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Heuristic evolutionary synthesis with non-sharp separators
Compurers chcm. Engng, Printed in Great Britain.
Vol. 13, No. I l/12, All rights reserved
HEURISTIC
pp.
1207-1219,
Copyright
EVOLUTIONARY SYNTHESIS NON-SHARP SEPARATORS W.-K.
Department
1989
of Chemical
CHAN
Engineering.
and
0098-l 354/89 $3.00 + 0.00 0 1989 Pergamon Press plc
WITH
R. G. H. PRINCE?
University
of Sydney, Sydney, NSW
2006, Australia
(Received for publication 19 June 1989) in undertaking a process separation task, such as separating the constituent minerals of an ore to the most profitable extent, we allow limited separation in each process unit (as against requiring a “perfect split”), the number and connection of the units have to be found. Here it is done by building up the required flowsheet unit by unit, stopping when further addition does not increase the profit. Recycle of streams is allowed, the extent being controlled by an arbitrary parameter over which a search may be made. Simulation is sequential modular, with convergence and the necessary process optimization of each intermediate flowsheet carried out by infeasible path successive quadratic programming. As heuristics guide the construction, “good”, not necessarily “the best”, flowsheets are obtained, which may be further ranked, such as by controllability and/or process flexibility. The approach is here developed for “homogeneous” Aowsheets (all process units identical), with minerals flotation separation as an example, using a very simple first-order kinetic flotation model. While countercurrent circuits are synthesized as limiting cases of the heuristics, flowsheets more profitable than these are also obtained. The paper reviews critically the heuristics and their application. Abstract-If,
1. INTRODUCTION
unit by unit. This evolution is achieved by assigning destinations to the outputs of the subsequently added units, one output at a time, according to a set of ordered heuristics. Specifically, the “richest” or “poorest” streams are never recycled, but sent to further processing. Other unassigned streams are “recycled” if their compositions are sufficiently “close” (to be defined later) to those of streams (usually already assigned) in the process; otherwise, new units are created to process All streams sent to further processing these further. are candidates for output, and will be output if not worth further processing and if product specifications are met; off-specification outputs are returned to the process. For each stream examined, a modification is made to the current flowsheet, and the new flowsheet is optimized to maximize a profit function. After this, the evolutionary process is repeated by examining another stream until destinations have been assigned to all the streams. At the final stage, countercurrent units are added to all proposed system outputs to further test these output synthesis decisions. The optimization of this structure then gives the final synthesized flowsheet. The four steps are summarized in Fig. 1, showing the three major elements of the method:
The synthesis of a chemical process flowsheet is usually decomposed into subproblems, such as the synthesis of heat exchange networks. Nishida et al. (198 1) reviewed the research efforts on these subproblems; and Westerberg (1985) more recently those on the separation subproblem. Work on non-sharp separation includes several papers (Siirola et al., 1971; Mahalec and Motard, 1977a,b; Lu and Motard, 1985; Muraki er al., 1986; Floudas, 1987) but many aspects, such as the optimal recycling of streams, are unresolved. This paper describes an evolutionary synthesis strategy whereby the flowsheet is built up unit by unit; heuristics eliminate many alternatives which are unlikely to be optimal; and each intermediate flowsheet is optimized, to maximize progressively a profit function. The work is one of few which considers each intermediate flowsheet at the unit level, thereby ensuring synthesis decisions are made on the basis of detailed simulations. We limit ourselves here to homogeneous flowsheets (i.e. ones made up of one kind of process unit only), and use as an example separation of a mineral ore by flotation.
2.
2.1. General
SYNTHESIS
METHOD
outline
The initially
basis of our method is that a flowsheet, consisting of only one unit, can be built up
tAuthor
to whom all correspondence should be addressed. I207
(i) the “ordering rule” which selects the stream to be examined; (ii) the “stream assignment scheme” which creates the flowsheet modifications; and (iii) the optimization of each intermediate flowsheet.
W.-K.
1208
CHAN and R. G. H. PRINCE final product streams; and (iii) the cost associated with each unit. The function used for the value of a stream includes contributions for both purity and valuable mineral recovery: value
(stream
s) = k, Z,st
(6)
where Z, is the flowrate of valuable mineral in stream s; X, is the valuable mineral weight fraction (dry basis) of stream s, and k,, 0 are cost constants. Valuable mineral recovery is represented by the Z, term while purity is represented by x,. For @ << 1, the benefit from achieving high purity is marginal. On the other hand, for 0 >> 1 the value of a stream is negligible until a high purity is attained. In this first attempt at auromatic synthesis F) = 1 was used. The cost associated with a unit was taken as proportional to capital charges only. modelled by a power law relationship: cost
+, i Fig.
1. General
outline
of heuristic strategy.
2.2.
Problem
sfaiement
evolutionary
synthesis
The flotation circuit synthesis problem examined can be stated as: “Given a feed of valuable mineral weight fraction (dry basis) X,, synthesize a flotation circuit to product a final concentrate and a final tailing of valuable mineral weight fractions (dry basis) s, and X, respectively, such that:
and
the
profit P = added
value
-
costs
(3)
the
P, = (value
partial
profit
of output - (value
of a unit
streams)
Rule
as:
- costs
Rule (4)
and using a single function to define the value of a stream, the profit function of any flowsheet of n units, whether final or intermediate, can thus be expressed as: P=
5 pi.
(5)
t-1
Note the terms cost of external
surviving in the summation are: (i) the feeds to the process: (ii) the value of
(7)
The synthesis strategy considers one stream assignment at a time, thereby building up the flowsheet unit by unit and recycling some streams as may bc appropriate. For the starting flowsheet. which consists of one unit recelvrng the external feed of concentration X,, there are two unassigned unit outputs: the concentrate and tailing. In general, there would be a number of units in the current intermediate flowsheet with outputs awaiting stream destinations. An ordering rule selects at the start of each stage of flowsheet evolution one of the unassigned unit outputs as the stream to bc examined. The results presented here use two rules:
streams) of input
,j) = k,(V,i50)ti’
where I;, is the pulp volume of cell i here expressed in ft’, and k,, w are cost constants. Given values of the cost constants k,, 0, k, and w, equations (6) and (7) can bc used to compute the partial profits of the units, and hence the profits of the intermediate and final flowsheets.
is maximized”. By defining
(unit
2.4.
A---Select the earliest unit with unassigned outputs; and examine its concentrate prior to its tailing. R--Select the earliest unit with unassigned outputs; and examine its tailing prior to its concentrate.
Stream
ussipwnenr
.rcheme
Having selected a stream, a set of ordered heuristics is used to create a “good” modification to the current flowshcct; and since the synthesis proceeds by one stream assignment at a time, these heuristics are described collectively as the “stream assignment scheme”. 2.4. 1. Se’et yf~olvsheet mod$cations. Let the unit output being examined be stream s and its unit of
Heuristic evolutionary synthesis origin be unit j, then one of six possible modifications may be chosen: 1. Assign stream s to feed an existing unit. 2. Join stream s with another unassigned unit output, stream p, and process both in a new unit n. 3. Assign stream s to further processing by a new unit n. 4. Remove unitj and assign its combined feed as a system output. 5. Remove unit j and assign each stream making up its combined feed to the unit whose feed composition is closest. 6. Implement 5 (above) first. Then progressively allow each stream originally feeding unit j to become a system output, beginning with that stream which least violates final product specifications,
flowsheet
[RECYCLE]
[JOIN] [NEWCELL] [REMOVE]
[RETURN]
1209
until a “structurally acceptable” flowsheet (explained later) is obtained.
These six flowsheet modifications are illustrated in Fig. 2. The actual one implemented on a current flowsheet depends on the variables of stream s, the design variables of unit j; and its interconnections with the rest of the flowsheet. These modifications are called “synthesis decisions” in this paper, and are named as shown. 2.4.2. Structural acceptability. Unlike the building of acyclic flowsheets (as in the perfect split problem), the building of cyclic flowsheets by the successive assignment of streams is complicated by the need to ensure that the structure of the flowsheet (or part thereof, called a “subsystem” here for convenience) is such that it can actually achieve a separation. For a binary system, the minimum requirements are: (i) each subsystem must have at least two outputs (to outside the subsystem); and
Decision
RECYCLE
JOIN
I
I Unit
i
[OFFSPEC]
I
I atraam
Decision
p
NEWCELL
Decision
REMOVE
Decision
RETURN
Decision
OFFSPEC
Fig. 2. Set of possible flowsheet modifications
Decision
W.-K.
1210
CHAN and
R. G. H. PRINCL H8--In
Fig. (ii) all the assigned
3. Structural outputs to the
of same
cannot
be
A flowsheet failing to meet the second of these criteria is shown in Fig. 3. Here, the outputs of the subsystem consisting of units 1. 2 and 3 are all assigned to feed unit 4, and the separation achieved by units 1. 2 and 3 would be nullified. This subsystem is said to be “structurally unacceptable”, because this conclusion can be discerned from the flowsheet topology. Hence in making synthesis decisions in which streams are recycled, structural acceptability of the new flowsheet must be ensured. 17.4.J. Oi-&rrd heuristics. The heuristics which determine the flowsheet modification to be made to the current flowsheet can be divided into three categories.
(a) The richest HI
(b)
or poorest
stream
heuristic
The “richest” or “poorest” current flowsheet should be processing. All other streams sidered for recycle.
Recycling
streams in the sent to further should be con-
heuristics
H2--A
“suitable” recycle destination for stream s is stream p if the composition of stream p is “near‘s that of s. WC dcfinc “nearness” as a (arbitrary) function of .Y;,:
H3-Of
the “suitable” streams p, feeds to existing units are preferred; and of these, Lhat which is closest in composition to stream .Yis chosen. Otherwise. stream s and the “first” stream p (a unit output) are joined and sent to further processing. If there are no “suitable” recycle destinations, stream s is sent to further processing.
H4~-
H5
(c) Heuristics constraints
for
termination
and
is
accepted
Heuristic 1 simply reserves the richest and poorest streams as the “best” candidates for becoming the final concentrate and tailing. Recycling is considered in two stages:
unacceptability. a subsystem unit.
the latter case. the stream temporarily as an output.
meeting
(i) When are two streams suitable for mixing? (ii) Of the streams that can mix suitably with the stream being examined. which should be chosen’? questlon (i) by a recycle range Heuristic 2 resolves of concentration are difficult function. Since extremes to achieve, 6(.u, j should vanish at the extremes. and attain a maximum at some Intermediate value of concentration. For a binary mixture a reasonable assumption is that (ir.v,) should atrain its maximum at a concentration of 50?.0, and that the function should bc symmetric about this value. A function which meets these criteria is: ii(.Y,‘l y= $Ci sin(7t.q
).
(8)
This heuristic recycle range function is illustrated in Fig. 4. The “optimal” extent to which streams should be recycled can then be found by a search over the arbitrary parameter rS. Heuristics 3. 4 and 5 then address question (ii). Further processing in its simplest form. that is feeding the stream .F to a new unit whose outputs are not connected with the rest of the flowsheet (a “feedforward” unit) is used in response to heuristic 6. However, a stream not worth further processing by a feedft>rwar-d unit may be worth processing in a new unit with one of its outputs tccyclcd. Since optimization will set the capacity of any feedforward unit to zero at this point in synthesis. there will be no information as to where the non-product outputs of
product
H6--All streams sent to further processing arc candidates for output, and will be so, if not worth further processing and if product specifications are met. H7-- -Otherwise. off-specification outputs are rcturned to the units closest in composition for further processing. unless forbidden for structural reasons.
recycle
lange
Of
xs_ &
Fig.
4. Eicuristic
rcc~cle
range
function
Heuristic evolutionary the new units may be “best” recycled. We here arbitrarily recycle to the immediately previous unit: leading to a simple countercurrent combination. The consequences of this decision are discussed later. Figure 5 shows a flowchart for these heuristics. 2.5. Optimality
of flowsheers
synthesized
Although heuristic in nature, the synthesis method described in this paper should produce “good” solutions to the binary non-perfect split problem. This is because: (i) the method is based on generally valid heuristics (e.g. streams should only be recycled to points of sufficiently similar composition); (ii) intermediate flowsheets are optimized, so a monotonically increasing path of the profit function is followed; and (iii) we can search over many arbitrary elements of the method (e.g. by use of a different ordering rule).
3. DATA
AND
RESULTS
For our test problem we modelled Botation separation very simply by first-order kinetics (Bull, 1966) requiring two rate parameters, which were kept constant over all units. The information then required was: (i) the component flowrates in the external feed; (ii) the minimum product specifications X, and X,; (iii) the parameters k,, tl, kc and CI> of the cost equations; and (iv) the physical properties (viz. densities, flotation rate constants and water/solid ratios) required in the flotation model. This information is summarized in Table 1 for four synthesis problems. The base case is called problem XF20; that with a feed grade of 7.5%, XF7.5; that with a difficult separation, DIFFSEP; and that with a high capital component, HIGHCAP. The base case consisted of a feed of 100,000 lb h-’ of solids, made up of valuable and gangue minerals. The volumetric flowrate (including water) was 6000 ft3 h-l. The valuable feed grade was 20% (i.e. 20,000 lb h-r of valuable). This feed was to be separated into a final concentrate of no less than 55% valuable on a dry basis, and a final tailing no richer than 2% valuable. The other problems reduced the feed grade (from 20 to 7.5%); made the separation more difficult (ratio of flotation rate constants changed from 6 to 3); 10 to and increased the separation costs (k, from 50 $ h-l). Simulation was sequential modular with optimization of the flowsheets by infeasible path successive quadratic programming (Powell, 1978; Berna et al., 1980; Locke ez ul., 1983; Biegler and Cuthrell, 1985).
synthesis
1211
Here, we maximized an economic function [equation (3)] with respect to the cell volumes. Derivatives were obtained by application of the chain rule of differentiation (Chan and Prince, 1986). Figure 6 shows the stage by stage synthesis of the flowsheets obtained for problem HIGHCAP using ordering rule A. As the recycle bandwidth 6 is varied, the varying extent of recycling gives flowsheets with different profitabilities. Figure 7 summarizes all the synthesized flowsheets for the four problems. 4. DISCUSSION
4.1. General
observations
The synthesis method may be seen to have responded appropriately to the variations of the problem specifications about the base case. In reducing the feed grade from 20 to 7.5%, the number of cleaners (units on the rich side of the feed) increased, whilst the number of scavengers (lean side of feed) decreased: more cleaning is required to achieve an acceptable final concentrate grade if the feed grade is low, but less scavenging. In reducing the ratio of rate constants, making the separation more difficult, the number of cleaners increased, as did the residence times of, particularly, the scavengers (not shown here). In increasing processing costs relative to recovered value (the ratio of capital charges to added value was increased five-fold) the number of both cleaners and scavengers (and their residence times) decreased. These changes in the structures synthesized then reflect the process kinetics and the economics. Bearing in mind that for large volume continuous processing (as is usual in the minerals industry) sustained changes in profit of a fraction of a pcrcent-.], even O.Ol%-may be significant, then for each of the four problems, the difference between the highest and lowest profits of the flowsheets, expressed as a percentage of the highest profit found for that problem, is quite marked (e.g. the difference between flowsheets XF20/A/Sf and XF20/B/3f is 6.6%). The difference between the more profitable flowsheets of a given problem is, of course, smaller (e.g. flowsheets XF20/B/lf, Zf, 3f, 4f and XF20/A/3f are all within 0.6% of each other). In addition to identifying a “best” flowsheet, the method then also finds sets of “good” flowsheets, for which differences in profit may not be the deciding factor in choosing the final design. A natural extension is then to include other design criteria (such as control) to further rank these more profitable flowsheets. We (Barton ei al., 1986) have presented this for a three-cell flotation circuit synthesis in which simple measures of controllability (Perkins and Wong, 1985; Lau et al., 1985) were used to distinguish between the more profitable flowsheets. Further criteria we have considered include process flexibility to uncertainties in cost or physical parameters.
CURRENT
(I’ INTERMEDIATE FLOWSHEET
stream 9 unit j
from
I
Remove unit j -_
;
F&$Q_A
SUtf&BJ,f=
RECYCLE
DESTINATION
* Exclude system outputs. * Exclude unit outputs assigned to feed units. * Exclude structurat!y unacceptable recycles. within the recycle range of * Find stream p beginning with the earliest unit stream s synthesized (concentrate before tailing).
RECYCl& DEClStQf$
-
ReCyCl@ stream 5% to Stream q , unit feed EloseSt in
system wtputs. begrnning with that slream which least violates product specifications, until a structurally acceptable is obtainerl. --
3
Fig. 5. Ordered heuristics for flowsheet evolution. 1212
Heuristic Table
I. Data
for
evolutionary
four
Problem name
Flotation
rate
constants
XF7.5
100,000 20 6000
(h-‘)
1213
flotation circuit svnthesis Droblems
XFZO
Feed Solid flowrate (lb h-‘) Valuable grade (O/o) Volumetric Rowrate (ft’ h-‘)
synthesis
100,000 7.5 6000
IO, 60
10, 60
DIFFSEP 100,000 20 6000 IO, 30
HIGHCAP
100,000 20 6000 10, 60
Cost data Value
of
stream,
k, ($ lb
‘)
0.2
e changes,
k, (% h-‘)
Product specifications Minimum grade of coricentrate A’, (%) Maximum grade of tailings A’, (%) Density of minerals = 187.5 Ib ft-’ Liquid/solid ratio in concentrate 0.04,
4.2.
Arbitrary
0.2
1.0
Capital 0
elements
in the synthesis
after
water
1.0
0.2
I.0
0.2
1.0
10.0 0.7
10.0 0.7
10.0 0.7
50.0 0.7
55 2
55 2
55 6
55 10
addition
method
4.2. I. Recycle bandwidth. The arbitrary parameter 6, here called the recycle bandwidth, controls the recycle decisions. As 6+0 acyclic flowsheets would be produced, and as 6 --P 03 all streams other than the “richest” or “poorest” will be recycled. These are recycled to the unit whose feed is closest in concentration, that is, the unit immediately up-stream. Countercurrent circuits are then produced, and that the method synthesized countercurrent circuits as the limiting case is a useful check of its validity. However, as Fig. 7 shows, flowsheets more profitable than countercurrent can be and were synthesized for intermediate values of 6. These results justify our more general approach. In particular, we note two types of substructure which deviate from the strict countercurrent, and are commonly used in practice. One is a “cleanerscavenger” which accepts the tailing of a cleaning unit and returns its concentrate to that cleaning unit. The tailing of the cleaner-scavenger is then recycled to a unit downstream of the cleaning unit. An example is unit 6 in flowsheet XF20/A/2f in Fig. 7. The other is the “two-cell cleaner bank” which is a two-cell bank performing a cleaning operation: the first cell receives the incoming feed and passes its tailing as feed to the second cell. The tailing of the second cell is the tailing from the bank whilst the concentrates of both cells are combined. Units 3 and 5 in flowsheets XF20/B/2f are an example. However, our heuristically-based approach cannot guarantee to generate all derivatives from countercurrent which may be of eventual interest. 4.2.2. Ordering rules. The difference between the two heuristic ordering rules A and B (both examine first the earliest synthesized unit with unassigned outputs) is that rule A assigns the concentrate first and rule B, the tailing. This difference between the two explains why rule B tended to synthesize two-cell cleaner banks, and rule A, cleaner-scavengers, when synthesis deviated from the countercurrent circuit. Here, a stream, otherwise recycled in countercurrent flow, was sent to a new unit. The tailing of the new
0.06
ft’ lb-‘.
unit would be recycled first if rule B was used, giving the two-cell cleaner. By contrast, rule A would recycle the concentrate of the new unit first, leading to the cleaner-scavenger. This observation and others show the types of flowsheets synthesized by the method are dependent on the heuristic ordering rule used, and although rule B was responsible for synthesizing the more profitable flowsheets, the results do not show that rule B is necessarily better than rule A. As it is not obvious what is the optimal order in which the streams should be assigned during evolution, this dependence of the method on the ordering rule may represent a weakness. A search over the possible types of ordering rules would then be required to ensure that the “best” flowsheet is synthesized. This approach could be exploited to obtain a wide set of “good” flowsheets, or to support empirically that one flowsheet is really “best’‘-that is, represents a global rather than a local optimum. 4.3. Premature
termination
The heuristics used here for flowsheet evolution may lead to premature termination. This is because new units introduced to further process the “richest” or “poorest” streams are added as feedforward units (i.e. outputs are initially not assigned). If optimization does not set the units to zero capacity, destinations will then be assigned to the outputs of these units. Otherwise, these “unprofitable” units are removed, and their feed streams (if meeting product specifications) assigned as system outputs. When all streams have been assigned it then appears no longer profitable to further build up the flowsheet by adding more units. However, it may be profitable to further process the system outputs in new units, if some of the outputs of these new units are recycled. The difficulty with this is that optimization will set the capacity of any feedforward unit to zero at this point in synthesis; so, there is no information upon which to decide as to where the outputs of these new units may be “best” recycled.
stream examined
-*
a
system
currently
output *
NEWCELL
NEWCELL
NEWCELL
NEWCELL
2.1
+g-g-c 2
_
-
REMOVE
REMOVE
REMOVE
2ze+*
REMOVE :.e-
2.5
a.3
. 4
RECYCLE
I-_
NEWCELL
REMOVE
Fig.
6.
Example
of stage by stage synthesis. 1214
*
P = 2697
= 0.1343+0.1599
PROBLEM
XF20
Fig. 7
(Base
Case)
B= 0.1217 +0.1592
P I 2707 B - 0.2988-he+
W.-K.
1216
CHAN and
PROBLEM Orderina
Rule
R. G. H. PRINCE
DIFFSEP
A
Orderina
Rule
F1
-Wr
3~i:7i 2-:::.,3
iz3. L2 P =
1935
P =
1971
B = 0.3478
-mm
PROBLEM Ordema
Rule
HIGHCAP Orderina
A
+
0.1929
Fig.
Rule
E$
0.1549
7-Conrinued.
Heuristic evolutionary
PROBLEM
6 B =
0.1893
-b
synthesis
1217
XF7.5
0.2635
=
0.3366
+
0.4567
P = 1041 6,
s B =
0.3083
+
=
0.2645
bO.3078
0.4567
+
ce
0.5415
P = 1042 6 B =
Notes
: P 6,
= Profit, = Recycle Fig.
0.5415
-+
-
Bandwidth. 7. Summary
of all flotation
To avoid the combinatorial problem of determining where these outputs may be optimally recycled, we have chosen arbitrarily to recycle to the unit immediately preceding, so that a countercurrent unit is added; and we have limited this to the addition of one countercurrent unit to each system output {when it was no longer profitable to add feedforward units). This avoids the combinatorial problem of adding the optimal number of countercurrent units to each output. The choice to add only countercurrent units is also arbitrary, and other recycle destinations for the new units should be tried. One might, however, argue that when it was no longer profitable to add feedforward units, there would be little room for improvement in the profit function by adding other types of new units. Consequently, extension of the synthesis program to add more units, and recycling their outputs to various trial-and-error locations so as to avoid premature termination, was deferred, pending the results of the numerical studies discussed here.
circuits synthesized
The present deficiency of the method is illustrated by the flowsheets DIFFSEP/A/lf and DIFFSEP/ Bjlf. Adding one more countercurrent unit to the final tailing, and removing the last cleaner (unit 9) in DIFFSEP/A/lf would lead to DIFFSEP/B/lf, resulting in an increase of 2.3% profit. As in any specific case, the different synthesis routes taken by variations on the arbitrary elements of the decision logic may lead to sufficient “good” solutions which have not been terminated prematurely, a more thorough combinatorial modification has not been implemented at the time of writing.
4.4. Model
implications
Flotation is here modelled as a first-order rate process, and the flotation cell as a continuous stirred tank reactor (CSTR). We have the well-known result that for a first-order irreversible reaction, carried out in a series of N CSTRs with identical residence time r,
W.-K.
1218 the fraction of unconverted reactant the Nth CSTR, is given by:
1 - X,,
CHAN
and R. G. H. PRINCE
leaving
1 -“=[&J which
can
also
be written
l-X,+=
as: .V
I-+& [
= [l -
RI-‘.
I
R is then equivalent to the fraction of a solid species floated to the concentrate in a flotation cell and (1 -~ R),’ the fraction of feed solid appearing in the final tailing from a bank of Ncells. Hence, several series smaller cells in may therefore be more profitable than a single large cell (just as several smaller CSTRs may require less reaction volume than a single large CSTR) if the separation is difficult or the cost of capital is high. This may be seen reflected in the presence of, fol- example, two-cell b an k.s in fiowsheets such as DIFFSEP/H/4f. On the other hand, the simple form of model here used, especially the parameters being held constant over all units of a fowsheet. limits the realism of the simulation and the correspondence with practice. Further work here will address the issue.
5.
CONCLUSIONS
Results obtained for four flotation problems show that “good” flowsheets can be synthesized, by using the heuristics for flowsheet evolution, and intermediate stage optimization. which allowed a monotonically increasing path of a profit function to be followed. Countercurrent circuits were synthesized as a limiting case, but flowsheets were found more profitable than these. Changes in the problem data were met with appropriately modified synthesized flowsheets. Some deviations found from strict countercurrent flow. such as cells in series and cleaner-scavengers, are known in flotation practice, but correspondence of our results with practice is limited by the simple model used. Of the arbitrary elements in our synthesis decisions, the recycle parameter itself could be varied over its full range. Others were here varied only in a limited way, to show how “good” flowsheets-rather than a “best”-would be generated. These might then be further examined by applying other criteria, such as controllability. However, while premature termination of some synthesis paths was observed, other paths would still yield “good” solutions. Further development of the synthesis program to add more units, recycling their outputs to various trial-and-error locations, may be a means to avoid this premature termination. Although applied here to flotation, the problem statement. the heuristics and the methodology described. apply to general binary separation synthesis
with non-sharp separators. The work can then be applied to separation technologies other than flotation. Components of the method such as the recycle range function and the ordering rule can be adapted heuristicaIly to the type of separation problem concerned. With additional heuristics. say to order the sequence of separation. it may be possible to extend the work here to non-sharp mullicomponent separatton. Acknowledgemurrrs -This work derives directly from Ph.D. research undertaken some years ago by Dr G. Fenton (1973) but not published at the time owing tc> the limitations in the then available mathenwtical tools. W.K.C. acknowledges the financiai support <>fCommonwealth Post Graduate Research and of Chemical Engincering Foundation Awards
NOMENCLATCRE
k,.
G = Flowrate
of gangue
k, = Cost constants
mineral
K, = Flotation
rate constant of species i Number of umts in current flowsheet Profit of flowsheet Partial profit 01’unit _j Pulp volume of cell j Flowrate of water Valuable mineral concentration (dry) Valuable mineral concentration of final concentrate and tailinp. respectively X, 7 Valuable mineral concentration of external feed A’,.: Minimum va!uahle mineral concentration in final concentrate
n P P, “; W x .x~, s,
= = = = = = =
% = Flowrate of valuable mineral co, 0 = Cost exponents 6 = Recycle range funcunn. recycle
bandwidth
REFXRF:NCES
Barton G. W., W.-K. Ghan, J. D. Perkms and R. G. If. Prince, Controllability analysis of alternative process designs. CHEMECA 36 141h Ausr. Chenr. EVIX. Cmf, pp. 200.-205 (1986). Berna T. J., M. H. Locke and A. W. Westerberg. A new approach to optlmlzatlon of chemical processes. AICIIE JI 26, 37 (1980). Biegler L. T. and J. E. Cuthrc-Il. Improved infeasible path optimization for sequential modular simulatosl---IT. The optimization algorithm. Conr/nr, i,hrr?r. E+q”R 9. 757 (1985). Bull W. R.. Flotation. Wirrrer Schoi~l ti,r,.bfincebs synthesis. Ph.D. Thesis. 1Jniversity of Sydney (1973). Floudas C. A., Separation synthesis ofmulticomponent feed streams into multicomponent product streams. AICJIE’ .I/ 33, 540 ( 1987). Lau H., J. Alvarez and K. F. Jensen, Synthesis of control structures by singular value analysis: dynamic measures of sensitivity and interaction. .4ICJ1E JI 31. 427 (1985). Locke M. H., A. W. Westerberg and R. H. Edahl, Improved successive qtradratrc programming optimization
Heuristic evolut.ionary synthesis algorithm for engineering design problems. AZChE JI 29, 871 (1983). Lu M. D. and R. L. Motard, Computer-aided total flowsheet synthesis. Cornput. &em. Engng 9, 431 (1985). Mahalec V. and R. L. Motard, Procedure for the initial design of chemical processing systems Compur. them. Engng 1, 57 (1977a). Mahalec V. and R. L. Motard, Evolutionary search for an optimal limiting process Aowsheet. Comput. them. Engng 1, 149 (1977b). Muraki M., K. Kataoka and T. Hayakawa, Evolutionary synthesis of a multicomponent multiproduct separation process. Chem. Engng. Sci 41, 1843 (1986).
1219
Nishida N., G. Stephanopoulos and A. W. Westerberg, A review of process synthesis. AIChE JI 27, 321 (1981). Perkins J. D. and M. P. F. Wong, Assessing controllability of chemical plants. Chem. Engng Res. Des. 63, 358 (1985). Powell M. J. D., A fast algorithm for nonlinearly constrained optimization calculations. Numerical Analysis, Dundee 1977 (G. A. Watson, Ed.), p. 144. SpringerVerlag, Berlin (1978). Siirola J. J., G. J. Powers and D. F. Rudd, Synthesis of system designs: III. Towards a process concept generator. AIChE
JI 17, 677 (1971).
Westerberg A. W., The synthesis of distillation-based separation systems. Cornput. them. Engng 9, 421 (1985).
Vol. 13, No. I l/12, All rights reserved
HEURISTIC
pp.
1207-1219,
Copyright
EVOLUTIONARY SYNTHESIS NON-SHARP SEPARATORS W.-K.
Department
1989
of Chemical
CHAN
Engineering.
and
0098-l 354/89 $3.00 + 0.00 0 1989 Pergamon Press plc
WITH
R. G. H. PRINCE?
University
of Sydney, Sydney, NSW
2006, Australia
(Received for publication 19 June 1989) in undertaking a process separation task, such as separating the constituent minerals of an ore to the most profitable extent, we allow limited separation in each process unit (as against requiring a “perfect split”), the number and connection of the units have to be found. Here it is done by building up the required flowsheet unit by unit, stopping when further addition does not increase the profit. Recycle of streams is allowed, the extent being controlled by an arbitrary parameter over which a search may be made. Simulation is sequential modular, with convergence and the necessary process optimization of each intermediate flowsheet carried out by infeasible path successive quadratic programming. As heuristics guide the construction, “good”, not necessarily “the best”, flowsheets are obtained, which may be further ranked, such as by controllability and/or process flexibility. The approach is here developed for “homogeneous” Aowsheets (all process units identical), with minerals flotation separation as an example, using a very simple first-order kinetic flotation model. While countercurrent circuits are synthesized as limiting cases of the heuristics, flowsheets more profitable than these are also obtained. The paper reviews critically the heuristics and their application. Abstract-If,
1. INTRODUCTION
unit by unit. This evolution is achieved by assigning destinations to the outputs of the subsequently added units, one output at a time, according to a set of ordered heuristics. Specifically, the “richest” or “poorest” streams are never recycled, but sent to further processing. Other unassigned streams are “recycled” if their compositions are sufficiently “close” (to be defined later) to those of streams (usually already assigned) in the process; otherwise, new units are created to process All streams sent to further processing these further. are candidates for output, and will be output if not worth further processing and if product specifications are met; off-specification outputs are returned to the process. For each stream examined, a modification is made to the current flowsheet, and the new flowsheet is optimized to maximize a profit function. After this, the evolutionary process is repeated by examining another stream until destinations have been assigned to all the streams. At the final stage, countercurrent units are added to all proposed system outputs to further test these output synthesis decisions. The optimization of this structure then gives the final synthesized flowsheet. The four steps are summarized in Fig. 1, showing the three major elements of the method:
The synthesis of a chemical process flowsheet is usually decomposed into subproblems, such as the synthesis of heat exchange networks. Nishida et al. (198 1) reviewed the research efforts on these subproblems; and Westerberg (1985) more recently those on the separation subproblem. Work on non-sharp separation includes several papers (Siirola et al., 1971; Mahalec and Motard, 1977a,b; Lu and Motard, 1985; Muraki er al., 1986; Floudas, 1987) but many aspects, such as the optimal recycling of streams, are unresolved. This paper describes an evolutionary synthesis strategy whereby the flowsheet is built up unit by unit; heuristics eliminate many alternatives which are unlikely to be optimal; and each intermediate flowsheet is optimized, to maximize progressively a profit function. The work is one of few which considers each intermediate flowsheet at the unit level, thereby ensuring synthesis decisions are made on the basis of detailed simulations. We limit ourselves here to homogeneous flowsheets (i.e. ones made up of one kind of process unit only), and use as an example separation of a mineral ore by flotation.
2.
2.1. General
SYNTHESIS
METHOD
outline
The initially
basis of our method is that a flowsheet, consisting of only one unit, can be built up
tAuthor
to whom all correspondence should be addressed. I207
(i) the “ordering rule” which selects the stream to be examined; (ii) the “stream assignment scheme” which creates the flowsheet modifications; and (iii) the optimization of each intermediate flowsheet.
W.-K.
1208
CHAN and R. G. H. PRINCE final product streams; and (iii) the cost associated with each unit. The function used for the value of a stream includes contributions for both purity and valuable mineral recovery: value
(stream
s) = k, Z,st
(6)
where Z, is the flowrate of valuable mineral in stream s; X, is the valuable mineral weight fraction (dry basis) of stream s, and k,, 0 are cost constants. Valuable mineral recovery is represented by the Z, term while purity is represented by x,. For @ << 1, the benefit from achieving high purity is marginal. On the other hand, for 0 >> 1 the value of a stream is negligible until a high purity is attained. In this first attempt at auromatic synthesis F) = 1 was used. The cost associated with a unit was taken as proportional to capital charges only. modelled by a power law relationship: cost
+, i Fig.
1. General
outline
of heuristic strategy.
2.2.
Problem
sfaiement
evolutionary
synthesis
The flotation circuit synthesis problem examined can be stated as: “Given a feed of valuable mineral weight fraction (dry basis) X,, synthesize a flotation circuit to product a final concentrate and a final tailing of valuable mineral weight fractions (dry basis) s, and X, respectively, such that:
and
the
profit P = added
value
-
costs
(3)
the
P, = (value
partial
profit
of output - (value
of a unit
streams)
Rule
as:
- costs
Rule (4)
and using a single function to define the value of a stream, the profit function of any flowsheet of n units, whether final or intermediate, can thus be expressed as: P=
5 pi.
(5)
t-1
Note the terms cost of external
surviving in the summation are: (i) the feeds to the process: (ii) the value of
(7)
The synthesis strategy considers one stream assignment at a time, thereby building up the flowsheet unit by unit and recycling some streams as may bc appropriate. For the starting flowsheet. which consists of one unit recelvrng the external feed of concentration X,, there are two unassigned unit outputs: the concentrate and tailing. In general, there would be a number of units in the current intermediate flowsheet with outputs awaiting stream destinations. An ordering rule selects at the start of each stage of flowsheet evolution one of the unassigned unit outputs as the stream to bc examined. The results presented here use two rules:
streams) of input
,j) = k,(V,i50)ti’
where I;, is the pulp volume of cell i here expressed in ft’, and k,, w are cost constants. Given values of the cost constants k,, 0, k, and w, equations (6) and (7) can bc used to compute the partial profits of the units, and hence the profits of the intermediate and final flowsheets.
is maximized”. By defining
(unit
2.4.
A---Select the earliest unit with unassigned outputs; and examine its concentrate prior to its tailing. R--Select the earliest unit with unassigned outputs; and examine its tailing prior to its concentrate.
Stream
ussipwnenr
.rcheme
Having selected a stream, a set of ordered heuristics is used to create a “good” modification to the current flowshcct; and since the synthesis proceeds by one stream assignment at a time, these heuristics are described collectively as the “stream assignment scheme”. 2.4. 1. Se’et yf~olvsheet mod$cations. Let the unit output being examined be stream s and its unit of
Heuristic evolutionary synthesis origin be unit j, then one of six possible modifications may be chosen: 1. Assign stream s to feed an existing unit. 2. Join stream s with another unassigned unit output, stream p, and process both in a new unit n. 3. Assign stream s to further processing by a new unit n. 4. Remove unitj and assign its combined feed as a system output. 5. Remove unit j and assign each stream making up its combined feed to the unit whose feed composition is closest. 6. Implement 5 (above) first. Then progressively allow each stream originally feeding unit j to become a system output, beginning with that stream which least violates final product specifications,
flowsheet
[RECYCLE]
[JOIN] [NEWCELL] [REMOVE]
[RETURN]
1209
until a “structurally acceptable” flowsheet (explained later) is obtained.
These six flowsheet modifications are illustrated in Fig. 2. The actual one implemented on a current flowsheet depends on the variables of stream s, the design variables of unit j; and its interconnections with the rest of the flowsheet. These modifications are called “synthesis decisions” in this paper, and are named as shown. 2.4.2. Structural acceptability. Unlike the building of acyclic flowsheets (as in the perfect split problem), the building of cyclic flowsheets by the successive assignment of streams is complicated by the need to ensure that the structure of the flowsheet (or part thereof, called a “subsystem” here for convenience) is such that it can actually achieve a separation. For a binary system, the minimum requirements are: (i) each subsystem must have at least two outputs (to outside the subsystem); and
Decision
RECYCLE
JOIN
I
I Unit
i
[OFFSPEC]
I
I atraam
Decision
p
NEWCELL
Decision
REMOVE
Decision
RETURN
Decision
OFFSPEC
Fig. 2. Set of possible flowsheet modifications
Decision
W.-K.
1210
CHAN and
R. G. H. PRINCL H8--In
Fig. (ii) all the assigned
3. Structural outputs to the
of same
cannot
be
A flowsheet failing to meet the second of these criteria is shown in Fig. 3. Here, the outputs of the subsystem consisting of units 1. 2 and 3 are all assigned to feed unit 4, and the separation achieved by units 1. 2 and 3 would be nullified. This subsystem is said to be “structurally unacceptable”, because this conclusion can be discerned from the flowsheet topology. Hence in making synthesis decisions in which streams are recycled, structural acceptability of the new flowsheet must be ensured. 17.4.J. Oi-&rrd heuristics. The heuristics which determine the flowsheet modification to be made to the current flowsheet can be divided into three categories.
(a) The richest HI
(b)
or poorest
stream
heuristic
The “richest” or “poorest” current flowsheet should be processing. All other streams sidered for recycle.
Recycling
streams in the sent to further should be con-
heuristics
H2--A
“suitable” recycle destination for stream s is stream p if the composition of stream p is “near‘s that of s. WC dcfinc “nearness” as a (arbitrary) function of .Y;,:
H3-Of
the “suitable” streams p, feeds to existing units are preferred; and of these, Lhat which is closest in composition to stream .Yis chosen. Otherwise. stream s and the “first” stream p (a unit output) are joined and sent to further processing. If there are no “suitable” recycle destinations, stream s is sent to further processing.
H4~-
H5
(c) Heuristics constraints
for
termination
and
is
accepted
Heuristic 1 simply reserves the richest and poorest streams as the “best” candidates for becoming the final concentrate and tailing. Recycling is considered in two stages:
unacceptability. a subsystem unit.
the latter case. the stream temporarily as an output.
meeting
(i) When are two streams suitable for mixing? (ii) Of the streams that can mix suitably with the stream being examined. which should be chosen’? questlon (i) by a recycle range Heuristic 2 resolves of concentration are difficult function. Since extremes to achieve, 6(.u, j should vanish at the extremes. and attain a maximum at some Intermediate value of concentration. For a binary mixture a reasonable assumption is that (ir.v,) should atrain its maximum at a concentration of 50?.0, and that the function should bc symmetric about this value. A function which meets these criteria is: ii(.Y,‘l y= $Ci sin(7t.q
).
(8)
This heuristic recycle range function is illustrated in Fig. 4. The “optimal” extent to which streams should be recycled can then be found by a search over the arbitrary parameter rS. Heuristics 3. 4 and 5 then address question (ii). Further processing in its simplest form. that is feeding the stream .F to a new unit whose outputs are not connected with the rest of the flowsheet (a “feedforward” unit) is used in response to heuristic 6. However, a stream not worth further processing by a feedft>rwar-d unit may be worth processing in a new unit with one of its outputs tccyclcd. Since optimization will set the capacity of any feedforward unit to zero at this point in synthesis. there will be no information as to where the non-product outputs of
product
H6--All streams sent to further processing arc candidates for output, and will be so, if not worth further processing and if product specifications are met. H7-- -Otherwise. off-specification outputs are rcturned to the units closest in composition for further processing. unless forbidden for structural reasons.
recycle
lange
Of
xs_ &
Fig.
4. Eicuristic
rcc~cle
range
function
Heuristic evolutionary the new units may be “best” recycled. We here arbitrarily recycle to the immediately previous unit: leading to a simple countercurrent combination. The consequences of this decision are discussed later. Figure 5 shows a flowchart for these heuristics. 2.5. Optimality
of flowsheers
synthesized
Although heuristic in nature, the synthesis method described in this paper should produce “good” solutions to the binary non-perfect split problem. This is because: (i) the method is based on generally valid heuristics (e.g. streams should only be recycled to points of sufficiently similar composition); (ii) intermediate flowsheets are optimized, so a monotonically increasing path of the profit function is followed; and (iii) we can search over many arbitrary elements of the method (e.g. by use of a different ordering rule).
3. DATA
AND
RESULTS
For our test problem we modelled Botation separation very simply by first-order kinetics (Bull, 1966) requiring two rate parameters, which were kept constant over all units. The information then required was: (i) the component flowrates in the external feed; (ii) the minimum product specifications X, and X,; (iii) the parameters k,, tl, kc and CI> of the cost equations; and (iv) the physical properties (viz. densities, flotation rate constants and water/solid ratios) required in the flotation model. This information is summarized in Table 1 for four synthesis problems. The base case is called problem XF20; that with a feed grade of 7.5%, XF7.5; that with a difficult separation, DIFFSEP; and that with a high capital component, HIGHCAP. The base case consisted of a feed of 100,000 lb h-’ of solids, made up of valuable and gangue minerals. The volumetric flowrate (including water) was 6000 ft3 h-l. The valuable feed grade was 20% (i.e. 20,000 lb h-r of valuable). This feed was to be separated into a final concentrate of no less than 55% valuable on a dry basis, and a final tailing no richer than 2% valuable. The other problems reduced the feed grade (from 20 to 7.5%); made the separation more difficult (ratio of flotation rate constants changed from 6 to 3); 10 to and increased the separation costs (k, from 50 $ h-l). Simulation was sequential modular with optimization of the flowsheets by infeasible path successive quadratic programming (Powell, 1978; Berna et al., 1980; Locke ez ul., 1983; Biegler and Cuthrell, 1985).
synthesis
1211
Here, we maximized an economic function [equation (3)] with respect to the cell volumes. Derivatives were obtained by application of the chain rule of differentiation (Chan and Prince, 1986). Figure 6 shows the stage by stage synthesis of the flowsheets obtained for problem HIGHCAP using ordering rule A. As the recycle bandwidth 6 is varied, the varying extent of recycling gives flowsheets with different profitabilities. Figure 7 summarizes all the synthesized flowsheets for the four problems. 4. DISCUSSION
4.1. General
observations
The synthesis method may be seen to have responded appropriately to the variations of the problem specifications about the base case. In reducing the feed grade from 20 to 7.5%, the number of cleaners (units on the rich side of the feed) increased, whilst the number of scavengers (lean side of feed) decreased: more cleaning is required to achieve an acceptable final concentrate grade if the feed grade is low, but less scavenging. In reducing the ratio of rate constants, making the separation more difficult, the number of cleaners increased, as did the residence times of, particularly, the scavengers (not shown here). In increasing processing costs relative to recovered value (the ratio of capital charges to added value was increased five-fold) the number of both cleaners and scavengers (and their residence times) decreased. These changes in the structures synthesized then reflect the process kinetics and the economics. Bearing in mind that for large volume continuous processing (as is usual in the minerals industry) sustained changes in profit of a fraction of a pcrcent-.], even O.Ol%-may be significant, then for each of the four problems, the difference between the highest and lowest profits of the flowsheets, expressed as a percentage of the highest profit found for that problem, is quite marked (e.g. the difference between flowsheets XF20/A/Sf and XF20/B/3f is 6.6%). The difference between the more profitable flowsheets of a given problem is, of course, smaller (e.g. flowsheets XF20/B/lf, Zf, 3f, 4f and XF20/A/3f are all within 0.6% of each other). In addition to identifying a “best” flowsheet, the method then also finds sets of “good” flowsheets, for which differences in profit may not be the deciding factor in choosing the final design. A natural extension is then to include other design criteria (such as control) to further rank these more profitable flowsheets. We (Barton ei al., 1986) have presented this for a three-cell flotation circuit synthesis in which simple measures of controllability (Perkins and Wong, 1985; Lau et al., 1985) were used to distinguish between the more profitable flowsheets. Further criteria we have considered include process flexibility to uncertainties in cost or physical parameters.
CURRENT
(I’ INTERMEDIATE FLOWSHEET
stream 9 unit j
from
I
Remove unit j -_
;
F&$Q_A
SUtf&BJ,f=
RECYCLE
DESTINATION
* Exclude system outputs. * Exclude unit outputs assigned to feed units. * Exclude structurat!y unacceptable recycles. within the recycle range of * Find stream p beginning with the earliest unit stream s synthesized (concentrate before tailing).
RECYCl& DEClStQf$
-
ReCyCl@ stream 5% to Stream q , unit feed EloseSt in
system wtputs. begrnning with that slream which least violates product specifications, until a structurally acceptable is obtainerl. --
3
Fig. 5. Ordered heuristics for flowsheet evolution. 1212
Heuristic Table
I. Data
for
evolutionary
four
Problem name
Flotation
rate
constants
XF7.5
100,000 20 6000
(h-‘)
1213
flotation circuit svnthesis Droblems
XFZO
Feed Solid flowrate (lb h-‘) Valuable grade (O/o) Volumetric Rowrate (ft’ h-‘)
synthesis
100,000 7.5 6000
IO, 60
10, 60
DIFFSEP 100,000 20 6000 IO, 30
HIGHCAP
100,000 20 6000 10, 60
Cost data Value
of
stream,
k, ($ lb
‘)
0.2
e changes,
k, (% h-‘)
Product specifications Minimum grade of coricentrate A’, (%) Maximum grade of tailings A’, (%) Density of minerals = 187.5 Ib ft-’ Liquid/solid ratio in concentrate 0.04,
4.2.
Arbitrary
0.2
1.0
Capital 0
elements
in the synthesis
after
water
1.0
0.2
I.0
0.2
1.0
10.0 0.7
10.0 0.7
10.0 0.7
50.0 0.7
55 2
55 2
55 6
55 10
addition
method
4.2. I. Recycle bandwidth. The arbitrary parameter 6, here called the recycle bandwidth, controls the recycle decisions. As 6+0 acyclic flowsheets would be produced, and as 6 --P 03 all streams other than the “richest” or “poorest” will be recycled. These are recycled to the unit whose feed is closest in concentration, that is, the unit immediately up-stream. Countercurrent circuits are then produced, and that the method synthesized countercurrent circuits as the limiting case is a useful check of its validity. However, as Fig. 7 shows, flowsheets more profitable than countercurrent can be and were synthesized for intermediate values of 6. These results justify our more general approach. In particular, we note two types of substructure which deviate from the strict countercurrent, and are commonly used in practice. One is a “cleanerscavenger” which accepts the tailing of a cleaning unit and returns its concentrate to that cleaning unit. The tailing of the cleaner-scavenger is then recycled to a unit downstream of the cleaning unit. An example is unit 6 in flowsheet XF20/A/2f in Fig. 7. The other is the “two-cell cleaner bank” which is a two-cell bank performing a cleaning operation: the first cell receives the incoming feed and passes its tailing as feed to the second cell. The tailing of the second cell is the tailing from the bank whilst the concentrates of both cells are combined. Units 3 and 5 in flowsheets XF20/B/2f are an example. However, our heuristically-based approach cannot guarantee to generate all derivatives from countercurrent which may be of eventual interest. 4.2.2. Ordering rules. The difference between the two heuristic ordering rules A and B (both examine first the earliest synthesized unit with unassigned outputs) is that rule A assigns the concentrate first and rule B, the tailing. This difference between the two explains why rule B tended to synthesize two-cell cleaner banks, and rule A, cleaner-scavengers, when synthesis deviated from the countercurrent circuit. Here, a stream, otherwise recycled in countercurrent flow, was sent to a new unit. The tailing of the new
0.06
ft’ lb-‘.
unit would be recycled first if rule B was used, giving the two-cell cleaner. By contrast, rule A would recycle the concentrate of the new unit first, leading to the cleaner-scavenger. This observation and others show the types of flowsheets synthesized by the method are dependent on the heuristic ordering rule used, and although rule B was responsible for synthesizing the more profitable flowsheets, the results do not show that rule B is necessarily better than rule A. As it is not obvious what is the optimal order in which the streams should be assigned during evolution, this dependence of the method on the ordering rule may represent a weakness. A search over the possible types of ordering rules would then be required to ensure that the “best” flowsheet is synthesized. This approach could be exploited to obtain a wide set of “good” flowsheets, or to support empirically that one flowsheet is really “best’‘-that is, represents a global rather than a local optimum. 4.3. Premature
termination
The heuristics used here for flowsheet evolution may lead to premature termination. This is because new units introduced to further process the “richest” or “poorest” streams are added as feedforward units (i.e. outputs are initially not assigned). If optimization does not set the units to zero capacity, destinations will then be assigned to the outputs of these units. Otherwise, these “unprofitable” units are removed, and their feed streams (if meeting product specifications) assigned as system outputs. When all streams have been assigned it then appears no longer profitable to further build up the flowsheet by adding more units. However, it may be profitable to further process the system outputs in new units, if some of the outputs of these new units are recycled. The difficulty with this is that optimization will set the capacity of any feedforward unit to zero at this point in synthesis; so, there is no information upon which to decide as to where the outputs of these new units may be “best” recycled.
stream examined
-*
a
system
currently
output *
NEWCELL
NEWCELL
NEWCELL
NEWCELL
2.1
+g-g-c 2
_
-
REMOVE
REMOVE
REMOVE
2ze+*
REMOVE :.e-
2.5
a.3
. 4
RECYCLE
I-_
NEWCELL
REMOVE
Fig.
6.
Example
of stage by stage synthesis. 1214
*
P = 2697
= 0.1343+0.1599
PROBLEM
XF20
Fig. 7
(Base
Case)
B= 0.1217 +0.1592
P I 2707 B - 0.2988-he+
W.-K.
1216
CHAN and
PROBLEM Orderina
Rule
R. G. H. PRINCE
DIFFSEP
A
Orderina
Rule
F1
-Wr
3~i:7i 2-:::.,3
iz3. L2 P =
1935
P =
1971
B = 0.3478
-mm
PROBLEM Ordema
Rule
HIGHCAP Orderina
A
+
0.1929
Fig.
Rule
E$
0.1549
7-Conrinued.
Heuristic evolutionary
PROBLEM
6 B =
0.1893
-b
synthesis
1217
XF7.5
0.2635
=
0.3366
+
0.4567
P = 1041 6,
s B =
0.3083
+
=
0.2645
bO.3078
0.4567
+
ce
0.5415
P = 1042 6 B =
Notes
: P 6,
= Profit, = Recycle Fig.
0.5415
-+
-
Bandwidth. 7. Summary
of all flotation
To avoid the combinatorial problem of determining where these outputs may be optimally recycled, we have chosen arbitrarily to recycle to the unit immediately preceding, so that a countercurrent unit is added; and we have limited this to the addition of one countercurrent unit to each system output {when it was no longer profitable to add feedforward units). This avoids the combinatorial problem of adding the optimal number of countercurrent units to each output. The choice to add only countercurrent units is also arbitrary, and other recycle destinations for the new units should be tried. One might, however, argue that when it was no longer profitable to add feedforward units, there would be little room for improvement in the profit function by adding other types of new units. Consequently, extension of the synthesis program to add more units, and recycling their outputs to various trial-and-error locations so as to avoid premature termination, was deferred, pending the results of the numerical studies discussed here.
circuits synthesized
The present deficiency of the method is illustrated by the flowsheets DIFFSEP/A/lf and DIFFSEP/ Bjlf. Adding one more countercurrent unit to the final tailing, and removing the last cleaner (unit 9) in DIFFSEP/A/lf would lead to DIFFSEP/B/lf, resulting in an increase of 2.3% profit. As in any specific case, the different synthesis routes taken by variations on the arbitrary elements of the decision logic may lead to sufficient “good” solutions which have not been terminated prematurely, a more thorough combinatorial modification has not been implemented at the time of writing.
4.4. Model
implications
Flotation is here modelled as a first-order rate process, and the flotation cell as a continuous stirred tank reactor (CSTR). We have the well-known result that for a first-order irreversible reaction, carried out in a series of N CSTRs with identical residence time r,
W.-K.
1218 the fraction of unconverted reactant the Nth CSTR, is given by:
1 - X,,
CHAN
and R. G. H. PRINCE
leaving
1 -“=[&J which
can
also
be written
l-X,+=
as: .V
I-+& [
= [l -
RI-‘.
I
R is then equivalent to the fraction of a solid species floated to the concentrate in a flotation cell and (1 -~ R),’ the fraction of feed solid appearing in the final tailing from a bank of Ncells. Hence, several series smaller cells in may therefore be more profitable than a single large cell (just as several smaller CSTRs may require less reaction volume than a single large CSTR) if the separation is difficult or the cost of capital is high. This may be seen reflected in the presence of, fol- example, two-cell b an k.s in fiowsheets such as DIFFSEP/H/4f. On the other hand, the simple form of model here used, especially the parameters being held constant over all units of a fowsheet. limits the realism of the simulation and the correspondence with practice. Further work here will address the issue.
5.
CONCLUSIONS
Results obtained for four flotation problems show that “good” flowsheets can be synthesized, by using the heuristics for flowsheet evolution, and intermediate stage optimization. which allowed a monotonically increasing path of a profit function to be followed. Countercurrent circuits were synthesized as a limiting case, but flowsheets were found more profitable than these. Changes in the problem data were met with appropriately modified synthesized flowsheets. Some deviations found from strict countercurrent flow. such as cells in series and cleaner-scavengers, are known in flotation practice, but correspondence of our results with practice is limited by the simple model used. Of the arbitrary elements in our synthesis decisions, the recycle parameter itself could be varied over its full range. Others were here varied only in a limited way, to show how “good” flowsheets-rather than a “best”-would be generated. These might then be further examined by applying other criteria, such as controllability. However, while premature termination of some synthesis paths was observed, other paths would still yield “good” solutions. Further development of the synthesis program to add more units, recycling their outputs to various trial-and-error locations, may be a means to avoid this premature termination. Although applied here to flotation, the problem statement. the heuristics and the methodology described. apply to general binary separation synthesis
with non-sharp separators. The work can then be applied to separation technologies other than flotation. Components of the method such as the recycle range function and the ordering rule can be adapted heuristicaIly to the type of separation problem concerned. With additional heuristics. say to order the sequence of separation. it may be possible to extend the work here to non-sharp mullicomponent separatton. Acknowledgemurrrs -This work derives directly from Ph.D. research undertaken some years ago by Dr G. Fenton (1973) but not published at the time owing tc> the limitations in the then available mathenwtical tools. W.K.C. acknowledges the financiai support <>fCommonwealth Post Graduate Research and of Chemical Engincering Foundation Awards
NOMENCLATCRE
k,.
G = Flowrate
of gangue
k, = Cost constants
mineral
K, = Flotation
rate constant of species i Number of umts in current flowsheet Profit of flowsheet Partial profit 01’unit _j Pulp volume of cell j Flowrate of water Valuable mineral concentration (dry) Valuable mineral concentration of final concentrate and tailinp. respectively X, 7 Valuable mineral concentration of external feed A’,.: Minimum va!uahle mineral concentration in final concentrate
n P P, “; W x .x~, s,
= = = = = = =
% = Flowrate of valuable mineral co, 0 = Cost exponents 6 = Recycle range funcunn. recycle
bandwidth
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