Ind. Eng. Chem. Fundam. 1985, 2 4 , 423-432
the shape of the velocity distribution is invariant with average velocity is only approximately correct. The simplified model is therefore open to improvement so that it may predict pressure losses in two-phase flow more accurately, but more data must become available before any "universal" constants for the model can be proposed. Although this model has been examined only for the case of a dispersed gas in a liquid continuum, the theory can be extended to fluid-solid systems and may prove useful in the fields of pneumatic and hydraulic transportation. Nomenclature B,, B, = dimensional constants in simplified model, N/(m4 S2)
B3 = ratio of Bz to B , Co = drift-flux constant CI-C13 = constants in the mixing length model g = acceleration due to gravity, m/sz K = mixing length constant 1 = mixing length, m m = void distribution exponent n = exponent for mixing length expression P = pressure, N/m2 R = pipe radius, m T = shear stress, N/m2 U = liquid velocity, m/s ll, = bubble rise velocity, m/s V , = drift velocity between bubbles and mixture, m/s W = superficial velocity, m/s x = height in pipe (axial distance), m y = distance from wall, m p = density, kg/m3 t = gas-phase voidage a = surface tension 6 = two-phase multiplier Subscripts c = at pipe center 1 = liquid 0 = evaluated at atmospheric pressure (except C,)
423
Superscript - = denotes quantity averaged over whole pipe cross section
Literature Cited Bankoff, S. G. J. Heat Transfer 1960,82, 265. Clark, N. N.; Flemmer. R. L. C. Chem. Eng. Sci. 1984a,39, 170. Clark, N. N.; Flemmer, R. L. C. Int. J. Muttiphase Flow I984b, 10, 737. 5 , 53. Clark, N. N.; Flemmer, R. L. C. J. Pipelines 1984~. Clark, N. N.; Flemmer, R. L. C. AIChE J . 1985,31, 500. Clark, N. N.; Flemmer, R. L. C.; Meloy, T. P. I n "Proceedings of the Technical Program-Powder and Bulk Solids Processing and Handling, Rosemont, IL, May 1964"; International Powder Institute: London, 1964; p 446. Davis, M. R. J. Fluids Eng. 1974,96, 173. Drew, D. A,: Lahey, R. T. J. Fluids Eng. I98la. 103, 583. Drew, D. A.: Lahey, R. T. J. Fluid Mech. 1961b, 117,91. Galaup, J. P. Ph.D. Dissertation, Scientific and Medical University of Grenoble, France, 1975. Govier, G. W.; Aziz, K. "The Flow of Complex Mixtures in Pipes"; Van Nostrand Reinhold: New York, 1972. Harmathy, T. 2. AIChE J. 1960,6 , 281. Herringe, R. A.; Davis, M. R. I n t . J. Multiphase Flow 1978,4 , 461. Hewitt, G. F. "Measurements of Two Phase Flow Parameters"; Academic Press: London, 1978. Hughmark, G. A.; Pressburg, B. S. AIChE J. 1961, 7 , 677. Lockhart, R. W.; Martineili. R. C. Chem. Eng. Prog. 1940,4 5 , 39. Mitchell, J.; Hanratty, T. J. J. Fluid Mech. 1966,26, 199. Nakoryakov, V. E.; Kashinsky, 0. N.; Burdukov, A. P.; Odnoral, V. P. Int. J. Multiphase Flow 1981, 7 , 63. Niino, M.; Kashinskil, 0. N.; Odnoral, V. P. J. Eng. Phys. 1978. 35, 1446. Nassos, G. P.; Bankoff, S. G. Chem. Eng, Sci. 1966,22, 661. Orkizewski, J. J. Pet. Technol. 1987, 19, 829. Oshinowo, T.; Charles, M. E. Can. J. Chem. Eng. 1974,5 2 , 25, 436. Petrick, M.; Kurdika, A. A. I n "Proceedings of the 3rd International Heat Transfer Conference, chicago, 1966"; American Institute of Chemical Engineers: New York, 1966; Vol. 4, p 184. Ros, N. J. C. J. Pet. Technol. 1961, 13, 1037. Schlichting, H. "Boundary Layer Theo:y"; McGraw-Hill: New York, 1966. Serizawa, A.; Kataoka, I.; Michlyoshi, I. Int. J. Multiphase Flow 1975, 2 , 235. Streeter, V. L.; Wylie, E. B. "Fluid Mechanics"; McGraw-Hill: New York, 1975. Wallis, G. B. "One Dimensional Two Phase Flow"; McGraw-Hill: New York, 1969. Yamazaki, Y.; Shlba, M. I n "International Symposium on Co-Current GasLiquld Flow, Waterloo, Ontario, Sept 1968"; Rhodes, E., Scott, S., Eds.; Plenum Press: New York, 1969; p 359. Yamazaki, Y.: Yamaguchi, K. J. Nucl. Sci. Technol. 1979, 16, 245. Zuber, N.; Findlay, J. A. Report GEAP-4592, General Electric Co., 1964. Zuber, N.; Findlay, J. A. J. Heat Transfer 1965,8 7 , 453.
sp = single phase tp = two phase 1, 2 = at vertical stations in the pipe (except C,, C,, B1,B,)
Received for review December 12, 1983 Revised manuscript received October 29, 1984 Accepted February 28, 1985
Effective Liquid-Phase Diffusivity in Ion Exchange Gloria R. S. Wlidhagen, Raad Y. Qasslm, and Krlshnaswamy Rajagopal COPPE Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
Khallqur Rahman' Department of Chemical Engineering, Bangladesh University of Engineering & Technology, Dhaka-2, Bangladesh, India
Various definitions of effective liquid-phase diffusivity in binary ion exchange systems have been tested against experimental data. Experiments were conducted using a shallow liquid-fluidized bed of ion exchange resins. I t is found that an effective diffusivity defined on the basis of simple film theory and a mass transfer coefficient based on equivalent fraction driving force can be correlated satisfactorily by using available mass transfer correlations. The theory has been extended successfully to a ternary system.
Introduction Mass transfer during ion exchange occurs due to coupled effects of diffusion and ionic migration. Unequal mobilities of ions tend to separate charges, creating an electric field 0196-4313/85/1024-0423$01.50/0
which in turn causes ionic migration to maintain local electrical balance. The flux of ions is given by the Nernst-Planck equation which contains, in addition to the Fickian component, a term for the electric field. The latter 0 1985 American
Chemical Society
424
Ind. Eng. Chem. Fundam, Vol. 24, No. 4. 1985
factor complicates the analysis of ion exchange kinetics compared to the case where driving force for mass transfer is solely due to concentration difference. Helfferich and co-workers (viz. Helfferich and Plesset, 1958) were the first to study the problem of particle-phase diffusion in binary ion exchange. Utilizing the condition of electroneutrality, these authors reduced the NernstPlanck equation to a form similar to Fick's law; i.e., the flux was expressed as a product of a diffusion coefficient and a concentration gradient. However, in this case, the diffusion coefficient is not a constant as is usually the case in Fick's law, but this depends on the individual ionic diffusivities, concentrations, and valences of the exchanging ions. The resulting nonlinear equation has to be solved numerically. Helfferich termed the coefficient an "interdiffusion coefficient", but we will refer to this as "effective diffusivity" because the latter is more commonly used. Using the concept of effective diffusivity, Helfferich was able to interpret the concentration profile within the particle phase. When kinetics is controlled by mass transfer in the liquid phase, the problem is further complicated compared to the particle-phase kinetics because of convection and the presence of mobile nonexchanging ions, these two factors being absent in the particle phase. Similar to Helfferichs approach, various authors have suggested different definitions of effective liquid-phase diffusivity. Most of these definitions were intended to yield appropriate diffusivity values which could be used to correlate data in a manner similar to correlations available for ordinary cases of solid-liquid mass transfer systems. Unfortunately, the various definitions of effective diffusivity differ in some important respects and it is not clear which particular choice of effective diffusivity should be used in ion exchange mass transfer calculations. This study was undertaken for a detailed examination of this issue. Ion exchange operations are dominantly carried out with dilute solutions, for which rates are liquid film controlled and for which an accurate knowledge of effective liquid-phase diffusivity is important. In this study a general case of binary and a somewhat restricted case of ternary (homovalent counterions) ion exchange systems have been investigated. Theory Let the exchanging counterions be designated as A, B, and C (where the counterion C is to be included for the ternary case only) and the nonexchanging co-ion as D. It is assumed that the solution is dilute, the sorption of coions is negligible, and the ion exchanger phase is fully swollen and spherical in shape and its size does not change during the exchange. The equation which governs the rate of mass transfer is dC,/at + UVC, + C.J1 = 0 (1) The diffusional flux, J , , is given by the Nernst-Planck equation
neutrality has previously been used to obtain the Nernst-Planck equations (Bird et al., 1960). The above equations may be manipulated to give the condition of no net current as
CZ,J, = 0
(5)
Equation 5 is not an independent condition, but it may appear to be so in some literature. Had this been so, the system would be overdetermined. Equation 5 is a derived condition and is helpful in simplifying subsequent algebra. It is easy to appreciate that the above equations are difficulat to solve analytically, even for steady flow and a known velocity distribution. Van Broklin and David (1972) obtained solutions for a general case of binary ion exchange for film, penetration, and boundary layer models (analogue computation for the latter two models), but these have not been tested against experimental data and also their results cannot be easily adopted for practical design calculations. Rahman and Streat (1978) presented an analytic solution for a homovalent binary case based on an integral method of solution and the assumption that the co-ion flux was zero everywhere in the film. Although the assumption of zero co-ion flux throughout the film is not serious, the limitation of the integral method of analysis is that the solution depends on the assumed concentration profile, and hence, this method does not guaranetee correct results. Other workers in this field (Schlogl and Helfferich, 1957; Copeland and Marchello, 1969; Turner and Snowdon, 1968; Kataoka et al., 1968) have invariably used the "stagnant film" model for mass transfer, and a similar approach will be adopted in the present work. In this work, it is assumed that a spherical particle is surrounded by a thin uniform liquid film across which concentration change occurs and that the solid-liquid interface is in equilibrium. It is further assumed that spatial variation of concentration is more important than time variation and one needs to consider the steady-state case only; this is referred to as pseudo-steady-state condition. Since the film is uniform and very thin, it is sufficient to consider diffusion in one direction only, its curvature may be neglected, the co-ion flux throughout the film may be taken as zero, and the condition of no co-ion flux across the interface given by eq 4 may be relaxed as JD =
0
(6)
Binary Exchange. Manipulating eq 2-6, we may eliminate the electric potential gradient from eq 2, and the flux of component A may be expressed in terms of its concentration gradient as J4
=-
DADB(ZA2CA+ ZB'CB + Z D ~ C DdCA ) D,ZA2CA + DB(ZB2CB + ZD2CD) dz
(7)
Defining a new variable called equivalent fraction, Y Las ,
In addition, we have the following conditions of electroneutrality in the liquid and no co-ion flux across the solid-liquid interface.
cz,c,= 0
13)
JD5 = 0
(4)
It may be mentioned here that the condition of electro-
dY A dz
- (9)
Integration of eq 9 between bulk and solid-liquid interface gives
Ind. Eng. Chem. Fundam., Vol. 24, No. 4, 1985
Equation 10 may be used to evaluate flux if the film thickness, 6, and the boundary conditions are known. Unfortunately, the film thickness cannot be obtained from the stagnant film model and thus the analysis is not directly useful for quantitative purposes. However, the above results may be used as a basis for defining effective liquid-phase diffusivity as seen below. The effective diffusivity may be in turn used for carrying out ion exchange rate calculations. In a manner similar to other common mass transfer processes, the mass transfer coefficient, KM, for ion A may be defined as ZAJA
KLAACA
(11)
where ACA is a characteristic driving force. In eq 11,valence has been used to express flux in equivalents so that fluxes of counterions remain equal. The choice of a single driving force in eq 11 is arbitrary. Since ion exchange occurs due to combined effects of diffusion and ionic migration, neither the concentration difference nor the electric potential difference alone constitutes the true driving force. The driving force may be expressed in various ways, but here we will consider the following two definitions. ZAJA= KLA'(CA~ - CA~) zAJA
= KLAYZDCDb(YAs-
(13)
YAb)
(14)
= DeA/6
where De, is called effective liquid-phase diffusivity for the ion A and is given by the expression (where equivalent fraction in eq 10 is replaced by concentration from eq A2 and A6) DeA
D A W Z B+ 2D)ZA ZB - Z D = (DAZA - DBZB) ZB + Z D
[--
CD, - CDb CAS- CAb
]
(15)
Equation 14 gives a result reminiscent of the classical film model where the mass transfer coefficient is given by the ratio of diffusivity to film thickness, and one brings this analogy to define an effective liquid-phase diffusivity for the case of ion exchange mass transfer. Similarly, use of eq 13 yields another value of effective diffusivity, given by DeA
DADB(ZB+ ZD)ZA = (DAZA - DBzB)ZD(yAs - YAb)
[ (K) YAb
Turner and Snowdon (1968) argued that the mass transfer coefficient must vary with the two-thirds power of effective diffusivity to bear conformity with results of other common solid-liquid systems. In the film theory they accommodated this by assuming that the film thickness was proportional to the one-third power of diffusivity, again a result reminiscent of convective diffusional boundary layer theory. These authors proposed that 6 a Dw where D is some average diffusivity. Arbitrarily, the authors used eq 9 to define an average diffusivity as (we have called this also an effective diffusivity) D = DeA = (ZB + 2,) + (2, - z B ) Y A DAZD A'
FA
(2, + ZD) + ((ob+ 1)(zA+ 2,) - (ZB + ZD))
(17)
where YA is the mean arithmetic value of equivalent fraction in the film. Replacing the value of film thickness in eq 10 in terms of one-third power of diffusivity given by eq 17 and assuming that the mass transfer coefficient defined by eq 13 varied with the two-thirds power of diffusivity, Turner and Snowdon (1968) expressed effective liquid-phase diffusivity for homovalent binary case as (they did not give a general solution)
(12)
The superscript on the mass transfer coefficient is used to remind us whether the mass transfer coefficient is based on the differences of concentration or equivalent fraction. Note that eq 12 and 13 are different as (CAS- CAb) # Z&Db(YAs - YAb), and the choice of the latter is a convenient one. Equating eq 10 and 12, we obtain KLAc
425
+q
( A ~ Y -A 1) ~ - (AzYAb - 1)
1
(16)
Kataoka et al. (1968) used eq 15 whereas Turner and Snowdon (1968) sought a modification of eq 16 to define effective diffusivity. It will, however, be seen that eq 16, as it stands, serves as the most useful expression of effective diffusivity.
The use of eq 9 to define an average diffusivity by these authors seems curious, and it is probable that when doing this they had Fick's law in mind. In fact, Copeland and Marchello (1969) used the analogy of Fick's law to define still another value of effective diffusivity. However, instead of eq 9 Copeland and Marchello based their definition on eq 7 in a manner equivalent to that of Helfferich for the particle phase. For homovalent exchange, the effective diffusivity takes on the form
where the bar indicates the arithmetic mean value of equivalent fraction in the film. The different definitions of effective diffusivity presented here have been based on intuitive and arbitrary considerations, and one must carry out suitable experiments to determine whether any of these serve as adequate definitions of effective liquid-phase diffusivity for a binary ion exchange system. Of the five expressions, eq 16-18 have the advantage that DeA = DeB, whereas this is not the case with eq 15 and 19. This is due to differences in the choice of driving force in these equations. A little thought will show that when driving force is based on equivalent fractions, as in eq 16-18, instead of concentrations, as in eq 15 and 19, the effective diffusivities for the two counterions, DeA and DeB, are equal. Ternary Exchange. For a ternary system (counterion valences equal) the integration of eq 2 gives
The analysis may now be extended to define effective diffusivity values similar to binary exchange. For example,
426
Ind. Eng. Chem. Fundarn., Vol. 24, No. 4, 1985
the De, value corresponding to eq 16 for equal valances of counterions and co-ions is given by the equation De, = 20,(0$Y,, + l)/[(@YA, + NEYB~+ 1) + ((aby.48 + ~PYBR + 1 ) ( a $ Y A b (YEYBb + 1)1'''] (21) Note that for the ternary system no two values of effective diffusivity need be the same as is the case in binary exchange. Similarly, other definitions of DeAare possible. The detailed derivations of eq 10 and 20 are shown in the Appendix. Further details of the theory may be found in references Wildhagen (1 979) and Rahman (1979). Experimental Section For a good test of the various definitions of effective diffusivity, it is desirable to work with a physicochemical system having the following features. (i) The counterions should have markedly different mobilities so that the effect of electric field is significant. (ii, The equilibrium relationship should be well-known. (iii) The solution should be dilute since the theory has been developed for this case. (iv) The system should lend itself to accurate analysis. (v) The fluid mechanics of the system should be well established. This condition is most difficult to meet and perhaps the most important one. The system chosen in this study were cation exchanges of H+, Na+, and K+ ions in a shallow fluidized bed of ion exchange resins. The choice of the chemical system was dictated by criteria i-iv. Uni-divalent exchange was not studied because of the difficulty of obtaining accurate description of the equilibria and also because this system is less interesting since equilibria is markedly favored in one direction. The binary pairs included H+/Na+,H+/K+, and Na+/K+with C1- or NO? as the co-ion and the ternary trios were H+/Na+/K+ions with C1 as the co-ion. In all cases dilute solutions having a concentration of about 0.03 N were used. The shallow fluidized bed was chosen principally for the following reason. Mass transfer in a shallow fluidized bed has been well studied and meets condition v listed above, as well as possible. Rahman and Streat (1981) carried out an extensive study of mass transfer in a shallow fluidized bed of ion exchange resins employing the H+/Na+ neutralization exchange. This is a special case of ion exchange where the effective diffusivity does not depend on the concentration of the ions and where the diffusional flux may be expressed similar to Fick's law with a constant value of the diffusion coefficient, Carrying out carefully controlled experiments, these authros correlated data for the special case of H+/Na+ neutralization exchange as Sh = 0-'--.Rel 86 t
'2Scl '7
(22)
Equation 22 has been found to apply very well in fixed and fluidized beds for systems having constant diffusivity over a wide range of conditions, and hence, eq 22 is chosen to correlate ion exchange mass transfer data. However, in the present case there is no unique value of diffusivity that may be used in eq 22. The choice of proper effective liquid-phase diffusivity out of the various definitions presented earlier is the issue of this paper. The mass transfer coefficient defined by either eq 12 or 13 may be obtained from a solution of the mass balance equation in the fluidized bed. It is believed that plug flow of liquid and complete mixing of solids give an adequate description of the profiles of the two phases in a fluidized bed (Church, 1962; Rahman, 1973; Slater, 1974). Based on these assumptions, the mass balance equation for an elemental height dh is
where the flux, JA,may be expressed in terms of the mass transfer coefficient from either eq 12 or 13. In eq 23 the accumulation term is usually very small, and it suffices to work with the pseudo-steady-state macroscopic mass balance equation. Neglecting the first term in eq 23, we may integrate t,he equation rising eq 12 to obtain
or using eq 13 to obtain
where in both the equations the mass transfer coefficient obtained is an average value in the bed. Note that in eq 24 concentration values have been replaced by an equivalent fraction using eq A2 and A 6 and that eq 24 i s to be integrated numerically. Experiments were performed by passing a dilute solution through a shallow bed (static bed height was
