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MODELING OF LIQUID LIQUID EXTRACTIONCOLUMN: A REVIEW

Swati Mohanty

Regional Research Laboratory C.S.I.R.),Bhubaneswar-751013, India

E-mail: [email protected]

ABSTRACT

Mathematical models are reviewed for different types of commonly used

extraction columns viz. pulsed sieve plate column rotating disc contactor

uhni column spray column Scheibel extractor packed columnOldshue-Rushton contactor and reciprocating plate column. In addition

numerical techniques process simulators and some estimation methods for

model parameters have also been included. The review cites over 120

references.

Keywords: mathematical modeling extraction column population balance

diffusion model solvent extraction

INTRODUCTION

Liquid-liquid extraction is the second most important separation process

after distillation in chemical industries. Its early application was in

oil-refining but it has now been extended to such varied applications as the

petrochemical industry Pharmaceuticals hydrometallurgy nuclear industry

and environmental protection. Despite years of study a design procedure

employing sound basic principles has not been developed and until today the

design of extraction column is based on pilot plant data and design experience

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Vol. 16 No. 3 2 Modeling of Liquid-Liquid Extraction Column:

A Review

and hence is far from the optimum. Polydispersity of turbulent drop swarm is

the main obstacle in designing liquid-liquid extractors.Mathematical models that adequately define the system lead to better

design of these equipments. However, adequate research has not been carriedout in this area, although attempts have been made to develop models basedon certain simplified assumptions. This is mainly due to the complexity o f thesystem where the dispersed phase constantly loses its identity due to breakage

and coalescence. Insufficient attention has been paid to the experimentaleffort required to verify the utility of the models. Much remains to be donebefore dispersed phase behaviour is adequately modelled and columnperformance can be confidently predicted. Recent books edited by Godfreyand Slater / / and Thorton 2 present detailed analysis of various types of

extraction equipment. Steiner and Hartland /3,4/ have discussed the

advantages and disadvantages o f two early models proposed by Sleicher 15 61

for liquid liquid extraction. One is the backmixing model, which assumes thatthe dispersed phase consists of uniform sized drops without breakage and

coalescence. The other is the forward mixing model, which takes intoconsideration the different drop size, drop breakage and coalescence, varyingvelocities of the drops due to varying size. Some of the stagewise and

differential models and their solution methods have also been reviewed byPratt and Baird Ð I and Steiner and Hartland /8/. Use of stochastic simulationtechniques (e.g., Monte Carlo techniques has been introduced for simulatingliquid liquid extraction, but these have been primarily applied to stirred tankcontactors and hence have no t been included in this review. The purpose o fthis paper is to present a general review of various models available for somemost widely used liquid-liquid extractors.

TYPES O MO ELS

Mathematical models for liquid-liquid extraction are categorised intothree basic types.a) Empirical: These are the simplest of all the models and are obtained by

fitting experimental data with empirical correlations which are functionsof liquid physical properties, column geometry and operating conditionsof the column. The drawback of this type of model is that this cannot be

extrapolated beyond the stipulated range of application. This type of

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Swafi Mohanty Reviews in Chemical Engineering

model has been proposed for predicting the mean drop size and hold-up in

various types of extraction columns. The hold-up, ø, or the mean dropdiameter, d, n, is given as

ö or dm =f z,,z2,.... zj 1)

where Z|, z 2, ... z„ are physical properties, column geometry and

operating conditionsb) Differential: These models are obtained by formulating differential

conservation equations for the column, both for the dispersed phase andthe continuous phase. Based on the representation of the dispersed phase,

the differential model is further classified into

i) pseudo-homogenous dispersion model Fig.la), applicable for both

single phase as well as dispersed multiphase system and are based on

the principles of physico-chemical laws. The system is represented as a

continuum with the dispersed phase treated as pseudo-homogeneous.

The general unsteady state mass balance equations for the continuous

and dispersed phase are given as

·£.= Vc +E C -L- koc C c -C c 2)a c o z c ^2 ° ° c c /

3

where Æ is measured in the direction of flow of the dispersed phase 1 5 1

ii) population balance dispersion model, applicable where the continuum

transport equation fails e.g. when a dispersed phase is discontinuous

and constantly undergoes changes and loses its identity due to

coalescence and breakage. Thus a differential drop population balance

for drops of different sizes in the dispersed phase is formulated for the

column Fig.lb). The general drop population balance equation is

written as

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Vol . 16 No. 3 2000 Modeling of Liquid-Liquid Extraction Column: Review

õ^

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Swati M ohanty Reviews in hemical Engineering

T / é — / j I E · ™ — / I D/ é r»/j\ /4\

where the birth, B d), and death, D d) of drops of size d, relates to

breakage and coalescence in the volume element of height, dZ A goodreview of population balance models applied to chemical processes is

presented by Ramakrishna 1 9 1

c) Stagewise: In this type of model, the column is described as a series of

completely mixed stages and the model equations are developed based on

the principle of mass conservation applied to each stage Fig.lc). Theseare similar in form to equations 2), 3) and 4) 161 The stages may be

real, as in agitated stagewise columns, or cascades with a large number of

hypothetical stages, as in differential extractors.

M AT H E M AT I C A L MO ELS

The first models of extraction column that were reported were simple innature. Both the phases were assumed to be in differential contact with eachother, with each phase assumed to be a continuum. The dispersed phase was

treated as pseudo-homogeneous and backflow was accounted for by the axialdispersion coefficient In the subsequent models, the dispersed phase was

represented by monodispersed or polydispersed drops without undergoingbreakage or coalescence. The complex behaviour of the dispersed phase,

consisting of swarms of droplets with a wide size distribution, varying

velocities relative to the continuous phase, and varying mass transfer rates,could be represented by the hydrodynamic and mass transfer behaviour of

some representatively-sized single drop, normally the surface-volume or

Sauter mean diameter drop. In the later models, the coalescence was

accounted for by introducing a coalescence height, which indicates the

average distance the drops actually travel before they coalesce and redisperseto maintain the original drop size distribution. The concentration of the

dispersed phase was assumed to be constant throughout a given cross-section

of the column due to sufficiently intensive coalescence and redispersion.More recently, population balance models have been developed to predictdrop-size distribution and hold-up profiles along the length of the columntaking into account drop breakage and coalescence. In addition, a number of






A Review

empirical correlations for predicting the hold-up in different types of

extractors have also been reported. A unified correlation which can predictthe hold-up in eight different types of extraction columns has been proposed

by Kumar and Hartland /l O/. A summary of d ifferent models for various types

of liquid liquid extractors is presented in Table 1.

Pulsed Sieve Plate xtraction olumn PSE)

This type of liquid liquid extractor finds application mainly in nuclear

processing industries. Most of the models reported for this type of extractorare based on population balance of drops. Models by Haverland et al l\ I/,

Dimitrova AI Khani et al I\2I and Zimmermann et al /13/ are based on thebasic population balance model given by equation 4). In the steady state

hydrodynamic model, Haverland et al /I I/ have neglected drop-drop

coalescence and have proposed a normalised Beta-distribution for the drop

size distribution which was found to fit best with the experimental data. A

correlation for break-up probability for drops of different sizes has been

proposed based on experimental data. Although the model predictions forhold up and drop size distribution agree quite well for the system

toluene-water and under operating conditions where coalescence is negligible,

it would not be applicable for systems and operating conditions where

coalescence is significant. Dimitrova A l Khani et al. I12I have made certain

assumptions for drop-drop interaction, such as that a mother drop breaks into

three equisized drops and the breakage rate is proportional to the eighth

power of the drop diameter. For coalescence, experimental values for the two

parameters, one for collision efficiency and another for coalescenceprobability, have been used but no detailed method of determination of these

two parameters has been reported. For the mass transfer model, a linear

equilibrium relationship and the mass transfer coefficient independent of the

drop diameter have been assumed. The model contains two fitting parameters,

one for the drop transport and the other for the drop breakage which limits the

applicability of the model. Zimmermann et al /13/ have applied the

population balance model to multi-component extraction. The model involves

too many parameters to be known. The authors have validated the model

against pilot plant data, but the values used for various model parameters

have not been given. Methods of determination of these parameters would

have been useful for applying the model to other sieve plate extractors.





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Swati M ohanty Reviews in hemical Engineering

Mohanty and Vogelpohl /14/ have simplified the differential population

balance model of equation 4) into a stagewise model which considerably

reduces the computational time. The space between two consecutive

sieve-plates is taken to be a single stage. Both breakage and coalescence have

been considered. A beta distribution for the daughter drop size distribution

has been used and the experimental breakage probability has been used.

However, the coalescence parameters have been taken as adjustable. Qian and

Wang 5 have developed a stagewise model where it is assumed that thedrop size distribution does not change throughout the column. At the end of

each stage, the drops coalesce to equalise the concentration and then

redisperse to maintain the original drop size distribution. For the mass

transfer coefficient of the dispersed phase, the Handlos and Baron /16/

correlation is used, while the Higbie penetration model is used for the

continuous phase. Although the model is quite simple, the predicted

concentration profile agrees well with the experimental data. The stagewise

population balance model by Garg and Pratt /17/ accounts for drop breakage

and coalescence and uses experimentally determined coalescence and

breakage parameters, which is unlike other models where these are used as

adjustable parameters. The breakage and coalescence rate constants have

been expressed as functions of hold-up and drop size. However, the

assumption of binary breakage of drops and coalescence occurring only

between adjacent and same drop sizes does not adequately represent the

actual system. Moreover, the colorimetric method used for determination of

coalescence rates, where drops of two different colours yellow and green)

coalesce to give rise to a third colour red), makes it difficult to distinguish

the colours when multiple coalescence takes place. Although the prediction of

drop size distribution at the outlet compared well with experimental data,

comparison of predicted number of coalesced red drops with experimental

data showed that the agreement was good only for the first few plates.

Blass and Zimmermann /18/ have presented a simple stagewise, backflowmass transfer model for a pulsed sieve-plate extraction column assuming a

constant holdup. A recirculation regime model developed by Prabhakar et al 9 compares well with experimental data obtained for mixer-settler andemulsion regime. Experimental determination of hold-up shows that in themixer settler region, the hold-up is affected by the hole diameter whereas inthe emulsion region it is affected mainly by the free area. Also, the drop

formation takes place in the mixer-settler region due to dispersed phase flow





Vol. 16. No . 3, 2000 Modeling of Liquid-Liquid Extraction Column :A Review

through the perfora tions and drop breakage takes place in the em ulsion region

due to pulsing action within the plate stack. Hold-up was found to be highwhen the solute transfer w as from the continuo us to the dispersed phase.

The empirical correlation fo r predicting th e hold-up in a PSE proposed b yKumar and H artland /20/, w hich was compared w ith other available empiricalcorrelations, is

Af - Af) m

(5)

where

Af)m =

O

(6)

The constants k b k2 an d k3 that appear above are given in Table 2 for thecase of no solute transfer, solute transfer from continuous to dispersed phase,and solute transfer from the dispersed to continuous phase. The correlation isbased on the available data for column diameters of 25-213 mm and isapplicable fo r prediction of hold-up in the mixer settler, transition andemulsion regions of operation within an average error of 17.8 . Theinfluence of hole diameter and column diameter w as insignificant.

ble

Parameter values for no solute transfer, solute transfer from continuous todispersed phase and solute transfer from dispersed to continuous phase.

no solutec-»d

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k,2.10xl0 6

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Swati Mohanty Reviews in Chemical Engineering

ot ting Disk ontactor RDC):

This type of extractor is commonly used for deasphalting of petroleum,for desulfurisation o f gasoline and for recovery of phenol from wastewater. Anew Open Turbine Rotating Disk Contactor OTRDC) has been developed by

Zhu et al I \l that has higher efficiency than the RD C and is also suitable fo rliquid-liquid-solid as well as liquid-solid systems. Modeling studies on the

RD C include pseudo-homogenous and population balance models as well as

those that take into account the interfacial reaction. The complexity of modelsaccounting for interphase chemical reaction increases since the observed rate

is a function of both chemical kinetics and diffusion.The differential pseudo-homogenous dispersion model by Chartres and

Korchinsky 22 includes a mass balance equation fo r each drop size fraction

for the dispersed phase and a mass balance equation for the continuous phase.Drop coalescence has been accounted for by introducing a coalescence heightto equalise the solute concentration and to then redisperse maintaining the

original drop size distribution. The drop size distribution has been estimatedusing Mugele-Evans functions /98/. The effect of drop size distribution was

found to decrease with a decrease in the coalescence height. The study alsoshows that the axial dispersion in the dispersed phase does not influence the

mass transfer significantly and can be neglected. However no verification of

the model equation has been made with experimental data.Korchinsky and Cruz Pinto /23/ have improved this model by introducing

a settling zone above the agitated section of the column, and by eliminatingthe assumption o f constant continuous phase concentration fo r estimating themass transfer coefficient by replacing it with rigid drop and turbulent

circulating drop model. The predicted number of transfer units was, on theaverage, ca. 10 higher than the experimental values. In order to improve the

agreement the mass transfer coefficient and the axial dispersion coefficient of

the continuous phase were adjusted. Thus four parameters, two for the

agitated zone and two for the settling zone, can be adjusted. Cruz-Pinto and

Korchinsky Ã Ë É have also solved the model equation of a RDC byincorporating one of the two diffusion models, i.e., either the Newman rigiddrop model 25 or the Handlos-Baron /16/ turbulent circulating drop model,

but taking into account the variation in the solute concentration of thecontinuous phase. This has been done by solving simultaneously the

hydrodynamic and mass transfer equation fo r swarm of liquid drops. Th e






A Review

model takes into consideration the axial mixing and the time dependent mass

transfer coefficient. The mass balance equation for the continuous phase issolved simultaneously by the drop diffusion model using a finite differencemethod and the results compared with that obtained by exact solution. The

predicted results have been compared with experimental data. The operating

condition of the RDC was so chosen that drop coalescence and drop breakage

were negligible. The discrepancy has been attributed to an inaccurate estimate

of the extent of mixing in the continuous phase and by accounting for therange of drop mass transfer that depends on the size.

A population balance model developed by Casamatta and Vogelpohl 7267

for a RDC accounts for drop coalescence and drop breakage with an

assumption that the drops break-up into three equal size daughter drops. The

factors accounting for the coalescence and breakage have been taken as

adjustable parameters.

The population balance model by Ghalehchian and Slater 7277 considers

drop breakage but no coalescence. The characteristic velocity predicted from

correlation from literature has been modified to minimise the discrepancy

between the predicted and experimental hold-up. Also the parameter, m, used

to account for the drop swarm, to predict the drop velocity has been taken as

the adjustable parameter. The beta function has been assumed for daughter

drop size distribution. The error in predicting hold-up is between ± 20 and

Sauter mean diameter between ± 18 .

A steady-state differential pseudo-homogenous dispersion model

involving extraction with interphase chemical reaction is given by Sarkar et al/28,29/. The model treats the disperse phase as pseudo-homogeneous. The

surface area for mass transfer is based on the average drop size measurement.

Expressions for extraction rates for very slow reactions, slow reactions, andfast reactions based on the film penetration, and Dankwerts models have been

given. The predicted extent of reaction has been compared with experimental

data for RDC with a maximum deviation being 20 . However, the model has

certain limitations, one being that it assumes the resistance to mass transfer

lies only in the aqueous phase and experimental values of mass transfer

coefficient without reaction under actual conditions of turbulence arerequired. Also the effect of droplet interaction has not been taken into

consideration.

The stagewise model by Azimzadeh-Khatayloo 7307 which is animprovement over Misek's model /31/ includes the possibility of entrainment

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Swat i Moh anty Reviews In Chemical Engineering

of smaller drops by the continuous phase. A mean diameter based on d 43 =

Znjdj4

/Znjdj3

has been used instead of the Sauter mean diameter for betterestimation of the characteristic velocity. The model incorporates into it the

influence of drop size distribution on mass transfer.

A new approach by Zhang et al 7327 uses stochastic process theory to

account for forward mixing, backmixing an d mass transfer behaviour of thedrops in the model. The drops move in the column in a stochastic manner an dthe displacement of the drops is described by the Fokker-Planck equation 7337

where q is the transition probability density, t and ô are time (1>ô), and both w

an d Æ are positions of the drop at time ô and t respectively. Also, U is theaverage velocity of the drop and Ed is the axial dispersion coefficient of thedispersed phase. The solution to this equation gives the age distribution of

drops at different heights. From the age distribution of different drops, the

RTD of drops at any height can be determined. The axial dispersion

coefficient is a function of the RDC rotor diameter, the height of a

compartment, the drop axial velocity and the free cross-sectional area. The

concentration of the dispersed phase is then calculated by solving the mass

transfer equation for a single drop an d then integrating over al l drops at anyheight to get the overall concentration of the dispersed phase. Both

coalescence and redispersion have also been neglected. The drop size

distribution was measured experimentally an d found to fit well with

Mugele-Evans upper limit log-normal distribution function. Three types of

model, namely, those of Newman 7257 Kronig an d Brink /34/, an d Handlos

and Baron /16/ have been used for estimating the mass transfer coefficient of

the dispersed phase, depending on the drop Reynold s number. The mass

transfer coefficient for the continuous phase is based on the Calderbank and

Moo Young /35/ correlation. For the axial dispersion coefficient, the

correlation suggested by Zhang et al 7367 has been used. The mass transfer

resistance in the continuous phase has been assumed to be negligible. A

comparison between the number of transfer units when the solute transfer is

from the continuous phase to the dispersed phase has been found to agree

well with experimental data, but the deviation was appreciable when the

transfer was in the opposite direction, i.e., from the dispersed to the





Vol. 16 No. 3 2000 Modeling of Liquid Liquid Extraction Column:A Review

continuous phase. This may be due to coalescence as coalescence is high

when the solute transfer is from the dispersed phase to the continuous phase.Zhu et al. I2\l have studied the hydrodynamics axial mixing and mass

transfer in OTRDC using the combined forward mixing and backflow model

fo r axial mixing.

In addition to the above discussed models there are some empirical

correlations that predict the hold-up in an extraction column. Sarkar et al /37/

have correlated the experimentally determined hold-up as a function of rotor

or impeller speed column height and dispersed phase velocity using the

toluene-water system. The correlation cannot be taken as a general one sincethe constants in the correlation may not be same for a RDC with a different

configuration and when a different system is used. However Kumar and

Hartland 738 39/ have suggested correlations for predicting the mean drop

size and hold-up in a RDC as a function of column geometry operating

conditions and physical properties.

ii ni Column

This type of column has been used for separation of aromatic and aliphatic

hydrocarbons. Modeling studies reported for this type of column are

relatively few in number an d include both pseudo-homogenous an dpopulation balance models. A dynamic pseudo-homogenous dispersion model

has been applied for simulating a Kiihni column by Hufnagl et al /40/ and

Hufnagl and Blass /41/ assuming constant flow rate of both the phases and

constant levels in the column. The hold-up and the Sauter mean diameter have

also been assumed to be constant along the length of the column. At thecolumn inlet and exit the model includes a mixer and a dead-time term to

describe the separation chamber. Correlations for estimating the Sauter mean

diameter relative phase velocity axial dispersion coefficient and mass

transfer coefficient have been taken from literature and in some cases thecorrelation constants have been modified to obtain a best fit with

experimental data. Correlations for the distribution coefficient and interfacial

tension have been suggested that are dependent on solute concentration. This

model is a component part of the DIVA simulator /42 43/. Simulated

responses fo r disturbances in feed concentration and flow rate have been

compared with experimental data. In the case of a disturbance in the feedconcentration the simulated curve at feed inlet attained steady state faster





Swati ohanty Reviews in Chemical Engineering

than the experimental curve. This may be due to the mixing in the feed line

that has not been accounted for by the model. When an increase in flow ratesof both the phases occurs, there is a sudden overshoot in the concentration

due to the level controller action to maintain constant interface level. The

deviation of the simulated values from the experimental value are due to the

assumption that the flow rate of both the phases are constant, whereas in

actual practice, due to the counteraction to maintain the interface level, there

is significant deviation in hydrodynamics. The model has not taken into

consideration the level controller.

In addition to the pseudo-homogenous dispersion models, several modelsbased on drop population balance have been reported. The differential

population balance model given in equation 4) was used to perform a steady

state simulation of the Kühni column by Haunold et al /44/. Coalescence has

been neglected whereas breakage has been described by a breakage

probability function function of impeller speed and Weber number) and a

daughter drop size distribution function. All the parameters have been

determined experimentally. However, comparison between the predicted and

experimental hold-up shows that the agreement is satisfactory only at lowflow rates. Steiner /45/ has presented a population balance model based on

geometrical droplet size classification for estimating the hold-up in Kühn i

column. The advantages and disadvantages of population balance models

have also been discussed.

The transient stagewise population balance model by Gerstlauer et al /46/

neglects coalescence and assumes a binary breakage of drops. The model

considers each drop in the dispersed phase to be characterised by the mass of

the solvent and the mass fraction of the solute. The breakage rate is predictedby the model proposed by Tsouris and Tavlarides /47/ and the daughter drop

distribution is predicted using the method of Coulaloglou and Tavlarides /48/.

Although the model considers polydispersity of the drops, all drops are

assumed to have the same velocity, which is not true in reality. The mass

transfer coefficient has been calculated using the Handlos and Baron /16/ and

Heeitjes et al /49/ models. The Gerstlauer et al /46/ model has been used to

simulate the startup behaviour of a Kühni column for drop number

distribution, dispersed phase hold-up, and solute mass fraction. No

comparison has been made with experimental data in order to check the

validity of the model. In their later work /50,51/, the same model has been

simplified by assuming that the solute concentration is same in all the drops.






A Review

The backflow of the continuous phase has been calculated by the wake

volume of the continuous phase carried away by the dispersed phase. Thesimulated start-up and step change behaviour of the column agreed well with

measured solute concentration profile. This model has been incorporated into

the DIVA simulator.

Spray Column

This is the simplest of all extractors and is used for washing treating and

neutralisation. The disadvantage of this type of column is its poorperformance due to intensive continuous phase backmixing. Not many models

have been reported for this type of column.

A hydrodynamic model for a spray column operating in the dense packing

region of drops has been developed by Noworyta and Kroti 1521 taking into

account the drop coalescence. The model assumes uniform sized drops across

any cross-section and constant flow rates of both the phases. The analytical

solution of the differential equation has been presented for predicting the

hold-up along the length of the column which is found to agree well withsystems with low coalescence.

Seibert and Fair 1531 developed mechanistic hydrodynamic and mass

transfer models based on fundamental principles fo r spray columns. The keyassumptions are that the drops are spherical and drop size can be represented

by the Sauter mean diameter. Models for determining the drop diameter drop

velocity drop-drop interaction hold-up and flooding velocity along with a

model for estimation of the mass transfer coefficient from fundamental

principles with limited number of empirical correlations have been presented.Comparison between experimental and predicted hold-ups shows that the

agreement is good at low hold-up but deviates significantly near flooding

regions. The predicted and experimental overall mass transfer coefficients

based on the continuous phase agreed quite well.

Schermuly and Blass /54/ have developed a model for a 3-component

mass transfer in a spray column. The system chosen is

glycerol-acetone-water. The model takes into account the variation of

concentration and mass flow rate along the column and liquid phase

backmixing The mass transfer coefficients are based on both the theoretical

and experimental results available in literature on mass transfer in saturated

and unsaturated phases. A correction for the eddy diffusivity parameter





Swati Mohanty Reviews in hemical Engineering

calculated using the Handlos Baron formula and the backmixing parameter

has been determined by adjusting these parameters using a best fit criterionbetween the calculated and experimental data. Since adjustable parameters

have been used, the model cannot be confidently used for predicting the

column performance adequately.

Kumar and Hartland 7557 have proposed a correlation for predicting the

hold-up for a spray column in the loose and dense packed region of operation

for with and without mass transfer. The correlation is a function of column

geometry, physical properties and known operating parameters.

Scheibe Extractor

Literature on modeling of Scheibel extractor is relatively sparse and

includes works by Alatiqi et al 756,577 and Pang and Johnson 7587. Alatiqi et

al 7567 have developed a stagewise, backflow model for mass transfer in a

Scheibel extractor. Each stage is assumed to consist of a mixing zone and a

settling zone. In the mixing zone, mass transfer takes place while coalescence

takes place in the settling zone. A correlation for hold-up which is a functionof agitation speed and phase ratio has been suggested for predicting the

hold-up at different heights of the column. The distribution coefficient is also

expressed as a function of solute concentration. Physical properties, such as

viscosity and surface tension, have been correlated as functions of solute

concentration. For estimating the continuous phase and drop-side mass

transfer coefficients, the correlations by Gamer and Tayeban 7597 and Rose

and Kintner 7607 have been used, respectively. The three parameters that are

adjusted to minimise the discrepancy between the simulated and experimentalresults are the two backmixing parameters one for the continuous and the

other for the dispersed phase) and the overall mass transfer coefficient. It was

found that the two backmixing coefficients were close to zero and hence were

neglected and the simulation was done with one adjustable parameter. This

shows that extrapolating a single droplet mass transfer correlation to droplet

swarms does not accurately predict the mass transfer coefficient. Steady-state

analysis studies were carried out and they found that the agitation speed and

the phase ratio had considerable influence on the control structure.

In their further studies, Alatiqi et al 7577 have studied two types of

dynamic stage-wise models, one equilibrium and the other non-equilibriumwith backflow, assuming constant mass transfer coefficient flow rates and





Vol. 16 No. 3. 2000 Mo deling o f Liquid-Liquid Extraction Co lumn:A Review

hold-up. Response to step disturbances in flow rates, solute concentration,

agitation speed and backmixing coefficients were studied. However, noverification has been m ade w ith experim ental data.

Transient response studies have been reported in simulation of

liquid-liquid extraction column. Pang and Johnson /58/ have developed a

stagewise dynam ic m odel for a m odified Scheibel ex tractor and have studiedthe frequency response to a pulse input at the feed concentration in the rangezero to 1.2 radian/minute. The model equations have been transformed to the

frequency domain to obtain the frequency response instead of the time

domain transient response for any disturbance in the feed concentration. Thestages were assumed to be non-identical i.e., the overall mass transfercoefficient was assumed to vary from stage to stage) and undernon-equilibrium conditions. The mass transfer coefficient, the time delay andthe number of mixing stages at both ends of the column, as well as thefraction of the feed and the solvent entrained at two ends of the column, weretaken as fitting parameters and were obtained by obtaining a best fit withexperim ental and predicted freq uency response am plitude ratio and phase

shift). When the predicted frequency response was transformed to the timedomain and compared with experimental transient response, the agreementwas good.

Packed Column

Packed columns are preferred to spray columns since they increase the

drop coalescence and breakage and decrease axial mixing, which results in

improved performance. Inspite of their popularity, not many modelingstudies have been reported in the literature. The mechanistic hydrodynamicand mass transfer model fo r spray columns developed by Seibert and Fair 1531

has also been extended to packed columns and has been applied to varioustypes of packings. However, the axial mixing of the continuous and dispersed

phases has been neglected and a tortuosity factor has been introduced in the

drop velocity equation. The discrepancy between the predicted hold-up and

overall mass transfer coefficient was higher than that for spray columns.

Steiner et áú . /61/ have presented simplified differential as well as stagewisemodels for predicting the hydrodynamic and mass transfer performance of an

extraction column filled with regular packings. For the differential model, thecoalescence and breakage time, the drop velocity, and the mass transfer






coefficient have been assumed constant and a linear equilibrium relationship

has been assumed. The drops are assumed to be monodispersed withnegligible axial dispersion. Analytical solutions for the drop volume and

concentration profiles for the continuous and dispersed phases have been

presented. The drop breakage-coalescence parameter and the inlet drop size

were taken as adjustable and were obtained by minimizing the differencebetween the measured and predicted drop size profile. For the stagewise

backflow model, the breakage and coalescence times, drop velocity and the

mass transfer coeffioient are estimated as functions of drop diameter. The

mean mass transfer coefficient was estimated using the Newman s function1251 and the age distribution of drops as given by the penetration theory of

Dankwerts I62I The diffusivity factor, R, which is the ratio of eddy to

molecular diffusivity was taken as adjustable in the estimation of mass

transfer coefficient from Newman s function by minimising th e differencebetween the predicted and experimental concentration profiles. The authors

have shown that there is significant change in the mass transfer coefficientalong the height of the column, hence the assumption of constant mass

transfer coefficient in the differential model is not justified. The modeldeveloped by Garg and Pratt /17/ for sieve plate extraction columns has been

extended to packed columns by Hamilton and Pratt 7637 and the coalescence

and breakage rate constants were obtained by the same -method of

colorimetry. The predicted column height has been presented for three

different cases, i.e., (1) when no breakage and coalescence takes place; (2)

when measured coalescence and breakage rates are used; and (3) when

infinite coalescence and monosized dispersion are assumed. However, no

comparisons have been made with experimental data. In determining thecoalescence and breakage constants, experimental data on the fraction of red

drops in different drop size intervals have been used. The experimental data

have been smoothed and correlations have been proposed for breakage and

coalescence rate constants as functions of drop diameter and hold-up. The

predicted rate constants vary significantly from experimental rate constants,

which shows that this method of determining the coalescence and breakage

rates may not be very accurate.

In addition to the above models, Salem et al

7647 have proposed a model

based on non-equilibrium stages. An overall and component balance is made

for the column. The flow rate of the extract stream in each stage is estimated

by Edmister group relationship 7657





Vol. 16 No. 3 2000 Modeling of L iquid-Liquid Extraction Column:

A Review

(8)

which is then used to calculate the raffinate flow rates in each stage. The

composition of the raffinate and the extract phases are then calculated and

compared with the experimental compositions. If the discrepancy is not

within the convergence limit, then a new number of stages is assumed and the

calculations repeated until convergence is achieved. The ratio of the

calculated extract phase solute concentration to raffinate phase solute

composition gives m . The m is therefore the non-equilibrium distribution

ratio of the solute in the two phases. The ratio of m to the experimental

equilibrium distribution ratio, K, gives the extraction efficiency. Only limited

work has been carried out and no general conclusions have been made as to

how the extraction efficiency could be affected with change in operating

conditions. Arimont et al I66I have tried to develop a model based on the

complete life-cycle of a single drop, i.e., formation, cleaving, coalescing or

departure from the column. The model equation is applied to each newly

formed drops. The model has been tested for the system n-butyl-

acetate/water/acetone and has been found to agree well with experimental

data.

Oldshue Rushton Contactor ORC), Disks an d Rings PulsedColumn:

The ORC s have been used in petrochemical, metallurgical,

pharmaceutical and fertilizer industries. Only limited studies on this type of

extractor have been reported in the literature. Lee et al. 1611 and Lee and Kim7687 have proposed a stagewise drop population model for this type of

extraction column where the five parameters accounting for drop breakage

an d coalescence have been taken as adjustable. These parameters were seen

to vary with varying operating conditions. Experiments on drop breakage in

different types of extractors have been reported in literature. To improve the

model, it would be desirable to obtain the breakup parameters experimentally.

A transient, stagewise, population balance model accounting for backflow and

forward flow has been developed by Tsouris et al I69I. The drop breakage

model is based on the eddy-drop collision frequency and breakage frequency,






while the drop coalescence model is based on collision frequency and

coalescence frequency. The convective terms have been accounted for by theforward and backward exit frequency of drops from one stage to another.

Good agreement between the experimental and predicted hold-up and Sauter

mean diameter has been reported. A single adjustable parameter has been

used in the model fo r forward an d backward exit frequency by fittingexperimental data with the population balance model solution. The model has

also been used for simulating model based predictive control. The

pseudo-homogenous dispersion model developed by Sarkar et al /28, 29/ for

RDC as described earlier has been validated for ORC also. Korchinsky andAzimzadeh-Khatayloo /70/ have proposed a stagewise model and used it for

predicting the concentration profile along an ORC. The continuous phase was

assumed to be completely mixed and coalescence was neglected in each

stage. The axial mixing was accounted for by assuming a backflow between

the stages. However no comparisons were made with experimental

concentration profile.

The population balance model given by equation 4) has also been used

for simulating the hydrodynamics and mass transfer in a Disks and Ringspulsed column by Dimitrova AI Khani et al Ð\É Assumptions made are

similar to that by Casamatta and Vogelpohl 1261 i.e., the drops break into

three equisized daughter drops and the breakage rate is proportional to the 8th

power of the drop diameter. The model involves three adjustable parameters

the mass transfer coefficient the breakage intensity and the coefficient for

calculating the characteristic drop velocity), which are obtained by fitting the

predicted and experimental hold-up and concentration profile. The

distribution coefficient is assumed to be constant and equal to 3.636 for thesystem water-acetic acid-isopropyl ether.

Reciprocating late Column RFC):

These types of extractors are used in pharmaceutical, petrochemical,

metallurgical and chemical industries. The specific types of RFC include

Karr, Prochazka and Tojo/Miyanami. Hafez et al n l have presented a

design model for a Karr reciprocating plate extraction column. The modelcalculates the diameter of the column at which flooding occurs and if the

degree of extraction is known then the height of the column. A hydrodynamic

and mass transfer model for a Karr RFC has been developed by Camurdan et





Vol. 16 No. 3 2 Modeling o f Liquid-Liquid Extraction Column:A Review

áú. 73Ë For estimating the hold-up, the following model is proposed:

1-Ö) 2 9)

where L = -=-;

where Kk =30 for rigid drop and 15 for circulating drops. The modelpredicted the hold-up with reasonable accuracy for agitation rates greater than3 cm/s. The model does not take into account transfer of solute from the

dispersed to the continuous phase which enhances the coalescence rate and

hence the model is not suitable fo r mass transfer from dispersed to continuous

phase. For the mass transfer, a stagewise model has been proposed with axialdispersion in both the phases. A linear equilibrium relationship has beenassumed and the aqueous phase mass transfer resistance has been neglected.The axial dispersion coefficient for both the phases were estimated using thecorrelation given by Hafez /74/. The overall mass transfer coefficient was

taken as the adjustable parameter to obtain a best fit between the experimentaland predicted aqueous phase outlet solute concentration. The system chosenwas water-acetic acid-kerosene. This shows that the correlation used for

estimating the mass transfer coefficient is not suitable for the system chosen.Sovov et al. Ð5 / have proposed a model for prediction of mean drop size

in any stage for a vibrating plate extractor VPE) or Prochazka RFC. In thismodel, an exponential rate of change of mean drop size has been assumed

12)

where the constant, Kp depends on the plate type and dm is the limiting dropsize. The total hold-up in any stage is the sum of the hold-up in the denselypacked layer zone and the hold-up in the loosely packed zone. A semi-






empirical correlation has been proposed for predicting the hold-up in the

densely packed zone

rv Vd

13

where k v is a fitting parameter.

For the loosely packed zone the hold-up ö; is obtained from the solutionof the nonlinear equation

Hence the total hold-up in any stage is the sum of the above two hold-up

The model parameters Kp and m were evaluated by min imiz ing the

deviation between the experimental and calculated hold-up profiles. For

predicting the thickness of the densely packed layer at the plate an extra

parameter is to be determined whereas for the sieve plate an extra parameter

is required to account for the instability near flooding.

G N R L

In addition to the above models for various types of columns there are a

few that are more general in nature and can be extended to different types of

columns.

A drop population model for agitated columns has been proposed by

Steiner et áú Ð6/ assuming a back flow model for the continuous phase andtwo different types of models i.e. forward mixing and mixing in both

directions for the dispersed phase. In the forward mixing model the drop

size distribution is assumed to remain constant and coalescence and





Vol. 16, No. 3, 2000 Modeling of Liquid-Liquid Extraction Column:A Review

redispersion are accounted for by assuming that all the drops coalesce and

redisperse thereby maintaining the original drop size distribution. The numberof such mixing stages introduced into the column is left as a free parameter

and is obtained from the measured concentration profiles. A logarithmic-

normal distribution of drop size is assumed. In addition, it is assumed that

coalescence takes place between two equal sized drops and a drop breaks into

two equal sized drops. The rate constants for the breakage and coalescence

have been taken as the adjustable parameters and were fitted to get good

agreement with experimental data obtained in a pilot-plant agitated column.

Caminos et al. ÐºÉ have developed a steady-state stagewise model forextraction columns assuming each stage to be in equilibrium using phase

equilibria obtained by the UN1FAC group contribution method /78/. A total

and component balance has been made for each stage. The model can be

applied to both stage extractors as well as sieve plates and has been applied to

industrial extractors. A solution method that requires less computation time

has been suggested. Sereno et al. /79f have assumed a constant hold-up along

the length of the column and have obtained a dynamic mass balance equation

fo r describing t he concentration in both phases o f the system. Instead o f using

a rigorous thermodynam ic model for describing the distribution ratios of the

ternary system, local non-linear models have been used that are valid over

moderate range of composition, which saves considerable computer time. The

distribution ratio, which is first evaluated by using a rigorous method such as

UNIFAC is then fitted by polynomial approximation in the follow ing form

In K-t = LljJc^f L2y?ef +L3, i = 1 2 3

where LI , L 2 and L 3 are adjustable empirical local model parameters, and xrefand yref are mole fractions of the reference component in the raffmate andextract phases, respectively. The dominant component is chosen as the

reference component. Introducing time derivative of the distribution ratios inthe general mass balance equation leads to a differential mass balance of

component i in the equilibrium unit. The differential equation is solved byGEMS General Equation Modelling System) /80/. The equation for a single

equilibrium unit has also been modified to account for a multistage model.Each stage is treated as an equilibrium stage and no backmixing between the

stages has been assumed. The model-predicted concentration profiles of the






raffmate and extract phases in multistage extraction columns ha s been

compared with those predicted by other workers for extraction of cyclohexane

from ç-heptane using furfural as the solvent and extraction of acetone from

water using 1 1 2-trichloroethane as the solvent. The model predictions were

in exact agreement with that of other workers.

PROCESS SIMUL TORS

Simulators with built-in models applicable to various process industriesare also available. Black /81/ gives a brief description of a process simulator

/82/ and its application to various petroleum and petrochemical industries for

simulating distillation columns reactors heat exchangers pumps mixers

liquid liquid extractors etc. However only stagewise simulation of

liquid liquid extractors is possible with this simulator. Marquardt et al /42/

have presented the tools available for dynamic sim ulation of chemical plants

with special reference to the DIVA simulator /43/ which was developed by

the authors. The DIVA simulator has the dynamic model developed by

Hufnagl et al /40/ as a component.

NUMERI L TECHNIQUES

Model equations range from simple to more complicated form. In some

cases the equations can be solved analytically while in others sophisticated

numerical techniques may be needed to solve the equations. Methods for

solving these equations over the years have been suggested by several

authors. Roche /83/ suggested a solution method for solving

multi-component multistage liquid liquid extraction column in early work.

Wang and Wang /84/ later reviewed the solution methods available fo rsimulation of separation process models. The three basic methods namely

equation decoupling relaxation and simultaneous solution methods

although discussed with reference to distillation column are also applicable

to any type of multistage countercurrent separation process involving mass

transfer between tw o fluid phases. A general calculation method fo rsimulation of a stagewise steady-state multi-solute countercurrent

extraction with axial dispersion ha s been suggested by Ricker et al 7857. In






A Review

this method, the convergence of the solution of the model equations is very

fast without making use of any special convergence-promotion techniques.Chatzi and Lee 7867 have suggested a numerical solution for thehomogeneous model developed for a batch agitated liquid liquid extractionprocess taking into account the drop breakage and coalescence. An algorithmfor solving stiff differential equations using a semi-implicit Runge Kutta

method and its application to liquid-liquid extraction has been proposed by

Michelsen/87, 887.Kronberger et al 7897 have suggested a new solution method for solving

the system of differential population balance equations. The system of eachIntegra Differential Equation IDE) mass balance and volume balanceequations) is discretised with respect to the drop diameter by applying the

Galerkins projection technique to give n Partial Differential Equations(PDE s) where n is the number of drop size intervals considered. The coupledPDE s are then space-time discretised by using finite-volume technique. The

spatial derivatives in the dispersive flux are approximated by centraldifferences and integration in time is performed by the Euler method. This

method ha s been implemented for simulation of an extraction column that iscalculated using the flux-extrapolation method with a time increment of 0.1 s.

Stable discretisation of the convective term has been incorporated. Ribeiro et

al . 7907 suggest an algorithm for solving the population balance equation pbe) that adequately predicts the drop distribution function for a perfectlymixed continuous or batch vessel with spatially homogeneous, stirred

turbulent dispersion. A first-order, explicit finite-difference method has beenused to solve the pbe. For the chosen systems, toluene-water, and methyl-isobutylketone-water, a logarithmic grid was used for discretization of the

pbe s. This algorithm has also been extended to simulate a trivariate (drop

volume, age and solute concentration distribution) with unsteady statebehaviour of the dispersed phase by Ribeiro et al 7917 and was found to be

fast enough to be applied to control of extraction columns. Regueiras et al

1921 have proposed a new simple algorithm for drop population balance thatfurther reduces the computation time and is therefore suitable for process

control. Instead of calculating the solute concentration distribution for each

drop class at every time interval, it calculates the mean solute concentrationand standard deviation for the solute concentration. However lack of

experimental verification limits confidence in this algorithm.





Swati Mohanty Reviews in hemical Engineering

MODEL P R METERS

Various extraction model parameters, such as drop size, drop size

distribution, slip velocity, mass transfer coefficient, and axial dispersion

coefficient play an important role in the modeling of extraction columns.

These parameters are either determined experimentally or estimated using

available correlations. orchinsky and Young /93/ have discussed the

computational techniques that can be used for estimation of key model

parameters, such as axial dispersion coefficient, mass transfer coefficient,

drop velocity and constriction factor required for a forward mixing model of

a RDC. The constriction factor and the drop velocity in the agitated and

settling zone are estimated from the hold-up and drop size distribution data,

whereas the axial dispersion coefficient is estimated from the concentration

profile data. Initially, the mass transfer coefficient k c) and the axial

dispersion coefficient E c) are estimated using available correlations. The kc

and E c for the settling zone are varied to optimize the parameters for the

agitated zone. The authors claim that the determination of the E c from the

concentration profile is a better method than using tracer techniques. They

found that the mass transfer coefficient increased with increasing drop size, a

trend that was correctly predicted by the Handlos-Baron model. In the tracer

technique, E c is determined in the absence of mass transfer. The mass transfer

experiments are carried using the same E c as determined without mass

transfer. The tracer technique is less reliable since the pseudo-homogenous

dispersion model does not accurately describe the dispersed phase, so the

solute concentration profile has been used to determine E c. The book edited

by Godfrey and Slater IM give a critical review of various correlations

available for estimation of these parameters for different types of extraction

columns. A review of various hydrodynamic parameters, such as hold-up,

drop size and slip velocity, is presented by Koganti and Srinivasulu /94/.

Below, only a few estimation methods that have been most commonly used

are discussed.

Drop Size

The dispersed phase, which consists of drops of varying sizes, is

characterised by a mean drop diameter. The Sauter mean diameter d 32 ),

defined as d 32 = Ó ç Ü^/Óç Ü?is the volume-surface mean diameter. For mass





Vol. 16, No. 3, 2000 Modeling of L iquid-Liquid Extraction Column:A Review

transfer studies, where the mass transfer rate depends on the surface area ofthe drop and the am ount of solute transferred into the drop volum ed

32 seems

to be most suitable and has been used by most of the authors. Howeverd43 defined as d43 = ZnidjV £n dj3 has been used for estimation of slip velocity.Misek /31/, discusses the error involved in characterising the dispersion bySauter m ean drop diameter and has justified the use of d43.

Purely empirical correlations based on column geometry and operatingconditions have been developed for predicting th e mean drop sizes in varioustype of extraction colum ns. Ku mar and H artland /38/ have critically reviewedthe available correlations and have proposed a correlation fo r predicting thedrop size in a RDC. Prediction of the drop size in a spray column from amulti-nozzle distribution has been suggested by Dalingaros et al. 1951. Forpulsed perforated plate columns the correlation proposed by Kumar an dHartland /95/ and Sovov Ð5É are more reliable. Kumar and Hartland /97/

have presented a unified correlation for predicting the drop size in eightdifferent types of extraction column.

rop Size istribution

Several au thors have tried to fit drop size distribution with different typesof statistical distributions of which the most commonly used ones are givenbelow.

The arithm etic normal distribution function has been used by Qian /15/ todescribe the drop size distribution for a pulsed plate extraction column. Thefunction is given as

8 d) = \ e Xp\- d-d 43 ) 2 /2a 2 ] 17)

\° ·4 3

< / 4 3 0.0123 Ì fA/;~°· 70 18)\Pc)

19)

where ó is the standard deviationThe basic equation of Mugele and Evans /98/ to describe the drop





Swati M ohanty Reviews in Chemical Engineering

distribution has been used to describe the drop size distribution in a rotating

disc contactor.

20

where v is the volume fraction of drops having diameter less than d, a is a

dimensionless constant, ä is the size distribution parameter and dm a x is the

maximum droplet diameter.

The log-normal distribution and Gamma distribution have also been used

frequently for theoretically predicting the drop size distribution based on

operating conditions.

Break Up Parameters

Cruz Pinto and Korchinsky 99 have also studied the drop breakage in a

RDC using a population balance technique. The operating conditions were

chosen so that coalescence was minimised. They have made use of a

breakage frequency function and breakage distribution function assuming

binary breakage. Three parameters have been optimised by minimising the

difference between the predicted and experimental values. However, since

these parameters are not generalised, the usefulness of this model will be

restricted only to the system and column configuration chosen by them.

Haunold et al /44/ have studied the break-up parameters for a Kiihni column.

A correlation for obtaining d m ,x as a function of Weber number (We = p 2 N 2

DR / ã), and impeller length has been suggested. The breakage probability

increases with mother drop diameter with a maximum of 100 . The system

used is water-butanol and water-succinic acid-butanol. Addition of solute

causes an increase of the number of daughter drops.

Mao el al /100/ have studied drop breakage for different types of

structured packings. They found that the drops broke at the edges and not

within the structures. A correlation for calculating the critical diameter above

which all drops break has been suggested. However, they have assumed that

the drops break into two equal-sized drops. The correlation was obtained by

making an energy balance before and after the breakage. The energy types

taken into account are kinetic energy, potential energy, surface energy, and

energy dissipation during deformation and breakage. The velocity of the

daughter drops after breakage is taken to be zero.





Vol. 16, No. 3, 2000 Modeling of L iquid-Liquid Extraction Column:A Review

Fang et al. /101/ have suggested correlations for estimating the breakage

probability, p, in a Kuhni column that is not in turbulent flow. One particularform is

21)1 p

where, We SLM = ñì™ (/V1 5 -N™ )p Rd / ã' 22)

W eSLM is the modified Schlichting laminar Weber number. The

correlation agrees well up to p=0.5 above which the divergence is large.Slater /102/ has reviewed drop breakage in various types of extraction

columns.

Slip Velocity

For countercurrent flow and constant velocities in two phase flow, the slip

velocity is defined as

Ñ*

d Ö + V 1-Ö) 23)

where VS|jp is related to the characteristic velocity V 0) by the equation

V*p = VJl-ö) 24)

The simplest form of this relationship assumes m=l . However, studies

have shown that m is a function of drop size and Reynolds number. Godfreyand Slater /103/ give a good review of various other slip velocity

relationships. Characteristic drop velocity is defined as the drop velocity

when the continuous phase velocity is zero and the dispersed phase hold-up

tends to zero. Correlations and methods for determining the characteristic

drop velocity for various type of extraction columns are available /103-105/.

xial Mixing

The axial mixing of the continuous phase is mainly due to channelling,

circulatory flow of the continuous phase and the continuous phase being

carried away by the rising drops in the form of wakes which then circulate





Swati M ohanty Reviews in Chemical Engineering

back to the column. The axial mixing of the dispersed phase is due to the

difference in drop sizes that results in difference in drop velocities. Severalcorrelations have been reported in literature for predicting the axial mixing

coefficients of the continuous and dispersed phases for different types of

extraction columns, but the percentage error in some cases is very high, lying

between 25-70%/ /.

. For continuous phase:

Kumar and Hartland /l 067 have presented a correlation for prediction of

the axial dispersion coefficients for RDC by making use of steady state tracerinjection and dynamic tracer injection technique. For the Kühni column,

Breysse et al /107/ have proposed a correlation for both single and two phase

flow. The correlation by Kumar and Hartland /108/ and Prvcic et al /109/ is

applicable to pulsed sieve-plate columns. A physical model has been

proposed by Stevens and Baird /1107 for single-phase axial mixing in a

single-stage Karr column. Two regimes, i.e., poorly mixed and well-mixed

regimes, have been considered to predict the overall axial dispersion

coefficient. The hydrodynamic model predicts the distance for which the

mixing is poor and the distance for which the mixing is good. Two

parameters must be estimated. The model is proposed based on flow rate,

frequency, amplitude, plate spacing, hole size, and fractional free area. These

authors give a good summary of work carried out by other workers on the

estimation of the axial dispersion in PSE and RFC. A correlation has been

proposed by Karr et al 71117 for axial mixing in a Karr column.

Steiner et al 71127 have suggested a method for experimental

determination of the axial dispersion co efficient for the continuous phase and

have fitted the data with an empirical correlation. For the dispersed phase, the

data seemed to be scattered with the correlation proposed.

Godfrey et al 71137 have reviewed the work carried out by various

authors on single-phase axial mixing in pulsed sieve plate extraction columns

and have proposed a correlation for stationary phase axial mixing with the

objective of getting better co-efficient required in calculating the mass

transfer performance. The phenomenon of axial mixing is generally described

by the usual effective diffusional model. They have used the idea and

technique developed by May I I147 Misek 71157 and Miyauchi and Oya 71167

for determining the stationary phase axial mixing coefficient.





Vol. 16 No. 3 2 Modeling of Liquid-Liquid Extraction Column:A Review

For dispersed phase:

Correlations available for predicting the dispersed phase axial mixingcoefficient are fewer in number and are unreliable as the error involved is

quite significant. Most of these correlations apply to specific situations.Kumar an d Hartland /106/ have proposed a correlation fo r axial mixingcoefficients in RDC and have also reviewed other available correlations. For

the pulsed sieve-plate extraction column, the correlation by Miyauchi and

Oya /Ð5/ is more commonly used.

ss Transfer oefficient

Mass transfer coefficient measurements are based on two theories:

a) Whitman s two-film theory and b) Higbie s penetration theory.For the continuous phase, the correlation proposed by Steiner /l 17/ is

recommended. The equation given by Lochiel and Calderbank IIW derivedfrom the diffusion continuity equation is recommended fo r intermediate and

high Reynolds numbers (Re = p d V / ì ).

For single drop mass transfer coefficient of the dispersed phase, theNewman s model 1251 for rigid drop and Kronig and Brink s model /34/ fo rcirculating drops have been used. For mass transfer in drop swarms, the

model by Yaron and Gal-Or /119/ is recommended. Korchinsky and

Cruz-Pinto /120/ have suggested a method fo r applying th e rigid drop an dHandlos-Baron drop models fo r evaluating the mass transfer co-efficientwhen th e continuous phase concentration is not constant and the drop sizes

are not uniform. Bahmanyar et al. /121/ have examined th e rate of masstransfer from a single drop in RDC s, pulsed sieve-plate columns, and packedcolumns and have compared these with those estimated by Handlos and

Baron model to incorporate necessary corrections.

ON LUSIONS

From the various types of models reviewed, it can be concluded that these

fall into tw o groups: 1) the diffusion model with turbulent back diffusion ofsolute superimposed on plug flow of both the phases, and (2) the backflowmodel with well-mixed non-ideal stages between which backflow occurs.The major difficulty lies in proper representation of the dispersed phase. The





Swati ohanty Reviews in hemical Engineering

pseudo-homogenous model assumes the dispersed phase to be pseudo-

homogenous whereas the population balance model treats it as a discreteelement. The drop population balance model takes into consideration the

previous history of each drop as it moves along the column, i.e., its breakage

and coalescence, to give birth to new drops of different sizes. The population

balance models have been found to represent the real system more closely

than the pseudo-homogenous model. As far as the population balance model

is concerned, the major difficulties lie in obtaining the break-up an dcoalescence parameters. Several researchers have assumed that the drops

break into 2 or 3 equisized daughter drops. In recent years, however, studieshave been carried out to determine the drop size distribution for various drop

sizes and fitting them to appropriate statistical functions. The constant

parameters depend on the system and operating conditions chosen. These

parameters for some systems are available in literature, but a lo t of

experimental data remain to be generated for determining the parameters.

Except for a few, the coalescence parameters in most models have been

estimated by obtaining best fi t with experimental drop size distribution data.

Even those who have used experimental values have made assumptions that

are not realistic, such as coalescence only between adjacent and same drop

sizes. However, none of the methods available to date is fully satisfactory and

more research in the area of d etermination of coalescence parameters for drop

swarms would be helpful in obtaining a better model. Applicat ion of

advanced modelling techniques, such as, Computational Fluid Dynamics

CFD), may lead to a better model, but may be extremely complex since the

dispersed phase continuously changes in size and composition.

K N O W L E D G E M E N T

The author gratefully acknowledges the f inancial assistance of the

Alexander von Humboldt Foundation, Bonn; Prof. Alfons Vogelpohl, Institutfü r Thermische Verfahrenstechnik, TU Clausthal, Germany, for his help an dcooperation; and the Director, Regional Research Laboratory C.S.I.R.),

Bhubaneswar, for permission to publish this paper.





Vol. 16, No. 3, 2000 Modeling o f Liquid-Liquid Extraction Column:A Review

N O M E N L T U RE

A = length of pulsation stroke, ma = asymm etrical distribution parameterB d) = rate of birth of drops of diameter d,m '.s '

C = concentration, kg.m3

C* = equilibrium concentration, kg.m3

d = droplet diameter, mD d) = rate of death of drops of diameter d, nf'.s 1

DK = turbine diameter, m

d32 = Sauter mean diameter, md43 = drop volume fraction mean diameter, mE = axial dispersion coefficient, m2.s '

/ = pulse frequency, s '

g d) = drop volume fraction distribution, m 1

h = centre to centre plate spacing, mH = height of the column, m = stage num berk = mass transfer coefficient, m.s1

ki, k2, k3 = constants in Eqn. (5)km = overall mass transfer coefficient based on

continuous phase, m.s*1

kv = parameter in Eqn.(13), m 1

K = distribution ratioKh p = constants in Eqn. (11) and (12) respectivelyL = extract flow rate, kg.hr1

LI L2,L3 = empirical parameters for local nodes in Eqn.(15)m = exponentm ratio of the solute composition in the raffmate to

that in the extract phaseW = turbine speed, s 1

Ë = number of stagesñ = break-up probabilityP d) = volume fraction of drops of diameter d, m 1

q = transition probability densityr = dimensionless function of drop diameter/ = time, su = drop velocity, m.s1





wati ohanty Reviews in hemical Engineering

U

yv

vyw

WestM÷

Ó Z 2 ... „

z

reek l tt rs

ä

Y

Ñ

ubscript

r

d

fm x

n

t

average velocity of the drop, m.s 1

superficial phase velocity, m.s1

volume fraction of the drops having diameterless than d

characteristic velocity, m.s 1

slip velocity, m.s 1

terminal velocity, m.s 1

as defined in Eqn. (9)

position of the drop at tim e ô, m

modified Schlichting laminar Weber numb ermole fraction of component in the raffinatemole fraction of component in the extractparameters in Eqn. (1)

position of drop at time t, m

fractional free area of perforated plateuniformity distribution parameter

surface tension, N.m 'viscosity, Pa.sdensity, kg.m 3

standard deviation of drop size distribution, mhold-up of dispersed phasetime, s

continuous phase riti l

dispersed phasefinalmaximumstage numberinitialtotaldensely packed region





Vol. 16 No. 3 2 Modeling o f Liquid-Liquid Extraction Column:A Review

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Swali ohanty Reviews in hemical Engineering

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A Review

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