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Feasible designs for separation networks: a selection technique
International Journal o f Mineral Processing, 32 ( 1991 ) 161-174 Elsevier Science Publishers B.V., Amsterdam
161
Feasible designs for separation networks: a selection technique M.C. Williams and T.P. Meloy Particle Analysis Center, 223 White Hall West Virginia University, Morgantown, WV26506, USA (Received June 8, 1990; accepted after revision February 13, 1991 )
ABSTRACT
Williams, M.C. and Meloy, T.P., 1991. Feasible designs for separation networks: a selection technique. Int. J. Miner Process.. 32:161-174. Presented is a non-mathematical, non-computer technique for obtaining feasible designs for simple separation networks. Once the transfer function of the unit operations of a network are known and the desired separation factor and beneficiation ratio specified, a set of graphs approximating network performance is used to design the network. The technique results in networks which closely meet target specification. These graphs specify: ( 1 ) the total number of network stages; (2) the stage at which the feed should enter the network; and (3) the specific arrangement of the stages, thereby specifying the entire network. An unexpected finding of this effort was the acknowledgement that the number of stages the recycle steam is fed back may be greater than one stage. This network designing method appears to be an excellent tool for teaching network design.
INTRODUCTION
This paper presents a graphical method of designing feasible designs for simple separation networks for the mineral, chemical, powder and other process industries. Networks are formed of stages composed of one or more unit operations spatially arranged to perform a better separation than can be done in a single unit. This paper presents the graphs, the method of designing networks using the graphs and an example. In the process industries, designing separation networks is a significant engineering activity. Improved process economies requires, among other things, on-line control, dynamic stability, and accurate cost analysis. These requirements have led increasingly to the use of computer programs in process network design. In the minerals field, many investigators, such as Harris et al. ( 1963 ), Davis (1964), Imaizumi and Inoue (1968), Arbiter and Weiss ( 1971 ), Agar and Kipkie (1978), King (1978), Hodouin and Everall (1980), Sutherland ( 1981 ), Gardener et al. ( 1981 ), Bascur and Herbst ( 1982 ), Meloy ( 1983 ), 0301-7516/91/$03.50
© 1991 Elsevier Science Publishers B.V. All rights reserved.
162
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Williams et al. (1986), and Williams and Meloy (1989), have successfully designed separation networks for the upgrading of fuels and minerals. However, in each case, the author(s) first selected a network, evaluated its performance primarily analytically, redesigned the network to make potential improvements, and then reanalyzed the network to see if an improvement had been affected. Networks and their unit operations have been designed and modeled using both analytical and numerical techniques, and presently there are available over fifty computer programs for use in the mineral industry. (see Sastry and Adel, 1984). Peterson et al. (1978a-d, 1979) have identified over 390 computer programs for the analysis of separation networks for feed components in solid, liquid and gas phases, e.g. solid-solid, solid-gas, solid-liquid, liquid-liquid, gas-gas separations. These include programs for unit operation simulation as well as flowsheet solvers, material balance packages and steady-state and dynamic simulators for process networks. In addition, simulation and process flowsheeting have been the subject of several texts (Himmelblau and Bischoff, 1970; Luyben, 1973; Westerberg et al., 1979; Denn, 1986). In addition to simulation, many widely applicable algorithms exist for network optimization (Fiacco and McCormick, 1968; Denn, 1969; Abadie, 1970; Mehotra and Kapur, 1974; Beveridge and Schechter, 1970; Taha, 1976; Lasdon and Waren 1978; Green, 1984; Yingling, 1990). Since there is no direct analytical method of finding the optimum separation network, various methods of optimization of networks are relied upon. These methods include linear and nonlinear methods that are both analytical and numerical. Nonlinear techniques being used include classical multivariable calculus, Lagrange multiplier methods, search and gradient methods, separable programming, and the Rayleigh-Ritz variational methods. Green (1984) has developed a two-step approach using linear programming. Stochastic dynamic programming is now being evaluated (Yingling, 1990). To emphasize network topology and its connectedness, Williams and Meloy ( 1992 ) are applying graph-theoretic (signal flow) techniques (Henley and Williams, 1973 ). Recently, Williams ( 1985 ) applied several of these optimization techniques to mineral separation networks and affected network optimization of stochastic module models using the SUMT algorithm. In 1985, Biegler and Grossman ( 1985 ) reviewed the literature on the strategies for the optimization of processes and discussed the combining of optimization with network simulation. Currently, engineers are optimizing and simulating networks by feasible path optimization with steady state, sequential modular simulation (Biegler and Hughes, 1985; Biegler, 1985; Biegler and Cuthrell, 1985). Using this method, process networks are divided into modules (unit operations ) connected by streams (feed or product streams). For each module, process behavior may be modeled individually and thus
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
163
simulated. With sequential modular simulation, calculation proceeds from module to module with trial-and-error iteration of recycle loops until convergence is achieved. For effective design of separation networks, along with the development of optimization and simulation algorithms, it is important to develop techniques for rapidly determining feasible solution ranges (Lasdon and Waren, 1978 ). These techniques such as the one developed in this paper improve the efficiency of optimization strategies such as GINO, LINDO, and other nonlinear programming algorithms that need help in finding solutions, by providing initial guesses in the feasible solution range close to optimums. Determination of the feasible solutions, in terms of separation networks, means that given the network recovery, separation and fractionation requirements, finding a feasible network design - number of stages and their spatial arrangement - that satisfy the fore mentioned requirements. In addition, feasible network designs have other uses such as serving as actual network design in the absence of optimal designs, and involving the user in the actual network design, leading to the development of valuable design insights. This paper's objective is to develop a technique for obtaining feasible designs for simple separation networks using stochastic stage transfer functions. It is assumed that the functions are linear - independent of feed composition, other unit operation variables such as temperature an pressure, and other transfer functions. Non-linear transfer functions would arise in separations processes such as distillation. While the technique is presented for the restriction of identical stage transfer functions, this restriction may be easily relaxed. In the derivation of the technique presented below, it was assumed that each unit operation or stage had only two outputs streams, however, this constraint also may relaxed and the method extended to multiple-output stages. (Note that one needs n outputs to fractionate a feed with n components. ) DESCRIPTION OF THE SELECTION TECHNIQUE
Transferfunctions A separation network consists of two or more stages; moreover, the network as well as each of its stages have feed, product and waste streams. In other words, the network has inputs and outputs (feeds and products), as well as various stages composed of unit operations such as cyclones, extractors, filters, impactors, reactors, classifiers, flotation cells and comminution units (crushers, mills and attriters). In the network, these unit operations normally perform solid-solid, solid-liquid, gas-solid, liquid-liquid, gas-gas, etc. separations. Models for individual unit operations in the stages may be developed having various degrees of complexity. For multi-phase, multi-component net-
164
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work inputs and outputs, distributed parameter models, incorporating conservation of mass, momentum and energy in three dimensions, provide detailed descriptions of the networks steady-state behavior. These steady state and dynamic descriptions take the mathematical form of systems of coupled, nonlinear differential equations together with other supporting or constitutive equations (Friedley, 1972; Denn, 1986). These systems may be linearized and simplified into mathematically tractable linear systems analyzable by matrix and Laplace transform methods. Less complex, lumped-parameter models incorporating less process detail, as well as empirical correlations among unit operation inputs and outputs, are widely used to predict the separation behavior of a given network. Moreover, stochastic models for the stage transfer functions are often used to predict a network's behavior. While these various types of models yield reasonable to excellent prediction ofa network's behavior, they do not show how a network should be designed. Steady-state models for a given stage ultimately relate conserved quantity inputs, such as mass, momentum and energy, to outputs, using transfer functions. A stage transfer function is a mathematical expression relating the amount of a conserved quantity or component, j (such as the mass of the jth component of a flowstream ), entering a stage to the amount leaving the stage at steady state. Transfer functions for one conserved quantity (mass, energy and/or momentum) may be coupled to those of another conserved quantity (e.g., through equilibrium expressions), uncoupled and linearized, or they may be expressed as a probability functions. In the development in this paper, it is assumed that the transfer functions are stochastic functions having a value between zero and one. This conservation relationship may be expressed for thejth component leaving the kth stage and entering the/th stage in the network as follows: Mju = T, uM,l
( 1)
where: T;k; is the transfer function, i.e., the fraction of conserved quantity, j, leaving the kth stage, Mk, that enters the/th stage, M; ; M;~ is the total amount of conserved quantity, j, entering the/th stage, Mr; Mjk; is the amount of conserved quantity, j, leaving the kth stage, Mk, for the/th stage, M;. At steady state, conservation o f j requires that:
~ M,u = Mjl
(2 )
k
ZL ,= 1.o /
where ~ indicates summation over k. k
(3)
FEASIBLEDESIGNS FOR SEPARATIONNETWORKS
165
The selection technique The following terms are defined: Q/= mass of component l in Q stream where Q=p,f, w, and l=i,j. From the data on the network waste and product stream compositions, one can readily compute the ratios (w~/wj), and (p,/pj),. The subscript n indicates a ratio of network rather than individual stage masses. The first design criterion is the criterion for determining the total number of stages required in the network. In effect this is to determine the number of stages required to upgrade the feed to the first stage to the product specification of the final concentrator stage. It can be shown (Williams, 1985) that the design which can accomplish this is a series of concentrator stages (see Fig. 1a) with the only unknown being the number of stages, N, required (in Fig. 1a the n subscript has been dropped). The task is to compute the number of stages that the feed to the first stage, f / ~ , must go through serially so that it is upgraded to the product specification, (p,/p~),. This leads to the following equation:
(p~/pj ) , = f /fj (T~/Tj) N
(4)
To calculate the number of stages required, the three ratios, (p,/pj)n, f / f , and T~/T, must be known. While the first and last of these ratios are known, the f / f j term, the feed to the 1st stage in Fig. 1a, is not known. However, since the (wi/wj), ratio is known, f / f j may be calculated from this waste stream ratio by the following equation:
f l f j = ( w,lwj),l [ ( 1 - T,) l ( l - Tj) ]
I
Fig. la. How many cells in the circuit? N = ?
(S)
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Substituting eq. 5 into eq. 4. yields:
(p,lp,).=(wilw,),,(T,17~)'/](1
- l ~ ) / ( 1 - 1})]
(6)
Separating terms and taking the common (base ten) logarithm of both sides yields the following equations specifying the total number of stages required in the separation network: N_lOg{ (p,/p: ) . [ (1 - T, ) / ( 1 - T~ ) ] / ( wi/w: ),, } log{ 7,/T~ ', •
(7)
Since all the terms on the right side of eq. 7 are known and dimensionless, the number of stage may be numerically calculated from this equation. In turn this means that the number of stages in Fig. 1a are specified, in this case 9. After determining the number of stages in the network, there remain the problems of locating where the feed stream should be placed and where the dangling waste streams from each stage fit into the network. See Fig. lb (in Fig. 1b the n subscript has been dropped). Consider next the placement of the feed stream. The second design criterion is the criterion for determining where the feed stream enters the network. It is embodied in the requirement that the feed steam be upgraded to the product stream grade by being processed through C cleaner stages. Mathematically stated, this becomes: (P,/P/),, = (I;I.I)),, (T~/T,) (
(8)
Taking the log of both sides and then solving for C, the number of cleaner stages, one obtains: C-log{ (p,/pj),,/ ( [ / £ ) . I Iog{T,/Tj}
(9 )
In the example being used, C, the number of cleaner stages that the network
{b)
Fig. lb. W h e r e are the u n d e r f l o w s t r e a m s c o n n e c t e d to the circuit? R = w h a t ? - 4?
167
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
feed must past through before it is upgraded to that of the network product, is equal to 4. Ergo, the feed stream should enter the network in the Cth stage from product end of the network, in this case the 6th stage - the 4th from the product end of the network. See Fig. 1c (in Fig. 1c the n subscript has been dropped). Once again, all terms on the right hand side of eq. 9 are known and dimensionless, thus C may be calculated numerically. As may be seen from Fig. 1c, all that remains to do to complete the design of the separation network is the placement of the dangling waste streams from the stages in the network. If a waste stream is fed to a stage with a higher primary feed stream grade, then the product grade from that stage will be lowered and the overall product grade of the network will be degraded. Consequently, the third design criterion states that: A recycle stream must have a grade that is equal to or higher than that of the primary feed grade to that same stage. Since a waste stream grade is:
wi/wj = f / f j [ ( 1 -- Ti)/ ( 1 - Tj) ]
(10)
The question arises as to how many stages must the waste stream, from a given stage, be recycled back (and thus pass through) such that the waste stream grade from the given stage will be upgraded to the original feed grade
(c)
Fig. 1c. Where does the dangling stream go? To cell 1, 2, 3, or 4.
Feed
,
~
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Fig. 1d. Where does the dangling stream go? To cell 1, 2, 3 or 4.
R
=
2 ~
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to that given stage. In the literature Agar and Kipkie ( 1976 ) have commented on the design or placement of these recycle streams and stated that any recycle stream going to a stage should be equal to or higher grade than the primary feed to that stage. If one defines R as the number of stages that waste must pass through to be upgraded, then the following equation expresses that relationship: f/f
=f/f[
( 1 - T,)/( 1 - T , ) ] ( T , / T / )
'~
(11)
Canceling terms, rearranging and solving for R, one obtains: R= l ° g { ( l - ~ ) / ( 1 - T ' ) l log{T,/L}
(12)
In the example being used, R, the number of cleaner stages that the recycle stream must past through before it is upgraded to that of the feed of the given stage, is equal to 2. This means that the recycle streams reenter the network to stages back, see Figure i d. That is, the recycle from the 9th stage is recycled back to the 7th stage, such that this recycled material, when processed (independently ) by the 7th and 8th stages and fed as a product to the 9th stage, will have a grade that is equal to or greater than the original feed to the 9th stage. Similarly, material recycled from the 8th, 7th, 6th, 5th, 4th and 3rd stages are recycled respectively to the 6th, 5th, 4th, 3rd, 2nd and 1st stages. In Fig. ld, the recycle of two streams are not shown: those from the 2nd and 4th stages. As defined above, the recycle from the 4th stage, enters the 2nd stage. However, none of the three design criteria define where, in the case where the recycle number is greater than one, the penultimate recycle stream should reenter the network. COMPARISONS
IN A N E T W O R K
For any stage having any number of output streams and feeds and components, comparisons can be made. Comparisons in a network are simply ratios of the mass (or other conserved quantities) of components in streams. Many useful comparisons can be generalized as follows:
G ( i,j; a,b ) = (x,/x,):/ (y,/y,)-
(13)
where:
a,b=p,f w Z=N,S Cs can be defined for a stage without reference to the total number of stages or components. However, it is sometimes useful to speak of the network feed or product. This presupposes one feed and one product. When Cs are defined
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
169
for a network with only two outputs, P and W, and a single feed F, the Cs are simply subscripted N. It will be obvious from the discussion that CNS will be between P and W streams and CNS will be between P and F streams. For any stage having only two output streams, P and W, simplified o u t p u t output comparisons, Cs, can be made. C o m m o n comparisons for a network include:
CN( i,j;P, W) = (Pi/Pj) N/ ( Wi/Wj) N
(14)
separation factor,
CN( i,j;P,F) = (Pi/Pj) N/ (fi/fj) N
( 15 )
beneficiation ratio,
CN( i,j; F) = ( f /fj ) u
(16)
feed grade, and,
C~( i;P,F) = (Pi ) x/ ~ ) U
( 17 )
recovery of/-component. From these three parameters, a feasible network design is directly and unambiguously specified. Each network designed by this technique has a unique three number designation - Network (N; C; R ). The designation open or closed is also indicated. A network is automatically closed only when R is one. In this case there are no unconnected recycle P streams. If R is not one, the network can be closed by connecting the recycle streams to some stage instead of letting them serve as additional network outputs. AN EXAMPLE
The values of N, C and R have been plotted versus various groups of terms in Fig. 2. These groups of terms are the fundamental comparisons most often used in a network. It is through these terms that design begins. It is possible t o p l o t all the curves on the same figure which results in much convenient. It is apparent that for large separation requirements, large values of N are necessary, especially at low Ti/Tj values. Large N, C and R values are required at low Ti/Tj values. Large values of N-C, large numbers of scavengers, are necessary for large recoveries, CN(i;P,F). Many values of N, C and R could be used to obtain the same network requirements, depending on the value of transfer functions used. Little change among N, C and R is required to achieve a very broad range of network requirements. Under such conditions little imagination or study would be necessary to come up with a satisfactory design. The increase in slope of the curves
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z,
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as Tff~ is decreased corresponds with network designs becoming less individualistic. For the example the network requirements are: a network/-component recovery of 0.90 and a network product stream separation factor, separation factor of 9. The T,/T] ratio for the stages (all transfer functions considered identical) was 2.0. This is a typical value for most separations. The value of the feed stream component ratio is again one. Figure 2 contains the calculated (graphical) N, C, and R values. The recovery was increased by placing scavenger stages on tailing streams in the scavenger section. From the results, a feasible design was obtained meeting all network requirements. The search for a solution could not be found for N equal 4. Satisfactory recoveries and separation factors were found for Network (5; 4; 1 ). Beneficiation ratios were used using open Network (5; 4; 2 ), but recovery decreased. Recovery was not substantially increased when Network (5;4;2) was closed forming Network (5;4; 1 ( W 2 ) , 2 ( W 3 - P 5 ) ). (The nomenclature use in this paper was developed by numbering the stages left to right. The final solution used was to use variable R choosing Network (5;4; 1 ( W2-W4),2 ( ( W5 ) ). The recycle of the product stage, W5, was sent back two stages instead of one stage. The solution network is shown in Fig. 3. This design is a feasible design for the specified network requirements as can
171
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
TABLE 1
Feasible designs for network specifications and conditions: CN(i,j;P,F)= 9.00; C~(i,j; P, W ) = 6.00; T~/~ = 2.00; CN(i,j; P, W) = 81.00; and C,v(i; P,F) = 0.90
N
C
R
4.75*
3.17"
1.58"
3 3 3 3 3 3 4 4 4 4
1 2 2 1 2 2 1 2 2 1/2
Target 4 4 4 5 5 5 5 5 5 5
C~v(i,j;P,F)ac, uaj
Cu(i;P,F)actual
Cjv(i,j;P,W)actual
9.00 6.12 8.68 7.78 4.31 9.24 6.65 7.79 20.98 16.62 9.66
0.90 0.75 0.69 0.88 0.98 0.83 0.86 0.93 0.66 0.86 0.93
81.00 21.73 25.62 55.99 215.78 49.76 41.23 110.07 60.46 112.25 134.83
closed open closed closed open closed closed open closed closed
9.56 9.86
0.92 0.93
75.43 129.41
open open
Variable R 6 7
3 3
2 2
*calculated.
FEED RODUCT
Fig. 3. Closed network (5; 4; 1 ( W 2 - W4 ),2 ( W5 ) ) Note R = 1 except for the last stage.
readily be determined. The closeness of the network to the design specifications suggests that the network may, per se, be a satisfactory network as well as the initial design for an optimization routine. DISCUSSION
Derived was a technique for designing feasible separation networks directly from a set of network specifications. Aside from some of the assumptions used in the derivation, the integer nature of unit operations and the characteristics of the feed impinge on the clarity and simplicity of the proposed design technique.
Among the most important findings of this study of designing feasible and near optimal networks are: ( 1 ) the recycle distance may be more than one stage back - instead of the recycle stream being fed back to the previous stage, it may be fed back two or more stages closer to the network's waste stream: and (2) if the recycle step back, R, is greater than one, then there arises as to where the recycle from the penultimate stage should be sent. Although quite unexpected, the finding that the recycle step back distance, R, can be an integer greater than one, means that at least some extant separation networks may be misdesigned. While there may exist separation networks where at some point the step back may be greater than one, the writers are not aware of such a network where the step back is consistently (systematically) greater than one. While it is natural to think that in separation networks a one stage step back is always optimal - that's what one's intuition says, actually when R is greater than one, a one stage step back lowers the grade for the same number of stages in the network's cascade. Until now, in the design of separation networks in all industries, R has been neglected, leading to the design of suboptimal networks. In the derivation it was assumed that the: unit operations behaved linearly and identically. If this assumption is not true, then N, R and C are dependent on the transfer functions of the individual unit operations used in the network design. However, using the same three criteria developed above, it is possible for the reader to develop the optimal network for stages with non-identical transfer functions. Optimization of networks to maximize an objective function, such as efficiency, is an arbitrary activity since there are at least as many objective functions as designers. In each case the values of the network design specifications which maximize an objective function can be determined first. These design specifications can then be used to design te appropriate, strongly-connected network given appropriate network conditions, that is, the Ts for each species in each stage. Of course, an objective function can also be a function of the N, C and R obtained and the Ts used, thus, introducing nonlinearities. In such a case the network obtained by the technique is just the first iteration in obtaining an exact solution. CONCLUSIONS
Presented is a non-mathematical, non-computer technique for obtaining feasible designs for simple separation networks. Once the transfer function of the unit operations of a network are known and the desired separation factor and beneficiation ratio specified, a set of graphs approximately defining the N, C and R required for the specified network performance is used to design the network. In addition it should be pointed out that:
FEASIBLEDESIGNSFORSEPARATIONNETWORKS
173
( 1 ) For given specifications, a feasible network may be designed directly no trial and error is needed. (2) In separation networks, the number of stages the recycle steam is fed back may be greater than one stage. (An unanticipated finding. )
REFERENCES Abadie, J., 1970. Integer and Nonlinear Programming. North-Holland, Amsterdam, 544 pp. Agar, C. and Kipkie, W., 1978. Predicting locked-cycle flotation test results from batch data. CIM Bull., 71:119. Arbiter, N. and Weiss, W., 1971. Design of flotation cells and circuits. Trans. AIME, 250: 340347. Biegler, L., 1985. Improved path optimization for sequential modular simulators-I: the interface. Comput. Chem. Eng., 9(3): 245-256. Biegler, L. and Cuthrell, J., 1985. Improved path optimization for sequential modular simulators-I: the optimization algorithm. Comput. Chem. Eng., 9 (3): 257-267. Biegler, L. and Hughes, R., 1985. Feasible path optimization with sequential modular simulators. Comput. Chem. Eng., 9(4): 379-394. Biegler, L. and Grossman, I., 1985. Strategies for the optimization of chemical processes. Rev. Chem. Eng., 3( 1): 1-47. Davis, W.J.W., 1964. The development of a mathematical model of the lead flotation circuit at the Zinc Corporation Limited. AIMM Trans., 212:61-80. Denn, M., 1969. Optimization by Variational Methods. McGraw-Hill, New York, NY, 419 pp. Denn, M., 1986. Process Modeling. Pitman, Marslafield, MA, 324 pp. Fiacco, A. ancl McCormick, G., 1968. Linear Programming, Sequential Unconstrained Minimization Techniques. Wiley, New York, NY, 600 pp. Friedly, J., 1972. The Dynamic Behavior of Processes. Prentice-Hall, Englewood Cliffs, N J, 590 pp. Gardener, R., Lee, H. and Yu, B., 1981. Development of radioactive tracer methods for applying the mechanistic approach to continuous flotation. In: P. Somasundaran (Editor), Fine Particle Processing, vol. 1. AIME, New York, NY, pp. 922-945. Green, J.C.A., 1984. The optimisation of flotation networks. Int. J. Miner. Process., 13: 83103. Harris, C.C., Jowett, A. and Ghash, S.K., 1963. Analysis of data from continuous flotation tests. Trans. AIME, 229: 444-447. Henley, E.J. and Williams, R.A., 1973. Graph Theory in Modern Engineering. Academic Press, New York, NY, 303 pp. Himmelblau, D. and Bischoff, K., 1970. Process Analysis and Simulation. Wiley, New York, NY, 348 pp. Houdouin, D. and Everell, M.D., 1980. A hierarchical procedure for adjustment and material balancing of mineral processes data. Int. J. Miner. Process., 7:91-116. Imaizumi, T. and Inoue, T., 1966. Kinetic considerations of froth flotation. J. Min. Metall. Inst. Jpn., 293(82): 17-24. King, R.P., 1978. A pilot plant investigation of a kinetic model for flotation. J.S. Afr. Inst. Min. Metall., 78: 325-338. Lasdon, L. and Waren, A., 1978. Generalized reduced gradient software for linearly and nonlinearly constrained problems. In: H. Greuenberg (Editor), Design and Implementation of Optimization Software. Sijthoff, Alphen a/d Rijn, pp. 45-60.
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M , ( . WIL,[.b~MS 4 N I ) T.P, MEI_()Y
Luyben W.L., 1973. Process Modeling, Simulation and Control lbr Chemical Engineers. McGraw-Hill, New York, NY. 558 pp. Mehrotra, S.P. and Kapur, P.C., 1974. Optimal-suboptimal synthesis and design of flotation circuits. Sep. Sci,, 9:167-184. Meloy, T.P., 1983. Analysis and optimization of mineral processing and coal cleaning circuit analysis. Int. J. Miner. Process., 10: 61-80. Peterson, J., Chau-Chyun, C. and Evans, L., 1978a. Computer programs for chemical engineers: 1978- Part 1. Chem. Eng.~ 85(11 ): 145-154. Peterson, J., Chau-Chyun, C. and Evans, L., 1978b. Computer programs for chemical engineers: I 9 7 8 - Part 2. Chem. Eng., 85(151: 69-82. Peterson, J., Chau-Chyun, C. and Evans, L., 1978c. Computer programs for chemical engineers: 1978 - Part 3. Chem. Eng., 85( 17): 79-86. Peterson, J., Chau-Chyun, C. and Evans, L., 1978d. Computer programs for chemical engineers: 1978 - Part 4. Chem. Eng., 85( 19): 107-116. Peterson, J., Chau-Chyun, C. and Evans, L., 1979. More computer programs for chemical engineers. In: Chem. Eng., May 21, 1979. Sastry, K. and Adel, G., 1984. A survey of computer simulation software for mineral processing systems. In: J.A. Herbst (Editor), Control '84. AIME, New York, NY, pp. 49-53. Sutherland, D.N., 1981. A study on the optimization of the arrangement of flotation circuits. Int. J. Miner. Process., 7: 319-346. Taha, H., 1976. Operations Research. McMillan, New York, NY, 648 pp. Westerberg, A.W., Hutchinson, H.P., Motard, R.L. and Winter, P., 1979. Process Flowsheeting. Cambridge University Press, New York, NY, 251 pp. Williams, M.C., 1985. Experimental investigation and mathematical modeling of multifeed flotation networks. Ph.D. Dissertation, Univ. California, Berkeley, CA, 274 pp. Williams, M.C. and Meloy, T.P., 1983. A dynamic model of flotation cell banks. Int. J. Miner. Process., 10:141 - 160. Williams, M.C. and Meloy, T.P., 1989. On the fundamental definition of process separation functions. Int. J. Miner. Process., 26: 65-72. Williams, M.C., Fuerstenau, D.W. and Meloy, T.P., 1986. Circuit analysis - General product equations for multifeed, multistage circuits containing variable selectivity functions. Int. J. Miner. Process., 17: 99-111. Williams, M.C., Fuerstenau, D.W. and Meloy, T.P., 1992. A graph-theoretic approach to network design. Int. J. Miner. Process., (in press). Yingling, J.C., 1990. Circuit analysis: optimizing mineral processing flowsheet layouts and steady state control specifications. Int. J. Miner. Process., 29:149-174.
161
Feasible designs for separation networks: a selection technique M.C. Williams and T.P. Meloy Particle Analysis Center, 223 White Hall West Virginia University, Morgantown, WV26506, USA (Received June 8, 1990; accepted after revision February 13, 1991 )
ABSTRACT
Williams, M.C. and Meloy, T.P., 1991. Feasible designs for separation networks: a selection technique. Int. J. Miner Process.. 32:161-174. Presented is a non-mathematical, non-computer technique for obtaining feasible designs for simple separation networks. Once the transfer function of the unit operations of a network are known and the desired separation factor and beneficiation ratio specified, a set of graphs approximating network performance is used to design the network. The technique results in networks which closely meet target specification. These graphs specify: ( 1 ) the total number of network stages; (2) the stage at which the feed should enter the network; and (3) the specific arrangement of the stages, thereby specifying the entire network. An unexpected finding of this effort was the acknowledgement that the number of stages the recycle steam is fed back may be greater than one stage. This network designing method appears to be an excellent tool for teaching network design.
INTRODUCTION
This paper presents a graphical method of designing feasible designs for simple separation networks for the mineral, chemical, powder and other process industries. Networks are formed of stages composed of one or more unit operations spatially arranged to perform a better separation than can be done in a single unit. This paper presents the graphs, the method of designing networks using the graphs and an example. In the process industries, designing separation networks is a significant engineering activity. Improved process economies requires, among other things, on-line control, dynamic stability, and accurate cost analysis. These requirements have led increasingly to the use of computer programs in process network design. In the minerals field, many investigators, such as Harris et al. ( 1963 ), Davis (1964), Imaizumi and Inoue (1968), Arbiter and Weiss ( 1971 ), Agar and Kipkie (1978), King (1978), Hodouin and Everall (1980), Sutherland ( 1981 ), Gardener et al. ( 1981 ), Bascur and Herbst ( 1982 ), Meloy ( 1983 ), 0301-7516/91/$03.50
© 1991 Elsevier Science Publishers B.V. All rights reserved.
162
M,(', WILLIAMS AN[) I P . M[]L(¢~
Williams et al. (1986), and Williams and Meloy (1989), have successfully designed separation networks for the upgrading of fuels and minerals. However, in each case, the author(s) first selected a network, evaluated its performance primarily analytically, redesigned the network to make potential improvements, and then reanalyzed the network to see if an improvement had been affected. Networks and their unit operations have been designed and modeled using both analytical and numerical techniques, and presently there are available over fifty computer programs for use in the mineral industry. (see Sastry and Adel, 1984). Peterson et al. (1978a-d, 1979) have identified over 390 computer programs for the analysis of separation networks for feed components in solid, liquid and gas phases, e.g. solid-solid, solid-gas, solid-liquid, liquid-liquid, gas-gas separations. These include programs for unit operation simulation as well as flowsheet solvers, material balance packages and steady-state and dynamic simulators for process networks. In addition, simulation and process flowsheeting have been the subject of several texts (Himmelblau and Bischoff, 1970; Luyben, 1973; Westerberg et al., 1979; Denn, 1986). In addition to simulation, many widely applicable algorithms exist for network optimization (Fiacco and McCormick, 1968; Denn, 1969; Abadie, 1970; Mehotra and Kapur, 1974; Beveridge and Schechter, 1970; Taha, 1976; Lasdon and Waren 1978; Green, 1984; Yingling, 1990). Since there is no direct analytical method of finding the optimum separation network, various methods of optimization of networks are relied upon. These methods include linear and nonlinear methods that are both analytical and numerical. Nonlinear techniques being used include classical multivariable calculus, Lagrange multiplier methods, search and gradient methods, separable programming, and the Rayleigh-Ritz variational methods. Green (1984) has developed a two-step approach using linear programming. Stochastic dynamic programming is now being evaluated (Yingling, 1990). To emphasize network topology and its connectedness, Williams and Meloy ( 1992 ) are applying graph-theoretic (signal flow) techniques (Henley and Williams, 1973 ). Recently, Williams ( 1985 ) applied several of these optimization techniques to mineral separation networks and affected network optimization of stochastic module models using the SUMT algorithm. In 1985, Biegler and Grossman ( 1985 ) reviewed the literature on the strategies for the optimization of processes and discussed the combining of optimization with network simulation. Currently, engineers are optimizing and simulating networks by feasible path optimization with steady state, sequential modular simulation (Biegler and Hughes, 1985; Biegler, 1985; Biegler and Cuthrell, 1985). Using this method, process networks are divided into modules (unit operations ) connected by streams (feed or product streams). For each module, process behavior may be modeled individually and thus
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
163
simulated. With sequential modular simulation, calculation proceeds from module to module with trial-and-error iteration of recycle loops until convergence is achieved. For effective design of separation networks, along with the development of optimization and simulation algorithms, it is important to develop techniques for rapidly determining feasible solution ranges (Lasdon and Waren, 1978 ). These techniques such as the one developed in this paper improve the efficiency of optimization strategies such as GINO, LINDO, and other nonlinear programming algorithms that need help in finding solutions, by providing initial guesses in the feasible solution range close to optimums. Determination of the feasible solutions, in terms of separation networks, means that given the network recovery, separation and fractionation requirements, finding a feasible network design - number of stages and their spatial arrangement - that satisfy the fore mentioned requirements. In addition, feasible network designs have other uses such as serving as actual network design in the absence of optimal designs, and involving the user in the actual network design, leading to the development of valuable design insights. This paper's objective is to develop a technique for obtaining feasible designs for simple separation networks using stochastic stage transfer functions. It is assumed that the functions are linear - independent of feed composition, other unit operation variables such as temperature an pressure, and other transfer functions. Non-linear transfer functions would arise in separations processes such as distillation. While the technique is presented for the restriction of identical stage transfer functions, this restriction may be easily relaxed. In the derivation of the technique presented below, it was assumed that each unit operation or stage had only two outputs streams, however, this constraint also may relaxed and the method extended to multiple-output stages. (Note that one needs n outputs to fractionate a feed with n components. ) DESCRIPTION OF THE SELECTION TECHNIQUE
Transferfunctions A separation network consists of two or more stages; moreover, the network as well as each of its stages have feed, product and waste streams. In other words, the network has inputs and outputs (feeds and products), as well as various stages composed of unit operations such as cyclones, extractors, filters, impactors, reactors, classifiers, flotation cells and comminution units (crushers, mills and attriters). In the network, these unit operations normally perform solid-solid, solid-liquid, gas-solid, liquid-liquid, gas-gas, etc. separations. Models for individual unit operations in the stages may be developed having various degrees of complexity. For multi-phase, multi-component net-
164
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work inputs and outputs, distributed parameter models, incorporating conservation of mass, momentum and energy in three dimensions, provide detailed descriptions of the networks steady-state behavior. These steady state and dynamic descriptions take the mathematical form of systems of coupled, nonlinear differential equations together with other supporting or constitutive equations (Friedley, 1972; Denn, 1986). These systems may be linearized and simplified into mathematically tractable linear systems analyzable by matrix and Laplace transform methods. Less complex, lumped-parameter models incorporating less process detail, as well as empirical correlations among unit operation inputs and outputs, are widely used to predict the separation behavior of a given network. Moreover, stochastic models for the stage transfer functions are often used to predict a network's behavior. While these various types of models yield reasonable to excellent prediction ofa network's behavior, they do not show how a network should be designed. Steady-state models for a given stage ultimately relate conserved quantity inputs, such as mass, momentum and energy, to outputs, using transfer functions. A stage transfer function is a mathematical expression relating the amount of a conserved quantity or component, j (such as the mass of the jth component of a flowstream ), entering a stage to the amount leaving the stage at steady state. Transfer functions for one conserved quantity (mass, energy and/or momentum) may be coupled to those of another conserved quantity (e.g., through equilibrium expressions), uncoupled and linearized, or they may be expressed as a probability functions. In the development in this paper, it is assumed that the transfer functions are stochastic functions having a value between zero and one. This conservation relationship may be expressed for thejth component leaving the kth stage and entering the/th stage in the network as follows: Mju = T, uM,l
( 1)
where: T;k; is the transfer function, i.e., the fraction of conserved quantity, j, leaving the kth stage, Mk, that enters the/th stage, M; ; M;~ is the total amount of conserved quantity, j, entering the/th stage, Mr; Mjk; is the amount of conserved quantity, j, leaving the kth stage, Mk, for the/th stage, M;. At steady state, conservation o f j requires that:
~ M,u = Mjl
(2 )
k
ZL ,= 1.o /
where ~ indicates summation over k. k
(3)
FEASIBLEDESIGNS FOR SEPARATIONNETWORKS
165
The selection technique The following terms are defined: Q/= mass of component l in Q stream where Q=p,f, w, and l=i,j. From the data on the network waste and product stream compositions, one can readily compute the ratios (w~/wj), and (p,/pj),. The subscript n indicates a ratio of network rather than individual stage masses. The first design criterion is the criterion for determining the total number of stages required in the network. In effect this is to determine the number of stages required to upgrade the feed to the first stage to the product specification of the final concentrator stage. It can be shown (Williams, 1985) that the design which can accomplish this is a series of concentrator stages (see Fig. 1a) with the only unknown being the number of stages, N, required (in Fig. 1a the n subscript has been dropped). The task is to compute the number of stages that the feed to the first stage, f / ~ , must go through serially so that it is upgraded to the product specification, (p,/p~),. This leads to the following equation:
(p~/pj ) , = f /fj (T~/Tj) N
(4)
To calculate the number of stages required, the three ratios, (p,/pj)n, f / f , and T~/T, must be known. While the first and last of these ratios are known, the f / f j term, the feed to the 1st stage in Fig. 1a, is not known. However, since the (wi/wj), ratio is known, f / f j may be calculated from this waste stream ratio by the following equation:
f l f j = ( w,lwj),l [ ( 1 - T,) l ( l - Tj) ]
I
Fig. la. How many cells in the circuit? N = ?
(S)
166
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Substituting eq. 5 into eq. 4. yields:
(p,lp,).=(wilw,),,(T,17~)'/](1
- l ~ ) / ( 1 - 1})]
(6)
Separating terms and taking the common (base ten) logarithm of both sides yields the following equations specifying the total number of stages required in the separation network: N_lOg{ (p,/p: ) . [ (1 - T, ) / ( 1 - T~ ) ] / ( wi/w: ),, } log{ 7,/T~ ', •
(7)
Since all the terms on the right side of eq. 7 are known and dimensionless, the number of stage may be numerically calculated from this equation. In turn this means that the number of stages in Fig. 1a are specified, in this case 9. After determining the number of stages in the network, there remain the problems of locating where the feed stream should be placed and where the dangling waste streams from each stage fit into the network. See Fig. lb (in Fig. 1b the n subscript has been dropped). Consider next the placement of the feed stream. The second design criterion is the criterion for determining where the feed stream enters the network. It is embodied in the requirement that the feed steam be upgraded to the product stream grade by being processed through C cleaner stages. Mathematically stated, this becomes: (P,/P/),, = (I;I.I)),, (T~/T,) (
(8)
Taking the log of both sides and then solving for C, the number of cleaner stages, one obtains: C-log{ (p,/pj),,/ ( [ / £ ) . I Iog{T,/Tj}
(9 )
In the example being used, C, the number of cleaner stages that the network
{b)
Fig. lb. W h e r e are the u n d e r f l o w s t r e a m s c o n n e c t e d to the circuit? R = w h a t ? - 4?
167
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
feed must past through before it is upgraded to that of the network product, is equal to 4. Ergo, the feed stream should enter the network in the Cth stage from product end of the network, in this case the 6th stage - the 4th from the product end of the network. See Fig. 1c (in Fig. 1c the n subscript has been dropped). Once again, all terms on the right hand side of eq. 9 are known and dimensionless, thus C may be calculated numerically. As may be seen from Fig. 1c, all that remains to do to complete the design of the separation network is the placement of the dangling waste streams from the stages in the network. If a waste stream is fed to a stage with a higher primary feed stream grade, then the product grade from that stage will be lowered and the overall product grade of the network will be degraded. Consequently, the third design criterion states that: A recycle stream must have a grade that is equal to or higher than that of the primary feed grade to that same stage. Since a waste stream grade is:
wi/wj = f / f j [ ( 1 -- Ti)/ ( 1 - Tj) ]
(10)
The question arises as to how many stages must the waste stream, from a given stage, be recycled back (and thus pass through) such that the waste stream grade from the given stage will be upgraded to the original feed grade
(c)
Fig. 1c. Where does the dangling stream go? To cell 1, 2, 3, or 4.
Feed
,
~
|
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'
I
m
~
Fig. 1d. Where does the dangling stream go? To cell 1, 2, 3 or 4.
R
=
2 ~
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to that given stage. In the literature Agar and Kipkie ( 1976 ) have commented on the design or placement of these recycle streams and stated that any recycle stream going to a stage should be equal to or higher grade than the primary feed to that stage. If one defines R as the number of stages that waste must pass through to be upgraded, then the following equation expresses that relationship: f/f
=f/f[
( 1 - T,)/( 1 - T , ) ] ( T , / T / )
'~
(11)
Canceling terms, rearranging and solving for R, one obtains: R= l ° g { ( l - ~ ) / ( 1 - T ' ) l log{T,/L}
(12)
In the example being used, R, the number of cleaner stages that the recycle stream must past through before it is upgraded to that of the feed of the given stage, is equal to 2. This means that the recycle streams reenter the network to stages back, see Figure i d. That is, the recycle from the 9th stage is recycled back to the 7th stage, such that this recycled material, when processed (independently ) by the 7th and 8th stages and fed as a product to the 9th stage, will have a grade that is equal to or greater than the original feed to the 9th stage. Similarly, material recycled from the 8th, 7th, 6th, 5th, 4th and 3rd stages are recycled respectively to the 6th, 5th, 4th, 3rd, 2nd and 1st stages. In Fig. ld, the recycle of two streams are not shown: those from the 2nd and 4th stages. As defined above, the recycle from the 4th stage, enters the 2nd stage. However, none of the three design criteria define where, in the case where the recycle number is greater than one, the penultimate recycle stream should reenter the network. COMPARISONS
IN A N E T W O R K
For any stage having any number of output streams and feeds and components, comparisons can be made. Comparisons in a network are simply ratios of the mass (or other conserved quantities) of components in streams. Many useful comparisons can be generalized as follows:
G ( i,j; a,b ) = (x,/x,):/ (y,/y,)-
(13)
where:
a,b=p,f w Z=N,S Cs can be defined for a stage without reference to the total number of stages or components. However, it is sometimes useful to speak of the network feed or product. This presupposes one feed and one product. When Cs are defined
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
169
for a network with only two outputs, P and W, and a single feed F, the Cs are simply subscripted N. It will be obvious from the discussion that CNS will be between P and W streams and CNS will be between P and F streams. For any stage having only two output streams, P and W, simplified o u t p u t output comparisons, Cs, can be made. C o m m o n comparisons for a network include:
CN( i,j;P, W) = (Pi/Pj) N/ ( Wi/Wj) N
(14)
separation factor,
CN( i,j;P,F) = (Pi/Pj) N/ (fi/fj) N
( 15 )
beneficiation ratio,
CN( i,j; F) = ( f /fj ) u
(16)
feed grade, and,
C~( i;P,F) = (Pi ) x/ ~ ) U
( 17 )
recovery of/-component. From these three parameters, a feasible network design is directly and unambiguously specified. Each network designed by this technique has a unique three number designation - Network (N; C; R ). The designation open or closed is also indicated. A network is automatically closed only when R is one. In this case there are no unconnected recycle P streams. If R is not one, the network can be closed by connecting the recycle streams to some stage instead of letting them serve as additional network outputs. AN EXAMPLE
The values of N, C and R have been plotted versus various groups of terms in Fig. 2. These groups of terms are the fundamental comparisons most often used in a network. It is through these terms that design begins. It is possible t o p l o t all the curves on the same figure which results in much convenient. It is apparent that for large separation requirements, large values of N are necessary, especially at low Ti/Tj values. Large N, C and R values are required at low Ti/Tj values. Large values of N-C, large numbers of scavengers, are necessary for large recoveries, CN(i;P,F). Many values of N, C and R could be used to obtain the same network requirements, depending on the value of transfer functions used. Little change among N, C and R is required to achieve a very broad range of network requirements. Under such conditions little imagination or study would be necessary to come up with a satisfactory design. The increase in slope of the curves
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z,
C,3, N:4, R-3
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---B- C-4, N-5, R-4 II
| n
Fig. 2. Log/log plot of the numbers N, C and R.
as Tff~ is decreased corresponds with network designs becoming less individualistic. For the example the network requirements are: a network/-component recovery of 0.90 and a network product stream separation factor, separation factor of 9. The T,/T] ratio for the stages (all transfer functions considered identical) was 2.0. This is a typical value for most separations. The value of the feed stream component ratio is again one. Figure 2 contains the calculated (graphical) N, C, and R values. The recovery was increased by placing scavenger stages on tailing streams in the scavenger section. From the results, a feasible design was obtained meeting all network requirements. The search for a solution could not be found for N equal 4. Satisfactory recoveries and separation factors were found for Network (5; 4; 1 ). Beneficiation ratios were used using open Network (5; 4; 2 ), but recovery decreased. Recovery was not substantially increased when Network (5;4;2) was closed forming Network (5;4; 1 ( W 2 ) , 2 ( W 3 - P 5 ) ). (The nomenclature use in this paper was developed by numbering the stages left to right. The final solution used was to use variable R choosing Network (5;4; 1 ( W2-W4),2 ( ( W5 ) ). The recycle of the product stage, W5, was sent back two stages instead of one stage. The solution network is shown in Fig. 3. This design is a feasible design for the specified network requirements as can
171
FEASIBLE DESIGNS FOR SEPARATION NETWORKS
TABLE 1
Feasible designs for network specifications and conditions: CN(i,j;P,F)= 9.00; C~(i,j; P, W ) = 6.00; T~/~ = 2.00; CN(i,j; P, W) = 81.00; and C,v(i; P,F) = 0.90
N
C
R
4.75*
3.17"
1.58"
3 3 3 3 3 3 4 4 4 4
1 2 2 1 2 2 1 2 2 1/2
Target 4 4 4 5 5 5 5 5 5 5
C~v(i,j;P,F)ac, uaj
Cu(i;P,F)actual
Cjv(i,j;P,W)actual
9.00 6.12 8.68 7.78 4.31 9.24 6.65 7.79 20.98 16.62 9.66
0.90 0.75 0.69 0.88 0.98 0.83 0.86 0.93 0.66 0.86 0.93
81.00 21.73 25.62 55.99 215.78 49.76 41.23 110.07 60.46 112.25 134.83
closed open closed closed open closed closed open closed closed
9.56 9.86
0.92 0.93
75.43 129.41
open open
Variable R 6 7
3 3
2 2
*calculated.
FEED RODUCT
Fig. 3. Closed network (5; 4; 1 ( W 2 - W4 ),2 ( W5 ) ) Note R = 1 except for the last stage.
readily be determined. The closeness of the network to the design specifications suggests that the network may, per se, be a satisfactory network as well as the initial design for an optimization routine. DISCUSSION
Derived was a technique for designing feasible separation networks directly from a set of network specifications. Aside from some of the assumptions used in the derivation, the integer nature of unit operations and the characteristics of the feed impinge on the clarity and simplicity of the proposed design technique.
Among the most important findings of this study of designing feasible and near optimal networks are: ( 1 ) the recycle distance may be more than one stage back - instead of the recycle stream being fed back to the previous stage, it may be fed back two or more stages closer to the network's waste stream: and (2) if the recycle step back, R, is greater than one, then there arises as to where the recycle from the penultimate stage should be sent. Although quite unexpected, the finding that the recycle step back distance, R, can be an integer greater than one, means that at least some extant separation networks may be misdesigned. While there may exist separation networks where at some point the step back may be greater than one, the writers are not aware of such a network where the step back is consistently (systematically) greater than one. While it is natural to think that in separation networks a one stage step back is always optimal - that's what one's intuition says, actually when R is greater than one, a one stage step back lowers the grade for the same number of stages in the network's cascade. Until now, in the design of separation networks in all industries, R has been neglected, leading to the design of suboptimal networks. In the derivation it was assumed that the: unit operations behaved linearly and identically. If this assumption is not true, then N, R and C are dependent on the transfer functions of the individual unit operations used in the network design. However, using the same three criteria developed above, it is possible for the reader to develop the optimal network for stages with non-identical transfer functions. Optimization of networks to maximize an objective function, such as efficiency, is an arbitrary activity since there are at least as many objective functions as designers. In each case the values of the network design specifications which maximize an objective function can be determined first. These design specifications can then be used to design te appropriate, strongly-connected network given appropriate network conditions, that is, the Ts for each species in each stage. Of course, an objective function can also be a function of the N, C and R obtained and the Ts used, thus, introducing nonlinearities. In such a case the network obtained by the technique is just the first iteration in obtaining an exact solution. CONCLUSIONS
Presented is a non-mathematical, non-computer technique for obtaining feasible designs for simple separation networks. Once the transfer function of the unit operations of a network are known and the desired separation factor and beneficiation ratio specified, a set of graphs approximately defining the N, C and R required for the specified network performance is used to design the network. In addition it should be pointed out that:
FEASIBLEDESIGNSFORSEPARATIONNETWORKS
173
( 1 ) For given specifications, a feasible network may be designed directly no trial and error is needed. (2) In separation networks, the number of stages the recycle steam is fed back may be greater than one stage. (An unanticipated finding. )
REFERENCES Abadie, J., 1970. Integer and Nonlinear Programming. North-Holland, Amsterdam, 544 pp. Agar, C. and Kipkie, W., 1978. Predicting locked-cycle flotation test results from batch data. CIM Bull., 71:119. Arbiter, N. and Weiss, W., 1971. Design of flotation cells and circuits. Trans. AIME, 250: 340347. Biegler, L., 1985. Improved path optimization for sequential modular simulators-I: the interface. Comput. Chem. Eng., 9(3): 245-256. Biegler, L. and Cuthrell, J., 1985. Improved path optimization for sequential modular simulators-I: the optimization algorithm. Comput. Chem. Eng., 9 (3): 257-267. Biegler, L. and Hughes, R., 1985. Feasible path optimization with sequential modular simulators. Comput. Chem. Eng., 9(4): 379-394. Biegler, L. and Grossman, I., 1985. Strategies for the optimization of chemical processes. Rev. Chem. Eng., 3( 1): 1-47. Davis, W.J.W., 1964. The development of a mathematical model of the lead flotation circuit at the Zinc Corporation Limited. AIMM Trans., 212:61-80. Denn, M., 1969. Optimization by Variational Methods. McGraw-Hill, New York, NY, 419 pp. Denn, M., 1986. Process Modeling. Pitman, Marslafield, MA, 324 pp. Fiacco, A. ancl McCormick, G., 1968. Linear Programming, Sequential Unconstrained Minimization Techniques. Wiley, New York, NY, 600 pp. Friedly, J., 1972. The Dynamic Behavior of Processes. Prentice-Hall, Englewood Cliffs, N J, 590 pp. Gardener, R., Lee, H. and Yu, B., 1981. Development of radioactive tracer methods for applying the mechanistic approach to continuous flotation. In: P. Somasundaran (Editor), Fine Particle Processing, vol. 1. AIME, New York, NY, pp. 922-945. Green, J.C.A., 1984. The optimisation of flotation networks. Int. J. Miner. Process., 13: 83103. Harris, C.C., Jowett, A. and Ghash, S.K., 1963. Analysis of data from continuous flotation tests. Trans. AIME, 229: 444-447. Henley, E.J. and Williams, R.A., 1973. Graph Theory in Modern Engineering. Academic Press, New York, NY, 303 pp. Himmelblau, D. and Bischoff, K., 1970. Process Analysis and Simulation. Wiley, New York, NY, 348 pp. Houdouin, D. and Everell, M.D., 1980. A hierarchical procedure for adjustment and material balancing of mineral processes data. Int. J. Miner. Process., 7:91-116. Imaizumi, T. and Inoue, T., 1966. Kinetic considerations of froth flotation. J. Min. Metall. Inst. Jpn., 293(82): 17-24. King, R.P., 1978. A pilot plant investigation of a kinetic model for flotation. J.S. Afr. Inst. Min. Metall., 78: 325-338. Lasdon, L. and Waren, A., 1978. Generalized reduced gradient software for linearly and nonlinearly constrained problems. In: H. Greuenberg (Editor), Design and Implementation of Optimization Software. Sijthoff, Alphen a/d Rijn, pp. 45-60.
174
M , ( . WIL,[.b~MS 4 N I ) T.P, MEI_()Y
Luyben W.L., 1973. Process Modeling, Simulation and Control lbr Chemical Engineers. McGraw-Hill, New York, NY. 558 pp. Mehrotra, S.P. and Kapur, P.C., 1974. Optimal-suboptimal synthesis and design of flotation circuits. Sep. Sci,, 9:167-184. Meloy, T.P., 1983. Analysis and optimization of mineral processing and coal cleaning circuit analysis. Int. J. Miner. Process., 10: 61-80. Peterson, J., Chau-Chyun, C. and Evans, L., 1978a. Computer programs for chemical engineers: 1978- Part 1. Chem. Eng.~ 85(11 ): 145-154. Peterson, J., Chau-Chyun, C. and Evans, L., 1978b. Computer programs for chemical engineers: I 9 7 8 - Part 2. Chem. Eng., 85(151: 69-82. Peterson, J., Chau-Chyun, C. and Evans, L., 1978c. Computer programs for chemical engineers: 1978 - Part 3. Chem. Eng., 85( 17): 79-86. Peterson, J., Chau-Chyun, C. and Evans, L., 1978d. Computer programs for chemical engineers: 1978 - Part 4. Chem. Eng., 85( 19): 107-116. Peterson, J., Chau-Chyun, C. and Evans, L., 1979. More computer programs for chemical engineers. In: Chem. Eng., May 21, 1979. Sastry, K. and Adel, G., 1984. A survey of computer simulation software for mineral processing systems. In: J.A. Herbst (Editor), Control '84. AIME, New York, NY, pp. 49-53. Sutherland, D.N., 1981. A study on the optimization of the arrangement of flotation circuits. Int. J. Miner. Process., 7: 319-346. Taha, H., 1976. Operations Research. McMillan, New York, NY, 648 pp. Westerberg, A.W., Hutchinson, H.P., Motard, R.L. and Winter, P., 1979. Process Flowsheeting. Cambridge University Press, New York, NY, 251 pp. Williams, M.C., 1985. Experimental investigation and mathematical modeling of multifeed flotation networks. Ph.D. Dissertation, Univ. California, Berkeley, CA, 274 pp. Williams, M.C. and Meloy, T.P., 1983. A dynamic model of flotation cell banks. Int. J. Miner. Process., 10:141 - 160. Williams, M.C. and Meloy, T.P., 1989. On the fundamental definition of process separation functions. Int. J. Miner. Process., 26: 65-72. Williams, M.C., Fuerstenau, D.W. and Meloy, T.P., 1986. Circuit analysis - General product equations for multifeed, multistage circuits containing variable selectivity functions. Int. J. Miner. Process., 17: 99-111. Williams, M.C., Fuerstenau, D.W. and Meloy, T.P., 1992. A graph-theoretic approach to network design. Int. J. Miner. Process., (in press). Yingling, J.C., 1990. Circuit analysis: optimizing mineral processing flowsheet layouts and steady state control specifications. Int. J. Miner. Process., 29:149-174.