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RIIERIIIITIOIR_Of

m i n E R A LP R O I E S S In G

ELSEVIER Int. J. Miner. Process. 53 (1998) 15-28

Spirals observed

P.C. Kapur a, * , T.P. Meloy b

a Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016,

India

b

Particle Analysis Center, 338 Comer, West Virginia University, Morgantown, WV 26506, USA

Accepted 15 Apri11997

Abstract

We first examine the three main components required to construct a realistic and tractable

model of the working of a spiral, namely (1) geometry of the spiral and its trough, (2) principal

forces acting on a particle which are gravity, centrifugal, hydrodynamic drag and lift and friction

forces, and (3) flow of fluid down the curvilinear tightly turning path of the spiral. Next we

combine these elements seamlessly by assuming that the particles eventually attain dynamicequilibrium in the forward longitudinal direction and static equilibrium in the transverse direction.

The resulting force function provides a spectrum of the particles' radial location on the trough

according to their size and density. Simulation results are highly encouraging. The model

faithfully reflects the present empirical spiral design philosophy for treating different feeds such as

coal or heavy density minerals. It seemingly has potential for computer-aided spiral design and in

control and optimization of spiral circuits. 1998 Elsevier Science B. V.

Keywords: spirals; gravity separators; coal cleaning; modeling and design

1. Introduction

After suffering a benign neglect for nearly half a century, spirals have begun to

attract somewhat greater attention of the researchers in recent years. The reason

presumably lies in the world-wide concern with the environmental damage which is

being wrought by the combustion of fossil fuels. Spirals are perceived as rugged,

compact, efficient and cost-effective gravity separators for cleaning coal in the size

Corresponding author. Present address: Tata Research Development and Design Centre 54 B, Hadapsar

Industrial Estate, Pune 411 013, India. E-mail: [email protected]

0301-7515/98/$19.00 1998 Elsevier Science B.V. All rights reserved.

PII S0301-7516(97)00053-7
mailto:[email protected]:[email protected]


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P.e. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 (1998) 15-28 17

Centre Column

Height, H

,

M ax . DePth_.-_~_~-_-_- -:, ...... ~,-lt Trough Slopea

Outer Radius, 'o'.--+',

Inner Radius, r i

Fig. 1. Schematic drawing of a spiral and its trough.

where N is twice the number of turns, H is spiral height and r is radial distance from

the centerline. From these relationships the longitudinal tangential slope S at any point

on the deck may be derived as:

uS=tan[a] =-

2 7 T r(4)

where u is pitch of the spiral and a is slope angle shown in Fig. 1. Most trough shapes

are of the form of a modified quarter-circular arc (Burt, 1984). Accordingly, these maybe represented, exactly or approximately, by an elliptical curve, also shown in Fig. 1.

For this, shape the local slope of the channel deck in radial or transverse direction is

given by:

(5)

where 1 I is local slope angle in the transverse direction, c y is the maximum depth of the

trough and rj and ro are, respectively, the inner and outer radii of the trough from the

centerline. Our choice of the elliptical contour for the trough is dictated primarily bymathematical convenience of realizing analytical or simpler expressions. We are of


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18 P.e. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 (1998) 15-28

course free to choose any arbitrary contour without introducing any new principle in the

development of our model.

3. Forces acting on particles

Because of the presence of mixed laminar, transitional and turbulent flow regimes,

particles on the spiral are subjected to a medley of randomly fluctuating, transient and

steady force fields. It is not easy to identify and quantify most of these forces precisely.

In general only rough estimates are possible of the five principal forces involved,

namely, gravity, centrifugal, hydrodynamic drag, lift and friction forces.

3.1. Gravity force

Force of gravity or weight of a spherical particle of diameter d and density (T

submerged in a liquid of density p is given by:

(6)

where g is the gravitational constant.

3.2. Centrifugal force

In view of the fast flowing and tightly turning flow obtaining in a spiral, centrifugal

force plays a significant role in its operation. A submerged particle moving with a

velocity v in a circular path of radius r is subjected to a centrifugal force of:

7T d3 v 2 ( (T - p )Fc=------

6 r(7)

In general, velocity of the particle will be different from that of the surrounding fluid, as

well as highly variable depending upon its size, density, position in the radial direction

and depth under water. A theoretical analysis of the particle velocity in a rotating

two-phase flow is not a trivial task. Consequently, we are obliged to adopt a forcebalance strategy for surmounting this problem, as elaborated in a later section.

3.3. Drag force

The drag force exerted on a body in a flowing stream of fluid due to skin friction and

eddies induced pressure difference between its upstream and downstream sides is given by (Allen, 1982a):

7T

F d = '4 p ghd? sin] a]

where h is depth of flow.

(8)


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P.C. Kapur, T.P. Meloy / Int. 1. Miner. Process. 53 (1998) 15-28 19

3.4. Lift force

Large-size particles in the vicinity of the channel deck are subjected to hydrodynamic

lift (Bagnold) forces that arise by the influences of nearby boundary, fluid shear and

particle spin or the Magnus effect (Bagnold, 1954, 1966; Allen, 1982a). In spirals liftforces assist in dilating and loosening the bed which facilitates segregation and sorting

of particles according to their size and density. As a first approximation lift forces can

be directly related to drag force as:

(9)

At present no consensus exists on the constant of proportionality k1; experimentally

determined values ranging from 1/7 to unity have been reported (Simon and Senturk,

1976). Since the spiral simulation results are not overly sensitive to its value, we

arbitrarily set kl = 0.33.

3.5. Friction forces

Resistance to the motion of the particle on spiral trough is proportional to the sum

total of the normal components of all forces FN acting on the particle. The constant of

proportionality is the coefficient of dynamic friction under water tan[j] which, accord-

ing to Bagnold, is virtually the same as the coefficient of static friction under water

(Allen, 1982a). Hence friction force is related to normal force by:

(10)

where based on the data provided by Allen and others (Chang. 1988), we take

tan[ c f > ] = 0.5.All gravity separation processes in essence entail countering the force of gravity on

particles by imposing some other forcers). Extensive computations of these forces

(Kapur and Meloy. 1996) reveal the following noteworthy points. One, with the

exception of gravity force, other forces are strongly dependent on the spiral geometry.

Two, the magnitude of gravity force is comparable to that of drag and centrifugal forces

in size ranges that are normally processed in spirals. And three, it is not so much the

magnitudle of individual forces as the rate of change of forces with particle size, density

and radial position which determines the efficiency of separation on spirals.

4. Fluid flow on spirals

Of all the different kinds of gravity separators available, spirals exhibit the most

complex flow regime. The action of centrifugal force on water as it flows down in a

spiral channel has two important consequences (Allen, 1982b; Chang, 1988). First, the

water level at the outer concave wall of the trough exceeds that at the inner convex

surface by an amount called superelevation. Second. a transverse secondary circulationis generated in the form of a vertically flattened helical spiral that moves forward in a


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2 0 P.e. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 (1998) 15-28

corkscrew fashion. The angles that the inward and outward bound flows makes with the

mean axial flow vary with depth and radial distance. Holtham (1990, 1992) and

Holland-Batt and Holtham (1991) have measured these angles in water only spirals with

rather large error bounds. It is however not clear if angles obtained in pure water are

meaningful for heavily loaded slurries. We find it convenient to utilize a widely quoted

expression in the hydrology literature by Rozovskii for the mean deviation angle 8

(Allen, I982b):

htan[ 8] = 11-

r(11 )

Analysis of fluid velocity and local flow profiles in a spiral by the Navier-Stokes

equations is an arduous task indeed. Consequently, we adopt an 'averaging' approach as

a matter of practical expediency and with no pretentions for describing the flow either

precisely or in detail. Flow of fluids and sediments in open channels are commonlydescribed by a plethora of power laws whose general form is:

( 12)

where V is mean flow velocity, R is hydraulic radius and K is a composite resistance

coefficient. The exponents a and b depend on the nature of the flow such as laminar,

Manning laminar, Lacey rough channel, Blasius turbulent, Bagnold suspension, transi-

tional or mixed type or a combination thereof (Dingman, 1984; Chen, 1991). We choose

the transitional or mixed flow equation because it is widely employed to describe the

sheet or overland flows of water during rainstorm where energy dissipation occurs, as it

does in spirals, by viscosity and turbulence frictions combined in relatively shallow

depths. Moreover, the mean flow velocities computed with this equation are comparable

with the measured data, even though the computed flow depths tum out to be greater

than the reported values. The power law for transitional or mixed flow is described as:

26.4V=_-RS1/2

dl/6p

(13)

where dp is a percentile size in the solid feed. Another expression for the mean flow

velocity is:

QV=-

A(14)

where Q is volumetric feed rate and A is cross-sectional area of flow. Next, we set Sin

Eq. (13) equal to Sm' the channel tangential slope at the midpoint and substitute

appropriate expressions for R and A in terms of the trough geometry and mean flow

depth hm and finally eliminate V between Eqs. (13) and (14) to arrive at:

_ 3 .3 7 7 ' 1.5 f E [e x e y - ( ex - hm) ( e y - hm) ] 2Q - 2 dl/ 6 _ '__~ '1T -: -;/ 2: ;- -;:: := = == = =: :: :- ''- -- -~

p ex 1 V(xI2+yI2)dt+hmo(15)


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P.C. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 (J998) 15-28 21

where x ' and y' are derivatives of x and y with respect to t and ex = To - Tj is theradial width of the trough. Eq. (15) can be solved for mean flow depth which is required

in Eqs. (8) and (I1) for implementing the force balance model.

5. Force eqmlibrium model

Fluid driven particles can travel by one or more of the following modes: sliding,

rolling, saltation (i.e, bounding), known collectively as the bedload transport, and in

suspension. Considering the few rom thick rotatory pulp flow obtained in a spiral, it is

reasonable to expect that, barring low density fines, movement of particles is by a

combination of mechanisms in which the bedload transport should play a dominant role.

In any case, it is not possible to track the trajectory of individual particles. We can at

best monitor the overall 'drift' by making some plausible assumptions. Ifand when a

steady state is attained, particles are in dynamic equilibrium in the longitudinal direction

and in static equilibrium in the transverse direction. The latter state is of primary interest

as it determines the degree and extent of sorting of particles according to their density

a Spi ra l S lope Ang le- - - - - - - - - ( - - - - - - -

TransverseDirection

D * a

g

"' ...... "" 6Direction

II

Fig. 2. Forces acting on a particle lying on the spiral trough. showing longitudinal and transverse slope angles

and the local shear deviation angle.


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P.e. Kapur, T.P. Meloy / Int. 1. Miner. Process. 53 (J998) 15-28 23

expression which for our purposes may be written in the form of a force equilibrium

function as follows:

(21)

Using this function we can compute, after inserting 'best' estimates of k, and c pavailable in the literature, the equilibrium radial position r on the trough of a particle of

size d and density (T for any spiral of design specifications rj, ro' C y and u.

6. Simulation results

The force equilibrium function in Eq. (21) can be readily solved for particle density

as a function of particle size and radial position or for particle size as a function of

C i iII:

1.5~--+-=rr:==1P;;;;::::::=f=~~~

1 ~ - - - - - - _ L - - - - - - ~ - - - - - - ~ - - - - - - ~ ~ ~ ~1 00 150 200 250 3 00

Eq. Posit ion, r mm

EE'0

~i : i 5

100 150 200 250 300

Eq. Pos it ion, r mm

Fig. 4. Parametric plots of simulation shown in Fig. 3.


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24 P.e. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 (1998) 15-28

0.8 ~--'~---,+----'''''=1-;----T-----j

ED.6~--~1-~~~q--~~+----~. g~0.4~---~----~-~~+--~-~Ut:til

0..0.2

100 150 200 250 300

Equilibrium Position, r [mm]

Fig. 5. Comparison of simulated results and experimental data for 0.85 XO.71, 0.71 XO.6 and 0.6XO.5 mmsize fractions of chromite and silica separated on a Multotec '21 degree' heavy mineral spiral (Loveday and

Cilliers, 1994).

particle density and radial position. Fig. 3 shows a 3-D plot of relative density for a

spiral of rj = 0.1 m, ro = 0.325 m, cy = 0.15 m and u = 0.35 with a slurry feed rate of 4m3/h. As observed in the actual spiral operation, here also the lighter particles in thesimulation are segregated in the outer region while heavier particles are concentrated in

the inner region of the trough. These results are perhaps more clearly illustrated in the

parametric plots in Fig. 4 of particle size or particle density versus the equilibrium radialposition. It is rather remarkable that the force equilibrium spiral model with its fairly

complicated expressions and without any fine tuning of the parameters culled from the

hydrology literature, nevertheless, yields realistic values for both particle size and

particle density.

Loveday and Cilliers (1994) separated chromite from silica on a Multotec '21 degree'

heavy mineral spiral of specifications rj = 0.1 m, ro = 0.3 m, cy = 0.125 m andu = 0.45 with a slurry feed rate of 3.5 m3/h. The simulated results and experimentaldata for 0.85 X 0.71,0.71 X 0.6 and 0.6 X 0.5 mm size fractions are compared in Fig. 5.

Li et al. (1995) measured the classification of color-coded plastic particles of different

densities on a LD2 spiral of specifications rj = 0.075 m, ro = 0.33 m, cy = 0.135 m andu = 0.308 with a slurry feed rate of 3 m3/h. Their data for three size fractions iscompared with the model results in Fig. 6. Considering the complexity of the spiral

system, the simplifications that we are obliged to introduce in the model, and the

unavoidable presence of not insignificant levels of errors in the measurements, we may

conclude that the model is able to mimic the spiral behavior reasonably well.

Over the years, some sort of empirical consensus has evolved on the design of spirals

for a given separation task. The spiral specifications for coal cleaning, for example,

according to Palowitch et al. (1991), are 0.6 m diameter and 0.254 m pitch with five andhalf turns. Holland-Batt and Holtham (1991) provided a table of different spirals in

which coal spiral LD2 has 0.33 m trough radius and 0.308 m pitch while high ash coalspiral LD9 has 0.35 m trough radius and 0.273 m pitch. Earlier, Burt (1984) listed


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P.e. Kapur, T.P. Meloy / Int. 1. Miner. Process. 53 (J998) 15-28 2 5

Feed Size 1xO.71 mm

1- -.


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26 P.e. Kapur, T.P. Meloy / Int. 1. Miner. Process. 53 (1998) 15-28

ParticleSize 1 mm

s r - - - - - - - , - - - - - - - , - - - - - - - ~ - - - - - - ~ - - ,

4.S~---__+----+___-- --t_----+------1

4~---__+----+___---_t_---

>-

~3.S~----~----t-------t-----T-~Q)

~ 3~~~~__+----i-----_t_---_T-~>

~w2.S~~~~~~~~-t_---_t_---_T-~c : :

1~------~-------~----~------~----~

1 00 1 S0 2 0 0 2S 0 3 0 0


Fig. 7. Effect of spiral pitch on the separation of I mm size particles in a spiral of specifications rj =0.1 m,

ro = 0.325 m and cy = 0.15 m. Pitch varies from 0.25 m for the bottom curve to 0.5 m for the top curve withan increment of 0.05 m.

slope, the greater is the efficiency. It will be seen that the optimum pitch for feed in the

range of 1.2 to 2.8 relative density lies between 0.25 and 0.3 m, which is the same as the

current industrial practice. Fig. 8 shows the effect of trough radius where ro is varied

from 0.25 m (left most curve) to 0.5 m (right most curve) in increments of 0.05 m.Again, it would seem that spirals with larger radii should exhibit superior performance

in coal cleaning.

ParticleSize 1 mm

S

4.S

4

~ i

~

+~ -I'-.-'\ j ",-- . . . . . . . . , ,~

~~3.SQ)

~ 3.~

:m2.5c : :

2

1.5

1

1 00 2 0 0 3 00 40 0 SO O


Fig. 8. Effect of trough outer radius on the separation of 1 mm size particles in a spiral of specifications

rj = 0.1 m, cy = 0.15 m and u = 0.35 m. Radius varies from 0.25 m to 0.5 m with an increment of 0.05 m.


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P.e. Kapur, T.P. Meloy / Int. J. Miner. Process. 53 ([998) 15-28 27

7. Summary and conclusions

The model presented here is at best only an exploratory foray and it undoubtedly has

considerable scope for improvement and refinement. Even though there is some

uncertainty regarding the attainment of a steady state in the process, the simulationresults suggest that the force equilibrium approach could provide a useful theoretical

frame-work for the model. The admittedly crude averaging approximation made for the

flow profile is justified by the need for developing a working model in an operational

sense in face of the sheer complexity of the hydrodynamic regime prevailing in the

system. Even then, further refinement of the flow description, without excessive and

elaborate computations, remains a challenging task for the future. No attempt was made

to fine-tune the various parameters required to implement the model, that is, tailor them

to a spiral operation in order to improve the model performance. In any case, this

exercise cannot be undertaken without a fairly extensive and reliable data base which

unfortunately is not available, at least in the public domain. Moreover, it is alsonecessary to account for the stochastic component in the working of the spiral which in

fact is inherent in all gravity separation processes. In spite of these drawbacks, we are

sufficiently encouraged by the simulation results to believe that the model presented

here has considerable potentials for analyzing spiral design and its separation character-

istics for different feed materials which may range from relatively light coal to heavy

high-grade minerals.

References

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dam, Chapters 2 and 7.

Allen, J.R.L., 1982b. Sedimentary Structures: Their Character and Physical Basis, Vol. II. Elsevier, Amster-

dam, Chapters 2 and 3.

Bagnold, R.A., 1954. Experiments on a gravity-free dispersion of large spheres in a Newtonian fluid under

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Burt, R.O., 1984. Gravity Concentration Technology. Elsevier, Amsterdam, pp. 261-287.

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