www.elsevier.com/locate/hydromet
Hydrometallurgy 74 (2004) 131–147
Design optimisation study of solvent extraction: chemical reaction,
mass transfer and mixer–settler hydrodynamics
Gilberto A. Pintoa, Fernando O. Duraob, Antonio M.A. Fiuzac,*,Margarida M.B.L. Guimaraesa, C.M. Novais Madureirac
aDepartment of Engineering Quımica, ISEP, IPP, R. S. Tome, 4100-Porto, PortugalbDepartment Engineering Minas, IST, UTL, Av. Duque de Avila, 1100-Lisbon, PortugalcDepartment Engineering Minas, FEUP, R. Dr. Roberto Frias, 4100-Porto, Portugal
Received 10 August 2003; received in revised form 13 February 2004; accepted 16 February 2004
Abstract
It is a well-known fact that a typical engineering design problem usually deals with more than one design criterion. If each
design criterion is stated as an objective function to be optimised, then the engineering design problem becomes a multicriterion
optimisation problem, requiring the simultaneous optimisation of more than one objective function.
In this paper, it is shown how the design of solvent extraction flow-sheets can be stated as a multicriterion optimisation
problem, using the positive weighted sum approach. This is used not only to obtain parametric optimisation (i.e., the best
operating conditions: agitation speed, residence time and phase flow ratio) but also to help in structural optimisation (i.e., to
synthesise the best process flow-sheet: number of stages, flow structure and phase recycle ratio). We demonstrate this over a
case study, namely, the selective separation of two chemically akin and hard to separate metals, zinc and cadmium, commonly
found together in the leaching liquor of complex ores.
With this case study, it is shown that the design solutions are richer and more wide-ranging when put together from the
vantage point of multicriterion optimization, whereas they become narrow-minded and/or biased if the starting point is a single
criterion point of view. Three other conclusions of less general validity were also obtained: (i) the opposite effects of feed phase
flow-rates on recovery and purity; (ii) the high sensitivity of short optimum residence times to variations in agitation speed; (iii)
the ability of counter-flow associations of a variable number of mixer–settler units to accommodate changes in metal purity and
overall recovery in response to drivers in market prices and environmental policies.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Solvent extraction; Optimization; Multiple criteria; Mixer– settler; Design
1. Introduction be defined in immediately functional terms and
In any design situation, two kinds of optimisa-
tion objectives should be considered: those that may
0304-386X/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.hydromet.2004.02.002
* Corresponding author.
E-mail addresses: [email protected] (G.A. Pinto),
[email protected] (A.M.A. Fiuza), [email protected]
(C.M.N. Madureira).
those that, by lack of a consistently solid definition,
are mediate in character. While the former usually
become entirely clear as soon as the original prob-
lem is defined, the latter (because they depend on
previous decisions about the detailed physical con-
tents of the object-in-project) can only become clear
when all relevant information about the system’s
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147132
behaviour is available. This implies an inescapably
recursive nature for the design problem. Most
environmental objectives of engineering design are
examples of this class. In turn, this means, of
course, that computer simulation, performed on
any available minimally realistic models of the
behaviour of the physical components of each
contemplated system, is an invaluable help to the
engineering design process.
When judiciously defined (e.g., a reaction’s yield,
a system’s performance or cost) the criteria that put
figures on the design objectives are usually
expressed as functionals of the behaviour of the
object-in-project. Although the nature of these func-
tionals is usually obvious in the case of the imme-
diate objectives, the same is not true for the mediate
objectives.
Analysis of the history of technological design
shows that a mere specification of intended behav-
iour, however thorough, is not enough to define
unique values for the parameters, or design variables,
of the physical components of the object-in-project;
this is the ultimate origin of design freedom (Madur-
eira, 1993), an important feature of engineering
design which allows for creativity and generates
diversity. However, this freedom may often be se-
verely curtailed, or even inhibited, by hasty, unwit-
ting, commitment to some apparently obvious
structural solution. In this case, the ‘‘optimum’’
solution to be found is merely parametrical, not all-
inclusive. The same author has shown that a design
is actually optimal only when it contemplates both
structural and parametric criteria and has discussed
the problem of the eventual separability of the two
types of criteria.
On the other hand, the same author has insisted on
the importance of the recently acknowledged fact that
imposing multiple conflicting objectives to a design
problem leads to the opening of new freedom spaces.
This means
(i) that the single objective optimisation solution to
a design problem is not an optimum solution but
a merely satisfactory solution in the technical
sense (Steuer, 1986), and
(ii) that the exploration of the total space of
satisfactory solutions is the ultimate kind of
sensitivity analysis, since it clearly exhibits what
may be won or lost in the different design
alternatives.
Costa (1988) had previously acknowledged, ‘‘Re-
ality is inherently multidimensional and its percep-
tion is multidisciplinary. So much so, that, even
when pertinent, single objective optimization is al-
ways preceded by some kind of implicit multiple
choice’’.
One possible way of solving a multi-objective
optimization problem is to abide by the protocol
(Steuer, ibid.):
(a) define the decision-promoting functional, or
decision-maker’s utility functional, U;
(b) solve the problem max{U(z1,z2,. . .zk} subject to
zi =fi(x), with 1 i k, xaS, where S is the
feasible region of the design variables space
and the zi are the design criteria.
Often, however, it is not possible to write down a
priori an adequate mathematical representation of the
decision-maker’s utility functional, U, so that the only
alternative is to use the information implicit in design
criteria: max{zi = fi(x)}. One solution technique for
dealing with such difficult decision situations is the
weighted sum approach, where U is represented as a
positively weighted convex combination of the design
criteria, i.e., U =Siaizi, where ai (with 0 ai 1 and
Siai= 1) is the relative weight (or importance) of the
design criterion zi.
A point x*aS is said to be Pareto optimal if and
only if there is no xa S such that zi(x)z zi(x*) for all
i: i = 1,2. . .,k, with at least one strict inequality (Ste-
uer, op. cit.). It can be proved that the maximizer,
x*(A) of this positively weighted sum function is a
Pareto optimal solution, also variously known as a
satisfactory, efficient, non-dominated or non-inferior
solution. It can also be shown that if x* is a Pareto
optimal point, then there exists a vector a* such that x*
is themaximizer of the functionalU in the feasible set S.
In the simplest (k = 2) case, the utility functional
isU = az1+(1� a)z2(0 a 1)andtheoptimalvalueismax{U(z1,z2)}.
max{U(z1,z2)}. The set {z1*(a), z2*(a)} resulting from
assigning different positive weights a to criterion z1describes, in the criterion space, the Pareto set or
Pareto curve of the different possible preferences of
the decision-maker.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 133
2. Problem identification
Solvent extraction in hydrometallurgy is clearly a
complex multi-component extraction problem: the
feed stream coming from upstream processes contains
an assortment of metals the span of which depends on
the mineralogical character of the processed ore.
The extractant used to capture certain chemical
species, usually chosen to maximize the recovery of
these species, will never exhibit perfect selectivity.
The specialised bibliography is thus full of works
designed to elucidate the operating conditions which
lead to the highest selectivity of a given extractant in a
given problem-mixture and/or comparative studies of
the efficiency of several functional groups in given
extraction processes (e.g. Rice and Smith, 1975;
Chapman, 1987; Micker et al., 1996; Goto et al.,
1996; Gloe et al., 1996; Tsurubou et al., 1996). Such
great R&D effort is, by itself, well revealing of the
economic importance of the problem and of the
phenomenological complexity of the subject. Indeed,
the problem is even more complex because optimal
selectivity seldom occurs at economically acceptable
cost and a priori commitment to some apparently cost-
effective extractant is one of the structural options that
may severely bias the final solution.
For the solvent extraction systems used in hydro-
metallurgical and environmental applications, the ratio
of aqueous to organic flow-rates is greater than one
for obvious economic reasons. If mixing is performed
with a phase ratio similar to the feed flow-rate ratio,
the continuous phase is aqueous and the dispersed
phase is organic. However, as it happens in many
extraction systems, if the organic is internally recycled
from each settler to the corresponding mixer, the
volume or phase ratio can be changed so that the
organic phase is prevalent and becomes continuous.
From the chemical point of view, there are no partic-
ular advantages or disadvantages. Physically, howev-
er, the system becomes more favourable: the phase
separation is easier with a lower height of dispersion,
the formation of emulsions is reduced and there is less
solvent loss by entrainment in the raffinate.
For these reasons, recycling of the organic from
settler to mixer within each stage is a common
procedure, leading to aqueous to organic phase ratios
in the mixer of 1:1 to 1:2. Obviously, this operating
procedure originates larger feed-rates to be processed
in the mixer and separated in the settler, leading in
turn to lesser residence times in both. This overall
balance is a delicate optimisation problem because the
phenomenology of phase inversion still is a poorly
known subject. Experimental studies have consistent-
ly shown that in stirred vessels the phase that is
present at less than about 0.3 volume fraction will
usually be the dispersed phase (Chapman and Hol-
land, 1966). This leaves a large region between the 0.3
and 3.3 volume fraction extremes where either phase
may be dispersed, called the ambivalent region
(Selker and Sleicher, 1965; Pacek et al., 1944). This,
of course, is the region where industrial separations
usually operate. Within this region, the precise phase
ratio leading to phase inversion depends on a number
of features, namely, the physical properties of the
dispersed phase (Selker and Sleicher, 1965), the
hydrodynamics of the mixer (Luhning and Sawistow-
ski, 1971), the surface area (Kumar et al., 1991), the
presence of solutes, additives and impurities (Brooks
and Richmond, 1991), and even the presence of mass
transfer itself (Godfrey and Slater, 1994, chap. 12).
Clark and Sawistowski (1978) have shown that a
significant reduction in the width of the ambivalent
region may occur in the presence of mass transfer.
Thus, sudden unpredictable phase inversion may
occur within the ambivalent region with possible
catastrophic consequences for the throughput and
efficiency of the process.
In hydrometallurgical industries, the need for prod-
ucts with high degree of purity thus calls for maxi-
mum selectivity in the mass transfer of the different
components not by means of the most selective
extractants, but by taking full advantage of slight
differences in the extraction kinetics of each one. This
may be achieved by means of judiciously chosen
operating conditions: maximum selectivity is obtained
at very short inter-phase contact times relative to the
kinetics of mass transfer (which requires accurately
defined residence times) but implies low recoveries.
Given both the scarcity and, consequently, the high
value of the substances to be extracted and the
objectionable character of their presence in the waste
streams, these short-lived contacts must be multiplied,
which may lead to process flow-sheets of great
topological and operational complexity.
Under these constraints, the mixer–settler combi-
nation is the usually favoured type of equipment in
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147134
solvent extraction hydrometallurgy because the me-
chanically agitated mixer unit affords high transfer
surface area under closely controllable residence times
and the settling unit swiftly quenches the mass trans-
fer (other advantages will shortly become evident).
Under these assumptions, the inter-phase mass trans-
fer occurs almost exclusively in the mixing vessel and
the extraction level obtained is essentially dependent
on three design variables, residence time, agitation
rate and phase volume ratio. Chapman (1987),
Thorsen (1991), Rydberg and Sekine (1992), Skelland
(1992), Cox (1992) and Godfrey and Slater (1994),
among others, have studied some of the challenging
problems in the use of this kind of equipment.
In such a separation process, two design criteria
are typically used to assess the process performance:
the weight recovery, or amount of component
extracted, and the purity of the component recov-
ered. Both these design criteria are functions of three
design variables, the mean residence time, s, in the
mixer, the dispersion phase hold-up, u, and the
agitation speed, N.
Defining the weight recovery of a given species as
Rjðs;u;NÞ ¼ Mass of the j�species in the extract
Mass of the j�species in the feed;
j ¼ 1; 2; . . . ; nspecies ð1Þ
and the purity as
Pjðs;u;NÞ¼ Mass of the j�species in the extractP
ðMasses of all species in the extractÞ ;
j ¼ 1; 2; . . . ; nspecies ð2Þ
we shall easily infer that an increase of the recovery of
the j-species will generally not entail an increase of its
purity because any factor which increases the recov-
ery of one particular species will usually increase to a
lesser but significant degree that of other similar
species. If two species have comparatively close
extraction kinetics, there will be a residence time
above which any attempt at increasing Rj will result
in a decrease of Pj, i.e., recovery and purity become
conflicting objectives.
In hydrometallurgical work, the extraction kinetics
of metals may sometimes be so fast that the range of
mean residence times in which the two criteria do not
conflict become so narrow that
(i) obtaining and steadily maintaining the resultant
operating conditions may become a tricky
proposition, and
(ii) the very low recovery obtained may call for a
disproportionate number of unit separation
stages.
Finding the operating conditions that maximise the
two conflicting objectives involves maximising an
arbitrary linear combination of them and the result is
a family of Pareto optimal solutions corresponding to
non-dominated design criterion vectors. At the design
stage, the optimisation is obviously to be performed
upon a mathematical model of the unit operation,
adequately developed, validated and implemented as
a computer algorithm.
Model solving and simulation of mixers as thor-
oughly agitated homogeneous vessels have been stud-
ied in depth by our group; more complex spatial
material and flow structures may easily be simulated
as combinations of these simple prototypes, as witness
the successful simulation of Kuhni-type columns by
Regueiras (1998); we have created and implemented
software which may be applied to control of equip-
ment, perhaps via neural networks to tune parameter
values, namely:
� an exceptionally fast algorithm for the steady-state
stirred vessel, using the method of moments for the
factored trivariate size-age-concentration drop dis-
tribution, which accommodates the sophisticated
Coulaloglou and Tavlarides (1977) drop-interaction
model aswell as itsmanyvariations, e.g., Tsouris and
Tavalarides (1994), Sovova (1981) and Guimaraes,
(1989). This is the algorithm used in this paper.� a computationally efficient algorithm for the
transient state of a non-factored distribution, using
the technique of integration over time of the
relevant population balance equations for a spa-
tially homogeneous domain (Ribeiro, 1995) allow-
ing the prediction of dynamic responses to changes
in operating conditions;� an accelerator for the computation of mass-transfer
effects (Regueiras et al., 1996) of particular interest
for the simulation of complex flow structures
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 135
requiring iteration for the computation of the
steady-state.
However, it must be stressed that this quite satis-
factory state of affairs as regards model solving does
not match either the present very sketchy understand-
ing of the hydrodynamic phenomena that the models
propose to describe, or the need for case-by-case
parameter tuning.
In contrast, our ability to model and simulate the
behaviour of a settler as a function of its operational
variables is at present in a very unsatisfactory state.
This is clearly demonstrated by the multitude of
models ably and exhaustively reviewed by Rommel
et al. (1992) and Hartland and Jeelani (1994). These
models are usually appropriate for specific physical
configurations of the equipment alone and, in most
cases, for the laboratory or pilot-plant scale only.
Ruiz (1985) and later Padilla et al. (1996) have
developed a model describing the hydrodynamics of
the settling chamber. Pinto et al. (2001) have created a
new algorithm coupling the two models and can
describe the combined steady-state operation of the
mixing and settling chambers. This algorithm was later
adequately modified and appended as a new unit
module for SIMUL, a modular sequential process
flow-sheeting program developed by Durao (1991),
thus creating a very convenient, fast and accurate
design and simulation tool for mixer–settler units
arrangements in any kind of topology and flow scheme.
The results to be presented below were obtained by
means of this computational set-up working on the
problem to be described in Section 3.
3. Case study
Zinc and cadmium are twometals that often occur in
association, both being economically valuable and
difficult to separate. See, for instance, Rice and Smith
(1975) about their extraction in 5 mM sulphuric acid
environment with 1 M naphthenic acid as extractant.
The pH values corresponding to 50% extraction (pH0.5)
of zinc and cadmium are, respectively, 5.36 and 5.48.
This means that, (i) even under optimal operating
conditions, the selectivity always remains very low;
(ii) under such conditions, impracticably accurate con-
trol of the pH becomes critical and (iii) the costs of a
suitably non-interfering neutralising agent easily
becomes uneconomical since feed streams are usually
very low in pH. Thorsen (1991) presents a few practical
cases of recovery of these metals. This system has been
chosen to illustrate qualitatively—in a particular case,
so not all results to be obtained may be generalised in a
quantitative way—how the proposed methodology is
to be implemented and the kind of benefits that may be
expected from it. This means that the results to be
exhibited, although informative, are not to be relied
upon in other specific situations and that experimental
verifications and parameter tuning of the model is
required in every application. This is mainly because
surfactant impurities may significantly change coales-
cence and breakage rates. However, it should be
noticed that the simulations used in the case study
described below were performed on a Coulaloglou-
Tavlarides model of the hydrodynamics of a stirred
vessel, the parameters of which have been tuned to a
pilot-scale mixer–settler unit purpose-built for the
Laboratory of Liquid–Liquid Systems at ISEP-IPP,
as described in Pinto (2004).
Anyhow, absolute realism and quantitative accuracy
is not in the scope of this paper, which unpretentiously
aims only at demonstrating the kind of results to be
expected from the proposed multi-objective optimisa-
tion technique with a view to (a) the conceptual design
of a process or (b) the on-line optimization of a process,
under the supervision of a neural network.
As a typical extractant, di-2-ethylhexylphosphoric
acid (D2EHPA) was considered in a 1 mM concen-
tration in kerosene and an aqueous phase 0.1 M in
KNO3. This is the system used by Micker et al. (1999)
as a standard for comparing different extractants. The
composition of the feed stream of the mixer–settler
unit was defined as
(i) an aqueous phase containing 20 g/l of both zinc
and cadmium metal ions and
(ii) an organic phase free from these metals.
Considering the pH value fixed at 5.0 and the equilib-
rium constants of reaction (4) as log(Kex(Zn)) =� 3.39
and log(Kex(Cd)) =� 4.30 (id., ibid.), the well-known
Eq. (4) allows the computation of the distribution
coefficients: 4.0 for Zn and 0.5 for Cd.
logD ¼ nðpHÞ þ logðKexÞ þ nðlogðCHLÞÞ ð3ÞMnþ
ðaqÞ þ nHLðorgÞ ¼ MLnðorgÞ þ nHþðaqÞ ð4Þ
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147136
The diffusivities of Zn2 + and Cd2 + in water
(7.02� 10� 10 and 6.02� 10� 10 m2 s� 1, respective-
ly) were used as approximations for those in the
aqueous phase. In the organic phase, given the
obscurity of the chemical composition of kerosene,
we took it to be equivalent to dodecane, as Ally et
al. (1996) have done. From Wilke and Chang’s
(1955) model, we estimated the diffusivities of the
complexes as Dcomplex Zn =Dcomplex Cd = 2.7� 10� 10
m2 s� 1.
We believe these approximations to entail only
moderate quantitative changes and keep qualitatively
intact the competitive nature of the two extraction
kinetics.
In such a separation unit, the design variables
which control the process may be reduced to three
(residence time, s, agitation power, N, and dis-
persed phase hold-up, u, in the mixing unit); the
values of these operating variables are given as
Pinto’s (2001) unit module’s input parameters to the
SIMUL program. The behaviour of the Zn recovery
and purity may then be studied as output of the
program for different combinations of the values of
the design variables.
3.1. A single separation unit
For three different values of the agitation speed
(100, 150 and 200 rpm) and five different values of
the dispersed phased hold-up (0.10, 0.20, 0.25, 0.30
and 0.40), the residence time was varied in small
equal increments from a minimum of 1 second up to
the residence time necessary to obtain mass transfer
equilibrium. This condition is computationally de-
tected as the point (which theoretically is at infinite
contact time) where extraction efficiency—defined as
the ratio between mass transferred to the dispersed
phase and the maximum mass transferable, EF( j)=
(mj�m0j)/(meq�m0j), mj being the mass actually
transferred, m0j the mass originally in the lean phase
and meq the mass transferred at the equilibrium has a
computational (i.e., within machine precision) unit
value.
Under these conditions, drop sizes are small and
drop lifetimes are short; therefore, molecular diffusion
may be considered the limiting kinetic step in a rigid
drop mass transfer model. However, the effects of the
chemical reaction between metal ions and the extrac-
tant must be computed because they determine the
concentration on the drop interface.
As an example, Figs. 1 and 2 show the behaviour
of Zn and Cd recoveries, and Zn purity as functions of
mean residence time for a fixed agitation speed of 200
rpm and different hold-ups. Fig. 3 shows the space
and the design criteria values.
It is obvious from these figures that both Zn and
Cd recoveries increase with increasing dispersed
phase hold-up, u, in the mixer; this may be under-
stood from Figs. 4 and 5, which show that increasing
hold-up results in decreasing average drop volume
and consequently increasing transfer surface. At the
same time, increasing number of drops means increas-
ing drop interactions, which promotes homogeneity of
the concentration inside the drops and increases the
global concentration gradient, thus increasing the
mass transfer rate (Fig. 6).
The change of purity of species A, PA=P (CA,
CB) =CA/(CA +CB), caused by a change of the resi-
dence time, may be described by
dPA ¼ BP
BCA
dCA þ BP
BCB
dCB ¼ CB
ðCA þ CBÞ2dCA
� CA
ðCA þ CBÞ2dCB ¼ CBdCA � CAdCB
ðCA þ CBÞ2
which means that PA will increase, while
CBdCA � CAdCB > 0Z CBdCA > CAdCBZdCA
CA
>dCB
CBor
dðlnðCAÞÞ > dðlnðCBÞÞ
This condition will only occur for short mean
residence times; whenever the two logarithmic differ-
entials become equal, the purity of the faster compo-
nent will have reached its highest value. This is an
important result, because it means that extraction
should continue while the purity of the extracted
increments is higher than the purity of the feed. The
corresponding value of the residence time would
represent the optimal operating conditions if the
optimization criterion were purity alone. Since, how-
ever, recovery must also be taken into account, the
overall optimal conditions will correspond to longer
Fig. 1. Change of Zn recovery (light lines) and purity (heavy lines) with residence time for different hold-ups at 200 rpm.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 137
residence times, the range of residence times leading
to the Pareto optimal solutions in Fig. 3.
Residence time, however, is not the only design
variable that affects the competition between extrac-
tion kinetics of the two species: the dispersed phase
Fig. 2. Change of Cd recovery (light lines) and purity (heavy lin
hold-up is another one that contributes to the quanti-
tative complexity of the process. Study of Figs. 1 and
2 has already shown that an increase in the dispersed
phase hold-up does not have a linear effect on the
amount extracted and does not have the same bearing
es) with residence time for different hold-ups at 200 rpm.
Fig. 3. Non-dominated values of the two design criteria for different hold-ups at 200 rpm, with residence time as a parameter (the weight
recovery RZn of Zn increases in the same direction as the residence time).
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147138
on the two competing species: rather, it has a moder-
ating effect on the advantage of the faster one.
Overall, it has a negative effect on the purity of the
favoured species, as illustrated in Fig. 1. The
Fig. 4. Change in the number of drops per unit volume w
conflicting objectives shown on Fig. 3 suggest a
serious difficulty in choosing the most favourable
combination of residence time and hold-up. A com-
bination favouring Zn recovery must have high hold-
ith residence-time for different hold-ups at 200 rpm.
Fig. 5. Change in the average drop size with residence time for different hold-ups at 200 rpm.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 139
up and residence time. A combination giving prefer-
ence to purity of the extract will have low values of
both Parameters (Fig. 7).
Let us now study the influence of agitation speed
on system performance: Figs. 8 and 9 show the
expected increase of recovery of both species with
increasing agitation speed, due to increased drop
interaction. In fact, for the same amount of dispersed
phase in the mixer, an increase of the dissipated
Fig. 6. Change of the mass transfer rate of Zn and Cd with
energy will bring about a larger number of drops
and a consequent increase in drop interaction frequen-
cy, which allows the equilibrium conditions to be
reached in shorter residence times. Under low resi-
dence times, the fast extracting species will feel this
effect in a more marked way: its kinetics will improve
more and purity will be favoured. If the residence time
increases, the slower species will tend to recuperate its
lag, now under better mass transfer conditions, thus
residence time for two hold-up values at 200 rpm.
Fig. 7. Change of the extraction efficiency of Zn and Cd with residence time at 200 rpm.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147140
promoting a decrease in the purity of the preferred
species. Purity may then reach lower values than it
would under lower agitation speeds. It should be
noticed that this effect is qualitatively different from
that of hold-up change: the purity vs. time curves
never intersected in the latter case (compare Figs. 3
and 11).
Fig. 8. Change of the Zn recovery with residence tim
Under changes in this parameter, the decision
function behaves as shown in Fig. 11: the effect of
the agitation speed is lost at longer residence times;
for short residence times, the effect is considerable.
Contrary to the hold-up effect, there is an obvious
gain when operating at high agitation speed, and this
gain is higher at shorter residence times.
e for different agitation speeds at 0.25 hold-up.
Fig. 9. Change of the Cd recovery with residence time for different agitation speeds at 0.25 hold-up.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 141
It must be mentioned that these inferences are quite
different from those to be obtained from a single
criterion optimization point of view: if high Zn
recovery alone was to be considered (with the atten-
dant low purity considered as an unavoidable catch),
Fig. 10. Change of Zn purity with residence time f
high hold-up, long residence time and low agitation
speed would be the most advantageous conditions
(Fig 10). Conversely, if purity were the objective,
low hold-up, short residence time and high agitation
speed would be preferred. Consideration of a merely
or different agitation speeds at 0.25 hold-up.
Fig. 11. Pareto curves in the criteria space for different agitation speeds at 0.25 hold-up.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147142
satisfactory, instead of optimal solution—based on
both criteria—allows an increased design freedom in
addition to opening up new opportunity for process
stability.
3.2. Phase recycling in a single separation unit
In order to determine the effects of flow structure
on both design criteria, recycling of both continuous
and dispersed phase was simulated as shown in Table
1 and Figs. 12–14. The values of the aggregate
objective function are shown in Fig. 15, where the
dotted parts of the curves refer to the residence time
stretches where the partial objectives do not conflict.
The conclusions from this study may be extended
to the consequences of imperfect flow homogeneity
within the vessel itself.
One inference should be immediately drawn from
Fig. 15: differences in the Pareto curves for the
Table 1
Recycle ratio and dispersed phase hold-up for four simulated
experiments
Continuous (aqueous)
phase recycle
Dispersed (organic)
phase recycle
Recycle ratio (R) 0.25 0.67 0.22 0.50
Hold-up (u) 0.20 0.10 0.30 0.40
different flow configurations are wiped out as resi-
dence time increases. While this was qualitatively
predictable from the previous results, the quantitative
behaviour is quite surprising: the utility function
significantly increases when the hold-up is increased
by dispersed phase recycling, whereas it hardly
changes when the hold-up is decreased by continuous
phase recycling (Fig. 16).
The implications of this result are important for the
structural optimization of complex arrangements of
extracting units as well as for the optimization of
single units.
A careful look at the variables implicated in this
singular phenomenon will help us to capture its under-
lyingworkings andwe believe the final interpretation is
as obvious as the previous result was surprising.
Because the values of the dispersed phase hold-up
are always kept below 0.5 (in order to prevent phase
Fig. 12. Original mixer–settler unit with variable volume and 0.25
hold-up.
Fig. 14. Mixer– settler with dispersed (organic) phase recycle.
Fig. 13. Mixer– settler with continuous (aqueous) phase recycle.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 143
inversion during uncontrolled transients), variations in
the opposite direction will have quantitatively differ-
ent effects on the process. A material balance at the
point of mixture of the fresh feed with three recycled
stream will make this point clear. In the equations
below, V, Qc and Qd stand for the vessel volume, the
continuous phase flow-rate and the dispersed phase
flow-rate, respectively:
(i) when the continuous phase is recycled, the
relation between residence time and hold-up
is scontinuous phase recycle ¼ V ð1�uÞQc
Z ds=du ¼ � VQc;
whereas,
(ii) when the dispersed phase is recycled, the
relation between residence time and hold-
up is sdispersed phase recycleVuQd
Z ds=du ¼ VQd;
This means that
½ds=ducontinuous phase recycle
½ds=dudispersed phase recycle
¼ � Qc
Qd
:
and, since Qc>Qd, the absolute value of [ds/du]continuous phase recycle is always less than that of
[ds/du]dispersed phase recycle. This difference in the
residence time variation impinges upon the useful
energy injected (i.e., the energy absorbed by the
dispersed phase), as Fig. 17 shows.
Notice that a sharp distinction must be made
between nominal residence time (as computed from
the fresh feed flow-rates), which is the same for both
phases, and actual residence time (as computed from
fresh + recycled flow-rates), because recycling a single
phase increases its actual residence time and decreases
the other.
Significant variations in the useful energy injection
will have a similar effect on the average drop volume
(see Fig. 17). In both kinds of recycling, a decrease in
energy density in the dispersed phase will cause an
increase in the average drop size. In the case of
dispersed phase recycling, this is an obvious effect
of the increase of the hold-up, as it was in the case of
the single unit. In the case of continuous phase
recycling, the qualitative effect is still the same,
although much smaller.
3.3. Counter-flow association of extracting units
Let us now study the effect on the utility function
of the counter-flow association of two and three equal
sized separation units, taking the single unit as a
reference. In all three cases, the total residence time
is the same, i.e., the total volume of the mixer units is
kept constant. Fig. 18 shows the results of these
simulations.
The first conclusion to be drawn from these
results is obvious: as the number of units grows,
the residence time corresponding to the highest
purity increases. The next conclusion is related to
the fact that the utility function curves become more
blunt-nosed with increasing number of separation
stages, which means higher total variations of the
recovery than those of the purity. Finally, whereas
there is, for any total residence time, a marked
increase in the recovery, the same does not obtain
for the purity. For the shorter residence times, up to
about 1 min, its value decreases. In the next resi-
dence time interval, the behaviour of the purity is no
longer uniform: it increases on passing from one to
two units and decreases for three. Ultimately, how-
ever, the purity decreases with a growing number of
separation units.
Obviously, from the results of these simulated
experiments another degree of freedom for the de-
Fig. 15. Criteria space and Pareto curves (heavy solid lines) for different hold-up (h) values induced by phase recycling.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147144
sign emerges: there is an interval of recoveries
corresponding to an interval of purities where the
best solution is obtained with three units, an inter-
Fig. 16. Mean residence time, mass of the dispersed phase, agitation pow
phase hold-up manipulated by phase recycling (values corresponding to a
mediate one where two are to be preferred, and
another one where the best is a single unit. This is
another important conclusion, since counter-flow
er, energy density and power density as functions of the dispersed
6-l mixer volume).
Fig. 17. Average drop volume as a function of dispersed phase hold-up manipulated by phase recycling.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147 145
association of a variable number of mixer–settler
units now emerges as a flexible solution, capable of
easy, instant response to fluctuating market prices
and/or contents of the crude ore in objectionable
impurities.
Fig. 18. Criterion space and Pareto curves for one,
4. Conclusions
It has been qualitatively shown that the design
solutions for the separation of similarly behaved
metals are richer and more wide-ranging when created
two and three extracting units in counterflow.
G.A. Pinto et al. / Hydrometallurgy 74 (2004) 131–147146
from the vantage point of multicriterion optimization,
whereas they become narrow-minded and lopsided if
the starting point is the single criterion point of view.
Another bonus of multicriterion optimization is the
fact that it incorporates automatic sensitivity analysis,
which is useful for predicting and/or avoiding the
effects of (i) inevitable departures of plant implemen-
tation from design specifications and (ii) uncontrolled
fluctuations either in feed composition or in flow-rate
or in environmental and/or market circumstances.
The advantages of multicriterion optimization as a
tool for technological design are, however, not limited
to parametric optimization. They extend to and may
become even more decisive for structural optimization,
as shown in the last section. The following findings are
cases in point: (a) the opposite effects of dispersed
phase hold-up changes on recovery and purity; (b) the
unexpected sensitivity of short optimum residence
times to variations in agitation speed; (c) the ability
of counter-flow associations of a variable number of
mixer–settler units accommodates changes in metal
purity and overall recovery in response to drivers in
market prices and environmental policies.
Notation
Ap(x) Average projected area of drops at the active
interface, L2
Cj Mean drop solute concentration of j species,
ML� 3
D Distribution coefficient
EF( j) Extraction efficiency of the j species
Kex Equilibrium constant
mj Actual mass transferred to organic phase, M
n(v,x)dv Number density per unit volume of the
dispersion of size v to v + dv, L� 3
Pj Purity of j species in the organic phase
Qc Volumetric flow-rate of continuous phase, L3
T� 1
Qd Volumetric flow-rate of dispersed phase, L3
T� 1
Rj Weight recovery of j species
u Dispersed phase hold-up in the mixer
chamber
k(v,vV) Drop–drop coalescence frequency, T� 1
k*(v) Drop–interface coalescence frequency, T� 1
k0 Parameter of the drop–interface coalescence
frequency model, T� 1
g(x) Volume fraction of dispersed phase or
dispersed phase hold-up
g*(x) Surface fraction of the dispersed phase at the
active interface, L2
g0 Volume packing efficiency of the dispersion
entering the settler
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