Prediction of mass transfer coefficients in an asymmetric rotating disk contactor using effective diffusivity

Meisam Torab-Mostaedi,Mehdi Asadollahzadeh*,Jaber Safdari

Nuclear Fuel Cycle Research School,Nuclear Science and Technology Research Institute,P.O.Box:11365-8486,Tehran,Iran

1.Introduction

Liquid–liquid extraction columns are widely used on commercial scale in the hydrometallurgical,pharmaceutical,chemical and nuclear industries.Of the available counter-current extraction columns,the rotating disk contactor(RDC)is widely used in industrial solvent extraction processes due to its high throughput,low investment, flexible operation,and easy maintenance.Due to limitation of its structure,the mass transfer performance of commercial RDC columns is low because of severe back-mixing[1–6].Therefore,modifications of the RDC have been attempted in order to reduce the extent of back-mixing which causes significant reduction in mass transfer efficiency.For this reason,the RDC has appeared with modified structure such as the RDC with perforated disks and the asymmetric rotating disk contactors.It goes without saying that,modified RDC columns are finding wide applications in different industries[7–10].Asymmetric rotating disk contactors(ARDCs)are widely used in pharmaceutical,petrochemical,and chemical industries because of dual advantages of a high mass transfer rates and reduced back-mixing in both phases[11].In this extractor,a shaft carrying the disks is mounted off-centre,and two sets of staggered stator plates are provided,connected by a vertical segmental baffle.By this means,the mixing chambers are enclosed,and are connected to one another via openings each side of the vertical baffle leading to chambers which partial coalescence occurs[11].Since the coalescence zone in ARDC is isolated from the mixing zone,the coalescence of the dispersed droplets and the transport of the separated phase occur more efficiently in this contactor as compared to those in the RDC.This feature not only reduces the back-mixing between the stages,but also increases the power consumption in the mixing zone and high efficiency per stage through higher values of the dispersed phase holdup and interfacial area[11].

Although the ARDC column has been used effectively for a number of separation processes,there are limited data in the literature on the performance of this type of extraction column[11].On this basis,pilot plant experiments on hydrodynamic and mass transfer characteristics of ARDC column is necessary for the purpose of establishing the optimal design procedure for this column.

The present research work has examined the influence of operating variables including the rotor speed as well as the dispersed and continuous phase velocities on the mass transfer performance in a pilot scale ARDC column.An empirical correlation for prediction of enhancement factor is developed and used for the estimation of overall mass transfer coefficient.

2.Previous Work

The reliability of the design of liquid–liquid extraction columns depends upon the theories or correlations used for the calculation of overall mass transfer coefficients.In the literature,a large number of such theoretical and empirical equations are reported for the dispersed and continuous phase,and each has a particular range of application.For the dispersed phase,theories rather than correlations have been mostly used for the estimation of the mass transfer coefficient[12].These theories have usually been considered in the light of three mechanisms of mass transfer inside drops.

The Gr?ber equation(Eq.(1))[13]concerns rigid drops which do not have any internal circulation and where mass transfer is more controlled by a transient molecular diffusion:

The Kronig–Brink model(Eq.(2))[14]assumes a laminar diffusion with an inner circulation inside the drop,induced by its relative motion,with respect to the continuous phase.

The Handlos–Baron model(Eq.(3))[15]deals with the case of drops with internal turbulent circulation and where mass transfer is controlled by turbulent diffusion.

Although mass transfer in to or out of drops have been investigated for many years,it is still not fully understood,since it depends on several factors.These factors include the fact that the dispersed phase mass transfer coefficient depends upon the nature,size and behavior of the drop.The presently available equations for calculation of the dispersed phase mass transfer coefficient are not usually valid over a range of drop sizes and behaviors in a typical extraction column.An attractive method uses an enhanced molecular diffusivity,RDd(also referred to as effective diffusivity,Deff.)in the equation for diffusion in rigid spheres(Eq.(1)).The enhancement factor contains the effects of all known and unknown parameters that influence the mass transfer coefficient.The resulting equation for the dispersed phase mass transfer coefficient,given by Johnson and Hamielec[16],is the following equation:

Several empirical correlations for estimation of the enhancement factor in extraction columns are described in the literature.These correlations are listed in Table 1.

The idea of using enhancement factor provides a very appropriate practical method for estimation of mass transfer coefficient for a variety of drop sizes in distribution,with different residence time in an extraction column.

3.Experimental

The pilot plant comprises a 36 compartments asymmetric rotating disk contactor of 113 mm diameter.The main section of the ARDC column consists of a 1430 mm long outer Pyrex glass shell and stainless steel internals.The disks are mounted onto a shaft and driven by an electric motor via a variable gearbox.The flow rates of both phases are controlled via rotameters.The interface is maintained at the required level by using an optical sensor,which has been previously described[22,23].Centrifugal pumps(Penax model)were used to circulate both liquid phases through the column.A scheme of the ARDC pilot scale unit used in the present study is shown in Fig.1.The column dimensions are listed in Table 2.

Two liquid–liquid test systems recommended by the European Federation of Chemical Engineering(E.F.C.E.),namely toluene-acetonewater andn-butyl acetate-acetone-water are used in the experiments.The physical properties of the liquid–liquid systems are given in Table 3.The equilibrium data were obtained from Mí?eket al.[24].

Before starting each run,the aqueous and organic phases were first mutually saturated,after which acetone was added to the dispersed(organic)phase to give a concentration of about 3.5 wt%acetone.The samples of each phase were taken at their inlets to the column and used for determination of the initial solute concentration.The rotor speed and the continuous phase flow rate were set at the desired valves and the dispersed phase gradually admitted into the column up to the desired volumetric flow rate.The interface location was then maintained at the desired height,and the system was allowed to reach steady-state.For all experiments the steady-state condition could be achieved after three times the residence times.At the end of each run,samples of the aqueous and organic phases were taken at their respective outlet.The solute concentrations were then determined by UV–visible spectroscopy.All experiments were performed far from flooding conditions.For each liquid–liquid system the operating variables were systematically varied to determine their influence on the volumetric overall mass transfer coefficient.

Drop size was determined by the photography method by means of a Nikon D3100 digital camera.Drop dimensions were then determined using Digimizer software.For elliptical drops both the vertical and horizontal axes were measured.In all cases,the stators thickness served as the reference for the drop size measurements.At least 400 drops were analyzed for each experiment to guarantee the significance of the determined Sauter mean drop diameter.The Sauter mean drop diameter was then calculated by the following equation:

wherenidenotes the number of drops of diameterdi.

The dispersed phase holdup was measured by the shut down(displacement)method where the continuous and dispersed inlet and outlet valves were shut simultaneously and the dispersion height between the initial and final interface was measured.

4.Modeling

In the design of extraction columns,an important step is to determine the required column height.Several models have been developed in recent years,since the effect of axial mixing on the performance of liquid–liquid extraction columns was recognized.The back flow and theaxial diffusion models are the most important practical approaches for considering the effect of axial mixing on the mass transfer performance of the column.In the present study,the mass transfer data are interpreted in terms of this model as described by Pratt and Stevens[25].Based upon the axial diffusion model and mass balance in the column,over the differential elements of the column with a total effective heightH,the equation set for the steady state process is established as follows,under the constant superficial velocitiesVcandVdat any given rotor speed:

Table 1Previous correlations for estimation the enhancement factor in extraction columns

Fig.1.Schematic flow diagram of the ARDC pilot plant.

Table 2Dimensions of the pilot plant ARDC

Table 3Physical properties of liquid-liquid systems investigated at 20°C[19]

In this study,the dispersed phase axial dispersion is assumed to be negligible,with the continuous phase axial mixing coefficient calculated by the following equation proposed by Kumar and Hartland[26]:

The three boundary conditions are as follows:

At the top of the column(Z=0):

By using the axial dispersion coefficient of the continuous phase,the measured continuous and dispersed phase concentrations and the boundary conditions together with equilibrium data,the continuous phase volumetric overall mass transfer coefficients(Koca)are calculated from Eqs.(13)and(14).

The dispersed phase volumetric overall mass transfer coefficients are then calculated as follows:

5.Results and Discussion

Fig.2 illustrates the effect of the rotor speed on the volumetric overall mass transfer coefficient for both studied liquid–liquid systems.This figure shows that the mass transfer performance of the column is markedly dependent on the rotor speed.As expected,the increase of agitation rate results in smaller drops.Moreover,a higher dispersed phase holdup is obtained upon increasing rotor speed.The value of the interfacial area increases with both effects.However,the overall mass transfer coefficient decreases with an increase in the rotor speed.The reduction of internal circulation and turbulence in drops leads to decrease mass transfer coefficients.The results show that the effect of interfacial area becomes more predominant than the overall mass transfer coefficient effect and consequently,the mass transfer performance will increase.At high values of the rotor speed,however,the overall mass transfer coefficient starts to fall significantly with the formation of the rigid droplets,whereby the molecular diffusion controls the drop's mass transfer rate.Forn-butylacetate–acetone–water system,the effect of interfacial area may compensated by the effect of overall mass transfer coefficient at high agitation rate and consequently,the column performance is not affected by the rotor speed in this range of operating conditions.Fig.2 also shows that the mass transfer performance is significantly influenced by the interfacial tension of the system.As seen in this figure,the column performance improves within the system of lower interfacial tension due to the formation of smaller drops and higher interfacial area.It is also found that the effect of the rotor speed on the column performance of the toluene acetone–water system(high interfacial tension)is greater than that of butyl acetate–acetone–water(medium interfacial tension),because the breakup of the drops into smaller ones is limited within the latter system due to its lower interfacial tension.

Fig.2.Variation in the volumetric overall mass transfer coefficient with rotor speed(V c=V d=1.33× 10?3 m·s?1).

As shown in Fig.3,the effect of the dispersed phase velocity clearly shows that the higher the dispersed phase flow rate,the better the mass transfer of the column becomes.Increasing the dispersed phase velocity tends to increase the mean drop size.A higher dispersed phase velocity results in not only to a lager drop formation but also the higher coalescence frequencies.As expected,an increase in the dispersed phase flow rate leads to the increased holdup because a greater volume of the organic phase is fed to the column.In the present work,it is observed that the effect of the dispersed phase holdup on the interfacial area is greater than that of the drop size and,consequently,the interfacial area increases with an increase in the dispersed phase velocity.Furthermore,the dispersed phase mass transfer coefficient increases with an increase in the drop size due to the internal circulation inside the drop.The column performance increases with the increase in both mass transfer coefficient and interfacial area.A comparison of Figs.2 and 3 shows that the effect of dispersed phase velocity on the mass transfer performance is less pronounced than rotor speed and interfacial tension.

Fig.3.Variation in the volumetric overall mass transfer coefficient with dispersed phase velocity(V c=1.33 ×10?3 m·s?1).

Fig.4.Variation in the volumetric overall mass transfer coefficient with continuous phase velocity(V d=1.33× 10?3 m·s?1).

The effect of continuous phase velocity on the volumetric overall mass transfer coefficient is illustrated in Fig.4.As seen in this figure,in the case of continuous phase velocity,no significant change in mass transfer performance is verified for the operating conditions continuous phase.Moreover,the experiments show that the mean drop size slightly increases with an increase in the continuous phase flow rate.As effects of the continuous phase velocity on the volumetric overall mass transfer coefficient,the contribution of holdup is positive and that of the mean drop size is negative because the interfacial area is directly proportional to the holdup and inversely proportional to the drop size.The results indicate that the former may be compensated by the latter and the volumetric coefficient is not affected by the continuous phase velocity within both systems.investigated in this research work.The dispersed phase holdup increases with an increase in the continuous phase velocity due to the increment of the drag force between the dispersed drops and the

Predictive correlation for enhancement factor

The overall dispersed phase mass transfer coefficient is determined by dividing the volumetric coefficient by the interfacial areaa(=6xd/d32).The experimental values of the overall mass transfer coefficient and the interfacial area are given in Table 4.

One of the main objectives of this investigation is to develop a correlation that is capable of predicting the mass transfer coefficient in ARDC columns.The experimental results for the dispersed phase overall mass transfer coefficient are compared with those of the theoretical models and the models suggested for other types of extraction columns.The values of the average relative deviation(ARD)of the calculated values of the overall mass transfer coefficient obtained by applying the previous correlations to the experimental results are summarized in Table 5.As can be seen in this table,none of the previous correlations give reasonable estimates of the overall mass transfer coefficients in ARDC column.

Therefore,the experimental values of overall mass transfer coefficient are used in Eq.(4)to define the enhancement factor.This equation is reduced to its first term in determining the R values.The experimental values ofRare also given in Table 4.

After calculating the experimental values of the enhancement factor for the investigated operating conditions,Eq.(20)is derived in terms of the Reynolds number by using the least squares method.

Table 4Experimental values of dispersed phase overall mass transfer coefficient,interfacial area,and enhancement factor

Table 5The ARD values in the predicted values of K od obtained by previous equations with respect to the experimental data

in whichVsis the slip velocity between the two phases through the column.The slip velocity between the phases is obtained as follows:

The predicted values ofRare used in Eq.(4)to calculateKodvalues.A comparison between the predicted and experimental values ofKodis depicted in Fig.5 where a good agreement is observed.The proposed method predicts the experimental values ofKodwith an average relative deviation of 7.13%.

Fig.5.Parity plot of experimental values of dispersed phase overall mass transfer coefficient against those predicted by Eqs.(4)and(20).

6.Conclusions

This paper presents an experimental study on the mass transfer performance of an asymmetric rotating disk contactor.The experimental findings show that the mass transfer performance is strongly dependent on the rotor speed and interfacial tension.Improved column performance is observed within the system of lower interfacial tension.The results show that the continuous phase velocity has little influence the value ofKoda,whileKodaincreases with an increase in the dispersed phase velocity.The results also show that the correlations developed in the other types of extractors cannot be used to predict the mass transfer performance of the ARDC column.An empirical expression for the enhancement factor as a function of Reynolds number is also proposed.The proposed correlation which predicts the enhancement factor in the ARDC column can be applied to estimate the column height in different separation processes.The present study has provided valuable information on the mass transfer characteristics of ARDC column about which there are currently limited pieces of information.

Nomenclature

ainterfacial area,m2·m?3

Bnnth coefficient in Eqs.(1)–(4)

Dmolecular diffusivity,m2·s?1

Dccolumn diameter,m

Deffeffective diffusivity,m2·s?1

DRimpeller diameter,m

d32Sauter mean drop diameter,m

Eaxial mixing coefficient,m2·s?1

E?E?tv?s number(=gΔρd232/σ)

efractional free cross-sectional area

gacceleration due to gravity,m2·s?1

Heffective height of the column,m

hccompartment height,m

Koverall mass transfer coefficient,m·s?1

mdistribution ratio

Nrotor speed,s?1

Noxnumber of‘true’transfer unit(=KocaH/Vc)

PPéclet number(=HV/E)

Peccontinuous-phase Péclet number(=d32Vs/Dc)

Qflow rate of the continuous or dispersed phase,m3·s?1

Renhancement factor for mass transfer

ReReynolds number(=d32Vsρc/ηc)

ScSchmidt number(=η/ρD)

ttime,s

Vsuperficial velocity,m·s?1

Vctrue velocity for continuous phase(=Vc/(1?xd)),m·s?1

Vsslip velocity,m·s?1

xmass fraction of acetone in continuous phase

xddispersed phase holdup

x* equilibrium mass fraction of acetone in continuous phase corresponding to dispersed phase

ymass fraction of acetone in dispersed phase

Δρ density difference between phases,kg·m?3

λnnth coefficient if Eqs.(1)–(4)

ρ density,kg·m?3

κ viscosity ratio(ηd/ηc)

η viscosity,Pa·s

σ interfacial tension,N·m?1

Subscripts

c continuous phase

d dispersed phase

o overall value

x x-phase(continuous phase in present case)

y y-phase(dispersed phase in present case)

Superscripts

* equilibrium value

° inlet to column

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