWR CBSPT 9.34

5. Roberts, A.W. and Manjunath, K.S. "Screw Feeder Design for Controlling the Gravity
Flow From Storage". Research Report, POSTEC Research, Porsgrunn, Norway,
October 1993.

6. Roberts, A.W. and Manjunath, K.S. "Volumetric and Torque Characteristics of Screw
Feeders". Proc. Powders and Bulk Solids Conference, Rosemont, Ill., USA, May 1994
(pp.189-208).

7. Roberts, A.W. and Willis, A.H. "Performance of Grain Augers", Proceedings of the
Institution of Mechanical Engineers, Vol. 176 (8), 1962. (pp.165-194)

8. Roberts, A.W., "An Investigation of Grain Vortex Motion with Relation to the
Performance within Vertical Grain Augers". Proc. of the Institn. of Mech. Engineers,
Vol. 178 (1), Nr. 12 (1963-1964).

9. Roberts, A.W., "Screw or Auger Conveyors Revisited", Proc. of 16th Annual Powder
and Bulk Solids Conference, Chicago, USA., May 1991.

10. Roberts, A.W. “Aspects of Attrition and Wear in Enclosed Screw Conveyors”. Proc.
Powders and Bulk Solids Conference, Chicago, USA, 1993.

11. Roberts, A.W., Ooms, M and Manjunath, K.S., "Feeder Loads and Power
Requirements in the Controlled Gravity Flow of Bulk Solids from Mass-Flow Bins"
Trans. I.E.Aust., Mechanical Engineering, Vol. ME9, No.1, April 1984.

12. Manjunath, K.S. and Roberts, A.W., "Wall Pressure-Feeder Load Interactions in Mass-
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