Design Models of Trommels for Resource Recovery Processing

DESIGN MODELS OF TROMMELS FOR RESOURCE
RECOVERY PROCESSING
HARVEY ALTER
Chamber of Commerce of the United States
Washington, D.C.
JEROME GAVIS
Department of Geography and Environmental Engineering
Baltimore, Maryland
MARC L. RENARD
National Center for Resource Recovery, Inc.
Washington, D.C.
terial riding on the barrel
ABSTRACT
surface
d
The design of a trommel requires knowledge of
the number of impingements necessary to achieve a
desired efficiency of separation of undersized ma­
E(xo,xm)
=
size of a spherical particle
=
efficiency of separation of
particles in the size range
terial from a feed stream and of the radius, length,
Xo EO;; x <,xm
inclination angle, and rotational speed necessary to
f
provide the required number of impingements for a
=
fraction of a barrel revolution
per particle impingement
specified mass feedrate. Expressions for predicting
the number of impingements required are derived,
f(x)
based on expressions for the probability of passage
of a particle through a hole, and for the particle
size distribution according to number, The dynamics
F(xo,xm)
of particle motion in the trommel are described and
=
size distribution of particles
=
according to number
"
,
fract on of articles in the
�
�
feed in the SIze
,
Xo EO;;x EO;;xm
used to derive equations for predicting the dimen-
g
=
gravitational constant
sions of the trommel and the rotational speed, Ex-
h
=
vertical height 'to which a
amples are given to show how the equations may be
particle rides on the barrel
applied.
=
trommel per impingement
NOTATION
Meaning
a
Dimensions
(mass
length,
time)
a round hole, short side of a
rectangular hole
=
size of the long side of a rec­
tangular trommel hole
b
=
n
= number of impingements
=
mass feedrate
needed to achieve a desired
efficiency
of a square hole, diameter of
A
= length of the trommel
L
M
= size of a trommel hole - side
p
=
Probability that a particle
passes through a hole per
L
impingement
p
L
=
probability that a particle
passes through a hole in n
impingements
thickness of the layer of ma-
361
L/t"
L
distance a particle moves
longitudinally in the
Symbol
L
L
L
L
M/t
P(X)
P(xo, xm)
=
=
Q
=
R
=
t
=
x
=
Xg
=
=
{3
=
=
=
Pb
=
=
r
=
probability that a particle
of size x passes through a
hole in n impingements
INTRODUCTION
probability that particles in
the size range Xo <:x <:xm
pass through a large number
of holes
ratio of hole area to screen
area, percent open area
inside radius of the trommel
L
time of flight during a particle trajectory
t
size of a particle in a feed
with a nonuniform size
distribution
L
geometric mean size of particles with a log-normal size
distribution
angle above the horizontal
diameter to which a particle
rides
•
angle of inclination of the
trommel
angle of impingement rela­
tive to the vertical diameter
of the barrel
angle of impingement rela­
tive to the normal to the
barrel surface at the point
of impingement
density of material in the
layer riding on the barrel
surface
MIL 3
logarithmic standard devia­
tion of sizes of particles
with a log-normal size dis­
tribution
residence time of particles
in the trommel
THE SIZE SEPARATION PROCESS
t
=a
derived factor, listed in
Table 1
=a
derived factor, listed in
Table 1
w
=
angular rotational velocity
of the trommel
Rotary screening is growing in acceptance as a
separation method in the' processing of municipal
solid waste (MSW) for resource recovery. The use
of rotary screens, commonly called "trommels," in
solid waste processing can be traced at least as far
back as 1928 [1] . In fact, a proposal to use a trom­
mel for such purposes was reported as early as
1924 [2]. More recently, trommels have been in­
corporated in various places in the processing
streams of resource recovery plants in Monroe
County, New York [3], Ames, Iowa [4], New
Orleans, L)Usiana [5], Baltimore, County, Mary­
land [6], and Bridgeport, Connecticut [7], in
North America, and in several plants in Europe [8].
Trommels have been used in minerals processing
for more than four centuries. Because their use in
resource recovery processing is recent, there is an
almost complete lack of reported methods of de­
sign and scaling factors for applications in resource
recovery processing. Recognizing this lack, several
investigators initiated experimental programs in
order to obtain empirical design and scaling criteria
for particular resource recovery applica tions [9-11].
None of the published reports, however, leads to an
understanding of the screening mechanism, nor
provides methods by which trommels can be de­
signed and scaled for resource recovery and other
applications.
The purpose of this paper is to present a quanti­
tative description of trommel screening and to pro­
vide at least an approximate method for the design
and scaling of trommels that will be generally ap­
plicable in resource recovery (and other) processing.
Lft
Once the maximum dimension of the under­
sized material has been decided and the screen hole
size fIxed, it is necessary to determine the number
of times the material to be screened must impinge
on the screen surface in order to achieve the desired
separation. The design is established when the dimen­
sions and rotational speed of the trommel that will
provide the required number of impingements are
specifIed.
The determination of the number of impinge­
ments needed is a problem in probability; the num­
ber is a function of the cumulative probability that
the desired number fraction of undersized material
passes the screen. When a spherical particle of diam-
362
eter d impinges on a square hole of side a(a > d),
the probability that it will pass through the hole is
p
=
d
I-­
a
2
(1)
Q
an equation originally given by Gaudin [12]. This
equation also describes the probability that the par­
ticle will pass through a circular hole of diameter a.
The factor Q is the ratio of the hole area to the
total screen surface area. As Gaudin pointed out,
this is a lower limit on the probability, since reflec­
tion from the edges of the hole is ignored. Equa­
tion (1) is strictly correct for particles impinging
normal to the screen surface, although there is only
a small change for small deviations from normal
impingement [12] . When the hole is rectangular,
the probability becomes [12]
p
=
l_d
a
1- � Q
(2)
A
where a and A are the rectangular dimensions of
the hole.
The cumulative probability that a particle will
pass through a hole after n impingements is
p
=
n
� p (1 - p)
j= 1
i-I
=
I- (1 - p)
n
(3)
In the absence of particle-to-particle interaction,
this is also the cumulative probability that a large
number of uniformly sized particles will pass
through a large number of holes after n impinge­
ments.
If the particles are not uniform in size, the
probability that particles of size x 1 will pass after
n impingements is
1-(1- p)
n
f (xd
(4)
where f(;c 1 ) is the number fraction of particles of
size XI, i.e., the size distribution function of the
particles evaluated at X I. The probability for all
particles equal to or greater than a minimum size
Xo and equal to or less than a maximum size xm is
the integral
n
xm
dx (5)
P (xo,xm) = J f (x) 1- (1- p)
Xo
an equation first given by Sucher [13] .
Sucher has also given an alternative form for
Eq. (5), suggesting that the probability of particles
passing at the nth impingement is a function of the
composition of the material after the nth impinge­
ment rather than of the material approaching the
nth impingement from the (n - I) st impingement.
Then
363
It is not evident, a priori, whiCh of these equations
better describes reality.
Real particles, of course, are not spherical. One
means of broadening the derivation to include non­
spherical particles is to alter the distribution func­
tion, f (x), to reflect an equivalent spherical size dis­
tribution by inclusion of suitable shape factors for
particles in each size range. An alternative is to ob­
tain an overall shape factor empirically for given
types of feed materials by comparison of the equa­
tions with experimentally determined probabilities.
The action of a trommel is to cause the feed
material to impinge n times on the screen surface
as it passes through; Eq. (6) expresses the cumula­
tive probability that particles ranging in size be­
tween Xo and Xm wil l be screened out during their
sojourn in the trommel. The total fraction of par­
ticles in this size range in the feed is
which is the cumulative size distribution between
Xo and Xm. The efficiency of the trommel is then
E (xo,xm)
=
P (xo, xm)/F (xo, xm)
(8)
In words, the efficiency is expressed as the ratio of
the number fraction of undersized material removed
to the number fraction of undersized material in
the feed. It is important to note that the efficiency
is expressed in terms of number fractions rather
than weight fractions, which are usually measured.
Because Eq. (5-8) allow the efficiency to be cal­
culated as a function of n, when the size distribu­
tion,! (x), is known, the number of impingements
necessary to achieve a desired fractional separation
of undersized material from the feed can be deter­
mined. Because the increase in probability resulting
from reflection of particles from the edges of the
holes has been neglected, the number of impinge­
ments needed to attain a specified effic.iency may
be somewhat lower than the calculated number in
practice. This is offset, however, by the decrease in
probability resulting from particle-to-particle in­
teraction, e.g., adhesion or deflection by collision,
or from blinding of holes. Because such interactions
increase as feedrate increases, the number of im­
pingements needed to achieve a specified efficiency
must increase as feed rate increases.
The difficulty, if not impossibili ty, of modeling
the effects of reflection from the hole edges and
particle-to-particle interactions for the complex
mixture of sizes and shapes of the materials that
are processed in resource recovery facilities pre­
vents a priori modification of the equations to
account for them. The calculation of the necessary
number of impingements by means of Eq. (5-8)
should, therefore, be viewed as a starting point to
which correction factors, obtained empirically for
different types of feed streams, may be applied in
order to obtain more accurate estimates.
At this point, the equations describe any screen­
ing action, whether in a trommel or other kind of
screen.
THE RIDING ANGLE
The angle above the horizontal diameter at
which a particle of mass m will fly from the sur­
face is a function of the angUlar velocity, w, and
the radius of the trommel, R. Figure 1 a shows that
as long as the particle remains on the surface, the
centrifugal force, w2R m, is equal to the sum of the
normal force, T, exerted by the surface and the
normal component of the gravitational force,
mg sin a. When T= 0, the particle will leave the sur­
face. The condition for this is
w2R
g
THE DYNAMICS OF TROMMEL ACTION
=
sma
•
(9)
The angul'ar velocity needed to reach the vertical,
The rotational motion of the barrel and its in­
a = 1T/2 (90 de g.), is the critical angular velocity
clination relative to the horizontal provide the
above which the particle will ride on the surface
means by which material fed to the trommel is
without falling. Thus, sin a is the square of the
made to impinge on the screen surface. The num­
fraction of the critical angular velocity for which
ber of impingements a mass of particles experiences the particle will ride to angle a before leaving the
during its residence in the trommel is a function of surface.
the trommel dimensions, its rotational speed, and
Actually, because the .barrel is inclined with re­
angle of inclination.
spect to the horizontal, it is necessary to multiply
In classical applications in minerals processing,
the denominator of the left side of Eq. (9) by the
trommels are rotated at angular velocities at which
cosine of the inclination angle, �. Because � is seldom
the material within them rides less than 1T/9
seldom more than 1T/36 (5 deg. ), however, cos� � 1
(20 deg.) above the horizontal diameter before fall­ and may be neglected.
ing back and impinging on the material riding be­
low [14]. Observation of trommels used in resource THE FLIGHT TRAJECTORY
recovery processing, however, indicates that they
The point-by-point description of the actual
are rotated at higher angUlar velocities so that ma­
trajectory is of little interest. What are needed are
terial rides higher above the horizontal diameter.
the relationship between the point of landing at the
Then, when the component of the gravitational
bottom of the barrel and the angle a, and the time
force normal to the surface becomes equal to the
interval, t, during which the particle is in flight.
centrifugal force, the particle leaves the surface,
For the case where the particle lands at the low­
rising at first because it still has vertical momentum, est point of the barrel, at the vertical diameter, the
but then arcing and dropping to the bottom of the
vertical distance from where the particl e leaves the
barrel. * While there does not appear to be any
surface to where it strikes the bottom of the barrel,
documentation of the reasons for such a mode of
as il lustrated in Fig. 1 b, is given by
operation, it is evident that this causes break-up of
gt2 - wRt cosa (10)
(R + h) R (1 + sina)
aggregated masses of particles, increases mixing,
and helps prevent blinding of screen holes. In fact,
if, as before, the small inclination of the barrel from
"lifters" are incorporated in many resource recov­
the horizontal is neglected. From this
ery processing trommels in order to enhance "flight"
112
2
2
wt = sina cos a + [sin a cos a + 2 sina (1 + sina)]
of particles from the surface. The quantitative de­
(11)
scription of the mechanism of trommel. action lead­
ing to design and scale-up criteria described below
The first term on the right is the time to reach the
is based on this mode of operation. Ufters, how­
apogee of the trajectory and the second is the time
ever. are not considered_
to fall from the apogee to the bottom of the barrel.
Having a horizontal velocity, wR sin a, the particle
travels the horizontal distance R cos a during this
The operation in minerals processing may be described as
time. When R cos a is set equal to wRt sin a, the
a "slumping mode" and in resource recovery as a "cascad­
ing mode."
result, after simplification, is
=
*
364
=i
mljl
h
'"
I
I
'1"'-Trajectory
R
R
o.
I
I
b.
I
I
I
;'
-.
Trajectory -WI(
I
"7
,
I
I
I
8
c.
lEGENDS FOR
. FIGURES
FLIGHT TRAJECTORIES IN A
TROMMEl
a. Forces Acting on a Particle Causing It to Ride on the
Barrel to Angle Ck
b. The Trajectory for Impingement at the Vertical Diam­
eter, 6 =0
c. The Trajectory for Impingement at Angle 6 from the
Vertical
FIG. 1
PARTICLE
cos3 a = [sin' a cos' a + 2 sin a {1 + sina)pl2 sin a
(12)
wt=sinacosa+ [sin'acos'a + 2 sina(coso +sina) f2
(13)
Equation (12) is transformed to
Because of Eq. (9) and the fact that sin' a + cos2
a = 1, Eq. (12) may be solved for sin a, to yield
cos3 eX + sin 0 = [sin' a cos2 a
sin a = 0.5, or a =n/6 (30 deg.).
11'
sin a (cos 0 + sin a)] sin a (14)
+
2
There is no a priori reason that the particle must
This has the solution
land at the vertical diameter. It is simple to show
that, if the particle is to land at an angle 0 from the
sin 0 = -cos 3a; cos 0 = sin 3a; 0 = 3a - n/2 (15)
vertical, as illustrated in Fig. lc, Eq. (11) becomes
365
TABLE 1 TROMMEL DESIGN PARAMETERS
-
f
3.141
0.758
2.42
0.5972
3.279
0.712
2.38
0.5630
3.340
0.677
2.31
sin
a
0.08687T
(15.6°)
0.6428
o
n
a
-
0.5495
0.06877T
(12.4 °)
3.355
0.663
2.27
0.5000
0.06057T
(10.9°)
3.366
0.609
2.12
0.05287T
(9.5°)
3.3l3
0.552
1.92
3.193
0.491
1.68
3.014
0.427
1.41
0.4488
0.1487T
(26.7°)
0.3961
0.1297T
(23.3°)
0.3421
The first three columns of Table 1 list values of 0:
and sin 0: for several values of {j.
Because the particle (in the event it does not
pass through a hole) has a horizontal velocity,
wR sin 0:, as well as a vertical velocity, given by
the product of g and the radical term on the right
side oJ Eq. (1 4), it impinges on the surface at an
angle, 1/, whose cotangent is the ratio of the vertical
to horizontal velocities. Thus, after simplification
112
1
os
{j
2
)
+
�
cot 1/ = .
(3 sin2 0:
(16)
sm 0:
sm0:
The angles 1/ are given in column 4 of Table 1 for
the listed values of 0: and {j. The angles 1/ and {j are
equal at 0.0711 7r (12.8 deg.) when 0: = 0.1 901T
(34. 3 deg. ). That is, the particle impinges normal
to the surface, and
w2 R
(17)
= 0. 563
g
I
an angle (3, as shown in fig. 2, the particle has a
horizontal component of velocity wR cos 0: sin (3.
During the time the particle is in flight it moves a
horizontal distance wRt cos 0: sin (3 and a distance
parallel to the inclined axis wRt cos 0: tan (3, where
t is given by Eq. (1 4) and tan (3 and sin (3 have been
approximated by (3, since (3 is a small angle in practice. In addition, the particle advances along the
barrel a distance eqUivalent to the product of total
distance the particle falls vertically and sin (3. Neg­
lecting cos (3, as before, the particle advances a dis­
tance R(3 (cos 0: + cos {j).
The total horizontal displacement per impinge­
ment is then
n
THE N UMBER OF IMPIN GEMENTS REALIZED
•
Q = R(3 (wt cos 0: + cos 0: + cos {j)
(18)
which may be written
Q
(3R
=
<I> =
(wt cos 0: + cos 0: + cos {j)
(1 9)
Values of <I> corresponding to the values of {j and 0:
given in columns 2 and 3 of Table 1, are listed in
column 5 of Table 1 . * The length, L, of the trom­
mel needed to produce n impingements is
Because the vertical velocity of a particle as it
leaves the barrel surface is actually a velocity per­
pendicular to the barrel axis which is inclined at
366
THE FEEDRATE
Equations
wRt coso
(9) and (20) do not completely
de­
termine the design of a trommel. They provide only
sinfJ
two relationships, given the hole size, among the
W�
COSOI
onll
four design parameters w, R, L, and (3. In particular,
they contain no dependence on feedrate.
At a cross-section of the barrel near its entrance,
material riding on the barrel in a layer of thickness
b along the angular section (8 + rr/2 + a) occupies a
FIG. 2 PARTICLE MOVEMENT ALONG THE INCLINED
BARREL SHOWING VELOCITIES AND DISTANCES
L
=
nQ
=
cross-sectional area bR (8 + rr/2 + a). Material in
flight occupies a cross-sectional area equivalent to
(20)
cI>n(3R
the material that rode on the barrel during the time
During each revolution of the barrel, the particle
of flight, t , or bRwt . The material is moving longi­
rests against the surface during the angular displace­
tudinally along the barrel at a velocity (w/2rr)Q/f
ment (8 + rr/2 + a). During the time the particle is
The product of the cross-sectional area occupied by
in flight, the angular displacement of the barrel is
the material and its velocity along the barrel near the
wt. Thus, the barrel makes the fraction of a revo­
entrance is the volumetric feedrate. When multiplied
by the density of the material in the layer, Pb *, this
gives the mass feedrate,M After substitution from
lution
(21)
Eqs.
by
where t is given by Eq. (13), per impingement of
the particle on its surface. The reciprocal of [ is the
'It = (sina)1I2 (wt cosa + sina + cos 8)
[, corresponding to the values of 8 and a given in
columns 1 and 3, are listed in column 6 of Table 1.
7 of Table 1 for corresponding values of 8 and a.
to the rotational frequency w/2rr. nlU�
n[
[
R
J
g sin a
Equation (23) provides another relationship
among the variables if Pb is known and b is speci­
fied. Although allowing b to increase permits larger
(22)
feedrates at a fixed R, doing so may decrease effici­
It has been assumed in the derivation of these
ency by allowing increased particle-to-particle inter­
equations that the center of mass of a particle
action, hole blinding, etc.
rides on the surface of the barrel at the radial dis­
tance, R, from the axis. The center of mass of a real
particle of finite size must actually ride at a distan'ce
smaller than R from the axis. Moreover, if material
rides several particles thick on th� surface, the dis·
is, of course, no way to account for the dynamics
of each particle. As an approximation R, in the
as 8 decreases. The gain from larger 8, however,
may be offset by the lower efficiency caused by the
mately averages about which the real values may
resulting departure from normal impingement when
be expected to scatter.
O.
0> 0.0711 rr (12.8 deg.). Experimental investiga-
Horizontal trommels are used but
or internal
scrolls forming an Archimedes
*
screw to "pump" the material through. Obviously, Eq.
(18)
8 decreases for a given R and (3. Alternatively,
larger radius is needed at rlXed (3 at a given feedrate
lated from the equations, will, then, be approxi­
=
than the average dimension of the feed material
Because 'It decreases as 8 decreases, M decreases
riding on the barrel surface. The quantities calcu­
f3
not evident, a priori, that magnitudes of b greater
as
less one-half the average thickness of the material
A special case is
mine the depth of material, b, that produces the
lead to more optimal designs.
equations, may be taken to be the trommel radius
with lifters
It is necessary to resort to experimental observa­
tion on different types of feed materials to deter­
greatest observed efficiency at the smallest R. It is
tances will be different for different particles. There
*
(24)
(3 because (3 is small. Values of 'It are listed in' column
mel is the ratio of the number of revolutions, nt,
= 2rrn[
= (
w/2rr)
(23)
with t given by Eq. (13), and with sin (3 replaced by
The residence time, T, of a particle in the trom­
T
(20) and Simplification, M is expressed
where
number of impingements per revolution. Values of
1I2
(9) and
does not hold in such cases. Presumably other rela­
Possibly Pb equals the bulk density of the feed, measured
in the usual way. Howaver, this is not certain and Pb will
have to be measured experimentally by observation of
the feed ena of the rotating trommel.
tions can b e developed to describe the movement.
367
tion is needed to ascertain the best values of {) to be
used with different types of feed materials. It is
reasonable, in the meantime, to specify normal im­
pingement, with {) = 0.071 rr (12.8 deg.) and
Q = 0.190 rr (34.3 deg.).
TO DESIGN A TROMMEL
The purpose of a trommel is to separate under­
sized material in the size range Xo .;; x .;; xm with
an efficiency E(xo, xm) from material fed to the
trommel at a mass rate,M In order to design a .
trommel to do so, it is necessary to specify the hole
shape and size, the fractional area of holes, the
rotational speed, radius,length, and angle of in­
clination.
Common practice in resource recovery processing
specifies circular holes (in order to minimize re.ten­
tion of textile and plastic material) and an angle of
inclination of rr/36 (5 deg.). The hole size is deter­
mined by the dimension xm. The fractional area of
holes should be as close to unity as possible con­
sistent with the structural strength of the barrel. Its
speCification is a function of structural design and
is not considered further here. Otherwise, the de­
sign is determined by Eq. (8) with Eqs. (7) and (5)
or (6), Eqs. (9), (20), and (23).
Equation (8), with Eqs. (7) and (5) or (6), allows
calculation of the number of impingements neces­
sary to attain the required efficiency. Equations
(9), (20), and (23), with sin Q = 0.5630, <I> = 3.34,
and '11 = 2.31 then determine w, R, and L for the
design feedrate and necessary number of impinge­
ments. This requires knowledge of the density Pb
and specification of the depth of material,b,riding
the trommel barrel surface. Until the results of ex­
periment and practice indicate otherwise, b may be
taken equivalent to the mean size of the feed ma­
terial.
N UMBER O F IMPINGEMENTS REQUIRED­
RAWMSW
Raw MSW is most often characterized according
to mass of material in different size ranges. Winkler
and Wilson. [15] , however, have given number histo­
grams for raw MSW from Cambridge, Massachusetts,
and Middlebury, Vermont, in different maximum
particle dimension ranges. While distribution accord­
ing to maximum dimension does not account for
particle shape, the distribution functions derived
from the histograms can be used at least for illus­
trative purposes.
The histograms for MSW from both cities may
be fit by the log-normal distribution
f (x)
=
1
...[2irri X 0
ln
2
(25)
..j2oln
The geometric mean size Xg = 6.0 in. (152 mm) and
the standard deviation 0ln = 0.56 for Cambridge,
Massachusetts, MSW. The number fraction of minus
4.75 in. (120 mm) material in MSW from that city
is given when Eq. (25) is inserted into Eq. (7).
2
4.75
F(O, 4.75) = 0.713
� exp
o
-
1.26 Qn
dx
6.�0
(26)
The probabilities, given by either Eq. (5) or (6),
add Eq. (1), with the insertion of Eq. (25), are
4.75
P(O, 4.75)
=
0.713
o
1
-
1
-
1
exp
x
-
2
x
1.26 Qn
6.00
x
Q I+
4.75
2
n
(27)
dx
4.7S
P(0, 4.75
=
0.713
o
APPLICATION
x
1
-exp - 1.26 Qn
6.00
x
x
1-1-Q 1+
4.75
As
examples of how the ideas developed may be
applied in resource recovery processing, Eqs. (5-8)
will be used to ascertain the number of impinge­
ments needed to separate undersized material from
raw MSW through 4.75 in. (120 mm) round holes,
and the number of impingements needed to separate
metal cans from a solid waste stream through holes
of the same size. Then Eqs. (9), (20), and (23) will
be used to illustrate the calculation of rotational
speed, radius, and length, for a typical feedrate.
exp -
X ,
QnXg
2
-n
2
(28)
dx
Integration of Eqs. (27) or (28) with different
values of n gives P(O, 4.75) as a function of n.
Division by F (O, 4.75) gives the efficiency of
separation,E (0, 4.75), as a function of n.
The integrals have been solved numerically by
means of Simpson's approximation method. Fig­
ure 3, where n has been plotted as a function of
Eon a log-normal grid,illustrates the results. Curves
368
A, B, and C show n as a function of E (0, 4.75),
determined for the probability calculated from
Eq. (27) with Q = 1 ,0.5, and 0.3 respectively.
Curve D shows n as a function of E(O, 4.75) for
the probability calculated from Eq. (28) with
Q = 0.5.
The most striking feature of the curves is that
they each consist of two straight lines of different
slope that are connected by a transition curve. In
this regard, they are similar to experimentally de­
termined curves of efficiency as a function of siev­
ing time on flat screens described by Whitby [16] .
The number of impingements necessary to achieve
a given efficiency is approximately inversely pro­
portional to Q, as a comparison of curves A, B, and
C show. The number of impingements at a given
efficiency for probabilities calculated from Eq. (28)
are slightly lower than those for probabilities calcu­
lated from Eq. (27) along the lower parts of the
curves, but differ negligibly along the upper parts
of the curves, as curves B and D illustrate.
If the upper integration limits of Eqs. (26) and
(27) or (28) are decreased, the integrations give the
probabilities of separating < 4.75 in. (1 20 mm)
material in a trommel with 4.75 in. (1 20 mm) holes.
Curve E of Fig. (3) is a plot of n against E (0, 4.5)
with Q = 0.75, while Curve F is a plot of n against
E (0, 4.25) with Q= 1 .' Remarkably, the efficiency
is log-normally distributed with respect to number
of impingements up to very high separation effici­
encies. Moreover, the number of impingements
needed to achieve a given efficiency is very much
smalle�, especially at high efficiencies, than when
undersized material up to the hole size is to be'
separated.
N UMBER OF IMPINGEMENTS REQUIRED­
100
U)
...
z
'"
:::E
'"
to
Z
Cl.
:::E
10
0.1
'l'o
80 9095
99
EFFICIENCY
FIG. 3 NUMBER OF IMPINGEMENTS NECESSARY TO
ACHIEVE SPECIFIED EFFICIENCIES FOR REMOVAL
OF UNDERSIZE MATERIAL RAW MSW IN A TROM·
MEL W ITH 4.75 IN. (120 MM) HOLES
A, B, C. Removal of Minus 4.75 in. (Minus 120 mm)
Material, with Probabilities Calculated According to Eq.
(27). for Q = 1,0.5, and 0.3 Respectively
D. Removal of Minus 4.75 in. (Minus 120 mm) Material,
with Probabilities Calculated According to Eq. (28) for
Q 0. 5
=
E. Removal of Minus 4.5 in. (Minus 114 mm) Material,
with Probabilities Calculated According to Eq. (27) for
Q = 0.75
F. Removal of Minus 4.25 in. (Minus 108 mm) Material,
with Probabilities Calculated According to Eq. (27) for
Q = 1
in. (1 20 mm). The distribution has the form
CANS
If recovery of food and beverage cans is an ob­
jective, a trommel may be an effective means of
separating them from raw MSW. If interference by
other materials is ignored emphasis may be placed
on efficiency of separation of cans if the equations
are written for distributions relative to cans alone.
For simplicity, only undamaged cans of one size
will be considered, 2.5 x 4 in. (64 x 1 02 rom ).
The probability of passage of a can through a
hole is dependent upon the orientation at which it
impinges. All orientations occur with equal proba­
bility, however. Therefore, the cans present them­
selves as particles uniformly distributed with re­
spect to a longest dimension that ranges between
that of its diameter and that of its diagonal, 4.75
10 20 40 60
1.0
x
[( )
=
1
-== 0.444 (29)
=
4.75 _ 2.5
-x-m -x-o
�
The number fraction F (2.5,4.75) is easily shown
to be unity. The probabilities are either
4.75
P (2.5, 4.75)= 0.444
or
P(2.5,4.75)= 0.444
2.5
1 -1-Q 1 -
X
n
4.75
4.75
2.5
2
1 - 1 - Q 1+
�
4 5
2 -n
dx
dx
(31 )
Here, the efficiencies, E (2.5, 4.75), are equal num­
erically to the probabilities.
Figure 4 illustrates the results of numerical in­
tegration of Eqs. (30) and (31 ). The curves are
369
is taken to be 5 Ib/ft3 (80 kg/m3), Eq. (23) with
'It = 2.31 and {3 = 1T/36 (5 deg.), gives R = 5.1 ft
(1.5 m). The reqUired rotation speed is 18 rev/min,
according to Eq. (9) with sin a: = 0.563.·
If 70 percent efficiency of removal of minus
4.75 in. (minus 120 mm) material by a trommel
with Q = 0.5 is specified, curve B of Fig. (3) shows
that n = 63. Then, Eq. (20) with <I> = 3.34, gives
L = 94 ft (28.5 m). If Q were 0.75, interpolation
between curves A and B of Fig. 3 would give
n = 45 for which the length would be 67 ft (20.4m).
Even smaller lengths would be specified if removal
of minus 4.5 in. (minus 114 mm) material through
4.75 in. (120 mm) holes were required. Thus, for
70 percent of efficiency of removal of such material
by a trommel with Q = 0.75 curve E of Fig. 3 shows
that n = 30, for which L = 45 ft (13.7 m).
Curve A of Fig. 4 shows that for Q = 0.5 and
n = 63 the efficiency of removal of cans is 67 per­
cent, while interpolation between curves A and B
of that figure shows that for Q = 0.75 and n = 30
the efficiency of removal of cans is about 60 per­
cent.
similar to those of Fig. (3). Curves A, B. C are for
probabilities calculated according to Eq. (30 for
Q = 1,0.5,0.3 respectively. The curve for proba­
bility calculated according to Eq. (31) for Q = 0.5
is undistinguishable from curve B.
100
(/)
fZ
'"
::E
'"
(!)
z
-
11.
::E
-
10
0.1
1.0
10 20 40 60
80 9095
99
WHAT HAS B E EN ACCOMPLISHED­
% EFFICIENCY
WHAT NEEDS TO BE DONE
FIG. 4 NUMBER OF IMPINGEMENTS NECESSARY T O
The development presented provides means for
ACHIEVE SPECIFIED EFFICIENCIES FOR REMOVAL
calculating:
OF 2.5 X 4 IN. (64 X 1 00 MM) CANS FROM RAW MSW
a. The number of impingements necessary to
IN A TROMMEL
separate undersized material from the feed to the
A, B. C. Removal with Probabilities Calculated According
to Eq. (30), Through 4.75 in. (1 20 mm) Holes, for Q = 1 ,
trommel, in any size range up to the hole diameter,
0.5, and 0.3 Respectively
as a function of separation efficiency. The latter is
D. Removal with Probabilities Calculated According to
defined as the ratio of the fraction of undersized
Eq. (30) through 5 in. (1 27 mm) Holes for Q 1
particles separated to that in the feed. In order to
calculate the fraction it is necessary to know the
Curve 0 illustrates how the number of impinge­
ments needed for a prescribed efficiency is marked­ distribution of particle size according to number in
ly decreased when holes larger than the can diagonal the feed. There is little to distinguish between the
are used. It was calculated for 5 in. (127 mm) holes two methods of calculating probabilities of passage
through the holes.
from the probabilities of Eq. (30) for Q = 1 with
b. The radius of the trommel necessary to pro­
the term (1 - 1/4.75) replaced by (1 - 1/5.00) in
cess material at a specified feedrate, if the thickness
the integral. Thus, a small increase in hole size re­
and density of the material riding the barrel surface
duces the number of impingements necessary to
attain any efficiency, especially at high efficiencies. are known and the angle of inclination is specified.
c. The length of the trommel needed to achieve
The efficiency is log-normally distributed with re­
the number of impingements calculated.
spect to number of impingements up to 0.99.
This provides the complete design for specific
applications e�cept for structural factors. There
TROMMEl DESIGN
are, however, lacks in information about materials
If, for example, the trommel is to process 60
tons/hr (55 t/h) of raw MSW and if an average den­
At this value of a, Ii = 11 or the particles impinge normal
sity for material riding the barrel surface at a depth
t o the screen surface which is the maximum size hal .. ,
equal to the geometric mean size, 6 in. (152 mm),
hence most probable and average value for d.
=
*
370
in resource recovery processing that introduce un­
certainty:
a. Size distributions of raw MSW and of other
process streams have usually been reported accord­
ing to weight fraction, but not according to num­
ber fraction.
b. Shape factors for the irregularly shaped ma­
terials fed to trommels in resource recovery pro­
cessing are not known.
c. Densities of the materials that ride on the
barrel surface have not been reported.
In addition, several important parameters in �e
derived design equations are imperfectly specified:
a. Because particle shape factors are unknown,
reflection from the hole edges has been omitted
from the derivation of efficiencies.
b. While normal impingement has been suggested
for use in design, optimal efficiency may occur at
other impingement angles.
c. The thickness of the material layer riding the
barrel surface that leads to the optimal design for a
given effici�ncy of separation and feedrate may not
be equivalent to a mean dimension of the feed ma­
terial.
There is need for better characterization of feed
materials with respect to size distribution according
to number and to shape factor, and for experimen­
tal programs in which optimal impingement angles
and thicknesses of materials riding the surface are
investigated for different feed materials. It is hoped
that the results presented here are suffiCiently pro­
vocative to inspire needed investigation. They are
but a first step toward a rational method for the
design of trommels for size separation in resource
recovery processmg.
•
ACKNOWLEDGMENTS
The beginnings of this work were encouraged by
support from Contract 68-03-2632 from the U.S.
Environmental Protection Agency, Office of Re­
search and Development, Mr. Carlton Wiles, Project
Officer.
The work was supported, in part, through Indus-
trial Research Participation Grant SPI-7907391
from the National Science Foundation.
REFERENCES
Anon., "A W estminster W onder," My Magazine
(July 1 924), reprinted, Solid Wastes, Vol. 66, 1976, pp.
536-539.
[1]
[2]
Anon., "The Govan Refuse Plant," Engineering,
Vol. 126, 1928, pp. 577, 641, 710.
[3] Carlson, D., Spencer, D., and Christensen, H.,
"Monroe County Resource Recovery Project," Proc. Fifth
Mineral Waste Utilization Symp., 1976, pp. 196-203.
[4] Funk, H. D. and Russell, S. H., "Energy and Ma­
terials Recovery System," Proc. F ifth Mineral Waste
Utilization Symp., 1 976, pp. 133-140.
[5] Bernheisel, J. F., Bagalman, P. M., and Parker,
W. S., "Trommel Processing of Municipal Solid W aste
Prior to Shredding," Proc. Sixth Mineral Waste Utilization
Symp., 1978, pp. 254-260.
[6] Willey, C. R., "The Maryland Environmental Ser­
vice/Baltimore County Resource Recovery Facility, Texas,
Maryland," Proc. Sixth Mineral Waste Utilization Symp.,
1978, pp. 280-285.
[7] Easterbrook, G. E., "The Acid Test for Bridge­
port," Waste Age, Vol. 9, No. 5, 1 978, pp. 46-52.
[8] Alter, H. and Dunn, J. J., Energy from Waste American and European Experiences, Marcel Dekker, NY ,
in press ( 1980).
[9] W oodruff, K. L., "Preprocessing of Municipal
Solid W aste for Resou rce Recovery with a Trommel,"
Trans. AIME, Vol. 260, 1976, pp. 601-604.
[1 0] Woodruff, K. L . and Bales, E. P., "Preprocessing
of Municipal Solid W aste for Resource Recovery with a
Trommel - Update 1977," Proc. 1978 National Waste
Processing Conf. ASME, New Y ork, pp. 249-257.
•
[1 1 ] Warren, J. L ., "The Use of a Rotating Screen as
a Means of Grading Crude Refuse for Pulverization and
Compression," Resource Recovery and Conservation,
Vol. 3, 1978, pp. 97-112.
[12] Gaudin, A. M., Principles of Mineral Dressing,
McGraw Hill, New Y ork, 1939, Chapter VII.
[13] Sucher, R. W ., Sieving - Theoretical and Ex­
perimental l nvestigations, Ph.D. Dissertation, Kansas
State Univ., 1 969.
[14] Taggart, A. F., Handbook of Mineral Dressing,
J. Wiley & Sons, New Y ork, 1954, Section 7.
[15] Winkler, D. F., and Wilson, D. G., "Size Charac­
teristics of Municipal Solid W aste," Compost Science,
Vol. 14, No. 5, 1 973, pp. 6-11.
[16] W hitby, K. T., "The Mechanics of Fine Sieving,"
Amer. Soc. for Testing and Materials, STP 234, 1958,
pp. 3-25.
Key Words
Classification
Concentration
Design
Engineering
Mathematical Model
Separator
371