A New Trommel Design

Resources, Conservation and Recycling, 6 ( 1991 ) 1-22 Elsev ier Science Publishers B.V. / Pergamon Press plc

A new t r o m m e l m o d e l Richard Ian Stessel Department of Civil Engineering and Mechanics, Universityof South Florida, Tampa, FL 33620-5350, USA (Received 26 November 1990, accepted 11 June 1991 )

ABSTRACT Stessel, R.I., 1991. A new trommel model. Resour. Conserv. RecycL, 6: 1-22. Rotary screens are in an important unit operation in materials processing. Much of their design has been empirical. A mathematical model was developed with three major components: particle rise on the screen, particle trajectory through the air, and screening of the particle while in contact with the screen. All were implemented on a computer using numerical methods, allowing the retention of all necessary mechanisms within the theoretical expressions. Particle rise incorporated friction; particle trajectory incorporated drag. The screening element made use of entirely new probabilistic theory differing from previous work by incorporating consideration of the depth of the bed. Results showed good predictive capabilities. Insights concerning the importance of bed layering were obtained. Further recommendations aiding design were obtained and analyzed.

INTRODUCTION

Trommels, or rotary screens, are an important unit operation in materials processing. They have a long history in the mineral dressing industry, and are of particular interest to those concerned with solid waste processing because of the difficulty of the material processing task. Trommels provide an elegant, low maintenance and operating cost, screening option that often justifies their higher initial cost. A trommel is a rotating, cylindrical screen, lying on its side, at a small angle from the horizontal. Material is fed in at its elevated end, and size-separation occurs as the material spirals down the drum at an axial speed governed by the angle of the drum to the horizontal. Behavior of the material normal to the drum axis is governed by the rotational velocity. An example of an important trommel application is the removal of small particulate material ("fines") from shredded waste. Trommels are favored for this task, but the separation is often inadequate [ 1-3 ]. Current design practice has evolved by experience. As waste processing options become more varied, roles for trommels (and other unit operations) become more specialized. As example is a trommel used to differentiate be0921-3449/91/$03.50 © 1991 Elsevier Science Publishers B.V./Pergamon Press plc. All rights reserved.

2

R.I. STESSEL

tween 2 1 plastic soda bottles, and cans and bottles, and a trommel used to separate humic material from a mined landfill. The work reported here attempts to broaden the applicability of trommels by enhancing the understanding of the underlying physical principals of operation. This paper presents a theoretical, descriptive or research model of trommel operation, described by equations of the mechanics of particles moving through the cylinder. The distinction must be made between a descriptive model that describes, in terms of meaningful output parameters, the behavior of a physical system based upon input parameters; and a prescriptive model, that allows the designer to input desired output as well as input parameters, having the model produce a finished design. Descriptive models allow a researcher to ask "open-ended" questions concerning the effect of varying input parameters, including those pertaining to design; a prescriptive model requires output parameters to be set, and adjusts the design accordingly. The descriptive model discussed here evolved from underlying physical principals, implemented in computer code in a manner not practicable for earlier researchers in the field. Two discrete efforts employing the model are discussed next. The first is its use to replicate experimental data from the literature, drawing conclusions from the required modifications of input parameters. The second is use of the model to draw conclusions by analyzing the effect of altered input parameters. MATHEMATICAL MODELING HISTORY AND DEVELOPMENT

As the speed of revolution of a trommel increases, it goes through different operating modes. At the slowest speed, the trommel contents tumble, called cascading. Mixing action is insufficient to adequately move the center of the tumbling mass to the outside so that it is presented to the screen openings. At the highest speed, particles are held against the outside of the drum by centrifugal force. In between is a motion called cataracting, when the drum rotates at a speed just slightly less than that needed for centrifugation. Particles are held on the wall of the drum against gravity until just before the top of the drum, where gravity overcomes the centrifugicity and the particles fall almost along the drum diameter to the bottom [ 4 ]. Under these conditions, particles fall the maximum distance possible, creating the greatest amount of tumbling, leaving few to cascade. These motions are shown in Fig. 1. Particle movement in cataracting has, most often, been divided into three basic phases: the particle rises along the screen; it departs from the screen and falls until it hits the screen again; while it rests on the screen, some of the material passes through the screen. Fig. 2 shows the key variables and coordinates. Each type of motion is addressed below in turn.

A NEW TROMMEL MODEL

3

BAND

@@0 (a)

(b)

(c)

Fig. l. (a) Cascade or kiln action; (b) cataract action showing band: (c) centrifuging.

(3

Fig. 2. Definition sketch.

Particle trajectory Most of the past reports addressing trommel behavior were geometrically oriented. Alter et al. [5] focused principally on departure location and an assumed location angle of impact. The actual trajectory of the particle in flight was of little significance to their analysis. Glaub et al. [6, 7] added more factors to Alter's paper. All these researchers specified the initial condition that forces on the particle are instantaneously balanced. The particle leaves the screen when the centrifugal reaction is no longer able to hold the particle on the screen against gravity; this gives the particle an instantaneous fixed velocity vector. None of these approaches fully included the effects of drag. Glaub et al. [6 ] began a discussion, but left the drag force coefficient unspecified. In the case of small particles, when, for example, minerals are pulverized m or in the case of light particles, such as paper or very small particles in solid waste - - the drag on a particle becomes an important component in the force balance. With trommel diameters of 1 to 2 m, it is quite possible that the aerodynamically lighter particles will have achieved terminal velocity in flight [ 8 ].

4

R.I. STESSEL

The present work uses the same initial conditions as in the Alter et al. and Glaub et al. above, which were the velocities instantaneously upon departure (see Fig. 2 ): 0Rcosa = z0

( 1)

ORsina=ko x and z are the horizontal and vertical locations of the particle. The weight of a particle, Fe, acting through the angle of inclination fl, is defined: F e = mcosflg

(2 )

where rn is the particle mass and g the acceleration of gravity. The inclination of the trommel, fl, is included because it determines the movement in the axial direction, i.e., down the trommel. When describing movement across the trommel, the small angle approximation (with fl< n/12, cosfl.~ 1 ) justified its omission [ 5-7 ]. The drag on a body is: FD=~CDpVIvl

n -~

(3)

V=(Vp--Va) where va is the velocity of air, and vp is the velocity of the particle. Drag is a function of the square of the difference between air velocity and particle velocity. To preserve the sense of direction, this is calculated as the net velocity multiplied by its absolute value. Particle trajectories come from a pair of equations balancing drag and gravity and neglecting buoyancy [9 ]. Solving for acceleration in the horizontal and vertical directions, respectively, these are: 1

J~=m[FD'x]

(4)

l

2 = m [ --rng+FD,:] using rn for mass. Calculation of the sphere Reynolds number, particularly in the case of the addition of airflow, shows that a drag coefficient correlation incorporating both the laminar and turbulent regimes is required; one of many statistical fits to the drag coefficient curve is employed [ 10 ]. These equations are solved parametrically using a fifth-order Runge-Kutta routine. Computation ends when the particle reaches the wall again, and R2=x2+z 2

(5)

The movement clown the trommel depends on both the axial component of

A NEW TROMMEL MODEL

3

the vertical departure velocity, and the angle of declination of the trommel. The z-direction shown in Fig. 2 is tilted into the page by the declination angle, ft. The vertical coordinates of the point of departure and impact, ZD and z~, respectively, are known, and the increment of axial movement during one turn in the particle's spiral path down the drum is: lA =

[ (ZD--Z 1) + ~ot]Sinfl

(6)

In addition to its importance in calculating overall trommel length, the increment of axial movement is required to determine the thickness of the bed during screening as discussed below. Together, these equations serve to determine the motion of the airborne particle within the trommel.

Particle rise The rise of particles in contact with the rotating screen gives the departure location of the particles and the area of screen occupied by the material. The contact are on the screen occupied by particles in one pass is defined as a band as shown in Fig. 2. The band area is critical to determining both the amount of screening that occurs, and the axial distance travelled, during each rotation. Screening occurs only when the particle contacts the screen, and is a function of the screen area contacted and the percentage of screen area devoted to holes. Ultimately, this yields the design length of the device. The traditional approach is to first determine a point of departure, then determine the rotational velocity, a, of the screen that would have the particle leave the screen at that point. This results from a force balance between the centrifugal reaction and the force of gravity [ 5-7]: (gsino~/R )~

(7 )

This analysis does not apply if the material slips against the screen. Glaub et al. [6] reported analyses incorporating friction. Separate force balances were required to calculate motion without slippage, motion with slippage, and the transition between slippage and sticking. It is necessary to distinguish between 0, which is the location of the particle, and o~, the location of a point on the screen. As with the simple force balance in Eq. 7, the mass terms cancel, leading to calculation of accelerations: AR in the radial direction and Av in the tangential direction:

AR= --R ( O)2--gsinO

(8)

A-r=RO= ~./dA R - - g c o s 0

(

9)

where/~ is the dynamic friction coefficient. Combining b = fld 0 2 - - g

[/~osin0 + cos0 ]

(1o)

0

R.1. STESSEL

This equation is solved by the same fifth-order Runge-Kutta method used in the trajectory calculations. Whether or not the particle is slipping is determined by checking the force balance between the tangential gravity force and static friction. This is done by determing ifEq. 9 is positive, after substituting Fts for #d. If the static force is larger, the particle velocity is that of the screen. Similarly, departure from the screen is determined by checking Eq. 8 at each time interval. The screening rate is a function of the area of the screen to which the particles are exposed as they rise. The inclusion of particle slippage means that the material rides over more screen openings than would be determined strictly by dividing the band area by the hole frequency. The increased number of holes is calculated by determining the time that it takes a point on the screen to cover the arc-length of the band area, the time that it takes the feed actually to rise through the arc-length, and taking the ratio of those times:

E= (trise)/(OI-OD)-¢

(1 1 )

0

/rise is calculate during the numerical integration. The capacity of the screen and the processing rate is a function of depth of material on the screen. The screen is conceptually divided into bands, as described above. The screen area occupied by one band is determined by multiplying the width of the band (see Eq. 6) by the arc-length of the band. The arc-length between the location of departure and the location of impact is: lr=R [2re- ( o q - O~D)]

(12)

The depth of bed, T, is then determined by dividing the flow into the trommel occuring during one rotation, VB, by the band area:

T= VB/IAlr

( 13)

Together, calculation of the particle trajectory and particle rise describe the movement of the particle. These are assembled into an independent computer model to analyze particle motion only. Full description of trommel operation requires incorporation of screening.

Screening While the material is moving on the screen, it is being separated. The model incorporates the screening of particles in conjunction with their rise up the screen, in the band. Trommel research in the literature has been probabilistic, considering individual layers, one particle thick. The central concept concerned the ratio of the opening size to particle diameter, attributed to Gaudin's handbook [ 11 ]

A NEW TROMMEL MODEL

7

by Alter et al. [ 5 ] (also derived in [ 12 ] ). The fraction of particles passing, f, was"

f( dh,d)=n[ d-dh/ d]

(14)

where d is the diameter of the trommel screen holes, dis the particle diameter, and H is the fraction of the screen area made up by holes. This expression was integrated to determine the total fraction of particles of diameter less than the hole diameter that pass, and made recursive to determine the effect of multiple passes. A single layer is a difficult concept to justify when the objective of screening is to separate a bed of material made of multiple particles sizes. Layers above that in immediate contact with the screen were not incorporated. Vorstman and Tels [ 12 ] took the thickness of the bed into account by determing surface concentrations arising from depletion of the bed, but were still bound by probabilities of single particles passing. Below, field data support surface concentration as an important phenomenon. A simple experiment was performed. A layer of granular material was placed on a plate with an obscured hole. All the particles were considerably smaller than the hole opening. When the hole was opened, a cone-shaped depression was formed, as one would expect. With a range of particle sizes, the number of particles passing is a function of the probability that a particle of any size larger than the hole would block the hole. An average number of particles passing through a hole is calculated by knowing the number-concentration of particles both larger and smaller than the hole: PN(dp)

(15)

NP= fdm'XPN(dp)ddp d dh

where Np is the number of particles passing, PN is the number particles size distribution (PSD), dh is the diameter of the trommel openings, and dmax is the diameter representative of the largest particles. The bottom integral gives the number-fraction of particles larger than the hole. For each particle size, Eq. ( 15 ) gives the average number that would pass before the hole is blocked by any of the particles larger than the hole. To obtain the volume PSD of particles passing:

Vr,=f, [PN(d)V(d)/ f2 o., PN(d)dd]dd

(16)

rain

where Vp is the total volume of particles passing, drain is the diameter of the smallest particles in the material, and V(d) is the volume of the particle of diameter d. Equation 16 gives an absolute volume of particles passing through a single opening of a given size, given input number-PSDs. Number-PSDs are easily obtained from the more common mass-PSDs by dividing by density,

8

R.I. STESSEL

summing, and dividing each interval's value by the total number of particles. Volume-PSDs and mass-PSDs are equal if the particle density is constant. Often, when better information does not exist; for example for homogeneous materials such as coal, the assumption is valid. With this model, a bulk density different from particle density may be specified to calculate bed dimensions. The total volume of a given feed that could pass through a hole is a function of material properties. In the case of the hard, granular material discussed above, a clearly-defined cone penetrates through the bed. The wall angles are the angle of repose. In the case of flake or soft material, an angle of repose could be greater than zc/2. Here, the wall angle is not so clearly related to the angle of repose because of bridging. Thus, a cone angle is defined that differs from the angle of repose. The cone angle can vary through n radians. For materials likely to have steep angles of repose, the cone angle should be measured with the material on the screen. This can be accomplished by running the trommel, stopping it, and introducing a slim ruler through an opening from underneath. Upon measuring the depth of penetration, the void volume could be taken as a cone, and the cone angle calculated. In the model, an algorithm was developed for calculation of frustum volume depending upon cone angle and depth of bed. The total volume of material passing the bed is:

lIT ,(/ VpB if VpB< Vh ~ Vh otherwise

VpB= VB

Pvdd

(17)

min

l/h (EHVp if Vp< Vc = ~EHVc otherwise where V~ is the volume of an individual frustum, Vh is the volume of particles calculated from passing through holes, Vpa is the total volume of particles in the band, and Pv is the volume PSD. This sequence of equations set maxima for the number of particles passing through the screen: first, the particles passing could not exceed the volume of the cones in the band; second, the particles could not exceed those contained in the band as a whole. Additionally, particle screening is calculated in the case of a striated bed, with particle diameter increasing as a function of distance from the screen. Originally, this was done to determine the effect of introducing airflow. However, the work of Vorstman and Tels [12] discussed above, and data from Barton [13], showed that striation is evident in normal trommel operation .The thickness of separate layers is calculated for each input particle size increment. A cone or frustum is defined as before, based solely upon the di-

A NEW TROMMEL MODEL

9

ameter of the trommel opening and the cone angle, defined as in Fig. 3. This is subdivided into frustums based upon the thickness o f each sequential layer, until a size increment is reached that is too large to pass through the trommel opening. All those frustums pass through the screen in their entirety. The PSDs of particles passing is determined by the volumes of the frustums containing the different particle size increments. These frustums are illustrated in Fig. 3. In determining the net volume and particle size distribution of particles, these mechanisms do not operate in isolation. Each mechanism was separately calculated and then, a parameter was included governing the fraction of particles passing obeying striation, with the remainder acting as a mixed bed.

Model operation A strength o f this work was the initial decision to implement full differential equations without simplifications using numerical methods for integration. This made the c o m p u t e r essential to implementation.* Input is divided into: general particle characteristics, particle size distribution, and trommel characteristics. Specifically, input variables are: General particle characteristics: • Mass feedrate; • bulk and materials densities;

TOP OF BED..........

T-----v- ............

LAYERSOF PARTICLES .......... LARGERTHAN OPENING

................ \ / iI

LAYERSOF PARTICLES SMALLERTHANOPENING

/ FRUSTUMS~_ j_ OF PARTICLES__~+

/ ;NG~/---

\ PASSING __~

CONE --SCREEN

SCREENOPENING

Fig. 3. Pattern of sequential frustums in striated bed. *The model is written in FORTRAN 77, and can be run on any computer with a suitable compiler. For this work, it was run on a variety of Intel-based microcomputers. Run time varies with the number of increments in the specification of the particle size distribution and the number of rotations requested. On microcomputers, a typical run for this work would range from about 8 hours for a PC to 2 rain for an Intel 80486 processor.

10

R.1. STESSEL

• static and dynamic coefficients o f friction, which can be set to 1 to model the effect of flights; and • cone angle, which became an important modeling variable since it was not measured in any of the work reported in the literature. Particle size distribution: • N u m b e r of increments; • whether the user desires to input a particle size distribution or have the model calculate a R o s i n - R a m m l e r distribution; • in the event the modeler wishes to input a distribution; for each particle size increment: • The representative particle diameter; • either the raw mass of that component, or its percent of the total. • In the event the modeler wishes the model to calculate the Rosin-Ram m l e r distribution: • Diameters representative o f the largest and smallest particle size increments; • shredder exponent; • critical size. T r o m m e l characteristics: • T r o m m e l diameter; • rotational velocity; • hole size; • n u m b e r of holes per unit area, which could be specified as a smaller number than the actual n u m b e r of holes in cases where effective screen area is less than the band area; • trommel incline; • the fraction of the particles obeying the striation p h e n o m e n o n discussed above. One complete set of output is provided for every rotation m a d e by a hypothetical particle spiraling down the trommel. For each revolution, output contains: rotation number, cumulative axial distance travelled, total time elapsed, mass of material remaining in the screen, cumulative mass in the unders, particle size distribution of the material remaining in the screen, particle size distribution of the unders passing through the screen on that revolution, angle of departure, and angles of impact for each particle increment. The appendix provides a sample output for a very limited n u m b e r of revolutions (limited for readability) with an idealized input. This model is run to produce data on trommel performance. Variation of the input parameters is combined with an understanding of how they affect the underlying theory to draw conclusions. USE O F T H E M O D E L

This model is used in two separate ways. In the first, published data from trommel tests are used. Parameters available in the model are adjusted, and

A NEW TROMMEL MODEL

11

the model is run so as to attempt to duplicate laboratory results. In some cases, only limited data are available concerning test conditions, requiring assumptions. Not only could model performance be demonstrated, but the settings of the parameters, and the procedure used to eventually produce the desired results, led to useful insights into trommel operation. The model was then made to perform analyses independent of laboratory data.

Performance of the model with respect to published data During the 1970s, detailed laboratory investigations on trommels were undertaken by investigators in the US and the UK. It was a primary impetus behind this work to be able to use the model to replicate the shapes of key performance curves in the literature. The most detailed data available in the comprehensive report of Glaub et al. dealt with exhaustion of fines in the overs as a function of axial distance travelled. Their data are graphed* with modeling results in Fig. 4. The trommel involved was of 2.36 m diameter, 6.1 m long, with 0.12 m circular openings covering 55% of the screen area. The trommel was reported to have em1.0~

"X 0.8

C

8

,g

0.6

L

0.4 "5 *5 O L u_

0.2

o.o 0

I

I

I

I

1

2

3

4

Distonce down t r o m m e l

5

(M)

Fig. 4. C o m p a r i s o n o f m o d e l results and data f r o m G l a u b et aL [ 6 ] for the retention o f fines as a function o f axial distance. *In this, as in subsequent graphs, lines connecting datapoints serve only to make the graphs more readable.

12

R.I. STESSEL

ployed flights. To model this, the coefficients of friction were set to 1, allowing no slippage, as would be the case with flights. The rotational velocity used in the test was given as 2. l rad/s (20 rpm). Calculations show the critical velocity to be approximately 2.9 rad/s ( 28 rpm ), for which their velocity becomes 70% of the critical value, which is according to standard design practice. When using the model, this produces cataracting, but with an impact location less than 3/2n (see Fig. 2). It does not seem reasonable that such operation would be allowed because it makes such poor use of available screen area. Flights influence the mode of departure from the screen in two ways. First, they would force the waste to remain on the screen to a higher departure location angle than would be achieved without flights; this is one of their key advantages, since it allows maintenance of a good cataract with reduced rotational velocities. Second, at these higher departure angles, the x-component of the screen velocity would be larger than it would be without flights, causing the material to land at a smaller impact angle, making better use of the screen. To model this, all that was needed was a slightly higher rotational velocity: 2.8 rad/s (27 rpm). A set of modeling runs shows the effect of varying rotational velocities. The range of rotational velocities yielding good cataracting is quite small; for a 2 m diameter trommel, the model shows a range of approximately 0.5 rad/s (4.8 rpm). The narrow range of suitable rotational velocities also shows why flights are so common: it is difficult to maintain good cataracting behavior by controlling rotational velocity. Constant adjustment of variable speed motors, possibly using electronic feedback systems, would be required. Model results also show that flights have a great effect on reducing available screen area under conditions of low speed and low feedrate. A low feedrate (8.5 kg/s) is used in these tests, making the effects very apparent. Flights transform the trommel into a quasi-batch operation: as each flight, carrying its load, reaches a certain height, it unloads. This results in material being dropped onto sequential, separate segments of screen below, as opposed to continuously laying the material on the screen. Furthermore, as the flights are used to raise material above the point that particles would ordinarily leave the screen, the material comes to rest on the flight, not on the screen area where it could still pass though the screen. Neither gravity nor centrifugal reaction can then drive particles through the screen. The slower the trommel turns, the less screen area is used for separation, even if the employment of flights forces good cataracting. This was apparent in the modeling conducted here, showing that only 3% of screen holes were used. In the areas where screening did occur, the shallowness of the bed was reflected in the use of a shallow cone angle of 0.3 rad ( 17 o ). This means that the cone always penetrated the bed, which was thin. Glaub et al. expected that depletion would occur in a straight line, as Fig. 4 shows it does not. Glaub et al. sought to explain the lack of a straight deple-

A NEW TROMMEL MODEL

13

tion by doing further laboratory analysis, obtaining a particle size distribution of each recovered sample of unders. They showed, and discussed, the increased recovery of smaller particles compared to larger within the unders component. For a slightly more revealing examination of a similar analysis, results from Warren Spring Laboratory (WSL) are analyzed. The test analyzed for this work involved a 6.6 m long trommel with 4.6 m of screen area length, and 1.53 m diameter. Their trommel used flights, which were again modelled by using coefficients of friction equal to 1. Material moved through their trommel by means of scrolls. These worked imperfectly: they calculated a residence of time of 13.5 s, and measured residence times from 67 s for bottles to 80 s for large cards; no measurement was made for fines remaining in the overs. Scrolls are modelled by adjusting the incline parameter to achieve a residence time near 60 s, since it was suspected that the smaller, more dense fines would be more likely to lie between the scrolls, and thus have shorter residence times. The modelled incline was 0.3 rad (17 ° ). Again, a rotational velocity of 2.8 rad/s (27 rpm) was used to achieve good cataracting behavior. The feed rate was 10.5 kg/s. While this was not that much greater than that of Glaub et al., the incline used in modeling WSL's work was 70% lower, the bulk density was lower, and the diameter was 65% of Glaub et al.'s. This resulted in a considerably larger bed depth. No reduction was required in the number of holes available for screening. The critical aspect of the WSL experimentation was that individual catch baskets were placed under each quarter of the trommel screen length. The material in the baskets was then analyzed to determine its particle size distribution. Their laboratory screening resolution only resolved two particle sizes completely contained within the trommel fines; the third-smallest particle size interval was partly larger, and partly smaller, than the trommel openings. This was modelled by further dividing WSL's third increment into two: one larger and one smaller than the hole. Figure 5 shows results obtained by WSL compared to the model output. In both cases, only the two small particle size intervals that are smaller than the holes are analyzed. Data are reported, not as retention in the material passing over the screen, but as fractions of the material passing through the holes. Further, the data are not cumulative. The critical result is that the smaller particles first pass through the screen at a greater rate, then a lesser rate, than the next larger particle size. This illustrates the effect that one would expect from concentration of fines on the screen reported by Voorstman and Tels [ 12 ]. Eventually, the fines, after being recovered more quickly, would diminish in concentration, and thus constitute a lower fraction of the screened material. To achieve the crossover behavior with the model, the capability to layer

14

R.I. STESSEL 0.6

c

i

,

i

0.5

~6

8

8 -~

O4

O

8

LL

0.3

, 0

1

,

'2

~

3

\~|4

Distance down trommel (M) Fig. 5. Modeling results showing the particle size distribution of the unders: circles are the smaller particles, squares are the larger; filled symbols indicate model results, and hollow indicate resuits from Warren Spring Laboratories [ 13 ].

particles is employed (see Fig. 3) together with adjustment of the cone angle to modify the amount of each layer passing through the screen at each hole. As one would expect with real waste and a reasonable bed depth, the cone angle is obtuse 2.6 rad ( 150 ° ), implying significant bridging. Layering of 10% of the particles produced the crossover as shown. Hasselriis [ 14 ] compiled an extensive collection of data. Figure 6 presents data obtained from a test of shredded wood, ranging from 0.4 m m to 4.75 mm, which is assumed for this modelling work to follow a Rosin-Rammler distribution. These data are thus distinguished from the above tests by representing a feed of uniform density, with little sheet material, breakage during trommelling, or other difficulties posed by a solid waste feed. 25% of the feed was smaller than the trommel hole openings, which were 2.36 mm. The RosinRammler parameters were adjusted accordingly. No further information concerning trommel configuration was given. For modeling, a radius of I m, a feedrate of 30 kg/s, and a particle density of 500 k g / m 3 (representative of pine) are chosen. No flights are assumed, so coefficients of friction less than 1 are required, and are taken from Glaub et al.: static of 0.8; dynamic of 0.7. A rotational velocity of 3.2 rad/s (31 rpm) produces a good cataract. It is to be noted that this is considerably higher than

A NEWTROMMELMODEL

15

1.0

~.:~-;-

0.8

E

0.6

~8 >~ L

>

0.4

0

0.2

0.0 0

~

,

10

20

Number

30

of r o t a t i o n s

Fig. 6. Recovery of small particles as a function of rotation; large circles are data from Hasselriis ( 1984); continuous line shows model results.

that used by Glaub et al., although their trommel diameter was similar. This shows that slippage considerably increases the required rotational velocity. Similar to the simulation of the WSL data, a cone angle of 2 radians ( 115 o ) was found suitable. No layering is necessary in the simulation; with this uniform feed, it is not a factor. The model does a good job at following the data. In Hasselriis' original graph, the data were plotted in semi-log fashion. This was not done here, because linear graphing allows greater visibility of the asymptotic behavior of the exhaustion that results from an ever-diminishing content of fines. Asymptotic removal was also reported by Wheeler et al. [ 15 ]. Hasselriis divided the curve into three distinct parts: the constant flow, decreasing flow, and probability portion. He described each with distinct curve fits. It is significant that the model produces the curve directly from the fundamental equations. From these results, it can be seen that the model performed well in echoing the mechanics of trommel operation. Some factors not clearly available from laboratory data remain to be analyzed as below.

Analysing using the model Once the basic ability of a predictive model to represent laboratory or field data has been demonstrated, it may be used to test variables not easily ad-

16

R.I. STESSEL

dressed in the laboratory. Researching questions that might occur to designers is shown below. Trommel design, as with all others, hinges on achieving the most performance with the minimum equipment. A parameter had to be developed that reflected the efficiency with which the trommel performed its screening job. To save costs, it is important to minimize residence time, which is addressed by two parameters: length and time. A recovery rate is defined: r=

I/pPBULK 7 IA

(18)

where Vpp is the mass recovered and 1A/t is the velocity with which particles have progressed down the trommel. The usefulness of this parameter is shown by comparing Figs. 7 and 8, where Fig. 8 presents the same data as Fig. 7, but using the recovery rate instead of unders. Simply reporting the recovery of fines to the unders produces a curve that is somewhat reverse-sigmoid. By contrast, recovery rate increased linearly. While increasing incline results in the obvious reduced recovery by the end of a fixed-length trommel, it does so by a mixture of complex phenomena. First, the residence time is reduced. Counteracting this is an increase in the band area, resulting in decreased bed-depth. From Fig. 7, it can be concluded that it would make no difference so long as one operated the trommel before the curve began to tail down, at about 0.7 rad. Economically, though, the operator also cares about throughput. Using the recovery rate to include throughput shows that one recovers with increasing efficiency with increasing

140

I

I

1.35

1.30

125

120

t 000

005

I 0 10

t 015

020

Fig. 7. The effectof inclineon the mass of unders recoveredafter 2 m of axial travel.

A NEW T R O M M E L MODEL

020

i

17

i

I

0.15 (9 klJ

<

010

>r~ LLI >

o o m

O7

f

005

000 000

i

I

I

005

010

015

020

INCLINE (RAD~

Fig. 8. Effect o f incline on recovery rate at 2 m axial travel.

incline. The design conclusion is then different; given a target recovery, the design decision is to buy the longest trommel that could be economically justified and give it the maximum incline. This would allow the maximum throughput. Although the conclusion is intuitive, it is difficult to show it by studying traditional parameters. Rotational velocity is further examined. Within the narrow permissible range of rotational velocities within which cataracting is maintained, the recovery rate at the end of 2 m of travel is calculated. Here in Fig. 9, a maximum is clearly demonstrated. Examining model output shows that, at low rotational velocities, the material departs the screen at very low angles, and reattaches at very great angles, leaving a small band area (see Figs. 1 and 2). This reduces the recovery rate by increasing bed depth and reducing the screen area available to the material. Conversely, at very high rotational velocities, angles of departure are quite high, and angles of impact are quite low (above the horizontal centerline of the trommel, n rad, see Fig. 2). As shown by Eq. ( 8 ), this reduces the axial distance travelled in every rotation, again increasing the bed depth and reducing the screen area available to the material. With a given waste, a designer might choose to examine several sequences of unit operations, potentially involving screening to remove different size fractions. The model is used to examine the effect of hole size on a feed of fixed composition. The input distribution is identical in all cases; the hole size is adjusted to allow passage of the first two, three, or four size increments into which the distribution was divided. Results are shown in Fig. 10. Significantly, the results follows the conclusions reached in examining WSL's parti-

18

R.I. S T E S S E L

0.14

0.12 (9 lJJ < rr >rr w :> O O LU dE

010

008

006 2.5

I

I

30

35

40

ROTATIONAL VELOCITY (RAD/8)

Fig. 9. The effect of rotational velocity on the recovery rate.

,

1.0

/F-

t. . . . . . .

0.8 >nw > O

o

0.6

z Q