- Email: [email protected]
Particle Residence Time in Column Flotation Based on Cyclonic Separation
Sept. 2007
Journal of China University of Mining & Technology
Vol.17
No.3
J China Univ Mining & Technol 2007, 17(3): 0349–0353
Particle Residence Time in Column Flotation Based on Cyclonic Separation 1,2
2
ZHOU Xiao-hua , LIU Jiong-tian 1
2
Department of Mining Engineering, University of Kentucky, Lexington, KY 40506, USA School of Chemical Engineering and Technology, China University of Mining & Technology, Xuzhou, Jiangsu 221008, China Abstract: The cyclonic static micro-bubble column flotation (FCSMC) is an effective separation device for fine particle treatment. The high mineralization rate and short flotation time of this equipment can be attributed to its unique cyclonic force field. It also has been observed that the presence of a cyclonic force field leads to a lower bottom separation size limit and a reduction of unselective entrainment. The collection zone of the column is considered to consist of two parts, a column separation zone and a cyclonic zone. Total recovery of the collection zone was developed. For our study, we analyzed the particle movement in the cyclonic zone. Particle residence time equations for the cyclonic zone were derived by force analysis. Results obtained in this study provide a theoretical foundation for the design and scale-up of the FCSMC. Key words: column flotation; cyclonic separation; residence time CLC number: TD 94
1
Introduction
Column flotation can be designed using the same principles applied to the design of chemical reactors. This requires an understanding of flotation kinetics and the residence time distributions of the solid and liquid phases. Hence it is clear that in the design of such columns a thorough investigation needs to be made in order to determine the appropriate rate and residence time distribution constants. An appropriate model was developed in our study to determine the residence time and describe the flow characteristics of the solid and liquid phases in both column flotation and cyclonic zones. Several researchers have conducted residence time distribution studies on column flotation cells. All the models proposed describe the behavior of the collection zone. However, data for the collection zone cannot be obtained using liquid tracers since very little of the feed water reports to this zone. In a study reported by Dobby and Finch non-floatable manganese dioxide was used as a tracer[1]; however, the concentration of solids was 2%–3%, which was too low to highlight any behavioral differences between different phases.
The most common approach to describe the collection zone residence time distribution is to use a vessel dispersion model. However, this model does not work well for column scale-up because, although it has been observed that for large columns the effect of the degree of mixing on the predicted performance of the column becomes less significant, it is, nevertheless, inappropriate to use vessel dispersion numbers (D/uL) greater than 0.2 because of the underlying assumptions of the model[2]. The approximation for the vessel dispersion model assumes that the end effects are negligible, which is clearly not the case at low aspect ratios. Secondly, in using the axial dispersion model the assumption is made that there is no radial mixing. This is unlikely to be valid in larger diameter columns. This model therefore underestimates the mixing in full-scale columns and leads to an overprediction of the required column size[3]. The use of a tanks-in-series model, or some other compartment model, is therefore preferred. Several researchers have derived correlations for the solid phase dispersion coefficient (D) as a function of the column diameter, the concentration of solids and superficial gas rates[1, 3–4]. Mills, et al have
Received 16 March 2007; accepted 20 May 2007 Project 50425414 supported by the National Outstanding Youth Science Foundation of China Corresponding author. Tel: +1-859-4202799; E-mail address: [email protected]
350
Journal of China University of Mining & Technology
shown that the use of liquid phase column flotation data, combined with three-phase correlations for bubble columns, to model the solid phase in column flotation systems, is incorrect[5]. These three-phase correlations are based on work done on bubble column systems under quite different conditions to those occurring in column flotation cells[6]. In our study, a dynamic analysis has been made of the particle residence time equation in a column flotation cell.
2 2.1
The overall flotation recovery in the collection zone is a function of both the column flotation zone and the cyclonic zone recovery. The flow diagram of the flotation process can be expressed as in Fig. 2. Concentrate to froth zone XRCZ
XRCX(1ˉRCZ) X (1ˉRCZ) Cyclonic zone X (1ˉRCZ) (1ˉRCX) Tailings
Fig. 2
Schematic figure. of collection zone recovery
Assume the amount of material reporting to the column flotation zone is X, the recovery of the column flotation zone RCZ and the recovery of the cyclonic zone RCX. Therefore, the amount of material reporting to the froth zone is: P=XRCZ (3) Likewise, the feed mass reporting to the tailings stream can be presented by:
RC
The flow in the column flotation zone is a plug flow, so the recovery of the column flotation zone in the FCSMC can be expressed by Equation (1) [9]:
1 exp(kCZW pZ )
Column flotation zone
T=X(ˉRCZ)(ˉRCX) (4) Since the overall recovery of the feed mass is equivalent to: P RC (5) P T Substituting Equations (3) and (4) into Equation (5) obtains:
Schematic of the cyclonic static micro-bubble flotation column
RCZ
X
Feed
Separation zone structure of flotation column
Fig. 1
No.3
2.2 Overall flotation recovery in the collection zone
Recovery Expession of Flotation Column
Cyclonic static micro bubble column flotation, FCSMC, consists of three parts which are a froth zone, a column flotation zone and a cyclonic zone, as shown in Fig. 1[7]. The collection zone includes the column flotation zone and the cyclonic zone[8]. The recovery of the collection zone, Rc, is a function of the flotation velocity rate (Kc) and particle residence time (IJp)[9].
Vol.17
(1)
where RCZ is the recovery of the column flotation zone, kCZ the flotation velocity rate in the column flotation zone and IJpZ the particle residence time in the column flotation zone. The flow is perfect mix in the cyclonic zone, so the recovery of the cyclonic zone in the FCSMC can be be expressed by Equation (2)˖
RCZ RCZ RCX 1 RCX
(6)
Substituting Equations (1) and (2) into Equation (6) gives: RC 1 exp(kCZWpZ ) (7) 1 1 >1 exp(kCZWpZ )@ ¬ª1 (1 kCXWpX ) º¼ 1 ª¬1 (1 kCXWpX ) º¼
In this model, two parameters need to be addressed including the flotation velocity rate and particle residence time. We studied particle residence time.
(2)
Particle Residence Time in Column Flotation Zone
where RCX is the recovery of the cyclonic zone, kCX flotation velocity rate in the cyclonic zone and IJpX particle residence time in the cyclonic zone.
For the column flotation zone, the flow is a plug flow in which the liquid residence time can be esti-
RCX
1 (1 kCX W pX )1
3
ZHOU Xiao-hua et al
Particle Residence Time in Column Flotation Based on Cyclonic Separation
mated by the following equation[10]:
Wl
L(1 H ) Vt
where L is length of the collection zone, Vt its superficial tailings rate and İ the fractional gas hold-up in the collection zone. The particle residence time IJp in the column flotation zone can be determined using Equation (9): §
Vt (1 H ) · ¸ © Vt (1 H ) U p ¹
Wp Wl ¨
Up
gd p2 ( U p Ul )(1 Is )2.7 18P (1 Rep 0.687 )
these particles under normal conditions. But the target particles collide and attach with airbubbles during which bubble mineralization occurs. In order to estimate the density of mineralized bubbles, assume that mineralized bubbles contains one bubble with diameter db and n particles with diameter dp. If the mass of the bubble and water film on the bubble surface can be ignored, the density of mineralized bubbles can be expressed by Equation (14):
(9)
Um
where Up is the slip velocity between the water and particle and can by calculated by:
nʌd p3 U p / 6 ʌd b3 / 6 nʌd p3 / 6
R ep
P
(10)
(11)
where dp is the particle diameter, ȡp the particle density, ȡl the liquid density, ijs the volumetric concentration of solids, ȝ the dynamic viscosity of the liquid and Rep particle Reynolds number. The flow in the column flotation zone is nearly a plug flow, so that the superficial tailings rate can be expressed by Equation (12): Vt
Qt Ac
(D Qf Qw )
4 ʌD 2
(12)
in which Qt is the tailing volume mass, Qf the feed mass, Qw the wash water mass and D the coefficient of feed reporting to tailings (normally in the range of 0.8–0.9; but if the amount is small, D can be taken as 1), Ac is the cross-sectional area of the column and D the diameter of the column. Substituting Equation (12) into Equation (9) provides the particle residence time in the column flotation zone IJpZ:
W pZ
LZ 4(D Qf Qw ) Up ʌD 2 (1 H )
(13)
where LZ is the length of the column flotation zone and Up is obtained from Equations (10) and (11).
4
Particle Residence Time in Cyclonic Zone
4.1
Particle movement in cyclonic zone
The particle movement in the cyclonic zone can be divided into two kinds of movement, i.e., the movement of the target mineral particles and that of the non-target particles. The movement of the non-target particles depends on particle density and size because no bubble attachment and mineralization occur for
(14)
which can be rearranged as˖
Um
and d pU p Ul (1 Is )
351
nd p3 U p d b3 nd p3
(15)
d b3 ) Up (16) d b3 nd p3 For a given mineral, the density is fixed. Therefore, according to Equation (15), for a given mineralized bubble, its density decreases with deceasing particle diameter and the number of particles attached to it. Similarly, the density of a mineralized bubble decreases by increasing the bubble diameter for a given particle. Assume the critical density occurs when the density of mineralized bubbles equals the density of slurry, that is, when ȡm˙ȡ. Thus, when ȡm<ȡ, the mineralized bubbles move upwards in the column. The separation of mineralized bubbles from tailings requires ȡm<ȡ and the moving-up speed of the mineralized bubbles be higher than the tailing rate. For a mineralized airbubble adhered to particles of density of ȡp, the smaller the mineral particle size, the easier the mineralized bubble reaches critical density and then moves inwards and upwards in the column. If the adhered mineral particles are too big, more bubbles are needed to obtain critical density. Therefore, small-sized target particles move inwards and upwards towards the froth zone at the top of the cyclonic zone when the circulating pulp is injected tangentially into the column. Big mineral particles are then subjected to further separation when reporting to the circulating pulp. However, mineral particles which are too large cannot adhere to airbubbles and therefore will discharge to tailings. The size of these particles is actually beyond the upper size limit which a flotation column can handle.
Um
(1
4.2 Forces exerted on particles in cyclonic zone
Assume a slick global particle in the cyclonic zone and that the velocity of the particle is Up at time t, as shown in Fig. 3. Assume also that Up is decomposed into three hefts upon the right angle coordinate system, i.e., a radial velocity Upx, an axial velocity Upz
352
Journal of China University of Mining & Technology
and a tangential velocity Upy.
Vol.17
ʌ 3 ʌ d p U l g 3ʌP d p u rz d p3 U p g 6 6
u pz
ʌ 3 d u pz dp Up 6 dt
dh / dt
No.3
(22) (23)
which can be rearranged as: ʌ 3 dh ʌ 3 d2h d p ( U l U p ) g 3ʌP d p (ufz ) d p U p 2 (24) 6 dt 6 dt The dynamic equation of particles in the cyclonic zone was shown as Equation (15), which was derived from force analysis[8]: Fig. 3
3D motion of particle in the cyclonic zone of FCSMC
d 2 h 18P dh 18P ufz Ul U p 2 g 0 (25) Up dt 2 d p2 U p dt dp Up where h is length of the cyclonic zone. When the velocity of the particle is close to the terminal settling velocity, we have,
d2h 0 (26) dt 2 Substituting Equation (26) into Equation (25) leads to Fig. 4
Forces exerted on a particle in the cyclonic zone
Forces exerted on a particle in the cyclonic zone are shown in Fig. 4, where three forces are presented in the radial direction, i.e., a centrifugal force Fc, a buoyancy force in the centrifugal field Fb and resistance Fdx. In order to study the particle residence time in the cyclonic zone, the forces are analyzed as follows: 1) Gravity fg fg
(ʌ / 6)d p3 Up g
mg
(17)
where m is the mass of the particle, dp the particle diameter and ȡp its density. 2) Buoyancy force in the centrifugal field f b (ʌ / 6)d p3 Ul g
fb
3ʌP d p urz
(19)
u rz
u fz u p z
(20)
where urz is the relative velocity between liquid and particle, ufz the radial velocity of liquid and upz the radial velocity of the particle. Axial (Z) dynamic equation of particle in cyclonic zone
Dynamics equations of particle in the axial direction can be expressed by: m (dupz / dt )
(21)
d p2 g ( U p U l ) 18P
(27)
The time t needed for moving vertically for a distance of h can be obtained by integrating Equation (24): h t (28) 2 d p g ( Up Ul ) u fz 18 P Both tailing and circulating pulp pass through the cyclonic zone of the FCSMC. As mentioned earlier, the amount of circulating pulp is about 2–3 times higher than the flotation feed rate and therefore the average speed of the liquid phase in the cyclonic zone, ufz, can be expressed as, ufz
Fdz
f b Fdz f g
u fz
(18)
where ȡl is liquid density. 3) Resistance (drag force) Fdz
4.3
dh dt
Qt Qx Ac
(29)
where Qt is the tailing rate, Qx the circulating pulp rate and Ac the cross-section area of column flotation. Qx is usually twice the feed rate, i.e.: Qx˙2Qf
(30)
Substituting Equations (29) and (30) into Equation (28) obtains: h t (31) Qt 2Qf d p2 g ( U p Ul ) Ac 18P Equation (31) is the particle residence time in the cyclonic zone which can be expressed further as:
ZHOU Xiao-hua et al
W pX
Particle Residence Time in Column Flotation Based on Cyclonic Separation
LX 2 4(Qt 2Qf ) d p g ( U p Ul ) ʌD 2 18P LX 2 4[Qw (2 D )Qf ] d p g ( U p Ul ) ʌD 2 18P
(32)
where LX is the length of cyclonic zone, Qt the tailing rate and Qf the feed rate. Therefore, the total particle residence time in the cyclonic static micro-bubble flotation column can be expressed as: LZ 4(D Qf Qw ) Up ʌD 2 (1 H ) LX 4[Qw (2 D )Qf ] d p2 g ( U p Ul ) ʌD 2 18P
353
with an increasing wash water rate. Therefore, a proper wash water rate is needed to increase the concentrate grade and yield. Moreover, for a given column flotation, the particle residence time is related to the height of the column flotation and cyclonic zones. If the height of the cyclonic zone is too high, the particle residence time will decrease. A higher turbulence degree has a negative effect on quiescent separation. Thus, a proper cyclonic zone height is beneficial in order to achieve a better separation performance.
5
Conclusions
1) The overall flotation recovery in the collection zone of the FCSMC is,
W p W pZ W pX
RC
(33)
Substituting Equations (13) and (32) into Equation (7) obtains the recovery of the column flotation in the collection zone. At a given feed rate, the particle residence time in the column flotation is, inter alia, also related to the wash water rate and the column diameter. The concentrate grade increases with an increasing wash water rate; however, the particle residence time decreases and as well, the concentrate yield decreases
1 exp(kCZWpZ ) . >1 exp(kCZWpZ )@ ª¬1 (1 kCXWpX )1 ¼º 1 ¬ª1 (1 kCXWpX )1º¼
2) The total particle residence time in the FCSMC can be expressed as: LZ 4(D Qf Qw ) U p ʌD 2 (1 H ) LX 4[Qw (2 D )Qf ] d p2 g ( U p Ul ) . ʌD 2 18P
W p W pZ W pX
References Dobby G S, Finch J A. Mixing characteristics of industrial flotation columns. Chem Eng Sci, 1985, 40(7): 1061–1068. Bischoff K B, Levenspiel O. Fluid dispersed-generalization and comparison of mathematical models - I Generalization of models. Chem Eng Sci, 1961, (17): 245–255. [3] Rice R G, Oliver A D, Newman J P, et al. Reduced dispersion using baffles in column flotation. Powder Technol, 1974, (10): 201–210. [4] Mavros P, Lazaridis N K, Matis K A. A study and modelling of liquid phase mixing in a flotation column. Int J Miner Proces, 1989, (26): 1–16. [5] Mills P J, O'Connor C T. The modelling of liquid and solids mixing in a flotation cell. Miner Eng, 1990, 3(6): 567–576. [6] Argo W B, Cova D R. Longitudinal mixing in gas sparged tubular vessels. Ind Eng Chem Process Des Dev, 1965, 4(4): 352–359. [7] Joshi J B, Sharma M M. A circulation cell model for bubble columns. Trans Inst Chem Eng, 1979, (57): 244–251. [8] Liu J T. Enhanced Separation Method and Apparatus of Static Micro-Bubble Column Flotation. Chinese patent: 97107091.1, 2002-3-13. (In Chinese) [9] Zhou X H. Cyclonic Separation Mechanism and Application of Column Flotation [Ph.D. dissertation]. Xuzhou: China University of Mining & Technology, 2005. (In Chinese) [10] Chen Z M. Review on mathematic model of column flotation. In: Proceedings of Mineral Processing Conference. Beijing: Metallurgical Industry Press, 1993: 248–291. (In Chinese)
[1] [2]
Journal of China University of Mining & Technology
Vol.17
No.3
J China Univ Mining & Technol 2007, 17(3): 0349–0353
Particle Residence Time in Column Flotation Based on Cyclonic Separation 1,2
2
ZHOU Xiao-hua , LIU Jiong-tian 1
2
Department of Mining Engineering, University of Kentucky, Lexington, KY 40506, USA School of Chemical Engineering and Technology, China University of Mining & Technology, Xuzhou, Jiangsu 221008, China Abstract: The cyclonic static micro-bubble column flotation (FCSMC) is an effective separation device for fine particle treatment. The high mineralization rate and short flotation time of this equipment can be attributed to its unique cyclonic force field. It also has been observed that the presence of a cyclonic force field leads to a lower bottom separation size limit and a reduction of unselective entrainment. The collection zone of the column is considered to consist of two parts, a column separation zone and a cyclonic zone. Total recovery of the collection zone was developed. For our study, we analyzed the particle movement in the cyclonic zone. Particle residence time equations for the cyclonic zone were derived by force analysis. Results obtained in this study provide a theoretical foundation for the design and scale-up of the FCSMC. Key words: column flotation; cyclonic separation; residence time CLC number: TD 94
1
Introduction
Column flotation can be designed using the same principles applied to the design of chemical reactors. This requires an understanding of flotation kinetics and the residence time distributions of the solid and liquid phases. Hence it is clear that in the design of such columns a thorough investigation needs to be made in order to determine the appropriate rate and residence time distribution constants. An appropriate model was developed in our study to determine the residence time and describe the flow characteristics of the solid and liquid phases in both column flotation and cyclonic zones. Several researchers have conducted residence time distribution studies on column flotation cells. All the models proposed describe the behavior of the collection zone. However, data for the collection zone cannot be obtained using liquid tracers since very little of the feed water reports to this zone. In a study reported by Dobby and Finch non-floatable manganese dioxide was used as a tracer[1]; however, the concentration of solids was 2%–3%, which was too low to highlight any behavioral differences between different phases.
The most common approach to describe the collection zone residence time distribution is to use a vessel dispersion model. However, this model does not work well for column scale-up because, although it has been observed that for large columns the effect of the degree of mixing on the predicted performance of the column becomes less significant, it is, nevertheless, inappropriate to use vessel dispersion numbers (D/uL) greater than 0.2 because of the underlying assumptions of the model[2]. The approximation for the vessel dispersion model assumes that the end effects are negligible, which is clearly not the case at low aspect ratios. Secondly, in using the axial dispersion model the assumption is made that there is no radial mixing. This is unlikely to be valid in larger diameter columns. This model therefore underestimates the mixing in full-scale columns and leads to an overprediction of the required column size[3]. The use of a tanks-in-series model, or some other compartment model, is therefore preferred. Several researchers have derived correlations for the solid phase dispersion coefficient (D) as a function of the column diameter, the concentration of solids and superficial gas rates[1, 3–4]. Mills, et al have
Received 16 March 2007; accepted 20 May 2007 Project 50425414 supported by the National Outstanding Youth Science Foundation of China Corresponding author. Tel: +1-859-4202799; E-mail address: [email protected]
350
Journal of China University of Mining & Technology
shown that the use of liquid phase column flotation data, combined with three-phase correlations for bubble columns, to model the solid phase in column flotation systems, is incorrect[5]. These three-phase correlations are based on work done on bubble column systems under quite different conditions to those occurring in column flotation cells[6]. In our study, a dynamic analysis has been made of the particle residence time equation in a column flotation cell.
2 2.1
The overall flotation recovery in the collection zone is a function of both the column flotation zone and the cyclonic zone recovery. The flow diagram of the flotation process can be expressed as in Fig. 2. Concentrate to froth zone XRCZ
XRCX(1ˉRCZ) X (1ˉRCZ) Cyclonic zone X (1ˉRCZ) (1ˉRCX) Tailings
Fig. 2
Schematic figure. of collection zone recovery
Assume the amount of material reporting to the column flotation zone is X, the recovery of the column flotation zone RCZ and the recovery of the cyclonic zone RCX. Therefore, the amount of material reporting to the froth zone is: P=XRCZ (3) Likewise, the feed mass reporting to the tailings stream can be presented by:
RC
The flow in the column flotation zone is a plug flow, so the recovery of the column flotation zone in the FCSMC can be expressed by Equation (1) [9]:
1 exp(kCZW pZ )
Column flotation zone
T=X(ˉRCZ)(ˉRCX) (4) Since the overall recovery of the feed mass is equivalent to: P RC (5) P T Substituting Equations (3) and (4) into Equation (5) obtains:
Schematic of the cyclonic static micro-bubble flotation column
RCZ
X
Feed
Separation zone structure of flotation column
Fig. 1
No.3
2.2 Overall flotation recovery in the collection zone
Recovery Expession of Flotation Column
Cyclonic static micro bubble column flotation, FCSMC, consists of three parts which are a froth zone, a column flotation zone and a cyclonic zone, as shown in Fig. 1[7]. The collection zone includes the column flotation zone and the cyclonic zone[8]. The recovery of the collection zone, Rc, is a function of the flotation velocity rate (Kc) and particle residence time (IJp)[9].
Vol.17
(1)
where RCZ is the recovery of the column flotation zone, kCZ the flotation velocity rate in the column flotation zone and IJpZ the particle residence time in the column flotation zone. The flow is perfect mix in the cyclonic zone, so the recovery of the cyclonic zone in the FCSMC can be be expressed by Equation (2)˖
RCZ RCZ RCX 1 RCX
(6)
Substituting Equations (1) and (2) into Equation (6) gives: RC 1 exp(kCZWpZ ) (7) 1 1 >1 exp(kCZWpZ )@ ¬ª1 (1 kCXWpX ) º¼ 1 ª¬1 (1 kCXWpX ) º¼
In this model, two parameters need to be addressed including the flotation velocity rate and particle residence time. We studied particle residence time.
(2)
Particle Residence Time in Column Flotation Zone
where RCX is the recovery of the cyclonic zone, kCX flotation velocity rate in the cyclonic zone and IJpX particle residence time in the cyclonic zone.
For the column flotation zone, the flow is a plug flow in which the liquid residence time can be esti-
RCX
1 (1 kCX W pX )1
3
ZHOU Xiao-hua et al
Particle Residence Time in Column Flotation Based on Cyclonic Separation
mated by the following equation[10]:
Wl
L(1 H ) Vt
where L is length of the collection zone, Vt its superficial tailings rate and İ the fractional gas hold-up in the collection zone. The particle residence time IJp in the column flotation zone can be determined using Equation (9): §
Vt (1 H ) · ¸ © Vt (1 H ) U p ¹
Wp Wl ¨
Up
gd p2 ( U p Ul )(1 Is )2.7 18P (1 Rep 0.687 )
these particles under normal conditions. But the target particles collide and attach with airbubbles during which bubble mineralization occurs. In order to estimate the density of mineralized bubbles, assume that mineralized bubbles contains one bubble with diameter db and n particles with diameter dp. If the mass of the bubble and water film on the bubble surface can be ignored, the density of mineralized bubbles can be expressed by Equation (14):
(9)
Um
where Up is the slip velocity between the water and particle and can by calculated by:
nʌd p3 U p / 6 ʌd b3 / 6 nʌd p3 / 6
R ep
P
(10)
(11)
where dp is the particle diameter, ȡp the particle density, ȡl the liquid density, ijs the volumetric concentration of solids, ȝ the dynamic viscosity of the liquid and Rep particle Reynolds number. The flow in the column flotation zone is nearly a plug flow, so that the superficial tailings rate can be expressed by Equation (12): Vt
Qt Ac
(D Qf Qw )
4 ʌD 2
(12)
in which Qt is the tailing volume mass, Qf the feed mass, Qw the wash water mass and D the coefficient of feed reporting to tailings (normally in the range of 0.8–0.9; but if the amount is small, D can be taken as 1), Ac is the cross-sectional area of the column and D the diameter of the column. Substituting Equation (12) into Equation (9) provides the particle residence time in the column flotation zone IJpZ:
W pZ
LZ 4(D Qf Qw ) Up ʌD 2 (1 H )
(13)
where LZ is the length of the column flotation zone and Up is obtained from Equations (10) and (11).
4
Particle Residence Time in Cyclonic Zone
4.1
Particle movement in cyclonic zone
The particle movement in the cyclonic zone can be divided into two kinds of movement, i.e., the movement of the target mineral particles and that of the non-target particles. The movement of the non-target particles depends on particle density and size because no bubble attachment and mineralization occur for
(14)
which can be rearranged as˖
Um
and d pU p Ul (1 Is )
351
nd p3 U p d b3 nd p3
(15)
d b3 ) Up (16) d b3 nd p3 For a given mineral, the density is fixed. Therefore, according to Equation (15), for a given mineralized bubble, its density decreases with deceasing particle diameter and the number of particles attached to it. Similarly, the density of a mineralized bubble decreases by increasing the bubble diameter for a given particle. Assume the critical density occurs when the density of mineralized bubbles equals the density of slurry, that is, when ȡm˙ȡ. Thus, when ȡm<ȡ, the mineralized bubbles move upwards in the column. The separation of mineralized bubbles from tailings requires ȡm<ȡ and the moving-up speed of the mineralized bubbles be higher than the tailing rate. For a mineralized airbubble adhered to particles of density of ȡp, the smaller the mineral particle size, the easier the mineralized bubble reaches critical density and then moves inwards and upwards in the column. If the adhered mineral particles are too big, more bubbles are needed to obtain critical density. Therefore, small-sized target particles move inwards and upwards towards the froth zone at the top of the cyclonic zone when the circulating pulp is injected tangentially into the column. Big mineral particles are then subjected to further separation when reporting to the circulating pulp. However, mineral particles which are too large cannot adhere to airbubbles and therefore will discharge to tailings. The size of these particles is actually beyond the upper size limit which a flotation column can handle.
Um
(1
4.2 Forces exerted on particles in cyclonic zone
Assume a slick global particle in the cyclonic zone and that the velocity of the particle is Up at time t, as shown in Fig. 3. Assume also that Up is decomposed into three hefts upon the right angle coordinate system, i.e., a radial velocity Upx, an axial velocity Upz
352
Journal of China University of Mining & Technology
and a tangential velocity Upy.
Vol.17
ʌ 3 ʌ d p U l g 3ʌP d p u rz d p3 U p g 6 6
u pz
ʌ 3 d u pz dp Up 6 dt
dh / dt
No.3
(22) (23)
which can be rearranged as: ʌ 3 dh ʌ 3 d2h d p ( U l U p ) g 3ʌP d p (ufz ) d p U p 2 (24) 6 dt 6 dt The dynamic equation of particles in the cyclonic zone was shown as Equation (15), which was derived from force analysis[8]: Fig. 3
3D motion of particle in the cyclonic zone of FCSMC
d 2 h 18P dh 18P ufz Ul U p 2 g 0 (25) Up dt 2 d p2 U p dt dp Up where h is length of the cyclonic zone. When the velocity of the particle is close to the terminal settling velocity, we have,
d2h 0 (26) dt 2 Substituting Equation (26) into Equation (25) leads to Fig. 4
Forces exerted on a particle in the cyclonic zone
Forces exerted on a particle in the cyclonic zone are shown in Fig. 4, where three forces are presented in the radial direction, i.e., a centrifugal force Fc, a buoyancy force in the centrifugal field Fb and resistance Fdx. In order to study the particle residence time in the cyclonic zone, the forces are analyzed as follows: 1) Gravity fg fg
(ʌ / 6)d p3 Up g
mg
(17)
where m is the mass of the particle, dp the particle diameter and ȡp its density. 2) Buoyancy force in the centrifugal field f b (ʌ / 6)d p3 Ul g
fb
3ʌP d p urz
(19)
u rz
u fz u p z
(20)
where urz is the relative velocity between liquid and particle, ufz the radial velocity of liquid and upz the radial velocity of the particle. Axial (Z) dynamic equation of particle in cyclonic zone
Dynamics equations of particle in the axial direction can be expressed by: m (dupz / dt )
(21)
d p2 g ( U p U l ) 18P
(27)
The time t needed for moving vertically for a distance of h can be obtained by integrating Equation (24): h t (28) 2 d p g ( Up Ul ) u fz 18 P Both tailing and circulating pulp pass through the cyclonic zone of the FCSMC. As mentioned earlier, the amount of circulating pulp is about 2–3 times higher than the flotation feed rate and therefore the average speed of the liquid phase in the cyclonic zone, ufz, can be expressed as, ufz
Fdz
f b Fdz f g
u fz
(18)
where ȡl is liquid density. 3) Resistance (drag force) Fdz
4.3
dh dt
Qt Qx Ac
(29)
where Qt is the tailing rate, Qx the circulating pulp rate and Ac the cross-section area of column flotation. Qx is usually twice the feed rate, i.e.: Qx˙2Qf
(30)
Substituting Equations (29) and (30) into Equation (28) obtains: h t (31) Qt 2Qf d p2 g ( U p Ul ) Ac 18P Equation (31) is the particle residence time in the cyclonic zone which can be expressed further as:
ZHOU Xiao-hua et al
W pX
Particle Residence Time in Column Flotation Based on Cyclonic Separation
LX 2 4(Qt 2Qf ) d p g ( U p Ul ) ʌD 2 18P LX 2 4[Qw (2 D )Qf ] d p g ( U p Ul ) ʌD 2 18P
(32)
where LX is the length of cyclonic zone, Qt the tailing rate and Qf the feed rate. Therefore, the total particle residence time in the cyclonic static micro-bubble flotation column can be expressed as: LZ 4(D Qf Qw ) Up ʌD 2 (1 H ) LX 4[Qw (2 D )Qf ] d p2 g ( U p Ul ) ʌD 2 18P
353
with an increasing wash water rate. Therefore, a proper wash water rate is needed to increase the concentrate grade and yield. Moreover, for a given column flotation, the particle residence time is related to the height of the column flotation and cyclonic zones. If the height of the cyclonic zone is too high, the particle residence time will decrease. A higher turbulence degree has a negative effect on quiescent separation. Thus, a proper cyclonic zone height is beneficial in order to achieve a better separation performance.
5
Conclusions
1) The overall flotation recovery in the collection zone of the FCSMC is,
W p W pZ W pX
RC
(33)
Substituting Equations (13) and (32) into Equation (7) obtains the recovery of the column flotation in the collection zone. At a given feed rate, the particle residence time in the column flotation is, inter alia, also related to the wash water rate and the column diameter. The concentrate grade increases with an increasing wash water rate; however, the particle residence time decreases and as well, the concentrate yield decreases
1 exp(kCZWpZ ) . >1 exp(kCZWpZ )@ ª¬1 (1 kCXWpX )1 ¼º 1 ¬ª1 (1 kCXWpX )1º¼
2) The total particle residence time in the FCSMC can be expressed as: LZ 4(D Qf Qw ) U p ʌD 2 (1 H ) LX 4[Qw (2 D )Qf ] d p2 g ( U p Ul ) . ʌD 2 18P
W p W pZ W pX
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