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Mass transfer coefficients in liquid–liquid extraction
Chemical Engineering and Processing 40 (2001) 477– 485 www.elsevier.com/locate/cep
Mass transfer coefficients in liquid–liquid extraction Radu Z. Tudose, Gabriela Apreotesei * Department of Chemical Engineering, Faculty of Industrial Chemistry, Technical Uni6ersity ‘Gh. Asachi’ Iasi, B-dul D. Mangeron, no. 71A, Iasi 6600, Romania Received 2 February 2000; received in revised form 6 September 2000; accepted 28 September 2000
Abstract The paper concerns with mass transfer in liquid–liquid extraction, considering that the experimental determination of the individual mass transfer coefficients during liquid–liquid extraction is still considered to be a difficult problem. In this work, the liquid–liquid extraction using a ternary system, water– acetone– carbon tetrachloride, was studied. The investigations were performed in an improved Lewis cell, in continuous and batch operation modes. The immiscible components (water–carbon tetrachloride) were mixed separately, while the acetone transfer was carried out in the two directions through an interface with a constant area of the surface, created between the two phases. The specific fluxes, the driving forces and the individual and overall mass transfer coefficients were determined by measuring the inflow and outflow acetone concentrations in the two phases, together with the equilibrium data. The values of the individual mass transfer coefficients in each phase were correlated and expressed in the form of criterial equation, as Sh= bRe pSc m. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Mass transfer coefficients; Acetone transfer; Criterial equation
1. Introduction The mass transfer of a solute between two immiscible liquids is an important method for separation and purification, having numerous industrial applications, even if the process has a very complex phenomenology. Different mechanisms and models were proposed to clarify the phenomenological aspects of the liquid –liquid extraction [1]. Based on the theoretical background, the researchers attempt to explain the specific aspects of the liquid –liquid extraction and to develop equations for the mass transfer coefficients, taken as a measure of mass transfer intensity. The most used models are based on film and penetration theories, and considered that the equilibrium is established at the interface, so that the interfacial resistance is negligible. Brenner and Leal [2,3] developed a theory of the interfacial resistance to the solute transfer between two liquids. Their theory is based on the Brownian transport of the molecules of the solute between the immis* Corresponding author. E-mail address: [email protected] (G. Apreotesei).
cible phases that argue the presence of such a resistance. Davies in 1961 and Chandrasekhar and Holscher [4] have noted the existence of an interfacial resistance in several systems, having a value comparable to that found in the liquid bulk. Also, there are numerous aspects elucidated incompletely regarding mass transfer between two liquids — mass transfer direction effects on the process performance, location of mass transfer resistance (in liquid bulk, near or at the interface). Colburn and Welsh [5] have determined the individual mass transfer coefficients experimentally, using two partial miscible liquids, which were reciprocating saturated. Often, the individual coefficients in both continuous and dispersed phase, respectively, were obtained using ‘single drop’ method. The experimental systems were characterised by very large or very small values for the partition coefficient [6]. The method has the disadvantage of not allowing the determination of the transfer resistance in both phases and the interfacial resistance. Also, the drop surface and velocity are dependent on its size, which is difficult to determine.
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R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
Under certain improvements, the Lewis cell [7] continues to be one of the most efficient methods to determine the mass transfer coefficients in any ternary system, as G.L. Standart [6] had already asserted. As a result, numerous researchers designed such modified cells [6]. The cell used in this work provides the mixing of the liquids apart, in its two compartments. Some devices, which were mounted inside the compartments of the cell, moderate the interfacial turbulence. Also, the continuous operation mode offers numerous operational possibilities for mass transfer coefficient evaluation. The present paper presents our modified Lewis cell and the results from extraction experiments, as well as the mass transfer coefficients calculated using the results.
2. Experimental apparatus and procedure The cell used for the study of the mass transfer in continuous and batch operation mode was made out of Cr – Ni stainless steel and glass (Fig. 1). The compartments (1) and (2), where the aqueous and organic phases are introduced, have the sidewalls made out of glass (3). The two covers are equipped with metallic seals (4), which support the bearings of the blade mixer (5), driven by separate engines, with changeable speed, between 80–1200 rpm. The compartments are separated by a metallic plate (6), which has a central hole with the diameter of 30 mm, where the interface between the two liquids was located (7). The hole-diameter can be modified using several rings. Several orifices were also made in the central plate to assure the feed the compartments with liquids.
To avoid the interface deformation by the turbulence, three baffles were placed inside each compartment. Each baffle has four holes of 4-mm diameter. Similar holes were made on the agitator blade surfaces. The experimental set-up, which contains the extraction cell, was equipped with feed tanks, and devices, which maintain a constant level, or indicated the flow rates, the temperature, the speed, as well as sample parts. The experiments were performed using the following ternary system, water–acetone–CCl4, in continuous operating mode. Several experiments were carried out in batch mode. Both liquids — water and CCl4 — were distilled previously. In the first set of experiments, the feeding of the cell was made with an aqueous solution containing 7 wt.% acetone and pure CCl4. In the second set of experiments, the feeding was made with CCl4 containing 7% acetone and distilled water as well. This way, the mass transfer of the solute in two directions has been studied. When the distillation of the mixture acetone–CCl4 was carried out, the azeotrope of CCl4 with acetone was considered. The feeding flow rates of the organic phase ranged between 1.5 and 6.6 kg/h, while those of the aqueous phase were 0.87–3.5 kg/h. In batch operation mode, the concentration dynamics were observed for values of the mean residence time ranged between 10 and 90 min. The acetone concentrations to the entrance and the exit for both phases were determined using the refractometric method. Acetone balance was used to verify the measuring accuracy. To establish the dominant resistance, the experiments occurred in the following conditions, mixing of the aqueous phase only; mixing of the organic phase only; mixing of both phases.
3. Theoretical concerns
Fig. 1. Extraction cell.
The Lewis cell is certainly one of the most recommended apparatus for the determination of the individual mass transfer coefficients in liquid–liquid extraction. Its advantage consists in the possibility to be applied for any system containing an initial solvent A and a solute B, from which the solute is extracted with the solvent S. The accuracy of the results is dependent on several conditions, each compartment of the cell must be equipped with stirrers, act by separate engines, having a variable speed; the stirrer configuration must be adequate to assure the homogenisation of the liquid in each compart-
R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
Fig. 2. The distribution of solute concentration between the two phases (X Y).
Fig. 3. The distribution of solute concentration between the two phases (Y X).
ment, before the perturbation of the interface takes place; each cell must be equipped with perforated blades, to assure mixing intensification and to prevent the rotation of phases and dead zones formation; the cell must operate continuously in order to attain steady-state conditions in both compartments; the cell can operate with mixing in both compartment, in one of the two compartments, or without mixing, but with continuous feed of the liquid phases.
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This cell type will allow to establish the phase where the mass transfer resistance is decisive, as well as the mixing conditions in one phase, when the resistance of the other phase becomes determinant. In this apparatus, the influences of mass transfer direction on the overall mass transfer rates of the solute can be evaluated. Also, it is possible to produce evidence for the presence of an interfacial resistance. The experimental results allow the accurate calculation of the individual and global mass transfer coefficients. When a single phase is mixed in a cell with steadystate conditions, the outlet solute concentration can vary up to a certain agitator speed, and then it remains constant. This independence between the solute concentration and the agitator speed marks the movement of the basic resistance of liquid–liquid extraction in the other phase. For example, in the system water–acetone–CCl4 when only the organic phase is stirred and the acetone transfer takes place from the aqueous to the organic phase, the distribution of solute concentration between the two phases is shown by the curves 1, in Fig. 2. In this situation, the transfer resistance is concentrated into the aqueous phase, while in the stirred compartment, the concentration becomes independent on the agitator speed, possibly as a consequence of achieving a perfectly mixed system. In these conditions, the value Y( S,1 can be reached momentarily in the stirred phase. The resistance to diffusion is concentrated near the interface, the driving force inside it being given by the difference (YS,e − Y( S,1). The driving force in the non-agitated phase is (XA,i − XA,e). Curve 2 show the distribution of the concentration between the two phases, when only the aqueous phase was stirred. In the stirred phase, the driving force is given by the difference (XA,i − XA,e), while in the organic phase by (YS,e −Y( S,2). When both phases are stirred at sufficiently high speeds (n\ 800 rpm), the solute distribution is that showed by curve 3, for the case of the independence of the concentration on the stirring intensity and for a flat interface. In both compartments of the cell, a perfectly mixed flow model with the transfer resistance concentrated in the vicinity of the interface can be assumed. Fig. 3 shows the distribution of the solute concentration when it is transferred from organic to aqueous phase. Curve 1 illustrates the situation when only the organic phase is stirred. Curve 2 is plotted for the situation when the aqueous phase is stirred only. If both phases are stirred (curve 3), the mass transfer resistance is concentrated in the vicinity of the interface and the driving force is given by the difference (XS,e − X( S,3) for the aqueous phase and by (YA,i − YA,e) for the organic phase, respectively. The equilibrium data for the system A–B–S are plotted in Fig. 4 (curve LF). A certain point P, located
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on the y axis represents the concentration of the solute in the phase A, with solvent pure (YA,i). The value of the equilibrium concentration (YA,e) is given by the point Q. PQ is the operating line for the transfer of the solute from the organic to the aqueous phase. A point belonging to the line PQ will give the concentration of the phases, which leave the components of the cell. It will be as close on the equilibrium curve as more intense the stirring is. In this situation, the equations for the specific transferred flux are the following, nB,YX = kX,YX(XS,e −XS,i) = kY,YX(YA,i −YA,e)
(1)
The slope of the operating line, PQ results from Eq. (1). −
1 lYX
=
kY,YX XS,i −XS,e = kX,YX YA,i −YA,e
(2)
The non-ideal systems do not respect the Nernst law, so that the partition coefficient will be dependent on the concentration. In these conditions, it is possible to have for the points L, Q, F the following co-ordinates, Y*A,i =iYX,LXS,i YA,e =iYX,QXS,e YA,i = iYX,FX*S,i
(3)
The equations for the specific flux of the transferred solute Eq. (4), containing the overall coefficients and the overall driving forces can be derived using Eqs. (1) and (3).
nB = KX,YX
iYX,F X* −XS,i iYX,Q S,i
= KY,YX YA,i −
iYX,Q Y* iYX,L A,i
(4)
In Eq. (4), the overall coefficients were calculated using the Eq. (5). 1 KX,YX
=
1 kX,YX
+
1 kY,YXiYX,Q
Fig. 4. The equilibrium curve and operating lines.
1 1 i = + YX,Q KY,YX kY,YX kX,YX
(5)
When the equilibrium curve is a straight line, it results that. iYX,L = iYX,Q = iYX,F = constant
(6)
So that the expression for the overall driving forces becomes less complex. The inversion of mass transfer direction implies the transport of the solute B from the initial solvent S to the extraction solvent A. The solute balance between the two phases leads to correlations alike Eq. (1). The operating line from Fig. 4 is given by the line MN. The concentration of the phase that leaves the cell will be given by a point located on this line. With the increasing residence time of phases in the cell, its position will be as close on the point N, which corresponds to the equilibrium.
4. Experimental results The cell used for experimentation offers a very small specific surface (1.4 m2/m3), calculated as a ratio between the interface area between phases and the volume of both compartments of the cell. In these conditions, it is expected that the specific flux of the transferred solute is small and dependent on the mean residence time of the phases in the cell compartments. This manner to relate the surface to both phases, volume is applied in the extraction columns of industrial scale. The dependence between the outlet phase concentrations and the mean residence time was determined in both batch and continuous operation modes. The experiments were carried out with an acetone– CCl4 solution, containing 7 wt.% acetone and distilled water. The concentrations were determined for the two situations, stirring of a single phase; and stirring of both phases. The results obtained in the continuous operation mode have shown that the outlet concentration becomes constant and independent of the agitator speed for a stirring speed of 850 rpm. The extraction was analysed at 850 rpm in both continuous and batch mode. Figs. 5 and 6 reveal the outlet concentration changes in both phases as a function on the mean residence time in the cell compartments. First of all, it is obvious that these concentrations tend to the equilibrium state very slowly, due to the small specific surface. Secondly, if only the organic phase is stirred, the outlet concentration diminution takes place much slower. This fact suggests that a dominant resistance is located in the non-stirred aqueous phase. The stirring of the aqueous phase will lead to a faster decreasing of the outlet concentration due to a greater transfer resistance and a
R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
Fig. 5. Acetone concentration changes in the organic phase with the cell mean residence time, in discontinuous operation mode, (1) the organic phase stirred; (2) the aqueous phase non-stirred; the aqueous phase stirred, the organic phase non-stirred; (3) both phases stirred.
Fig. 6. The variation of the acetone concentration in the organic phase with the cell means residence time, in continuous operation mode.
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librium state can be reached slowly because the specific surface is very small. This fact leads to the conclusion that it is possible to work at different mean residence time values, at steady-state conditions. Afterwards, the experimental investigations were performed in steady state only, having in view the dependence of the solute concentration on the stirrer speed and considering the mass transfer in both directions. Also, the stirring of a single or both phases was considered. The dependence of the outlet solute concentration on stirrer speeds was represented in Figs. 7 and 8. In all situations, the starting concentration in the liquid phase decreases with the agitator speed. The mass transfer intensity in a single or both phases shows a limiting value. When this limit is reached, the concentration remains constant. The independence of the concentration on the agitator speed indicates that the basic resistance is transferred in the other phase. When both phases are stirred, the decrease in the concentration solute is faster, and then remains constant for a certain value of the agitator speed. This constant value proves that the hydrodynamic changes do not affect the mass transfer and that the resistances in both phases can be concentrated near the interface [3] (Fig. 3). The specific fluxes of the transferred solute from aqueous to organic phase, (Fig. 9) and vice-versa (Fig. 10) will increase continuously with the agitator speed. This increase shows a limiting value, beyond which the flux transferred becomes independent on the agitator speed. When only the organic phase is stirred at speed values ranged between 500 and 900 rpm (9000BReag B 10 000), the flux of solute transferred from the aqueous to the organic phase remains constant. Its value is of 20 kgAc per hm2 when the solute is transferred from the aqueous to the organic phase, and 70 kgAc per hm2 when this transfer takes place from the organic to the aqueous phase. These values are less than those obtained when only the aqueous or both phases (organic
Fig. 7. Dependence of the outlet acetone concentration on the stirrer speed (X Y).
reduced value of the molecular diffusion coefficient in this phase, compared with the organic one. As was expected, the mass transfer intensity increased by stirring both phases and the outlet concentration tends close to the equilibrium value. From the above figures, it is evident that the equi-
Fig. 8. Variation of the acetone concentration on mixing intensity (Y X).
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Fig. 9. Dependence of the specific fluxes of the transferred solute on mixing intensity, for mass transfer from the aqueous phase to the organic one.
Fig. 10. Variation of the specific flux of solute on mixing intensity, for the acetone transfer from the organic to the aqueous phase.
and aqueous) are agitated. These significant differences between the acetone transferred fluxes in the above — mentioned mixing conditions can be the result of several influences, such as. The different values of the diffusion coefficient in the aqueous phase (0.923×10 − 5 cm2/s) and the organic one (1.763× 10 − 5 cm2/s). The values of the partition coefficient between the two phases and, the effect Marangoni, but in a less measure. The values of the partition coefficient for the transfer from the aqueous to the organic phase ranged between 0.175 and 0.52 and are dependent on the concentration. For our experimental conditions, the partition coefficient is 0.25. The results obtained using the following ternary systems, 1. water–acetone–carbon tetrachloride (present paper); 2. water–acetone–chloroform (unpublished results); allowed us to conclude that the smallest specific flux always results when the mixing is performed only in that phase were the equilibrium concentration of the solute has a low value (CCl4, for the first system; water,
for second one). Of course, when the mixing of the phases occurs, the influence of the molecular diffusion coefficients is negligible. Also, analysing the cell construction details, the phase contacting mode as well as the shape and the position of the interface, we concluded that it is less probable that the Marangoni effect influences the transfer of the solute. The specific flux of the solute increases when only the aqueous is stirred up to a Reynolds number of Re $ 10 000 and then the solute concentration remains constant. Also, a higher specific flux is observed when the transfer occurs from the organic to the aqueous phase. When both phases are stirred, the dependence between the specific flux and the agitator speed is almost analogous to that resulted when only the aqueous phase is stirred. The absolute values of this flux are, however, higher than those in the above-mentioned situation. For agitator speeds higher than that corresponding to the point G, the specific flux shows a significant increase, as a result of the interface deformation. Visually, it can be observed that, the surface of the interface is increased as a consequence of waving for speeds higher than 1000 rpm, that is Reag \ 14 000. The experimental values of the specific fluxes as well as the individual or overall driving forces resulted from the equilibrium diagram allow for the calculation of the individual Eq. (1) or the overall Eq. (4) mass transfer coefficients. The driving force values result from the plot of the equilibrium curve together with the operation lines for both directions of the solute transfer (see Fig. 4). The equilibrium curve (LF) and the operating lines (PQ and MN) having the slope given by a relation alike Eq. (2), were plotted in Fig. 11. The points representing the phase outlet concentrations in the domain DG (see Figs. 9 and 10) are located on the operating lines PQ and MN, respectively. For example, the point 1 represents the outlet acetone concentration when the solute transfer occurs from the organic to the aqueous phase and the organic phase is stirred only; the point 2 results when only the aqueous phase is stirred while point 3 corresponds to both phases stirring. The individual and global driving forces are given by the relations from Table 1. The position of the points 1–3 tends to the equilibrium state (the point Q), if the mean residence times in the cell compartments increase. This statement was verified using the points located on the three curves from Fig. 6. All points are located on the line PQ (not represented in Fig. 11). The flow inside the cell can be assumed mixed perfectly at higher agitator speeds (Reag \ 10 000) located between the positions D and G (Figs. 9 and 10). In these conditions, the initial concentration of the solute suddenly decreases at a value, which corresponds to the
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483
Fig. 11. The operating lines and the equilibrium curves for acetone transfer in both directions.
outlet concentration and remains constant in steadystate conditions. The corresponding driving forces will have smaller values, so that the mass transfer coefficients increased. The actual driving forces can be evaluated with the equations in Table 2. The statistical analysis of the experimental data allowed for the development of several criterial equations, which can be used to calculate the individual and overall mass transfer coefficients. The individual coefficient values result from the ratio between the specific transferred solute flux (Figs. 9 and 10) and the individual driving forces, given in Table 1. The criterial equations have the following general form, Sh = bRe pagSc 1/3 where Sh = dk/DB, Reag =nd2z/p and Sc =k/DB. Four sets of criterial equation result, for each direction of the mass transfer (Table 3). The equations can be applied for Reag B11 000, if the transfer of the solute occurs from the organic to the aqueous phase. For Reag \11 000, the value of the individual mass transfer coefficients remains constant (the domain DG in Figs. 9 and 10). k%X,YX =4.57×10 − 4 m/s and k%Y,YX = 4.11× 10 − 4 m/s If the transfer of the solute takes place from the organic to the aqueous phase, the equations hold for Reag B10 000. For Reag higher than 10 000, the individual mass transfer coefficient values remain constant (domain DG).
k%X,XY = 11.11×10 − 4 m/s
and
k%Y,XY = 14.01×10 − 4 m/s Based on the assumption that an intense agitation occurs and the perfectly mixed flow model is valid, the individual mass transfer coefficients were calculated using the driving forces obtained from Table 2. These Table 1
X Y
YX
Theoretical o6erall dri6ing forces DX tg,XY =XA,i DY tg,XY =Y*S,i
DX tin,YX =X*S,i DY tin,YX =YA,i
Theoretical indi6idual dri6ing forces DX tin,XY =XA,i−XA,e DX tin,YX =XS,e DY tin,XY =YS,e DY tin,YX =YA,i−YA,e
Table 2
The actual driving force X Y
YX
DX rXY,1 =XA,1−XA,e DX rXY,2 =XA,2−XA,e DX rXY,3 =XA,3−XA,e DY rXY,1 =YS,e−YS,1 DY rXY,2 =YS,e−YS,2 DY rXY,3 =YS,e−YS,3
DX rYX,1 =XS,e−XS,1 DX rYX,2 =XS,e−XS,2 DX rYX,3 =XS,e−XS,3 DY rYX,1 =YA,1−YA,e DY rYX,2 =YA,2−YA,e DY rYX,3 =YA,3−YA,e
R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
484 Table 3
Criterial equations Stirred phase Aqueous phase Organic phase Both phases
XY 1/3 ShX,XY = 1.441Re 1.6013Sc X 0.7097 1/3 ShY,XY = 1665.613Re ag,Y Sc Y 1/3 ShX,Y,XY = bRe 1.56 Sc when b= 2,3 for organic phase ag,X,Y X,Y and 3 for aqueous phase
will be great values, independent of the agitator speed, and for a perfectly plane interface. The overall mass transfer coefficients were calculated directly, from the transferred specific fluxes (Figs. 9 and 10, curve ODG) and the overall driving forces, given by the relations Eq. (4). These coefficients were also calculated using the equations, for the individual mass transfer coefficients from Table 3 and by substituting then in the relations Eq. (5). The results plotted in Fig. 12 demonstrate the validity of the Eq. (5). The overall coefficients resulted from the specific fluxes and theoretical overall driving forces were represented on the y-axis, while the values calculated using the experimental individual coefficients were represented on the x-axis. The overall mass transfer coefficient values in both directions of transfer calculated using the two procedures mentioned above are very close, being placed on lines, which partially superpose one to another.
5. Conclusions
The experimental installation contains an improved Lewis cell as a central element, which can be exploited in continuous and discontinuous modes, also allowing for the steady-state operation for a short time when continuous running is used. The mass transfer surface can be rigorously measured. The cell construction allows for individual or simultaneous stirring of the phases, for a large range of speeds. Logically, the mass transfer intensity in one phase shows significant increases, which were dependent on the agitator speed. This increase occurs up to a certain limit beyond the specific flux and becomes independent on the speed. The specific flux is greater when the organic phase is stirred only; comparative to the situation wen the mixing of the aqueous or both phases occurs, irrespective of transfer direction. The flux of the transferred solute is higher if the
Y X 1/3 ShX,YX =155.927Re 0.9889 ag,X Sc X 1.1431 1/3 ShY,YX =13.824Re ag,Y Sc Y 1/3 ShX,Y,YX =bRe 0.91 ag,X,YSc X,Y when b =354 for organic phase and 333 for aqueous phase
transfer occurs from the organic to the aqueous phase that is in the direction of the increasing acetone solubility. The individual and overall mass transfer coefficients in both phases were calculated. The individual coefficients were correlated in the form of criterial equations. The experimental values of the overall coefficients are very close to the calculated values using the correlations developed for the individual coefficients and the partition constants.
Appendix A. Notations
Ac d D k k%
acetone agitator diameter diffusion coefficient individual mass transfers coefficient when only one phase is stirred individual mass transfer coefficient when both phases are stirred
Fig. 12. Comparison of the experimental and calculated values of the overall mass transfer coefficient.
R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
K l n nB t X X X* Y Y Y* z k p i
overall mass transfer coefficient the reverse of the slope for the operating line stirrer speed the specific flux of the transferred solute residence time inside the compartments of the cell acetone concentration in the aqueous phase acetone concentration in the aqueous phase, when perfectly mixed model can be assumed acetone equilibrium concentration in the aqueous phase acetone concentration in the organic phase acetone concentration in the organic phase, when perfectly mixed model can be assumed acetone equilibrium concentration in the organic phase density of solution kinematic viscosity dynamic viscosity partition coefficient
Subscripts ag agitation A initial solution B solute e equilibrium exp experimental g overall i initial in individual r actual S solvent
.
485
t theoretical X aqueous phase XY the direction of the transfer is from aqueous to organic phase Y organic phase YX the transfer direction is from organic to aqueous phase 1 the organic phase is stirred only 2 the aqueous phase is stirred only 3 both phases are stirred
References [1] J.T. Davies, E.K. Rideal, Interfacial Phenomena, Academic Press, New York, 1961. [2] H. Brenner, L.G. Leal, Interfacial resistance to interphase mass transfer in quiescent two-phase systems, Am. Inst. Chem. Eng. J. 24 (2) (1978) 246 – 254. [3] H. Brenner, L.G. Leal, Conservation and constitutive equations for adsorbed species undergoing surface diffusion and convection at a fluid– fluid interface, J. Colloid Interf. Sci. 88 (1982) 136 –184. [4] S. Chandrasekhar, L.G. Leal, Mass transfer studies across liquid/ liquid interfaces (use of analytical ultracentrifuge), Am. Inst. Chem. Eng. J. 21 (1) (1975) 103 – 109. [5] A.P. Colburn, D.G. Welsh, Mass transfer coefficients, Trans. Am. Inst. Chem. Eng. 38 (1942) 179 – 186. [6] H.R.C. Pratt, Interphase mass transfer, in: Handbook of Solvent Extraction, Wiley/Interscience, New York, 1983. [7] J.B. Lewis, The mechanism of mass transfer of solutes across liquid – liquid interfaces, Chem. Eng. Sci. 3 (1954) 248 – 259.
Mass transfer coefficients in liquid–liquid extraction Radu Z. Tudose, Gabriela Apreotesei * Department of Chemical Engineering, Faculty of Industrial Chemistry, Technical Uni6ersity ‘Gh. Asachi’ Iasi, B-dul D. Mangeron, no. 71A, Iasi 6600, Romania Received 2 February 2000; received in revised form 6 September 2000; accepted 28 September 2000
Abstract The paper concerns with mass transfer in liquid–liquid extraction, considering that the experimental determination of the individual mass transfer coefficients during liquid–liquid extraction is still considered to be a difficult problem. In this work, the liquid–liquid extraction using a ternary system, water– acetone– carbon tetrachloride, was studied. The investigations were performed in an improved Lewis cell, in continuous and batch operation modes. The immiscible components (water–carbon tetrachloride) were mixed separately, while the acetone transfer was carried out in the two directions through an interface with a constant area of the surface, created between the two phases. The specific fluxes, the driving forces and the individual and overall mass transfer coefficients were determined by measuring the inflow and outflow acetone concentrations in the two phases, together with the equilibrium data. The values of the individual mass transfer coefficients in each phase were correlated and expressed in the form of criterial equation, as Sh= bRe pSc m. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Mass transfer coefficients; Acetone transfer; Criterial equation
1. Introduction The mass transfer of a solute between two immiscible liquids is an important method for separation and purification, having numerous industrial applications, even if the process has a very complex phenomenology. Different mechanisms and models were proposed to clarify the phenomenological aspects of the liquid –liquid extraction [1]. Based on the theoretical background, the researchers attempt to explain the specific aspects of the liquid –liquid extraction and to develop equations for the mass transfer coefficients, taken as a measure of mass transfer intensity. The most used models are based on film and penetration theories, and considered that the equilibrium is established at the interface, so that the interfacial resistance is negligible. Brenner and Leal [2,3] developed a theory of the interfacial resistance to the solute transfer between two liquids. Their theory is based on the Brownian transport of the molecules of the solute between the immis* Corresponding author. E-mail address: [email protected] (G. Apreotesei).
cible phases that argue the presence of such a resistance. Davies in 1961 and Chandrasekhar and Holscher [4] have noted the existence of an interfacial resistance in several systems, having a value comparable to that found in the liquid bulk. Also, there are numerous aspects elucidated incompletely regarding mass transfer between two liquids — mass transfer direction effects on the process performance, location of mass transfer resistance (in liquid bulk, near or at the interface). Colburn and Welsh [5] have determined the individual mass transfer coefficients experimentally, using two partial miscible liquids, which were reciprocating saturated. Often, the individual coefficients in both continuous and dispersed phase, respectively, were obtained using ‘single drop’ method. The experimental systems were characterised by very large or very small values for the partition coefficient [6]. The method has the disadvantage of not allowing the determination of the transfer resistance in both phases and the interfacial resistance. Also, the drop surface and velocity are dependent on its size, which is difficult to determine.
0255-2701/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 5 - 2 7 0 1 ( 0 0 ) 0 0 1 4 6 - X
478
R.Z. Tudose, G. Apreotesei / Chemical Engineering and Processing 40 (2001) 477–485
Under certain improvements, the Lewis cell [7] continues to be one of the most efficient methods to determine the mass transfer coefficients in any ternary system, as G.L. Standart [6] had already asserted. As a result, numerous researchers designed such modified cells [6]. The cell used in this work provides the mixing of the liquids apart, in its two compartments. Some devices, which were mounted inside the compartments of the cell, moderate the interfacial turbulence. Also, the continuous operation mode offers numerous operational possibilities for mass transfer coefficient evaluation. The present paper presents our modified Lewis cell and the results from extraction experiments, as well as the mass transfer coefficients calculated using the results.
2. Experimental apparatus and procedure The cell used for the study of the mass transfer in continuous and batch operation mode was made out of Cr – Ni stainless steel and glass (Fig. 1). The compartments (1) and (2), where the aqueous and organic phases are introduced, have the sidewalls made out of glass (3). The two covers are equipped with metallic seals (4), which support the bearings of the blade mixer (5), driven by separate engines, with changeable speed, between 80–1200 rpm. The compartments are separated by a metallic plate (6), which has a central hole with the diameter of 30 mm, where the interface between the two liquids was located (7). The hole-diameter can be modified using several rings. Several orifices were also made in the central plate to assure the feed the compartments with liquids.
To avoid the interface deformation by the turbulence, three baffles were placed inside each compartment. Each baffle has four holes of 4-mm diameter. Similar holes were made on the agitator blade surfaces. The experimental set-up, which contains the extraction cell, was equipped with feed tanks, and devices, which maintain a constant level, or indicated the flow rates, the temperature, the speed, as well as sample parts. The experiments were performed using the following ternary system, water–acetone–CCl4, in continuous operating mode. Several experiments were carried out in batch mode. Both liquids — water and CCl4 — were distilled previously. In the first set of experiments, the feeding of the cell was made with an aqueous solution containing 7 wt.% acetone and pure CCl4. In the second set of experiments, the feeding was made with CCl4 containing 7% acetone and distilled water as well. This way, the mass transfer of the solute in two directions has been studied. When the distillation of the mixture acetone–CCl4 was carried out, the azeotrope of CCl4 with acetone was considered. The feeding flow rates of the organic phase ranged between 1.5 and 6.6 kg/h, while those of the aqueous phase were 0.87–3.5 kg/h. In batch operation mode, the concentration dynamics were observed for values of the mean residence time ranged between 10 and 90 min. The acetone concentrations to the entrance and the exit for both phases were determined using the refractometric method. Acetone balance was used to verify the measuring accuracy. To establish the dominant resistance, the experiments occurred in the following conditions, mixing of the aqueous phase only; mixing of the organic phase only; mixing of both phases.
3. Theoretical concerns
Fig. 1. Extraction cell.
The Lewis cell is certainly one of the most recommended apparatus for the determination of the individual mass transfer coefficients in liquid–liquid extraction. Its advantage consists in the possibility to be applied for any system containing an initial solvent A and a solute B, from which the solute is extracted with the solvent S. The accuracy of the results is dependent on several conditions, each compartment of the cell must be equipped with stirrers, act by separate engines, having a variable speed; the stirrer configuration must be adequate to assure the homogenisation of the liquid in each compart-
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Fig. 2. The distribution of solute concentration between the two phases (X Y).
Fig. 3. The distribution of solute concentration between the two phases (Y X).
ment, before the perturbation of the interface takes place; each cell must be equipped with perforated blades, to assure mixing intensification and to prevent the rotation of phases and dead zones formation; the cell must operate continuously in order to attain steady-state conditions in both compartments; the cell can operate with mixing in both compartment, in one of the two compartments, or without mixing, but with continuous feed of the liquid phases.
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This cell type will allow to establish the phase where the mass transfer resistance is decisive, as well as the mixing conditions in one phase, when the resistance of the other phase becomes determinant. In this apparatus, the influences of mass transfer direction on the overall mass transfer rates of the solute can be evaluated. Also, it is possible to produce evidence for the presence of an interfacial resistance. The experimental results allow the accurate calculation of the individual and global mass transfer coefficients. When a single phase is mixed in a cell with steadystate conditions, the outlet solute concentration can vary up to a certain agitator speed, and then it remains constant. This independence between the solute concentration and the agitator speed marks the movement of the basic resistance of liquid–liquid extraction in the other phase. For example, in the system water–acetone–CCl4 when only the organic phase is stirred and the acetone transfer takes place from the aqueous to the organic phase, the distribution of solute concentration between the two phases is shown by the curves 1, in Fig. 2. In this situation, the transfer resistance is concentrated into the aqueous phase, while in the stirred compartment, the concentration becomes independent on the agitator speed, possibly as a consequence of achieving a perfectly mixed system. In these conditions, the value Y( S,1 can be reached momentarily in the stirred phase. The resistance to diffusion is concentrated near the interface, the driving force inside it being given by the difference (YS,e − Y( S,1). The driving force in the non-agitated phase is (XA,i − XA,e). Curve 2 show the distribution of the concentration between the two phases, when only the aqueous phase was stirred. In the stirred phase, the driving force is given by the difference (XA,i − XA,e), while in the organic phase by (YS,e −Y( S,2). When both phases are stirred at sufficiently high speeds (n\ 800 rpm), the solute distribution is that showed by curve 3, for the case of the independence of the concentration on the stirring intensity and for a flat interface. In both compartments of the cell, a perfectly mixed flow model with the transfer resistance concentrated in the vicinity of the interface can be assumed. Fig. 3 shows the distribution of the solute concentration when it is transferred from organic to aqueous phase. Curve 1 illustrates the situation when only the organic phase is stirred. Curve 2 is plotted for the situation when the aqueous phase is stirred only. If both phases are stirred (curve 3), the mass transfer resistance is concentrated in the vicinity of the interface and the driving force is given by the difference (XS,e − X( S,3) for the aqueous phase and by (YA,i − YA,e) for the organic phase, respectively. The equilibrium data for the system A–B–S are plotted in Fig. 4 (curve LF). A certain point P, located
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on the y axis represents the concentration of the solute in the phase A, with solvent pure (YA,i). The value of the equilibrium concentration (YA,e) is given by the point Q. PQ is the operating line for the transfer of the solute from the organic to the aqueous phase. A point belonging to the line PQ will give the concentration of the phases, which leave the components of the cell. It will be as close on the equilibrium curve as more intense the stirring is. In this situation, the equations for the specific transferred flux are the following, nB,YX = kX,YX(XS,e −XS,i) = kY,YX(YA,i −YA,e)
(1)
The slope of the operating line, PQ results from Eq. (1). −
1 lYX
=
kY,YX XS,i −XS,e = kX,YX YA,i −YA,e
(2)
The non-ideal systems do not respect the Nernst law, so that the partition coefficient will be dependent on the concentration. In these conditions, it is possible to have for the points L, Q, F the following co-ordinates, Y*A,i =iYX,LXS,i YA,e =iYX,QXS,e YA,i = iYX,FX*S,i
(3)
The equations for the specific flux of the transferred solute Eq. (4), containing the overall coefficients and the overall driving forces can be derived using Eqs. (1) and (3).
nB = KX,YX
iYX,F X* −XS,i iYX,Q S,i
= KY,YX YA,i −
iYX,Q Y* iYX,L A,i
(4)
In Eq. (4), the overall coefficients were calculated using the Eq. (5). 1 KX,YX
=
1 kX,YX
+
1 kY,YXiYX,Q
Fig. 4. The equilibrium curve and operating lines.
1 1 i = + YX,Q KY,YX kY,YX kX,YX
(5)
When the equilibrium curve is a straight line, it results that. iYX,L = iYX,Q = iYX,F = constant
(6)
So that the expression for the overall driving forces becomes less complex. The inversion of mass transfer direction implies the transport of the solute B from the initial solvent S to the extraction solvent A. The solute balance between the two phases leads to correlations alike Eq. (1). The operating line from Fig. 4 is given by the line MN. The concentration of the phase that leaves the cell will be given by a point located on this line. With the increasing residence time of phases in the cell, its position will be as close on the point N, which corresponds to the equilibrium.
4. Experimental results The cell used for experimentation offers a very small specific surface (1.4 m2/m3), calculated as a ratio between the interface area between phases and the volume of both compartments of the cell. In these conditions, it is expected that the specific flux of the transferred solute is small and dependent on the mean residence time of the phases in the cell compartments. This manner to relate the surface to both phases, volume is applied in the extraction columns of industrial scale. The dependence between the outlet phase concentrations and the mean residence time was determined in both batch and continuous operation modes. The experiments were carried out with an acetone– CCl4 solution, containing 7 wt.% acetone and distilled water. The concentrations were determined for the two situations, stirring of a single phase; and stirring of both phases. The results obtained in the continuous operation mode have shown that the outlet concentration becomes constant and independent of the agitator speed for a stirring speed of 850 rpm. The extraction was analysed at 850 rpm in both continuous and batch mode. Figs. 5 and 6 reveal the outlet concentration changes in both phases as a function on the mean residence time in the cell compartments. First of all, it is obvious that these concentrations tend to the equilibrium state very slowly, due to the small specific surface. Secondly, if only the organic phase is stirred, the outlet concentration diminution takes place much slower. This fact suggests that a dominant resistance is located in the non-stirred aqueous phase. The stirring of the aqueous phase will lead to a faster decreasing of the outlet concentration due to a greater transfer resistance and a
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Fig. 5. Acetone concentration changes in the organic phase with the cell mean residence time, in discontinuous operation mode, (1) the organic phase stirred; (2) the aqueous phase non-stirred; the aqueous phase stirred, the organic phase non-stirred; (3) both phases stirred.
Fig. 6. The variation of the acetone concentration in the organic phase with the cell means residence time, in continuous operation mode.
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librium state can be reached slowly because the specific surface is very small. This fact leads to the conclusion that it is possible to work at different mean residence time values, at steady-state conditions. Afterwards, the experimental investigations were performed in steady state only, having in view the dependence of the solute concentration on the stirrer speed and considering the mass transfer in both directions. Also, the stirring of a single or both phases was considered. The dependence of the outlet solute concentration on stirrer speeds was represented in Figs. 7 and 8. In all situations, the starting concentration in the liquid phase decreases with the agitator speed. The mass transfer intensity in a single or both phases shows a limiting value. When this limit is reached, the concentration remains constant. The independence of the concentration on the agitator speed indicates that the basic resistance is transferred in the other phase. When both phases are stirred, the decrease in the concentration solute is faster, and then remains constant for a certain value of the agitator speed. This constant value proves that the hydrodynamic changes do not affect the mass transfer and that the resistances in both phases can be concentrated near the interface [3] (Fig. 3). The specific fluxes of the transferred solute from aqueous to organic phase, (Fig. 9) and vice-versa (Fig. 10) will increase continuously with the agitator speed. This increase shows a limiting value, beyond which the flux transferred becomes independent on the agitator speed. When only the organic phase is stirred at speed values ranged between 500 and 900 rpm (9000BReag B 10 000), the flux of solute transferred from the aqueous to the organic phase remains constant. Its value is of 20 kgAc per hm2 when the solute is transferred from the aqueous to the organic phase, and 70 kgAc per hm2 when this transfer takes place from the organic to the aqueous phase. These values are less than those obtained when only the aqueous or both phases (organic
Fig. 7. Dependence of the outlet acetone concentration on the stirrer speed (X Y).
reduced value of the molecular diffusion coefficient in this phase, compared with the organic one. As was expected, the mass transfer intensity increased by stirring both phases and the outlet concentration tends close to the equilibrium value. From the above figures, it is evident that the equi-
Fig. 8. Variation of the acetone concentration on mixing intensity (Y X).
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Fig. 9. Dependence of the specific fluxes of the transferred solute on mixing intensity, for mass transfer from the aqueous phase to the organic one.
Fig. 10. Variation of the specific flux of solute on mixing intensity, for the acetone transfer from the organic to the aqueous phase.
and aqueous) are agitated. These significant differences between the acetone transferred fluxes in the above — mentioned mixing conditions can be the result of several influences, such as. The different values of the diffusion coefficient in the aqueous phase (0.923×10 − 5 cm2/s) and the organic one (1.763× 10 − 5 cm2/s). The values of the partition coefficient between the two phases and, the effect Marangoni, but in a less measure. The values of the partition coefficient for the transfer from the aqueous to the organic phase ranged between 0.175 and 0.52 and are dependent on the concentration. For our experimental conditions, the partition coefficient is 0.25. The results obtained using the following ternary systems, 1. water–acetone–carbon tetrachloride (present paper); 2. water–acetone–chloroform (unpublished results); allowed us to conclude that the smallest specific flux always results when the mixing is performed only in that phase were the equilibrium concentration of the solute has a low value (CCl4, for the first system; water,
for second one). Of course, when the mixing of the phases occurs, the influence of the molecular diffusion coefficients is negligible. Also, analysing the cell construction details, the phase contacting mode as well as the shape and the position of the interface, we concluded that it is less probable that the Marangoni effect influences the transfer of the solute. The specific flux of the solute increases when only the aqueous is stirred up to a Reynolds number of Re $ 10 000 and then the solute concentration remains constant. Also, a higher specific flux is observed when the transfer occurs from the organic to the aqueous phase. When both phases are stirred, the dependence between the specific flux and the agitator speed is almost analogous to that resulted when only the aqueous phase is stirred. The absolute values of this flux are, however, higher than those in the above-mentioned situation. For agitator speeds higher than that corresponding to the point G, the specific flux shows a significant increase, as a result of the interface deformation. Visually, it can be observed that, the surface of the interface is increased as a consequence of waving for speeds higher than 1000 rpm, that is Reag \ 14 000. The experimental values of the specific fluxes as well as the individual or overall driving forces resulted from the equilibrium diagram allow for the calculation of the individual Eq. (1) or the overall Eq. (4) mass transfer coefficients. The driving force values result from the plot of the equilibrium curve together with the operation lines for both directions of the solute transfer (see Fig. 4). The equilibrium curve (LF) and the operating lines (PQ and MN) having the slope given by a relation alike Eq. (2), were plotted in Fig. 11. The points representing the phase outlet concentrations in the domain DG (see Figs. 9 and 10) are located on the operating lines PQ and MN, respectively. For example, the point 1 represents the outlet acetone concentration when the solute transfer occurs from the organic to the aqueous phase and the organic phase is stirred only; the point 2 results when only the aqueous phase is stirred while point 3 corresponds to both phases stirring. The individual and global driving forces are given by the relations from Table 1. The position of the points 1–3 tends to the equilibrium state (the point Q), if the mean residence times in the cell compartments increase. This statement was verified using the points located on the three curves from Fig. 6. All points are located on the line PQ (not represented in Fig. 11). The flow inside the cell can be assumed mixed perfectly at higher agitator speeds (Reag \ 10 000) located between the positions D and G (Figs. 9 and 10). In these conditions, the initial concentration of the solute suddenly decreases at a value, which corresponds to the
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Fig. 11. The operating lines and the equilibrium curves for acetone transfer in both directions.
outlet concentration and remains constant in steadystate conditions. The corresponding driving forces will have smaller values, so that the mass transfer coefficients increased. The actual driving forces can be evaluated with the equations in Table 2. The statistical analysis of the experimental data allowed for the development of several criterial equations, which can be used to calculate the individual and overall mass transfer coefficients. The individual coefficient values result from the ratio between the specific transferred solute flux (Figs. 9 and 10) and the individual driving forces, given in Table 1. The criterial equations have the following general form, Sh = bRe pagSc 1/3 where Sh = dk/DB, Reag =nd2z/p and Sc =k/DB. Four sets of criterial equation result, for each direction of the mass transfer (Table 3). The equations can be applied for Reag B11 000, if the transfer of the solute occurs from the organic to the aqueous phase. For Reag \11 000, the value of the individual mass transfer coefficients remains constant (the domain DG in Figs. 9 and 10). k%X,YX =4.57×10 − 4 m/s and k%Y,YX = 4.11× 10 − 4 m/s If the transfer of the solute takes place from the organic to the aqueous phase, the equations hold for Reag B10 000. For Reag higher than 10 000, the individual mass transfer coefficient values remain constant (domain DG).
k%X,XY = 11.11×10 − 4 m/s
and
k%Y,XY = 14.01×10 − 4 m/s Based on the assumption that an intense agitation occurs and the perfectly mixed flow model is valid, the individual mass transfer coefficients were calculated using the driving forces obtained from Table 2. These Table 1
X Y
YX
Theoretical o6erall dri6ing forces DX tg,XY =XA,i DY tg,XY =Y*S,i
DX tin,YX =X*S,i DY tin,YX =YA,i
Theoretical indi6idual dri6ing forces DX tin,XY =XA,i−XA,e DX tin,YX =XS,e DY tin,XY =YS,e DY tin,YX =YA,i−YA,e
Table 2
The actual driving force X Y
YX
DX rXY,1 =XA,1−XA,e DX rXY,2 =XA,2−XA,e DX rXY,3 =XA,3−XA,e DY rXY,1 =YS,e−YS,1 DY rXY,2 =YS,e−YS,2 DY rXY,3 =YS,e−YS,3
DX rYX,1 =XS,e−XS,1 DX rYX,2 =XS,e−XS,2 DX rYX,3 =XS,e−XS,3 DY rYX,1 =YA,1−YA,e DY rYX,2 =YA,2−YA,e DY rYX,3 =YA,3−YA,e
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484 Table 3
Criterial equations Stirred phase Aqueous phase Organic phase Both phases
XY 1/3 ShX,XY = 1.441Re 1.6013Sc X 0.7097 1/3 ShY,XY = 1665.613Re ag,Y Sc Y 1/3 ShX,Y,XY = bRe 1.56 Sc when b= 2,3 for organic phase ag,X,Y X,Y and 3 for aqueous phase
will be great values, independent of the agitator speed, and for a perfectly plane interface. The overall mass transfer coefficients were calculated directly, from the transferred specific fluxes (Figs. 9 and 10, curve ODG) and the overall driving forces, given by the relations Eq. (4). These coefficients were also calculated using the equations, for the individual mass transfer coefficients from Table 3 and by substituting then in the relations Eq. (5). The results plotted in Fig. 12 demonstrate the validity of the Eq. (5). The overall coefficients resulted from the specific fluxes and theoretical overall driving forces were represented on the y-axis, while the values calculated using the experimental individual coefficients were represented on the x-axis. The overall mass transfer coefficient values in both directions of transfer calculated using the two procedures mentioned above are very close, being placed on lines, which partially superpose one to another.
5. Conclusions
The experimental installation contains an improved Lewis cell as a central element, which can be exploited in continuous and discontinuous modes, also allowing for the steady-state operation for a short time when continuous running is used. The mass transfer surface can be rigorously measured. The cell construction allows for individual or simultaneous stirring of the phases, for a large range of speeds. Logically, the mass transfer intensity in one phase shows significant increases, which were dependent on the agitator speed. This increase occurs up to a certain limit beyond the specific flux and becomes independent on the speed. The specific flux is greater when the organic phase is stirred only; comparative to the situation wen the mixing of the aqueous or both phases occurs, irrespective of transfer direction. The flux of the transferred solute is higher if the
Y X 1/3 ShX,YX =155.927Re 0.9889 ag,X Sc X 1.1431 1/3 ShY,YX =13.824Re ag,Y Sc Y 1/3 ShX,Y,YX =bRe 0.91 ag,X,YSc X,Y when b =354 for organic phase and 333 for aqueous phase
transfer occurs from the organic to the aqueous phase that is in the direction of the increasing acetone solubility. The individual and overall mass transfer coefficients in both phases were calculated. The individual coefficients were correlated in the form of criterial equations. The experimental values of the overall coefficients are very close to the calculated values using the correlations developed for the individual coefficients and the partition constants.
Appendix A. Notations
Ac d D k k%
acetone agitator diameter diffusion coefficient individual mass transfers coefficient when only one phase is stirred individual mass transfer coefficient when both phases are stirred
Fig. 12. Comparison of the experimental and calculated values of the overall mass transfer coefficient.
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K l n nB t X X X* Y Y Y* z k p i
overall mass transfer coefficient the reverse of the slope for the operating line stirrer speed the specific flux of the transferred solute residence time inside the compartments of the cell acetone concentration in the aqueous phase acetone concentration in the aqueous phase, when perfectly mixed model can be assumed acetone equilibrium concentration in the aqueous phase acetone concentration in the organic phase acetone concentration in the organic phase, when perfectly mixed model can be assumed acetone equilibrium concentration in the organic phase density of solution kinematic viscosity dynamic viscosity partition coefficient
Subscripts ag agitation A initial solution B solute e equilibrium exp experimental g overall i initial in individual r actual S solvent
.
485
t theoretical X aqueous phase XY the direction of the transfer is from aqueous to organic phase Y organic phase YX the transfer direction is from organic to aqueous phase 1 the organic phase is stirred only 2 the aqueous phase is stirred only 3 both phases are stirred
References [1] J.T. Davies, E.K. Rideal, Interfacial Phenomena, Academic Press, New York, 1961. [2] H. Brenner, L.G. Leal, Interfacial resistance to interphase mass transfer in quiescent two-phase systems, Am. Inst. Chem. Eng. J. 24 (2) (1978) 246 – 254. [3] H. Brenner, L.G. Leal, Conservation and constitutive equations for adsorbed species undergoing surface diffusion and convection at a fluid– fluid interface, J. Colloid Interf. Sci. 88 (1982) 136 –184. [4] S. Chandrasekhar, L.G. Leal, Mass transfer studies across liquid/ liquid interfaces (use of analytical ultracentrifuge), Am. Inst. Chem. Eng. J. 21 (1) (1975) 103 – 109. [5] A.P. Colburn, D.G. Welsh, Mass transfer coefficients, Trans. Am. Inst. Chem. Eng. 38 (1942) 179 – 186. [6] H.R.C. Pratt, Interphase mass transfer, in: Handbook of Solvent Extraction, Wiley/Interscience, New York, 1983. [7] J.B. Lewis, The mechanism of mass transfer of solutes across liquid – liquid interfaces, Chem. Eng. Sci. 3 (1954) 248 – 259.