Ind. Eng. Chem. Fundam.
388
Subscripts
1981, 20, 368-371 coddl,R. B.; En , A. J. A I C E J .
max = maximum min = minimum Superscripts (‘1 = d ( ) l d t T = transpose * = evaluated at the optimal path
Literature Cited
1971, 17, 220. LeVenSplel, 0. #emical Reaction Engineering”; Wky: New York, 1962; Chapter 6. Lund, M. M.; Seagrave. R. C. Ind. Eng: Chem. Fundam. 1071, 70, 494. Lund, M. M.; Seagrave, R. C. AIChEJ. 1071, 77, 30. Pontryagin, L. S.; Bdtyanskli, Y. G.; Gamkrelidre, R. V.; Mlshchenko, E. F. “The Mathematical Theory of Optimal Proc~sses”;Wy-Intersclence: New York, 1962. Ridlehoover, G. A.; Seagrave, R. C. Ind. Eng. Chem. Fundam. 1078, 12, 444. Roth, D. D.; Basaran, V.;Seasrave, R. C. Ind. Em. Chem. Fundam. 1070, 18, 376.
Received for review July 24, 1980 Accepted June 4, 1981
Bryson, A. E.; Ho, YuChi “Applied Optimal Control”, Wiley: New York, 1975; Chapter 8.
Simpllfied Evaluations of Mass Transfer Resistances from Batch-Wise Adsorption and Ion-Exchange Data. 1. Linear Isotherms Shlgeo Goto,” Motonobu Goto, and Hldeo Teshlma & p r t m n t of Chemical Engimdng, Nagoya University, Chikusa, Nagoya, 464, Japan
A_ method of sknmneous evaluation of the interphase mass transfer coefficient, k,, and the intraparticle diffushrky, D, from batchwise sorption rate data is proposed. An apparent mass transfer coefficient,kao,which is determined by neglectrng intraparticle diffusion resistances, ts used to evaluate the individual coefficients. Adsorption isotherms or equiilbriwn relations of ion exchange are assumed to be linear. Values of k, and Dare determined from published sorption rate data, and the values agree with the cited values.
Introduction Rates of liquid-phase adsorption and ion-exchange reaction with particles suspended in batch-wise stirred tanks may generally be Wuenced by both intra- and interphase mass transfer resistances. Furusawa and Suzuki (1975) derived moment equations and evaluated the liquid-toparticle mass- transfer coefficient, k,, and the intraparticle diffusivity, D, from experimental adsorption data using linear isotherms. However, there may be some uncertainty in the results if tailing parts of the rate data are long. Huang and Li (1973) obtained the value of D from isotopic ion-exchangedata for high stirring speeds, where the resistance of fluid-to-particle mass transfer was negligible. Then they determined values of k, from the data for lower speeds, where two resistances were not negligible. In this paper, a semiempiricalmethod of simultaneous evaluation of k, and D from adsorption and ion-exchange rate data in batch-wise stirred tanks is proposed by defining an apparent mass transfer coefficient, kd, which is determined by neglecting intraparticle diffusion resistances. It is assumed that adsorption isotherms or equilibrium relations of ion exchange are linear. The data of Huang and Li (1973) on ion-exchange reactions are used to evaluate the values of k, and D by this method. Definition of k,O The apparent liquid-to-particlemass transfer coefficient, kd, may be defined by assuming that diffusion resistances inside the resin phase are negligible and the concentration in the particle is uniform. On the basis of the film theory, mass balance equations in a batch-wise stirred tank yield 0196-4313/81/1020-0368$01.25/0
-VdC/dt = k,oaV(C - C*)
(1)
V(C0 - C) = PC
(2)
e,
where the resin phase concentration, is initially zero and the liquid phase concentration at the surface of particle is assumed to be in equilibrium with that in the particle, C*, because intraparticle diffusion resistances are negligible. On the assumption of a linear isotherm, we have
C = KC*
(3) If it is assumed that ka is independent of time and concentration, eq 1 can be integrated by using eq 2 and 3.
where C, is the concentrationin the liquid phase at infiiite time. Since both C and C* at infinite time are equal to C,, eq 2 and 3 give C, = VC,/(KP+ V) (5) If the assumption that the diffusion resistances inside the resin phase are neglible is not valid, eq 1 should be replaced by -VdC/dt = k,aP(C - C,) (6) where k, is the liquid-to-particlemass transfer coefficient and C, is the liquid phase concentration at the surface of the particle. Since local concentrations in the particle are decreased as the radial position moves from the surface 0 1981 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 569 1.0
1 .o
03
0.7
0.5
$0.5
‘ P
0.4 \ 9
0.4
0.3
0.3
2
t
e.2
O’* 01
0
01
02
03
04
05
06
07
00
09
01
IO
ks0 _ - -a In (1 -Fj k, 35T(1 + a)
(7)
The fraction, F , of adsorbates or ions taken up by particles can be obtained analytically from mass balance equations where both intra- and interphase mass transfer resistances are considered (Huang and Li, 1973)
co - c F=i=1
6t2(1+ a) exp(-g?T) (9/a + ~ r g ?+ 9)t2 - (6 + a)g?[
+ agt
(8)
where tan -- gi gi
-
35 - agt
(5 - 1)agF + 35
02
03
04
05
06
07
08
09
10
Figure 2. The effect of P on ka/k, at a = 2 / 9 and E = 9.
to the center, C, in eq 6 is higher than C* in eq 1, which corresponds to the average value of local concentrations. Therefore, k, in eq 6 is usually higher than ka in eq 1 and the difference between k, and ka0becomes greater as the concentration profile in the particle is sharper. From batch-wise sorption rate data, the concentration in the liquid phase, C, can be obtained as a function of time, t , for given experimental conditions. Thus, ka can also be determined from eq 4 for various values of t. If the value of ka is independent of time and constant throughout an experimental run, it is verified that intraparticle diffusion resistances are negligible and that the apparent liquid-to-particle mass transfer coefficient, ka, can reduce to the “true” liquid-to-particle mass transfer coefficient, k,. On the other hand, if the value of keOdecreases with time, the assumption that intraparticle diffusion resistances are negligible is not valid and k, in eq 6 should be considered. Even if in this case, ka at t = 0 must be equal to k, because the concentration in the particles is uniform, that is, (3 = 0 at t = 0. It is well known, however, that concentration measurements at early stages in batch-wise operations are often inaccurate. Therefore, k , must be obtained by extrapolating ka at any time to that at t = 0. An extrapolation method will be developed in the following sections. Theory The ratio k,,/k, can be derived from eq 4 and the dimensionless terms, T = D t / R 2 , a = V / K V , and 5 = k$/DK.
1-E
01
F”
Figure 1. The effect of P on ka/k, at a = 2/a and 5 = 25.
co - c,
11111111111
0
F”
(9)
Therefore, for specified values of a and 5 at a given system, F can be calculated from eq 8 and 9 as a function
1 .o
I
1
I
0.3
0.4
0.5
I
I
I
1
0.6
0.7
0.8
0.9
0.7
~
3
0.5
0.4
0.3
0.2
0.1
0.1
0.2
F”
Figure 3. The effect of P = k a / k , at a = 2/9 and E = 3.
of dimensionless time, T. Then, the ratio ka/k, may be determined as a function of F from eq 7. Figures 1,2,and 3 show ka0/k,vs. F plots for 5 = 25, 9,and 3. In these figures, the ratio of total ions in the solution to those in the particles, a,is fixed at 2/3, which corresponds to the experimentaiconditions described in the next section. The exponent, n,is an arbitrary number in the range of 0.5 5 n I1. It is evident from these figures that the curves for n = 1 have an S-shape and that the extrapolation to the limit (F = 0) has uncertainty when the usual semilogarithmic plots are adopted. The exponent, n,is introduced in an attempt to obtain straight lines, which are easy to extrapolate. If n is equal to 0.6, the linea are approximately straight in the region of 0.1 < F < 0.6 and the point of (kn/kJ = 1at F = 0 may be nearly on the straight lines. If a is different from 2/3, the same procedure is repeated to find straight l i e s in semilogarithmic plots of ka/k, w. F. In general, straight lines are obtained when n = 0.5 for 50 > a > 1, n = 0.6 for 1 1 a 1 0.1, and n = 0.7 for 0.1 > a > 0.05. When a > 50 or a < 0.05, we could not find straight lines for any values of n. In these regions, however, the volume of solid phase, V is either much smaller or much greater than that of liquid phase, V. Thus, experiments in these regions should be avoided, while those in the region of 1 1 a 1 0.1 may be useful. As seen in Figures 1,2,and 3,the slopes of straight lines become steeper as the values of 5 increase. The values of (k,O/kB)F=l,that is, (ka0/k,)extrapolated to F = 1, can be obtained from these figures for E = 25,9,and 3 at a = 2/3. Thus,as shown in Figure 4, the relationship between 5 and (kBO/ks)F=l for various values of a may be determined by
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981
370
Table I. The Values of k, and 5 Huang and Li
this work
-
DX DX mesh R X k , x l o 5 , l o 1 ' , k,X l o 5 , lo", no. IO4, m m/s m/s m*/s m2/s 40 2.1 0.520 1.05 0.440 0.967 25 3.55 1.40 1.57 1.29 1.43 20 4.2 2.90 1.35 1.41 1.57 20
7
H u n g and LI ( 1973)
5
$
07
3
~~
1.0
I
I
I
I
1
1
I
I
I
I
I
0
01
02
03
0 4
0 5
06
07
08
09
I O
( kso
0 05
$-
/kS
Figure 4. The relationship between 6 and (klO/ka)P1.
04
03
2o 0
T E
01
10
0
B
01
02
03
04
05
06
07
08
09
IO
F
Figure 6. k,, vs. F plot of data of Huang and Li on isotopic ionexchange reaction for 25-mesh particles.
05 04
03
t
I
07
"2
1 I
2
t-
0 2 0 mcs
(1973). Since the diameter of the resin spheres was apparently miswritten as the radius in their table, their values of k, and D were recalculated. The values obtained by both methods are in reasonable agreement. Discussion In published papers on the measurements of mass transfer rate in the beds packed with ion-exchange resins (Tien and Thodos, 1960; Kataoka et al., 1965),the values of the apparent mass transfer coefficient, klo, were plotted vs. F and the value extrapolated to F = 0 by a straight line was taken as the value of the true mass transfer coefficient, k,. The data of Huang and Li (1973) were plotted €or kd) vs. F as shown in Figure 6. A straight line could be drawn through the data and k, was determined from the extrapolation to F = 0. The result may be inaccurate, however, because there is no theoretical basis for the extrapolation. The dotted lines of Figure 6 show the values of k,,,calculated from eq 5,7, and 8, with values of k, and D in Table I for Huang and Li (1973) and from this work. It is impossible to draw a theoretical curved line such as dotted lines of Figure 6 unless the values of k, and D are known a priori. No information on intraparticle difbivity, D,can be obtained from this plot. In conclusion, the proposed method in which values of both k, and 15 can be evaluated from one experimental run by the linear extrapolation of the plot, kd)vs. F may be faster and nearly as accurate as other methods where at least two runs are required. Nomenclature a = outer surface area per unit volume of particles, l / m C = concentration in the liquid phase, mol/m3 in the resin hase, mol/m3 Dcf == concentration intraparticle diffusivity, m /s F = fraction of ions uptaken by particles K = equilibrium constant k , = liquid-to-particlemass transfer coefficient, m/s kso = apparent mass transfer coefficient, m/s R = radius of particle, m T = dimensionless time, D t / R 2
O 21 t -7 4 0 mesh
01
0
01
02
03
04
05
06
07
08
09
IO
F
Figure 5. k,, vs. Eo.6 plot of data of Huang and Li on isotopic ion-exchange reaction.
the repetition of this procedure. Evaluation of k, and D Huang and Li (1973) measured isotopic ion-exchange rates between calcium chloride solutions and a calciumform cationexchangeresin, Dowex 50W-X8in a batch-wise stirred tank. The value of CY was 2/3 in their experiments. The plot of log (1- F)vs. reaction time, t , at 300 rpm and 25 "C was shown in a figure (Figure 9 in Huang and Li, 1973) for three sizes of resin. The values of kd were calculated from eq 4 and plotted vs. F'.6 as shown in Figure 5. Since Figures 1,2, and 3 indicate linearity of this plot in the region of 0.1 < F ' s 6 < 0.6, straight lines are drawn to evaluate k, and D for data of 20,25, and 40-mesh particles. The liquid-to-particlemass transfer coefficient, k,, is then determined from the value of kd)extrapolated to F = 0. The ratio (kd/ks)FP1 was obtained from kd extrapolated to F = 1as shown in solid lines in Figure 5. The value of f could be determined from the plot of E vs. (kllo/Qbl for CY = 2/3 in Figure 4 and then D was evaluated from the relation of D = k a / f K . The value of K was calculated from Table I in Huang and Li (1973) as 35.3. To check the reliability for drawing straight lines in Figure 5, the values of kd)were recalculated by using eq 5,7, and 8 with the above values of k,, D,and K. As shown by the dotted lines in Figure 5, the recalculated values of kso are in good agreement with straight lines at least within 0.1 < PSC 0.6. The values of k, and D determined in this manner are given in Table I together with those in Huang and Li
F
Ind. Eng. Chem. Fundam. 1981, 20, 371-375
a7 1
* = infinite Literature Cited
t = time. s = volume of liquid phase, m3 V = volume of resin phase, m3
Furusawa, T.; Suruki, M. J . Chem. Eng. Jpn. 1971, 8 , 119. Huang,T. C.; Li, K. Y. I d . Eng. Chem. fundam. 1973, 12, 50.
Greek Letters
Kataoka, T.; Takashima, K.; Fwute, I.; Ueyama, K. Kageku Kogaku 1965, 29, 368. Tien, C.; Thodos, G. Chem. Eng. Scl. 1980, 73, 120.
Subscripts 0 = initial
Received for review January 26, 1981 Accepted August 4,1981
Simplified Evaluations of Mass Transfer Resistances from Batch-Wise Adsorption and Ion-Exchange Data. 2. Nonlinear Isotherms Motonobu Goto, Shlgeo Goto,' and Hldeo Teshlma Depadmnt of chemical Engineering, Nagoya Unlverstty, Chlkusa, Nagoya, 464, Japan
The proposed method in our companion paper is extended to the case of nonlinear isotherms, and mass balance equations are solved numerically. The rate of exchange of H+ and Na+ ions with a strong-acid ion exchanger in aqueous solution was measured in a batch-wise stirred tank. The values of mass transfer coefficients and intraparticle diffusivlties were evaluated from experimental results by the proposed method. These values are In good agreement with those estimated from published correlations.
Introduction As described in our companion paper (Goto et al., 1981), the method of simultaneous evaluation of the interphase mass transfer coefficient, k,, and the intraparticle diffusivity, D,has been proposed for linear isotherms. Analytical solutions of mass balance equations in a batch-wise stirred tank were obtained; the procedure to estimate the values of k, and D was rather simple. Generally, however, adsorption isotherms and equilibrium relations of ion exchange are not linear. In most of the published papers on batch-wise adsorption and ion exchange with nonlinear isotherms, one rate-controlling step was considered, that is either intraparticle diffusion or interphase mass transfer. For the case where intraparticle diffusion was controlling, Dryden and Kay (1954) studied the adsorption of acetic acid on carbon with the Freundlich isotherms and proposed a method which approximated a nonlinear isotherm by a linear equation. Hashimoto et al. (1975) presented diagrams to estimate intraparticle diffusion using both the Freundlich and the Langmuir isotherms and investigated adsorption of DBS and phenol on activated carbon. On the other hand, for a case in which the interphase mass transfer was controlling, Smith and Dranoff (1964) measured the rate of exchange of H+and Na+ ions with Dowex 50 resin and compared the experimental data with predictions based on the Nernst-Planck equations. Few papers, however, treated the case where both intraparticle diffusion and interphase mass transfer were controlling. Kamemoto et al. (1977) studied adsorption of phenol on activated carbon. They assumed that the rates were governed by interphase mass transfer at the early stage of adsorption and intra article diffusion at the later stage. The values of k, and were determined from observation in the extreme regions but data in the inter-
8
0196-4313/81/1020-0371$01.25/0
mediate region could not be used. In this paper, the proposed method (Goto et al., 1981) is extended to the case of nonlinear isotherms and mass balance equations are solved numerically. The rate of exchange of H+ and Na+ ions with Dowex 5OW-Xl2 resin in aqueous solution was measured in a batch-wise stirred tank. The values of k , and D were evaluated from experimental results by the proposed method. Theory Univalent ions between A-ion in the resin phase and B-ion in the liquid phase are exchanged as follows.
+
(1) R-A+ + B+ e R-B+ A+ Figure 1 shows the schematic drawing of concentration profiles where both intra- and interphase mass transfer resistances are significant. The diffusive fluxes across the liquid film and in the resin phase may be expressed by the Nernst-Planck equations (Helfferich, 1962). Mass balance equations are given as follows.
(4)
where the interdiffusion coefficient Dm is given as
The equivalence of ion-exchange reactions leads to 0 1981 American Chemical Society
