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A contribution to the theory of jigging, Part I: Similarity criteria of the motion of jig layers
Minerals Engineering, Vol. 9, No. 6, pp. 675-686, 1996
Pergamon Plh S0892-6875(96)00055-6
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0892-6875/96 $15.o0+0.o0
A CONTRIBUTION SIMILARITY
T O T H E T H E O R Y OF JIGGING, PART I: CRITERIA OF THE MOTION OF JIG LAYERS
H.J. STEINER Instimt fiir Aufbereitung und Veredlung (Department of Mineral Processing), Montanuniversit~it Leoben (University of Mining and Metallurgy), A-8700 Leoben, Austria (Received 16 June 1995; accepted 18 February 1996)
ABSTRACT
A harmonicpulsion superimposed by a constant hutch waterflow serves as the model case in the derivation of similarity criteria of the motion of jig layers. At first the decisive portion of the jig bed is identified and characterized by two hydraulic properties, i.e. the initial acceleration and the terminal settling velocity of the separating layer at the cut position. A concise differential equation of the motion ofjig layers is formulated and solved for the velocity function and the displacement function. In the normalized versions of the said functions, two main dimensionless similarity criteria of the motion of a jig layer are identified. One of the criterions is the lift-off angle, i.e. the phase angle at the beginning of the displacement of the jig layer. The second criterion comprises a modified Froudenumber. The latter relates the maximum acceleration of thefluid to the initial acceleration of the separating layer, thus combining a characteristic of the pulsion with a characteristic of the fig bed. Additional dimensionless parameters characterize the flow regime and the degree of suction. Empirical knowledge about the average settings of jigs is summarized in average values of the said similarity criteria. All the equations are derived from basic hydraulics and observe the requirement of dimensional homogeneity. Keywords Mineral processing; gravity separation; modelling
INTRODUCTION An attempt has been made to predict the performance of jigging operations from basic data of the feed without referring to empirical partition curves. It was clear from the beginning that the goal in view could only be reached by structuring the complex process into sub-problems of less complicated nature. Due to the fundarctental difference between the time of layer displacement and the time of dilation, a separate mathematical analysis of the kinematics of jig layers and of the kinetics of particle segregation seemed justified. I1: was expected that the link between the two issues could be provided by adequate dimensionless numbers. Thus the identification of relevant similarity critera was the prime concern in developing a physical model of the motion of pulsed particle layers. A future contribution will deal with the segregation kinetics in a jig bed. Presented at Minerals Engineering '95, St. Ives, Cornwall, England, June 1995 675
676
H.J. Steiner CHARACTERIZATION OF THE FLOW OF DILATION WATER
The particular motion of the particle bed on the supporting screen is caused by the flow of the "dilation water" passing upwards or downwards through the openings of the screen. Thus the most relevant information about the water flow in a jig is the velocity of the dilation water as measured directly below the screen. The time profile of that velocity may be divided into a periodic pulsion and a constant portion related to a part of the hutch water addition. The periodic pulsion may be reduced to the motion of an imaginary plunger reciprocating close to the screen as sketched in Figure 1.
U_(300~dOC_ °_cd-OkV 6-o
U
l> -_
-_--
I -g i Fig. 1 Simplified model of a jig bed under the influence of a harmonic pulsion superimposed by a constant hutch water flow. Phase angle a and amplitude r of the pulsion. Peripheral velocity c of the imaginary crank-drive. Hutch water U The motion of a plunger is usually described by the length of the strokes ("throw") and their frequency. However, the model of a harmonic pulsion results in more concise equations of the bed displacement if the motion of the imaginary plunger is characterized by the peripheral velocity and by the radius of an imaginary crank-drive connected to the plunger by a rod of infinite length as indicated in Figure 1. Thus the amplitude of the pulsion--i.e, half the water stroke--is fixed by the crank radius. The peripheral velocity c of the crank ("crank-velocity") is a trivial function of the crank radius r and of the frequency f of the strokes: Eq. (1). The frequency equals the reciprocal of the cycle time te: Eq. (2).
c=2-z .rf
(1)
f=llt ,
(2)
The lowest position of the crank and of the plunger respectively defines the zero-points on the scales of
A contributionto jigging theory--I
677
the crank-angle o~ ( = phase angle) and of the time t. The relationship between a time differential dt and a corresponding plhase angle differential dc~ is expressed by Eqs (3) and (4):
d&ldt=2.r~'f
(3)
d&/dt=c]r
(4)
The velocity vf of the dilation water is to be understood as the velocity of approach or return with respect to the screen surf~Lce. It is elsewhere called "superficial velocity", well to distinguish from the velocity within the voids of the particle bed. The velocity vf splits into a periodic portion and a constant portion u ("hutch water velocity"). The normalized version of the functions of the velocity vf and of the displacement sf of the dilation water are listed below: Eqs (5) (6). See also Figure 2.
vslc=sin~+(u/c)
(5)
sf/r = 1 -coset + a.(u/c)
(6) 3
~ //,."
\ 1
.,~//'/" .~oJ*
0 0
2.
.
,
~
3
\,,, \,
"~. ,
2~
phase angle Fig. :2. Normalized (= amplitude related) displacement of the dilation water and of a jig layer under the influence of a harmonic pulsion. Curve 1 :Displacement of a particle layer; Curve 2:Displacement of the dilation water due to the pulsion; Curve 3:Displacement of the dilation water due to the pulsion and a constant hutch-water flow The quotient "cran'k velocity squared through crank radius" represents both the constant centrifugal acceleration of the circular motion of the crank as well as the maximal acceleration afmax of the flow of dilation water through the screen into the particle bed: Eq. (7).
a~ax=c2/r
(7)
The relevant kinematic parameters of the flow of dilation water can be derived from the real motion of the pulsion mechanism, the geometry of the jig, the time profile of the hutch water addition and the portion of hutch water discharged into the underflow. The movement of the water level in the jig does not correspond exactly to that of the dilation water, since the latter is partly diverted from the vertical direction into a horizontal one at each up-stroke.
678
H.J. Steiner IDENTIFICATION OF THE DECISIVE PORTION OF T H E JIG B E D
The jig bed, being inhomogeneous both in vertical and horizontal direction, may be roughly divided into a top layer, a central or intermediate layer, and a bottom layer. The easily separable particles of the feed are rather rapidly recovered either in the top layer or in the bottom layer. Their properties are not of prime interest, since the efficiency of separation--as reflected by the partition curves--depends primarily on the hydraulic behaviour of the separating layer at the cut position. This layer--though inhomogeneous with respect to particle size and particle densitiy--is characterized by rather uniform hydraulic properties. Its composition with respect to the distribution of particle size and particle density may be estimated from the density values at the cut positions of the various particle size classes (as indicated by the partition curves) and the frequencies of the said particle size classes in the feed. A restriction of the mathematical treatment to the analysis of the motion of the separating layer signifies a considerable simplification as well as focusing the attention to the most relevant portion of the jig bed. The concept of a central layer also corresponds with Taggart's view on the separation in a jig bed [1].
CHARACTERIZATION OF THE HYDRAULIC PROPERTIES OF A COHERENT JIG LAYER The pressure drop of a fluid passing upwards through the voids of a coherent bed of particles exerts a lifting force which is counteracted by the apparent weight of the particles, i.e. the difference "weight minus buoyancy". The state of a diminishing resultant force marks the transition from a particle bed at rest to one lifted off the support. The velocity vf of the approaching fluid in the said state of equilibrium corresponds in amount (though opposite in sign) to the imaginary settling velocity v e of the particle bed in quiet water. If the particle bed had the opportunity to settle in a quiet fluid from a position at rest on a suddenly withdrawn support, it would start off at an initial acceleration ao, which is determined by the average density if, of the particles, the density pf of the fluid and the acceleration g of gravity. The initial acceleration is quite independent of the geometry of the voids, since the fluid resistance equals zero at zero velocity: Eq. (8). ao=g.(1 -pil'~)
(8)
The terminal settling velocity v e and the initial acceleration ao represent the basic hydraulic properties of a coherent jig layer. In this respect, there exists a perfect hydraulic analogy between the properties of a single particle and those of a multi-particle bed. The ratio of the terminal settling velocity v e to the intial acceleration ao may be interpreted as a nominal time of acceleration t e (see Figure 3): Eq. (9). The ratio of that characteristic time span t e to the duration t c of one jigging cycle represents the normalized time r e of acceleration. It is expressed by Eq. (10) as a fraction of the cycle time, and converted by Eq. (11) into radians. t=v lao
(9)
%=t,/t~
(lO)
~e=2.~ ..c e
(11)
A contributionto jiggingtheory--I
679
te Ve :> ° , ~
0 0 oD :>
0 time t Fig.3 Accelerated motion of a coherent particle layer in quiet water from a position at rest. v e..... terminal settling velocity; te ..... nominal time of acceleration An alternative explession of the normalized time of acceleration follows from substituting the Eqs (1) (2) (10) in Eq. (11):
¢:e=(ve.c)/ (ao.r) The dimensionless parameter ~
(12) combines characteristics of both the particle bed and the pulsion. It
proves useful in the mathematical treatment of the jig layer kinematics.
MATHEMATICAL ANALYSIS OF THE KINEMATICS OF A JIG LAYER
Lift-off angle A jig layer of uniform hydraulic properties will be lifted off its support and move upwards as a coherent set of particles when the velocity vf of the dilation water compensates the terminal settling velocity v e of the particle layer. "lqaecorresponding phase angle of the jigging cycle may be called the lifl-offangle o~o. It can be calculated on substituting the restraints as defined by Eq. (13) in the velocity function of the dilation water. In the model case of a harmonic pulsion superimposed by a constant hutch water velocity u, the said velocity fimction is given by Eq. (5). It leads to Eq. (14), expressing the sine of the lift-off angle % by the difference of two dimensionless velocity quotients (see Figure 4.)
v/=%: =, ~x=~zo
(13)
sin% =(v~[c)-(u/c)
(14)
General differential equation The balance of the various forces (gravity, buouyancy, inertia and fluid resistance) acting on a jig layer of homogeneous hydraulic properties can be reduced to a quite concise relationship of the relevant kinematic quantities, i.e. accelerations ax and velocities Vx: Eq. (15). The mathematical procedure leading
680
H.J. Steiner
to that remarkable structural simplicity is rather analogous to the method applied in deriving a concise differential equation of the accelerated motion of single particles (to be described in a future part II of the publication).
(a/ao)+(vr/%) m = 1
(15)
The first term of Eq. (15) relates the variable acceleration (= a) of the jig layer to its constant initial acceleration ao. The second term in brackets relates the variable relative velocity v r of the jig layer to its constant terminal settling velocity v e. The relative velocity represents the velocity difference of the particle layer with respect to the fluid. The dimensionless exponent m to the (Vr/Ve)-term is identical to the velocity exponent in the equation of the fluid resistance. Thus the value of m depends on the flow regime in the voids of the particle layer. The theoretical limits of that "flow regime parameter" are m --- 1 (laminar flow) and m = 2 (fully turbulent flow). Under normal conditions of jigging, the m-value will be found in the range 1.5 to 2.
O~0 "
Lift-off angle
~7 i
I i
I
"1o
I t~
E
i i
2
/
/
I ,
I
E (D
N \
/
J
r-
f
/
/
\
/
water ~\
/
.i
o
_ "
r I1¢ t/ i c l e ' I '// layer Ia
u} , m
"o
0
\\
\ \
i
0
n
2~
phase angle o~ Fig.4 Illustration of the term "lift-off angle"
Transformation of the general differential equation
The accelerations and velocities appearing in Eq. (15) are counted positive when their vectors point in the direction of gravity. On the other hand it seems appropriate to attribute positive signs to upward directions of the velocity v of the particle layer and to the velocity vf of the dilation water. The trivial Eqs (16), (17) are in accordance with the said sign convention:
a= -dv]dt
(16)
Vr=-(V-V/)
(17)
During the period of layer displacement, the relative velocity is never far off the terminal settling velocity: Eq. (18). On that condition, Eq. (19) represents an acceptable simplification, introduced for the benefit of an explicit final solution of both the velocity function and the displacement function of a jig layer.
A contributionto jiggingtheory--I
681
(v,/v,)-I:
08)
(v,/v~)m'.l-m-(l - v ] v )
(19)
For the benefit of a concise differential equation of the model case under investigation, a dimensionless parameter z is defined by the ratio of the normalized time ~ of acceleration to the flow regime parameter m: Eq. (20)
z =~•/m
(20)
A combination of the Eqs (1) (2) (4) (5) (9) thru (11) and (14) thru(20) yields the approximate differential equation of the motion of a jig layer of homogeneous hydraulic properties under the influence of a harmonic pulsion and a constant hutch water flow. Equation (21) presents the compact version, structured into dimensionless terms. (v/c) =sin c~ - sin % - z
.d(v/c)/da
(21)
Integration and auxiliary functions Due to the inhomogeneous character of Eq. (21), a rather complicated expression of the velocity function is obtained on rigorous integration. However, on account of the generally prevailing condition z2,1, an approximate solution of quite sufficient accuracy may be derived. The displacement s of the particle layer from the support as a function of the phase angle c~is obtained on integrating the velocity function v=ds/dt. Prior to that second integration, the phase angle differential da has to be substituted for the time differential dt. The Eqs (22)-(25) define auxiliary functions, introduced for the benefit of more concise final expressions. A &=~-ao
(22)
Asina =sin~ -sh,%
(23)
Acosa =cos~ -o)s%
(24)
A =z'cos%'[1 -exp(-A =]z)]
(25)
Velocity function The normalized version of the approximate velocity function relates the variable velocity v of the particle layer to the const~at peripheral velocity c of the imaginary crank-drive: Eq. (26).
v/c =Asin ~ -z'A cos a -,4
P£ 9-6-D
(26)
682
H.J. Steiner
Displacement function The normalized version of the approximate displacement function relates the variable displacement s of the particle layer to the constant amplitude r of the pulsion, i.e. the radius of the imaginary crank-drive: Eq. (27)
s/r=-Acosa -z.A sina -A &.sinao+z.A
(27)
Figures 5 and 6 illustrate the motion of a particle layer at varying values of the parameters (or0, z) acccording to Eq. (27).
2 112 n
e~
E
03
¢..
E O
03
tD
0,1
0 0
~ phase angle
2~
Fig.5 Normalized (= amplitude related) displacement of jig layers according to Eq. (27) at a constant lift-off angle
ao = 0.4. Parameter: z-value
2 "10 Q.
E
03
0'3
,4\
E t,.) n
0,5 0,6
03 Q.
"0
0 0
7g
2~
phase angle Fig.6 Normalized (= amplitude related) displacement of jig layers according to Eq. (27) at a constant z-value (z = 0.15). Parameter: lift-off angle
ao in radians.
A contribution to jigging theory--I
683
In contrast to the displacement function derived by Finkey [2], Eq. (27) does not result in physically impossible (i.e. negative) values of the layer displacement in the vicinity of the lift-off angle.
SIMILARITY CRITERIA OF JIG LAYER MOTIONS Equation (27) describes the dimensionless ratio "vertical displacement of the particle layer divided by the amplitude of the pulsion" as a function of the variable phase angle c~ and of just two numerical values, i.e. the lift-off angle c~o and the parameter z. Thus the conclusion--already suggested by the constituents of the differential Eq. (21)--reads: Constant values of oto and z determine similar jig bed motions. From the mathematical point of view, the two parameters (%, z) fulfill the requirements of similarity criteria. The lift-off angle c~0 represents a self-explaining term. The z-parameter however calls for an interpretation from the technical point of view. A thorough analysi:s of all the equations involved revealed a connection between the z-value and the degree of suction in a jig bed. Experienced jig operators are familiar with the phenomenon of suction, they know about the detrimental effects of an excessive degree of suction and are aware of the significance of an adequate suction control. Mathematically, suction is signified by negative values of the relative velocity v r of the particle layer with respect to the dilation water (see Eq. (17)). According to the results of the analysis, negative Vr-Values occur if the z-value exceeds the (Ve/C)-ratio. Thus the degree of suction is controlled by the dimensionless ratio S as defined by Eq. (28). Theoretically, the transition from nonsuction to suction takes place at S = 1. Increasing S-values signify increasing degrees of suction.
S=ZI(Ve/C)
(28)
The elimination of z from Eq. (28) by substituting the Eqs. (12) (20) yields Eq. (29):
S=c2[(r'ao'm)
(29)
On considering Eq. (7), the c2/r-term of Eq. (28) is identified as the maximum acceleration afmax of the dilation water. The dimensionless ratio of the maximum acceleration a/maxof the dilation water to the initial acceleration a o of the particle layer represents a modified Froude-number: Eqs (30) (31)
Fr=ahnox/ao
(30)
Fr=c2/(r.ao )
(31)
Finally it follows from substituting Eq. (31) in Eq. (29) that the Froude-number is also determined by the flow regime parameter m and the suction parameter S: Eq. (32) Fr=m.S
(32)
The significance of the c2/r-term has been already recognized by previous authors [3] [4] [5], presumably on the basis of statistical data on jig settings. However, the c2/r-term has been related in the past to the constant acceleration of gravity instead of relating it to the case-specific initial acceleration a0 of the separating layer as recommended by Eq. (31) (see Figure 7).
TYPICAL, VALUES OF THE SIMILARITY CRITERIA IN JIGGING PRACTICE The identification of the main similarity criteria of jigging (i.e. the lift-off angle o~o and the modified Froude-number Fr) is followed by the question which of their possible values signify favourable conditions
684
H.J. Steiner
of separation. The formulae of the similarity criteria cannot answer that question, since a stated similarity of two jig operations may correspond either to good or badly chosen settings in both cases. 100
/ 10 ms z
o
o o o
% E E d
"0
ooo o o :o~ °°°° o 0000 0 1 m s -2 °o ooo o/
10
0 0 0 / O0 000 / 00 0 /
Q.
E t~
0 0 0
o°o o/ 000 /
°:/
11 ° 0,1
cycle time, s Fig.7 Amplitude of the pulsion vs. cycle time of jigs in a number of ore processing plants. Data source [4]. Parameter: (c2/r) A well founded answer on this matter could be expected from a detailed analysis of the results of technical jig operations accompanied by additional laboratory investigations. However, for a rough estimate it seems quite sufficient to draw on already available empirical knowledge about the average settings of jigs. According to the result of an evaluation of plant data, the S-values, controlling the degree of suction as explained in the preceding chapter, crowd near S --, 0.75, thus restricting the modified Froude-number to the range Fr < 1.5. The second restraint, required to fix both the length and the frequency of the water strokes, may be expressed in terms of the lift-off angle. Its average value can be derived from the empirical rule that the conditions of separation are favourable in general if the maximum displacement of the separating layer equals approximately the amplitude of the pulsion, i.e. half the water stroke as measured directly below the screen. A sensitivity analysis of the path function assigns the said condition to a lift-off angle of about 20 degrees. However, it should be born in mind that the average values of the similarity criteria do not necessarily characterize the optimal settings. Within certain limits, the length of the strokes and their frequency will have to deviate from a general recommandation in order to pursue the specific goals of the process, e.g. either a high recovery in roughing operations or a high grade of the concentrate in a cleaning circuit. A narrow sized feed will tolerate comparatively longer strokes to the benefit of a high throughput while rather short strokes at the expense of a lower throughput will reduce the detrimental effect of a wide particle size distribution on the overall separation results.
A contribution to jigging theory--I
685
SUMMARY The fundamental difference between the time of jig layer displacement and the time of dilation justifies a separate mathematical analysis of the kinematics of jig layers and of the kinetics of particle segregation. Similarity criteria related to the motion of the separating layer at the cut position are expected to provide the link between the flow characteristics of the dilation water and the phenomenon of par'tide segregation. The hydraulic properties of a coherent jig layer are characterized by its initial acceleration, its terminal settling velocity and a dimensionless flow regime parameter. Concise equations of the kinematics of jig layers are obtained on characterizing the pulsion by the motion of an imaginary crank-driven plunger. The lift-off angle and a modified Froude-number represent the main similarity criteria of the kinematics of jig layers. The lift-off angle ,depends on the hutch water velocity, on the peripheral velocity of the imaginary crank, and finally on the terminal settling velocity of the jig layer in view. The modified Froude-number relates the maximal acceleration of the dilation water to the initial acceleration of the separating layer. The degree of suction in the separating layer is determined by the modified Froude-number and the flow regime parameter. The average values of the said similarity criteria, calculated from the settings of industrial jigs, crowd in a rather narrow range. NOMENCLATURE a
a0 a f max C
f Fr g m r
sf t
t~ te u v
ve
v/ vr
acceleration of a coherent particle layer initial acceleration of a coherent particle layer maximal acceleration of the dilation water peripheral velocity of the imaginary crank, equal to the maximal velocity of the imaginary plunger frequency of the pulsions modified Froude-number, defined by Eq. (31) acceleration of gravity flow regime parameter radius ("eccentricity") of the imaginary crank, equal to the amplitude of the pulsion displacement of the jig layer in view displacernent of the dilation water time cycle time (= duration of a jigging cycle) nominal time of acceleration, defined by Eq. (9) hutch water velocity velocity of the jig layer in view terminal settling velocity of a coherent particle layer velocity of the dilation water relative velocity of a jig layer with respect to the dilation water dimensionless parameter in the equations of jig layer motions
Greek symbols ot
6t
phase angle of a jigging cycle phase angle expressed in radians
Ps
lift-off angle, i.e. phase angle at the beginning of the displacement of the jig layer density of the fluid (= dilation water) average density of the particles in the separating layer
Ze
normalized time of acceleration, defined by Eq. (10)
o~o
686 ¢
H.J. Steiner normalized time of acceleration, expressed in radians
REFERENCES .
2. .
4.
.
Taggart, A.F., Handbook of Mineral Dressing, John Wiley, New York, 11-07f (1945). Finkey, J., Die wissenschafllichen Grundlagen der Erzau3'bereitung. Springer-Verlag, Berlin (1924). Wiard, E.S., The Theory and Practice of Ore Dressing, Mc Graw Hill, New York, 326 (1915). Kizevalter, B.V., Vlijanie ~ isla i rasmacha kolebanija ~ idkosti v processe otsadki, in: Voprosy teorii gravitacionnych metodov obogascenija poleznych iskopaemych. Gosgortechizdat, Moscow, 11-21 (1960). Karantzavelos, G.E. & Frangiscos, A.Z., Contribution to the Modelling of the Jigging Process. In: Herbst, J.A. (Ed.). Control '84 Mineral~Metallurgical Processing. AIMME, New York (1984).
Pergamon Plh S0892-6875(96)00055-6
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0892-6875/96 $15.o0+0.o0
A CONTRIBUTION SIMILARITY
T O T H E T H E O R Y OF JIGGING, PART I: CRITERIA OF THE MOTION OF JIG LAYERS
H.J. STEINER Instimt fiir Aufbereitung und Veredlung (Department of Mineral Processing), Montanuniversit~it Leoben (University of Mining and Metallurgy), A-8700 Leoben, Austria (Received 16 June 1995; accepted 18 February 1996)
ABSTRACT
A harmonicpulsion superimposed by a constant hutch waterflow serves as the model case in the derivation of similarity criteria of the motion of jig layers. At first the decisive portion of the jig bed is identified and characterized by two hydraulic properties, i.e. the initial acceleration and the terminal settling velocity of the separating layer at the cut position. A concise differential equation of the motion ofjig layers is formulated and solved for the velocity function and the displacement function. In the normalized versions of the said functions, two main dimensionless similarity criteria of the motion of a jig layer are identified. One of the criterions is the lift-off angle, i.e. the phase angle at the beginning of the displacement of the jig layer. The second criterion comprises a modified Froudenumber. The latter relates the maximum acceleration of thefluid to the initial acceleration of the separating layer, thus combining a characteristic of the pulsion with a characteristic of the fig bed. Additional dimensionless parameters characterize the flow regime and the degree of suction. Empirical knowledge about the average settings of jigs is summarized in average values of the said similarity criteria. All the equations are derived from basic hydraulics and observe the requirement of dimensional homogeneity. Keywords Mineral processing; gravity separation; modelling
INTRODUCTION An attempt has been made to predict the performance of jigging operations from basic data of the feed without referring to empirical partition curves. It was clear from the beginning that the goal in view could only be reached by structuring the complex process into sub-problems of less complicated nature. Due to the fundarctental difference between the time of layer displacement and the time of dilation, a separate mathematical analysis of the kinematics of jig layers and of the kinetics of particle segregation seemed justified. I1: was expected that the link between the two issues could be provided by adequate dimensionless numbers. Thus the identification of relevant similarity critera was the prime concern in developing a physical model of the motion of pulsed particle layers. A future contribution will deal with the segregation kinetics in a jig bed. Presented at Minerals Engineering '95, St. Ives, Cornwall, England, June 1995 675
676
H.J. Steiner CHARACTERIZATION OF THE FLOW OF DILATION WATER
The particular motion of the particle bed on the supporting screen is caused by the flow of the "dilation water" passing upwards or downwards through the openings of the screen. Thus the most relevant information about the water flow in a jig is the velocity of the dilation water as measured directly below the screen. The time profile of that velocity may be divided into a periodic pulsion and a constant portion related to a part of the hutch water addition. The periodic pulsion may be reduced to the motion of an imaginary plunger reciprocating close to the screen as sketched in Figure 1.
U_(300~dOC_ °_cd-OkV 6-o
U
l> -_
-_--
I -g i Fig. 1 Simplified model of a jig bed under the influence of a harmonic pulsion superimposed by a constant hutch water flow. Phase angle a and amplitude r of the pulsion. Peripheral velocity c of the imaginary crank-drive. Hutch water U The motion of a plunger is usually described by the length of the strokes ("throw") and their frequency. However, the model of a harmonic pulsion results in more concise equations of the bed displacement if the motion of the imaginary plunger is characterized by the peripheral velocity and by the radius of an imaginary crank-drive connected to the plunger by a rod of infinite length as indicated in Figure 1. Thus the amplitude of the pulsion--i.e, half the water stroke--is fixed by the crank radius. The peripheral velocity c of the crank ("crank-velocity") is a trivial function of the crank radius r and of the frequency f of the strokes: Eq. (1). The frequency equals the reciprocal of the cycle time te: Eq. (2).
c=2-z .rf
(1)
f=llt ,
(2)
The lowest position of the crank and of the plunger respectively defines the zero-points on the scales of
A contributionto jigging theory--I
677
the crank-angle o~ ( = phase angle) and of the time t. The relationship between a time differential dt and a corresponding plhase angle differential dc~ is expressed by Eqs (3) and (4):
d&ldt=2.r~'f
(3)
d&/dt=c]r
(4)
The velocity vf of the dilation water is to be understood as the velocity of approach or return with respect to the screen surf~Lce. It is elsewhere called "superficial velocity", well to distinguish from the velocity within the voids of the particle bed. The velocity vf splits into a periodic portion and a constant portion u ("hutch water velocity"). The normalized version of the functions of the velocity vf and of the displacement sf of the dilation water are listed below: Eqs (5) (6). See also Figure 2.
vslc=sin~+(u/c)
(5)
sf/r = 1 -coset + a.(u/c)
(6) 3
~ //,."
\ 1
.,~//'/" .~oJ*
0 0
2.
.
,
~
3
\,,, \,
"~. ,
2~
phase angle Fig. :2. Normalized (= amplitude related) displacement of the dilation water and of a jig layer under the influence of a harmonic pulsion. Curve 1 :Displacement of a particle layer; Curve 2:Displacement of the dilation water due to the pulsion; Curve 3:Displacement of the dilation water due to the pulsion and a constant hutch-water flow The quotient "cran'k velocity squared through crank radius" represents both the constant centrifugal acceleration of the circular motion of the crank as well as the maximal acceleration afmax of the flow of dilation water through the screen into the particle bed: Eq. (7).
a~ax=c2/r
(7)
The relevant kinematic parameters of the flow of dilation water can be derived from the real motion of the pulsion mechanism, the geometry of the jig, the time profile of the hutch water addition and the portion of hutch water discharged into the underflow. The movement of the water level in the jig does not correspond exactly to that of the dilation water, since the latter is partly diverted from the vertical direction into a horizontal one at each up-stroke.
678
H.J. Steiner IDENTIFICATION OF THE DECISIVE PORTION OF T H E JIG B E D
The jig bed, being inhomogeneous both in vertical and horizontal direction, may be roughly divided into a top layer, a central or intermediate layer, and a bottom layer. The easily separable particles of the feed are rather rapidly recovered either in the top layer or in the bottom layer. Their properties are not of prime interest, since the efficiency of separation--as reflected by the partition curves--depends primarily on the hydraulic behaviour of the separating layer at the cut position. This layer--though inhomogeneous with respect to particle size and particle densitiy--is characterized by rather uniform hydraulic properties. Its composition with respect to the distribution of particle size and particle density may be estimated from the density values at the cut positions of the various particle size classes (as indicated by the partition curves) and the frequencies of the said particle size classes in the feed. A restriction of the mathematical treatment to the analysis of the motion of the separating layer signifies a considerable simplification as well as focusing the attention to the most relevant portion of the jig bed. The concept of a central layer also corresponds with Taggart's view on the separation in a jig bed [1].
CHARACTERIZATION OF THE HYDRAULIC PROPERTIES OF A COHERENT JIG LAYER The pressure drop of a fluid passing upwards through the voids of a coherent bed of particles exerts a lifting force which is counteracted by the apparent weight of the particles, i.e. the difference "weight minus buoyancy". The state of a diminishing resultant force marks the transition from a particle bed at rest to one lifted off the support. The velocity vf of the approaching fluid in the said state of equilibrium corresponds in amount (though opposite in sign) to the imaginary settling velocity v e of the particle bed in quiet water. If the particle bed had the opportunity to settle in a quiet fluid from a position at rest on a suddenly withdrawn support, it would start off at an initial acceleration ao, which is determined by the average density if, of the particles, the density pf of the fluid and the acceleration g of gravity. The initial acceleration is quite independent of the geometry of the voids, since the fluid resistance equals zero at zero velocity: Eq. (8). ao=g.(1 -pil'~)
(8)
The terminal settling velocity v e and the initial acceleration ao represent the basic hydraulic properties of a coherent jig layer. In this respect, there exists a perfect hydraulic analogy between the properties of a single particle and those of a multi-particle bed. The ratio of the terminal settling velocity v e to the intial acceleration ao may be interpreted as a nominal time of acceleration t e (see Figure 3): Eq. (9). The ratio of that characteristic time span t e to the duration t c of one jigging cycle represents the normalized time r e of acceleration. It is expressed by Eq. (10) as a fraction of the cycle time, and converted by Eq. (11) into radians. t=v lao
(9)
%=t,/t~
(lO)
~e=2.~ ..c e
(11)
A contributionto jiggingtheory--I
679
te Ve :> ° , ~
0 0 oD :>
0 time t Fig.3 Accelerated motion of a coherent particle layer in quiet water from a position at rest. v e..... terminal settling velocity; te ..... nominal time of acceleration An alternative explession of the normalized time of acceleration follows from substituting the Eqs (1) (2) (10) in Eq. (11):
¢:e=(ve.c)/ (ao.r) The dimensionless parameter ~
(12) combines characteristics of both the particle bed and the pulsion. It
proves useful in the mathematical treatment of the jig layer kinematics.
MATHEMATICAL ANALYSIS OF THE KINEMATICS OF A JIG LAYER
Lift-off angle A jig layer of uniform hydraulic properties will be lifted off its support and move upwards as a coherent set of particles when the velocity vf of the dilation water compensates the terminal settling velocity v e of the particle layer. "lqaecorresponding phase angle of the jigging cycle may be called the lifl-offangle o~o. It can be calculated on substituting the restraints as defined by Eq. (13) in the velocity function of the dilation water. In the model case of a harmonic pulsion superimposed by a constant hutch water velocity u, the said velocity fimction is given by Eq. (5). It leads to Eq. (14), expressing the sine of the lift-off angle % by the difference of two dimensionless velocity quotients (see Figure 4.)
v/=%: =, ~x=~zo
(13)
sin% =(v~[c)-(u/c)
(14)
General differential equation The balance of the various forces (gravity, buouyancy, inertia and fluid resistance) acting on a jig layer of homogeneous hydraulic properties can be reduced to a quite concise relationship of the relevant kinematic quantities, i.e. accelerations ax and velocities Vx: Eq. (15). The mathematical procedure leading
680
H.J. Steiner
to that remarkable structural simplicity is rather analogous to the method applied in deriving a concise differential equation of the accelerated motion of single particles (to be described in a future part II of the publication).
(a/ao)+(vr/%) m = 1
(15)
The first term of Eq. (15) relates the variable acceleration (= a) of the jig layer to its constant initial acceleration ao. The second term in brackets relates the variable relative velocity v r of the jig layer to its constant terminal settling velocity v e. The relative velocity represents the velocity difference of the particle layer with respect to the fluid. The dimensionless exponent m to the (Vr/Ve)-term is identical to the velocity exponent in the equation of the fluid resistance. Thus the value of m depends on the flow regime in the voids of the particle layer. The theoretical limits of that "flow regime parameter" are m --- 1 (laminar flow) and m = 2 (fully turbulent flow). Under normal conditions of jigging, the m-value will be found in the range 1.5 to 2.
O~0 "
Lift-off angle
~7 i
I i
I
"1o
I t~
E
i i
2
/
/
I ,
I
E (D
N \
/
J
r-
f
/
/
\
/
water ~\
/
.i
o
_ "
r I1¢ t/ i c l e ' I '// layer Ia
u} , m
"o
0
\\
\ \
i
0
n
2~
phase angle o~ Fig.4 Illustration of the term "lift-off angle"
Transformation of the general differential equation
The accelerations and velocities appearing in Eq. (15) are counted positive when their vectors point in the direction of gravity. On the other hand it seems appropriate to attribute positive signs to upward directions of the velocity v of the particle layer and to the velocity vf of the dilation water. The trivial Eqs (16), (17) are in accordance with the said sign convention:
a= -dv]dt
(16)
Vr=-(V-V/)
(17)
During the period of layer displacement, the relative velocity is never far off the terminal settling velocity: Eq. (18). On that condition, Eq. (19) represents an acceptable simplification, introduced for the benefit of an explicit final solution of both the velocity function and the displacement function of a jig layer.
A contributionto jiggingtheory--I
681
(v,/v,)-I:
08)
(v,/v~)m'.l-m-(l - v ] v )
(19)
For the benefit of a concise differential equation of the model case under investigation, a dimensionless parameter z is defined by the ratio of the normalized time ~ of acceleration to the flow regime parameter m: Eq. (20)
z =~•/m
(20)
A combination of the Eqs (1) (2) (4) (5) (9) thru (11) and (14) thru(20) yields the approximate differential equation of the motion of a jig layer of homogeneous hydraulic properties under the influence of a harmonic pulsion and a constant hutch water flow. Equation (21) presents the compact version, structured into dimensionless terms. (v/c) =sin c~ - sin % - z
.d(v/c)/da
(21)
Integration and auxiliary functions Due to the inhomogeneous character of Eq. (21), a rather complicated expression of the velocity function is obtained on rigorous integration. However, on account of the generally prevailing condition z2,1, an approximate solution of quite sufficient accuracy may be derived. The displacement s of the particle layer from the support as a function of the phase angle c~is obtained on integrating the velocity function v=ds/dt. Prior to that second integration, the phase angle differential da has to be substituted for the time differential dt. The Eqs (22)-(25) define auxiliary functions, introduced for the benefit of more concise final expressions. A &=~-ao
(22)
Asina =sin~ -sh,%
(23)
Acosa =cos~ -o)s%
(24)
A =z'cos%'[1 -exp(-A =]z)]
(25)
Velocity function The normalized version of the approximate velocity function relates the variable velocity v of the particle layer to the const~at peripheral velocity c of the imaginary crank-drive: Eq. (26).
v/c =Asin ~ -z'A cos a -,4
P£ 9-6-D
(26)
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H.J. Steiner
Displacement function The normalized version of the approximate displacement function relates the variable displacement s of the particle layer to the constant amplitude r of the pulsion, i.e. the radius of the imaginary crank-drive: Eq. (27)
s/r=-Acosa -z.A sina -A &.sinao+z.A
(27)
Figures 5 and 6 illustrate the motion of a particle layer at varying values of the parameters (or0, z) acccording to Eq. (27).
2 112 n
e~
E
03
¢..
E O
03
tD
0,1
0 0
~ phase angle
2~
Fig.5 Normalized (= amplitude related) displacement of jig layers according to Eq. (27) at a constant lift-off angle
ao = 0.4. Parameter: z-value
2 "10 Q.
E
03
0'3
,4\
E t,.) n
0,5 0,6
03 Q.
"0
0 0
7g
2~
phase angle Fig.6 Normalized (= amplitude related) displacement of jig layers according to Eq. (27) at a constant z-value (z = 0.15). Parameter: lift-off angle
ao in radians.
A contribution to jigging theory--I
683
In contrast to the displacement function derived by Finkey [2], Eq. (27) does not result in physically impossible (i.e. negative) values of the layer displacement in the vicinity of the lift-off angle.
SIMILARITY CRITERIA OF JIG LAYER MOTIONS Equation (27) describes the dimensionless ratio "vertical displacement of the particle layer divided by the amplitude of the pulsion" as a function of the variable phase angle c~ and of just two numerical values, i.e. the lift-off angle c~o and the parameter z. Thus the conclusion--already suggested by the constituents of the differential Eq. (21)--reads: Constant values of oto and z determine similar jig bed motions. From the mathematical point of view, the two parameters (%, z) fulfill the requirements of similarity criteria. The lift-off angle c~0 represents a self-explaining term. The z-parameter however calls for an interpretation from the technical point of view. A thorough analysi:s of all the equations involved revealed a connection between the z-value and the degree of suction in a jig bed. Experienced jig operators are familiar with the phenomenon of suction, they know about the detrimental effects of an excessive degree of suction and are aware of the significance of an adequate suction control. Mathematically, suction is signified by negative values of the relative velocity v r of the particle layer with respect to the dilation water (see Eq. (17)). According to the results of the analysis, negative Vr-Values occur if the z-value exceeds the (Ve/C)-ratio. Thus the degree of suction is controlled by the dimensionless ratio S as defined by Eq. (28). Theoretically, the transition from nonsuction to suction takes place at S = 1. Increasing S-values signify increasing degrees of suction.
S=ZI(Ve/C)
(28)
The elimination of z from Eq. (28) by substituting the Eqs. (12) (20) yields Eq. (29):
S=c2[(r'ao'm)
(29)
On considering Eq. (7), the c2/r-term of Eq. (28) is identified as the maximum acceleration afmax of the dilation water. The dimensionless ratio of the maximum acceleration a/maxof the dilation water to the initial acceleration a o of the particle layer represents a modified Froude-number: Eqs (30) (31)
Fr=ahnox/ao
(30)
Fr=c2/(r.ao )
(31)
Finally it follows from substituting Eq. (31) in Eq. (29) that the Froude-number is also determined by the flow regime parameter m and the suction parameter S: Eq. (32) Fr=m.S
(32)
The significance of the c2/r-term has been already recognized by previous authors [3] [4] [5], presumably on the basis of statistical data on jig settings. However, the c2/r-term has been related in the past to the constant acceleration of gravity instead of relating it to the case-specific initial acceleration a0 of the separating layer as recommended by Eq. (31) (see Figure 7).
TYPICAL, VALUES OF THE SIMILARITY CRITERIA IN JIGGING PRACTICE The identification of the main similarity criteria of jigging (i.e. the lift-off angle o~o and the modified Froude-number Fr) is followed by the question which of their possible values signify favourable conditions
684
H.J. Steiner
of separation. The formulae of the similarity criteria cannot answer that question, since a stated similarity of two jig operations may correspond either to good or badly chosen settings in both cases. 100
/ 10 ms z
o
o o o
% E E d
"0
ooo o o :o~ °°°° o 0000 0 1 m s -2 °o ooo o/
10
0 0 0 / O0 000 / 00 0 /
Q.
E t~
0 0 0
o°o o/ 000 /
°:/
11 ° 0,1
cycle time, s Fig.7 Amplitude of the pulsion vs. cycle time of jigs in a number of ore processing plants. Data source [4]. Parameter: (c2/r) A well founded answer on this matter could be expected from a detailed analysis of the results of technical jig operations accompanied by additional laboratory investigations. However, for a rough estimate it seems quite sufficient to draw on already available empirical knowledge about the average settings of jigs. According to the result of an evaluation of plant data, the S-values, controlling the degree of suction as explained in the preceding chapter, crowd near S --, 0.75, thus restricting the modified Froude-number to the range Fr < 1.5. The second restraint, required to fix both the length and the frequency of the water strokes, may be expressed in terms of the lift-off angle. Its average value can be derived from the empirical rule that the conditions of separation are favourable in general if the maximum displacement of the separating layer equals approximately the amplitude of the pulsion, i.e. half the water stroke as measured directly below the screen. A sensitivity analysis of the path function assigns the said condition to a lift-off angle of about 20 degrees. However, it should be born in mind that the average values of the similarity criteria do not necessarily characterize the optimal settings. Within certain limits, the length of the strokes and their frequency will have to deviate from a general recommandation in order to pursue the specific goals of the process, e.g. either a high recovery in roughing operations or a high grade of the concentrate in a cleaning circuit. A narrow sized feed will tolerate comparatively longer strokes to the benefit of a high throughput while rather short strokes at the expense of a lower throughput will reduce the detrimental effect of a wide particle size distribution on the overall separation results.
A contribution to jigging theory--I
685
SUMMARY The fundamental difference between the time of jig layer displacement and the time of dilation justifies a separate mathematical analysis of the kinematics of jig layers and of the kinetics of particle segregation. Similarity criteria related to the motion of the separating layer at the cut position are expected to provide the link between the flow characteristics of the dilation water and the phenomenon of par'tide segregation. The hydraulic properties of a coherent jig layer are characterized by its initial acceleration, its terminal settling velocity and a dimensionless flow regime parameter. Concise equations of the kinematics of jig layers are obtained on characterizing the pulsion by the motion of an imaginary crank-driven plunger. The lift-off angle and a modified Froude-number represent the main similarity criteria of the kinematics of jig layers. The lift-off angle ,depends on the hutch water velocity, on the peripheral velocity of the imaginary crank, and finally on the terminal settling velocity of the jig layer in view. The modified Froude-number relates the maximal acceleration of the dilation water to the initial acceleration of the separating layer. The degree of suction in the separating layer is determined by the modified Froude-number and the flow regime parameter. The average values of the said similarity criteria, calculated from the settings of industrial jigs, crowd in a rather narrow range. NOMENCLATURE a
a0 a f max C
f Fr g m r
sf t
t~ te u v
ve
v/ vr
acceleration of a coherent particle layer initial acceleration of a coherent particle layer maximal acceleration of the dilation water peripheral velocity of the imaginary crank, equal to the maximal velocity of the imaginary plunger frequency of the pulsions modified Froude-number, defined by Eq. (31) acceleration of gravity flow regime parameter radius ("eccentricity") of the imaginary crank, equal to the amplitude of the pulsion displacement of the jig layer in view displacernent of the dilation water time cycle time (= duration of a jigging cycle) nominal time of acceleration, defined by Eq. (9) hutch water velocity velocity of the jig layer in view terminal settling velocity of a coherent particle layer velocity of the dilation water relative velocity of a jig layer with respect to the dilation water dimensionless parameter in the equations of jig layer motions
Greek symbols ot
6t
phase angle of a jigging cycle phase angle expressed in radians
Ps
lift-off angle, i.e. phase angle at the beginning of the displacement of the jig layer density of the fluid (= dilation water) average density of the particles in the separating layer
Ze
normalized time of acceleration, defined by Eq. (10)
o~o
686 ¢
H.J. Steiner normalized time of acceleration, expressed in radians
REFERENCES .
2. .
4.
.
Taggart, A.F., Handbook of Mineral Dressing, John Wiley, New York, 11-07f (1945). Finkey, J., Die wissenschafllichen Grundlagen der Erzau3'bereitung. Springer-Verlag, Berlin (1924). Wiard, E.S., The Theory and Practice of Ore Dressing, Mc Graw Hill, New York, 326 (1915). Kizevalter, B.V., Vlijanie ~ isla i rasmacha kolebanija ~ idkosti v processe otsadki, in: Voprosy teorii gravitacionnych metodov obogascenija poleznych iskopaemych. Gosgortechizdat, Moscow, 11-21 (1960). Karantzavelos, G.E. & Frangiscos, A.Z., Contribution to the Modelling of the Jigging Process. In: Herbst, J.A. (Ed.). Control '84 Mineral~Metallurgical Processing. AIMME, New York (1984).