design optimisation study of solvent extraction: chemical

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


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


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


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.


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).


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.


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.


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.


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.


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.


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.


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.


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.


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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