Ind. Eng. Chem. R e s . 1990,29,949-955
Literature Cited Coulson, J. M.; Richardson, J. F.; Backhurst, J. R.; Harker, J. H. Chemical Engineering, 3rd ed.; Pergamon Press: Oxford 1978; VOl. 2. Degner, V. R.; Sabey, J. B. WEMCO/Leeds Flotation Column Development. In Column Flotation ‘88,Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 2528,1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 267. Dell, C. C.; Jenkins, B. W. The Leeds Flotation Column. Seventh International Coal Preparation Congress, Sydney, 1976. Dobby, G. S.; Finch, J. A. Mixing Characteristics of Industrial Flotation Columns. Chem. Eng. Sci. 1985,40, 1061. Dobby, G. S.; Finch, J. A. Flotation Column Scale-up and Modeling. CZM Bull. 1986, 79, 89. Egan, J. R.; Fairweather, M. J.; Meekel, W. A. Application of Column Flotation to Lead and Zinc Beneficiation at Cominco. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 19. Geankoplis, C. J. Transport Processes and Unit Operations, Allyn and Bacon: Boston, 1978. Hucko, R. E.; Gala, H. B. Promising Advanced Coal Preparation Technologies for Reducing SO2 Emissions. Prepr. Pap.-Am. Chem. Soc., Diu. Enuiron. Chem. 1988,28 (No. l), 211-215. Idlas, S. A.; FitzPatrick, J. A.; Slattery, J. C. Conceptual Design of Staged Flotation Columns. Znd. Eng. Chem. Res. 1990, following pap& in this issue. Jameson, G. J.; Nam, S.; Young, M. M. Physical Factors Affecting Recovery Rates in Flotation. Miner. Sei. Eng. 1977, 9, 103. Kawatara,-S. K.; Eisele, T. C. Column Flotation of Fine Coal. In Fine Coal Processing; Mishra, S. K., Klimpel, R. R., Eds.; Noyes Publications: Park Ridge, NJ, 1987; p 414. Kawatara, S. K.; Eisele, T. C. Studies Relating to Removal of Pyritic Sulfur from Coal by Column Flotation. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 213. Killmeyer, R. P.; Hucko, R. E. Interlaboratory Comparison of Advanced Froth Flotation Processes. Preprint, Society of Mining Engineers Annual Meeting, Las Vegas, NV, Feb 27-March 2, 1989; Society of Mining Engineers: New York, 1989; No. 89-137. Laplante, A. R.; Yianatos, J.; Finch, J. A. On the Mixing Characteristics of the Collection Zone in Flotation Columns. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 69. Li, C. Process Analysis of a Static Tube Flotation System for Fine Coal Cleaning. M.S. Thesis, Michigan Technological University, Houghton, 1987. Luttrel, G. H.; Adel, G. T.; Yoon, R. H. Modeling of Column Flotation. Preprint, AIME Annual Meeting, Denver, CO, Feb 24-27,
949
1987; AIME: New York, 1987; No. 87-130. Misra, M.; Harris, R. Column Flotation of Fine Coal from Waste Coal Refuse. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28,1988, Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988, p 235. Moon, K. S.; Sirois, L. L. Theory and Industrial Application of Column Flotation in Canada. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, New 1988: Sastry, K. V. S., Ed.; Society of Mining- Engineers: York, 1988; p 91. Mori. S.: Okamoto. H.: Hara. T.: Aso. K. Kinetics Studies of Fluorite Flotation. Proc. 15th Znt. A h.hocess. Cong., Cannes 1986,3, 155-162. Nicol, S. K.; Roberts, T.; Bensley, C. N.; Kidd, G. W.; Lamb, R. Column Flotation of Ultrafine Coal: Experience at BHP-Utah Coal Limited’s Riverside Mine. In Column Flotation ‘88,Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 7. Sastry, K. V. S.; Lofftus, K. Mathematical Modeling and Computer Simulation of Column Flotation. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 57. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Subramian, K. N.; Connelly, D. E. G.; Wong, K. Y. Commercialization of a Column Flotation Circuit for Gold Sulfide Ore. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 13. Szatkowski, M.; Freyberger, W. L. Model Describing Mechanism of the Flotation Process. Trans. Znst. Min. Metall. 1985,94, C61C70. Treybal, R. E. Mass-Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980. Wheeler, D. A. Historical View of Column Flotation Development. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 3. Yang, D. C. Development and Demonstration of a Static Tube Flotation System for Producing Superclean Coal. Eleventh Quarterly Progress Report to USDOE under Contract De-AC22-85PC87210, July 13, 1988. Ynchausti, R. A.; McKay, J. D.; Foot, D. G. Column Flotation Parameters-Their Effects. In Column Flotation ‘88, Society of Mining Engineers Annual Meeting, Phoenix, AZ,Jan 25-28,1988, Society of Mining Engineers: New York, 1988; p 157.
Received for reuiew September 18, 1989 Revised manuscript received February 23, 1990 Accepted March 6, 1990
Conceptual Design of Staged Flotation Columns Scott A. Idlas,?Joseph A. FitzPatrick,*and John C. Slattery*pt Department of Chemical Engineering and Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208-3120
A new analysis is presented for staged flotation columns, of which the WEMCO-Leeds flotation column, which is staged above the feed, appears to be the only current example. When the column height is fixed, we find that reflux significantly improves the product grade at the expense of decreased recovery. Increasing the number of stages while maintaining the height of the column constant can help, only if there is a mechanism for scouring gangue particles from the froth as it passes from one stage to the next. More generally, one must solve an optimization problem in which both the excess interfacial area available for flotation and the height of the column below the feed may be increased. The recent round-robin test conducted by the Department of Energy (Killmeyer and Hucko, 1989; Hucko and *Current address: Department of Chemical Engineering, Texas A&M University, College Station, T X 77843-3122. t Department of Chemical Engineering. t Department of Civil Engineering.
Gala, 1988) suggests that columns have an inherent advantage over standard mechanical/pneumatic flotation machines in cleaning coal. Three types of columns have been suggested for use in flotation: open, packed, and staged (Kawatara and Eiele, 1987). In a companion paper, Idlas et al. (1990) give a brief summary of the literature concerned with open columns as well as an in-depth dis-
0888-5885/ 90 / 2629-0949$02.50/ 0 0 1990 American Chemical Society
950 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990
I
GAS
is returned to the column as reflux. The barrier (tray or plate) defining a stage maintains a pool of pulp above it. It permits (episodically) a portion of this pulp to pass through to the stage below, while allowing a portion of the froth or foam generated on the stage below to pass up. As the froth and pulp move countercurrently through the barrier, the pulp flows through the froth or foam, washing down the less pure pulp carried up from the stage below. There are no separate downcomen for the pulp as in a staged distillation column. The WEMCO-Leeds barrier (Dell and Jenkins, 1976; Degner and Sabey, 1988) may be thought of as a model for this tray design. Why isn't an open column with froth washing sufficient? There are at least two reasons. First, backmixing (dispersion) may cause impure pulp to rise, decreasing grade, and it may cause pulp containing the valuable mineral to sink, decreasing recovery (Dobby and Finch, 1985; Laplante et al., 1988). Compared with an open column, axial concentration gradients are more likely to develop in a staged column. Second, because of imperfect separation of the minerals during the crushing and grinding preceding flotation, the surfaces of some particles will be heterogeneous, preventing even the best reagent systems from providing perfect selectivity. The staged column may facilitate more intimate contact between the rising bubbles and the descending pulp, giving the sorting process further opportunities to operate. One can visualize the staged column as a stack of (short) open columns. But there are some important differences. Each stage is not equivalent to an independent open column with foam washing, since the height of a stage must be considerably less than that of an independent open column (typically 20-40 ft). It is likely that the reagent system must be chosen so as to create a relatively small amount of froth or foam on each stage; the layer of froth in an open column may be relatively thick. An open column can take greater advantage of foam washing, if the washing solution used does not collapse the foam, since the washing solution contains no solids and since the layer of froth is much thicker. On the other hand, the staged column allows virtually unlimited multiple staging with little additional plumbing or pumping. We will make the following assumptions in our analysis. (1)The column will be considered to operate at steady state, although in reality the flow of the froth or foam through the barriers is likely to be episodic. (2) We will not make any assumption about the relative amounts of froth and pulp on each stage. We will simply say that the character of the rising bubbles does not change as they pass from pulp to froth and from stage to stage. This means that we are neglecting both coalescence and splitting of the bubbles as they pass through the column. This assumption can be eliminated. (3) The feed is a pulp or suspension rather than a foam or froth. It contains the surfactant system. (4) Any make-up solution added to the reflux (see Figure 1) contains sufficient surfactant to avoid dilution of the reagent system. (5) Entrainment of pulp with the rising gas bubbles is neglected. The analysis does not recognize that liquid is carried from stage to stage with the rising gas bubbles; the only solid carried with the rising gas bubbles is that which is attached to the bubble. We suggest accounting for entrainment is much the same manner that entrainment is treated in a staged distillation column (Treybal, 1980).
I r"";""""""
SEPARATOR 1
1
-PRODUCT
STAGE N
-STAGE 2
-STAGE 1 I
GAS
TAILINGS
Figure 1. Staged flotation column containing N stages.
cussion of the design of packed columns. The WEMCO-Leeds flotation column (Dell and Jenkins, 1976; Degner and Sabey, 1988) appears to employ stages in the cleaning section of the column above the feed. The feed "stage" is a relatively standard flotation cell. There is no staged scavenging section in their column, nor do they employ reflux, in which a portion of the product is recycled to the top of the column. In commercial-scale tests,the WEMCO-Leeds column is reported to be more efficient than a standard cell (Degner and Sabey, 1988). Until recently, coal and other minerals were typically ground too coarsely to be used in packed or staged columns without serious plugging problems. Particularly in the case of coal, where there is an increased emphasis on the physical removal of as much of the pyrites as possible, this limitation appears to have been removed. In what follows, we discuss the scale-up of the number of stages in staged flotation columns to increase product purity and product recovery. Our approach is influenced by the design of staged columns for distillation, gas adsorption, and liquid-liquid extraction (Coulson et d., 1978; Geankoplis, 1978; Sherwood et al., 1975; Treybal, 1980). But there is at least one important difference. In these latter processes, equilibrium between the phases is approached on each stage; flotation is not an equilibrium process. Statement of the Problem Our objective is to outline a design algorithm for the staged flotation column sketched in Figure 1. The vertical column contains N stages separated by barriers (plates or trays), with the feed entering on one of the intermediate stages. We refer to the stages above the feed as the cleaning section of the column: the concentration of the mineral being floated increases as the froth or foam ascends through this portion of the column, with the purified product being removed from the top of the column after being separated from the gas. The stages below the feed are referred to as the recovery of scavenging section of the column: the concentration of the mineral being floated decreases as the pulp descends through this portion of the column, with the tailings leaving from the bottom of the column. Gas is introduced at the bottom of the column and is rapidly dispersed. A portion of the product pulp
Mass Balances With reference to Figure 2, the mass balance for solid
Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990 951 G. Y,(i)
I
SCAVENGING
L
,
I
u STAGE N
I STAGE
1
CLEANING SECTION
Figure 3. Scavenging section of column below the feed.
STAGE n + 1
G. Y,(')
L, xn+,(i)
Figure 2. Cleaning section of column above the feed.
species i over stage n in the cleaning section of the column requires
G(Yg)- ytLl) = L(X;il - X:))
where L is the mass rate of flow of the continuous liquid phase in the scavenging section of the column (below the feed) per unit cross-sectional area of the column. This can also be known as the operating line for solid species i in the scavenging section of the column. The mass balance for the continuous liquid phase in this section of the column is E = LT (6) Here the subscript T denotes the stream of tailings. The mass balance for species i over the mixer shown in Figure 1 gives
(1)
This can also be referred to as the operating line for solid species i in the cleaning section of the column. Here Y g) is the mass of solid species i per unit mass of gas associated with the bubbles rising from stage n, X !il is the mass of solid species i per unit mass of the continuous liquid phase in the pulp descending from stage n + 1,G is the mass rate of flow of gas per unit cross-sectionalarea which is assumed to be a constant, and L is the mass rate of flow of the continuous liquid phase in the cleaning section of the column (above the feed stage) per unit cross-sectional area. In the same manner, the mass balance for the continuous liquid phase in this section of the column requires
where R is the reflux ratio (fraction of the pulp stream exiting from the separator that is diverted to reflux), the subscript P denotes the overhead product stream from the separator, and the subscript R denotes the reflux stream returned to the top of the column. It is important to recognize that the product stream exiting overhead from the column is a froth, the quality of which is
(3)
and that different amounts of solids are carried by the bubbles and pulp in this froth (4)
Referring to Figure 3, the mass balance for solid species i over stage n in the scavenging section of the column takes the form
(7) The mass balance for the continuous liquid phase requires LR =
Lp+M 1-R
in which we have introduced M as the mass rate of flow of make-up solution added per unit cross-sectional area. A t stage 2 on which the feed is introduced, the mass balance for species i requires L L . LF Yp -xy = Y E l + ,xp+ 1 + -gxp G and the mass balance for the continuous liquid phase demands L - L + LF = 0 (10) So far we have developed the equations that ensure that mass is conserved. Next we must describe the attachment process. These equations, unlike the preceding, will be very dependent on the tray design and operating conditions.
+
Rate of Attachment In order to describe the transfer of particles from the pulp to the gas on a stage, we must describe the functional form of the attachment/detachment rate expression. The kinetics of flotation processes have been reviewed by Jameson et al. (1977) and Mori et al. (1986). We will assume, consistent with standard flotation practice, the following. (6) The rate of collisions between the particles and bubbles is proportional to the concentration of particles in the pulp. (7) The probability of a successful collision is proportional to the uncovered area on a bubble. A slightly more complicated relationship, which recognizes that particles attach on the front of bubbles and slide to the rear, has
952 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990
been used by Szatkowski and Freyberger (1985);however, a simple linear relationship is adequate for our purposes. (8) The rate of particle detachment is proportional to the surface concentration of particles on the bubbles. Little work has been done on detachment rates in flotation; this is one of the simplest choices. An equation that describes the rate of attachment on stage n and that is consistent with assumptions 6-8 is d(GY"))/dz =
Illustration In order to illustrate how a staged column may work, let us restrict ourselves to the following idealizations. (a) The number of stages above and below the feed are the same. (b) Thinking in terms of an application in which we wish to remove ash and pyrite from coal, we will assume that there are only two solid species present, coal and gangue. (c) With respect to eq 11, we will assume that detachment does not occur:
k(i)a n n Here Kg) and kg) are the rate constants that take into account the effects of hydrodynamics, of surface chemistry, and of gas volume fraction on the attachment/detachment processes, respectively; a, is the interfacial area per unit volume of column on stage n; and c
cU
= X I;t)p(W(1 + C X L k ) p ( L ) / p ( k ) ) - l
=o
(15)
Under these conditions, the analysis of the system requires that we solve 29 equations in 29 unknowns as suggested above. In presenting the results, it will be convenient to define the dimensionless rate constant for the attachement of coal to the bubbles
(12)
k=l
is the mass of species i in the pulp per unit volume of pulp on stage n. The quantity
characterizes the difference between the interfacial area b per unit mass of gas and the interfacial area per unit mass of gas currently occupied by all of the various species present; V ( k )is the volume of species i per unit area of interface occupied by species i. This must be greater than zero for attachment to continue. In view of assumption 2, b is assumed to be a constant throughout the column. Equation 11, applied successively to each of the C species of solid on each of the N stages of the column, must be integrated consistent with the conditions at z = z,
We recognize that these rate constants are products of the dimensional rate constants and a gas-phase residence time on stage n. By
K$)*/KLd*
(17)
-
we define the selectivity of the reagent system. Perfect selectivity corresponds to 8, m . By grade
we mean the ratio of coal volume to total volume of solids in the overhead product; by recovery
(13)
y(i1 = ,(i)y(i) n-1
yCi) = y!) (14) in which di)is the fraction of the particles of species i that remain attached to the interface after passin up through a tray. This requires that we know how Xd varies with z. For simplicity, we recommend that it be assumed to be independent of axial position on a stage. This is equivalent to assuming that the pulp phase is perfectly mixed or there is no backmixing (axial dispersion). at z = z , + ~
(5
Summary of Equations Let us assume that there are C species of solid in the system, that the properties of the feed (LF, X 1))are given, that there are N - 2 stages above the feed stage, that there are 2 - 1 stages below the feed stage, that the gas flow rate G, reflux ratio R, and make-up solution flow rate M are known, that the quality Qp of the froth exiting overhead from the column is known, that the fraction di)of the particles of species i that remain attached to the interface after passing up through a tray has been specified, and that we have reviously determined the rate constants for each stage K ,tif a, and #) as well as b and V('). In order to determine Lp and X$,we must solve 2CN + 2C + 5 equations for each of the C solid species [eq 1 for N - 2 cleaning stages above the feed, eq 5 for 2 - 1 scavenging stages below the feed, eqs 4, 7, 9, and 11 (sub'ect to eqs 13 and 14, with eq 12 substituted to eliminate c, ) for each of the N stages in addition to eqs 2 , 3 , 6 , 8 , and 101 in 2CN i-2C f 5 unknowns for each of. the C solid species [ N Y)! ( Y -= 0), N X Q), X #), and X @ as well as L p , LT, LR,L, and L ] .
we mean the ratio of coal volume in the product to coal volume in the feed; and by A
3
IC
I"
bG CXp-(')LF/(p(')V')) - 1 ;=l
(20)
we mean the excess interfacial area required to float all of the solids in the feed, both coal and gangue. Except where noted otherwise, the standard parameters used in these example computations are as follows. (i) Mass percent solids in feed = 6.9%,mass percent coal in feed solids = 60%, mass percent gangue in feed solids = 40%, bubble radius R b = 1 mm, particle radius Rp = 10 hm, V ( ' )= 4/3Rp = 13.3 pm, QP = 0.74, p(c) = 1.5 g/cm3, p(g) = 2.5 g/cm3, p @ ) = 1.225 X lo4g/cm3, p(L)= 1 g/cm3, b = 3/(Rbp(G))= 2.45 X lo4 cm2/g, S = Sn for all n, and K (i)* K ( I ) * for all n. a.. (11) The volumetric flow rate of gas is twice the volumetric flow rate of the feed slurry:
(iii) The volumetric flow rate of make-up solution is
ti,
This means that, as R is varied, the volume rate of flow of the slurry returned to the top of the column remains constant and equal to twice the volume rate of flow of
Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990 953 1.0
l\
w
0.7
::~i 0.6
0
1
2
-.
3
K IC) * Figure 4. Product grade 9 as a function of K(c)*for various values of selectivity 8,the excess interfacial area A = 1, the reflux ratio R = 0, the number of stages N = 5, and a@ = 1.
;
0
2
(C)
3
*
Figure 7. Product recovery R as a function of K(c)*for various values of the reflux ratio R,the excess interfacial area A = 1, the selectivity 8 = 10, the number of stages N = 5, and a(g)= 1.
0.6
R
0.4
0.2 0
0
1
2
0.6 O 0
3
K IC) * Figure 5. Product recovery R as a function of K(c)*for various values of selectivity 8, the excess interfacial area A = 1,the reflux ratio R = 0, the number of stages N = 5, and ab) = 1.
72
a 1
31
K IC) * Figure 8. Product grade 9 as a function of K(C)*for various numbers of stages N , the excess interfacial area A = 1,the selectivity 8 = 10, the reflux ratio R = 0, and a@) = 1. This comparison assumes columns having equal heights.
0.8 l
'
O
r
RTL 0.2
0.6
0
1
2
3
K IC) * Figure 6. Product grade 9 as a function of K(c)*for various values of the reflux ratio R,the excess interfacial area A = 1,the selectivity 8 = 10, the number of stages N = 5, and ak) = 1.
liquid and solid in the froth exiting the column as overhead product. In the absence of reflux, Figure 4 shows that the grade of the product increases as the selectivity increases. Figure 5 indicates that the recovery of coal in the product is relatively independent of the selectivity, although it does increase as K(c)*increases. The increase in 9 with increased reflux ratio is illustrated in Figure 6. The corresponding decrease in 93 is shown in Figure 7. The large decrease in recovery for R = 0.8 can be attributed to a limitation in the interfacial area available to float the solids. Only 20% of the interfacial area is used to remove product. Figures 8-11 illustrate the changes in 9 and 73 for equal height columns with different numbers of stages, which means, for example,
00
1
K IC) *2
::jy
Figure 9. Product recovery R as a function of K(c)*for various numbers of stages N, the excess interfacial area A = 1, the selectivity 8 = 10, the reflux ratio R = 0, and ab)= 1. This comparison asaumes columns having equal heights.
(7
0.7
0.6 0
1 IC1
*
2
3
Figure 10. Product grade 0 as a function of WC)* for various numbers of stages N, the excess interfacial area A = 1, the selectivity 8 = 10, the reflux ratio R = 0, and a@ = 0.9. This comparison asaumes columns having equal heights.
954 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990
R
O.7
generally, one must solve an optimization problem in which both the excess interfacial area available for flotation and the height of the column below the feed may be increased. As might be expected, both operating costs and capital costa must be increased, in order to improve the product grade while maintaining the product recovery constant.
i
Acknowledgment 0.4! 0.2
We are grateful for the financial support received from the Center for Research on Sulfur in Coal, Carterville, L.
; 0 0
1
3
2
K (' * Figure 11. Product recovery W as a function of K(c)*for various numbers of stages N , the ex= interfacial area A = 1,the selectivity S = 10, the reflux ratio R = 0, and cy@) = 0.9. This comparison assumes columns having equal heights.
As seen in Figures 8 and 9, 9 increases slightly and 3 remains virtually unchanged, when there is no mechanism for scouring gangue as it passes from one stage to the next (ab) = l),as suggested by Degner and Sabey (1988). For a(g)= 0.9, Figures 10 and 11 indicate that there is a significant increase in 9, while R is nearly unchanged as the number of stages is increased. Discussion In order to implement this analysis for scale-up, experimental studies are required to determine the rate constants K t)u, for attachment and k$, for detachment of each species i on each stage n,as well as the interfacial area per unit mass of gas, the volume of species i per unit area of interface occupied by species i, and the fraction of particles of species i that remain attached to the interface after passing up through a tray. All of these parameters can be expected to be functions of a variety of variables affecting local column conditions as well as the chemistry of the system. They would, for example, certainly be functions of the tray configuration chosen. Once these parameters are known, scale-up is straightforward, at least in principle. For the illustrative computation presented above, eqs 16 and 17 permit one to determine the height of each stage and consequently the height of the column as functions of the reflux ratio and of the excess interfacial area. With given objectives for product grade and recovery, the column height as well as R and A would be chosen to be consistent with minimization of capital and operating ex enses. This of course assumes that the rate constants K i a, are independent of the operating parameters. The range of validity of such an assumption can be determined only experimentally. The primary conclusion here is that, just as in the case of the packed flotation column (I& et al., 19901, product grade can be improved by using reflux. This provides additional opportunities for filling the liquid-gas interface with the desired product and for displacing any undesired species by competition. This does not appear to have been previously suggested in the context of flotation, although it is common practice in the context of distillation, liquid-liquid extraction, and gas adsorption (Coulson et al., 1978; Geankoplis, 1978; Sherwood et al., 1975; Treybal, 1980). However, there is a penalty. If all other operating variables are held fixed, product recovery decreases as the reflux ratio and product grade are increased. Increasing the number of stages while maintaining the height of the column constant can help, only if there is a mechanism for scouring gangue particles from the froth as it passes from one stage to the next (Degner and Sabey, 1988). More
P
Nomenclature a, = interfacial area per unit volume of column on stage n A = excess interfacial area, defined by eq 20 b = interfacial area per unit mass of gas c;) = mass of solid species i in the pulp per unit volume of pulp on stage n, defined by eq 12 G = mass rate of flow of gas per unit cross-sectional area 9 = grade, defined by eq 18 k!) = rate constants that take into account the hydrodynamics and surface chemistry of the detachment processes for stage n, introduced in eq 11 K 1) = rate constants that take into account the hydrodynamics and surface chemistry of the attachment processes for stage n, introduced in eq 11 K(i)*= K:)* = for all n in the example computations K t ) * = defined by eq 16 L = mass rate of flow of the continuous liquid phase in the cleaning section of the column (above the feed stage) L = mass rate of flow of the continuous liquid phase in the scavenging section of the column (below the feed) per unit cross-sectional area of the column M = mass rate of flow of make-up solution added per unit cross-sectional area QP = quality of froth exiting overhead from column, volume of gas per unit volume of froth as defined by eq 3 R = reflux ratio (fraction of the pulp stream exiting from the separator that is diverted to reflux) 3 = recovery, defined by eq 19 S = 8, for all n in the example computations 8, = selectivity for stage n, defined by eq 17 V'" = volume of species i per unit area of interface occupied by species i Xx i1 = mass of solid species i per unit mass of the continuous liquid phase in pulp descending from stage n + 1 Y = mass of solid species i attached to rising bubbles per unit mass of gas rising from stage n z = axial position measured between the stages Greek Letters
di)= fraction of particles of species i that remain attached to interface after passing up through a tray p(i) = 7;:)
mass density of phase i
= gas residence time on stage n, defined by eq 16
Others
F = subscript denotes feed stream P = subscript denotes overhead product stream from separator R = subscript denotes reflux stream returned to the top of column T = subscript denotes tailings stream i = superscript denotes species i G = superscript denotes gas phase L = superscript denotes continuous liquid phase
Literature Cited Coulson, J. M.; Richardson, J. F.; Backhurst, J. R.; Harker, J. H. Chemical Engineering, 3rd ed.; Pergamon Press: Oxford 1978; V O l . 2. Degner, V. R.; Sabey, J. B. WEMCO/Leeds Flotation Column Development. In Column Flotation 88, Society of Mining Engineers
Znd. Eng. C h e m . Res. 1990,29,955-967 Annual Meeting, Phoenix, AZ, Jan 25-28,1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 267. Dell, C. C.; Jenkins, B. W.The Leeds Flotation Column. Seventh International Coal Preparation Congress, Sydney, 1976. Dobby, G. S.; Finch, J. A. Mixing Characteristics of Industrial Flotation Columns. Chem. Eng. Sci. 1985,40,1061. Geankoplis, C. J. Transport Processes and Unit Operations; Allyn and Bacon: Boston, 1978. Hucko, R. E.; Gala, H. B. Promising Advanced Coal Preparation Technologies for Reducing SO2 Emissions. Prepr. Pap.-Am. Chem. Soc., Diu. Environ. Chem. 1988,28 (No. l), 211-215. Idlas, S. A.; FitzPatrick, J. A.; Slattery, J. C. Conceptual Design of Packed Flotation Columna. Znd. Erg. Chem. Res. 1990,preceding paper in this issue. Jameson, G. J.; Nam, S.; Young, M. M. Physical Factors Affecting Recovery Rates in Flotation. Miner. Sci. Eng. 1977,9,103. Kawatara, S.K.; Eisele, T. C. Column Flotation of Fine Coal. In Fine Coal Processing; Mishra, S . K., Klimpel, R. R., Eds.; Noyes Publications: Park Ridge, NJ, 1987; p 414. Killmeyer, R. P.; Hucko, R. E. Interlaboratory Comparison of Advanced Froth Flotation Processes. Society of Mining Engineers
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Annual Meeting, Las Vegas, NV, Feb 27-March 2,1989; Preprint 89-137. Laplante, A. R.; Yianatos, J.; Finch, J. A. On the Mixing Characteristics of the Collection Zone in Flotation Columns. In Column Flotation 88, Society of Mining Engineers Annual Meeting, Phoenix, AZ, Jan 25-28, 1988; Sastry, K. V. S., Ed.; Society of Mining Engineers: New York, 1988; p 69. Mori, S.; Okamato, H.; Hara, T.; k o , K. Kinetics Studies of Fluorite Flotation. h o c . 15th Znt. Min. Process. Cong., Cannes 1986,3, 155-162. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. Szatkowski, M.; Freyberger, W. L. Model Describing Mechanism of the Flotation Process. Trans. Znst. Min. Metall. 1985,94,C61C70. Treybal, R. E. Mass-Transfer Operations, 3rd ed.; McGraw-Hill: New York, 1980.
Received for review September 18, 1989 Revised manuscript received February 23, 1990 Accepted March 6,1990
Rate of Collection of Particles by Flotation Dongming Li,+Joseph A. FitzPatrick,*and John C. Slattery*pt Department of Chemical Engineering and Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208-3120
The capture of a single spherical particle by a bubble in froth flotation is analyzed to estimate both the induction time and the rate of flotation. Both London-van der Waals forces and electrostatic double-layer forces are recognized. The induction time is the time required for the thinning and rupture of the liquid film between the spherical particle and the bubble. With the assumption that the effects of the electrostatic double layer can be neglected, the predicted result for the induction time describes the trends seen in prior experimental studies. When the effects of the electrostatic double layer cannot be neglected, better selectivity between particles having different surface potentials is predicted at intermediate values of the electrolyte concentration. When the effects of the electrostatic double layer can be neglected, an expression for the rate of flotation can be derived up to a proportionality factor. This allows us to draw several qualitative conclusion regarding the rate constant that are supported by experimental observations. Froth flotation is a process in which particles are captured selectively from a suspension by air bubbles. The capture of a particle by a bubble can be thought of as occurring in a series of stages: bubble-particle approach, thinning and rupture of the liquid film between them, and formation of a stable particle-bubble aggregate. The attachment of mineral particles to air bubbles is the most fundamental requirement for successful flotation. When as a result of mixing a solid particle is brought into near contact with an air bubble for a sufficiently long period of time, a thin liquid film forms between them and begins to drain (Schulze, 1984). The thin film is not bounded by parallel planes. As a drop or bubble approaches an interface, it develops a dimple: the film is thicker at ita center than a t its rim (Derjaguin and Kussakov, 1939; Allan et al., 1961;Platikanov, 1964;Hartland, 1967, 1969; Hodgson and Woods, 1969;Hartland and Woods, 1973;Burrill and Woods, 1973). As the thickness of the draining film becomes sufficiently small (about 10oO A), the effects of the disjoining pressure attributable to the London-van der Waals forces and to any electrostatic double layer become significant. The liquid film drains *Current address: Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122. Department of Chemical Engineering. t Department of Civil Engineering. 0888-5885/90/2629-0955$02.50/0
until coalescence occurs, and the particle is attached to the bubble. We usually refer to the time required for the thinning and rupture of the liquid film between particle and bubble as the i n d u c t i o n time. The induction time depends not only upon the disjoining pressure but also upon many other variables including the particle size, the bubble size, the surface tension, and the viscosity of the continuous phase. Particles having a shorter induction time will be more easily captured, and they will be said to have a higher selectivity. Experimentally, there have been attempts to characterize the induction time by measuring the particles of different species captured by an air bubble on the tip of a capillary tube that is forced against a bed of particles for a short period of time (Laskowski and Iskra, 1970; Schulze, 1984;Yordan and Yoon, 1985;Yoon and Luttrell, 1985;Ye and Miller, 1988). From the viewpoint of theory, Ralston (1983)and Schulze (1984)have pointed out that an analysis of the induction time which assumes a parallel-plane film misses the strong influence of film dimpling upon film drainage. In what follows, we develop a more complete hydrodynamic theory for the thinning of a dimpled liquid film between a bubble and a solid particle under the influence of disjoining pressure attributable both to London-van der Waals forces and to an electrostatic double layer. Our primary objective is to determine the induction time. 0 1990 American Chemical Society
