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
372
Ind. Eng. Chem. Fundam., Vol. 20,No. 4, 1981
CA
+ C B = CO
(6)
+ C B = C,
(7) The instantaneous equilibrium at the resin-film interface may be represented by a selectivity coefficient, KAB. CA
Initial and boundary conditions are given as CA = C, and C A = 0 (at t = 0) aC,/ar = 0 (at r = 0 )
(9) (10)
Equations 2 through 10 can be reduced to dimensionless equations.
= 1 and
YA
= 0 (at TA = 0)
(15)
azA/ax = 0 (at x = 0) (16) The dimensionless concentration of A at infinite time, Y A , ~ , may be derived from eq 6, 7, 8 and 11. YA,-
--
((1 + 6) - [(l + E)‘ - 4t(l - 1/K.4B)]’/2}/241- 1/Km) (17) Equations 12-16 may be reduced to finite difference equations and can be solved numerically for given values of 8, Y, XA, and KAB. Derivation of (k& The apparent liquid-to-particlemass transfer coefficient, &A)&, may be derived on the assumption of film diffusion-controlled binary ion exchange where intraparticle diffusion resistances are neglected. Smith and Dranoff (1964) developed the theory for this case and obtained the following equations ~ Y A Vu = -JA dt VCo
where
=
LIQUID FILM
RESIN PHASE
= ’f-7
[
i
CA
‘* JC,’
CA
and the concentrations at infinite time can be related to VCA,, = P(C0 - C A , a ) (11)
ZA
, BULK SOLUTION
(18)
I
R+ L
R
7” 0
Figure 1. Concentration profiles of ion-exchange reaction.
9
U
‘
W‘
1
I
“
4
Figure 2. Experimental apparatus: 1, reactor; 2, thermoetat; 3, buffle; 4, impeller; 6, torque meter; 6, conductivity cell; 7, conductivity meter; 8, recorder; 9, thermometer; 10, motor.
Since the equation for &A), (eq 30) in their paper, was miswritten in the third and fourth terms on the right hand-side, eq 19 has been corrected. When eq 18 is integrated, (k& may be represented by
From batch-wise ion-exchange data of t vs. YA, (k& can be @culated from the numerical integration in eq 22. As described in our companion paper (Goto et al., 1981), if the value of is independent of time throughout an experimental run,the apparent coefficient, (kA)& becomes the “true” mass transfer coefficient, (k& On the other decreases with time, the intraparticle difhand, if fusion resistance must be significant. Equation 22 can be rewritten to obtain the ratio of the apparent to the true coefficients in dimensionless terms.
Experimental Section A schematic drawing of the apparatus is shown in Figure 2. The reactor (1)was a glass vessel 8.4 cm in diameter and 12.5 cm high which was immersed in the thermostat at 30 “C (2). Four evenly spaced vertical baffles (3) 0.9 cm wide were fitted on the wall of the reactor. The impeller (4) was of the turbine type with 6-bladed paddles; it had a diameter of 4.4 cm. The rotating speed, N,, was varied from 200 to 900 rpm and the torque was measured
Ind. Eng.
Chem. Fundam., Vol.
20, No. 4, 1981
373
IO
09
-
08
-
t-
07 -
06
i
-
IS \05-I
IZO&-
, I'
03-
-
,' o / O
-
'0
'O,', /
0
' 0 01
-
,,'oP
02-
0 1 -,,'o'
0'
02
03
04
05
06
07
08
09
10
t
$
5
Figure 4. Changes of H+ion concentration with time for W = 7 g. 10 09 08 07
-3
06 05
\
3
-
4,
04
03
i
02
f
l
i I
04tI
02
0
01
02
03
04
05
06
07
08
09
10
Fn
Figure 6. The effect of F on ( k ~ ) a / ( kat~ )c ,= 2.52 and AA = 6.
Similar results were obtained for W = 2,5, and 10 g. Solid lines in Figure 4 indicate calculated results as described later. Numerical Analysis. The relation of yAvs. T A can be obtained from numerical solutions of eq 12 through 16 for specified values of 8, y, XA, e, and K B Thus, the value of ( k A ) & / ( k A ) # may be determined as a function of YA from eq 23. The ratio of the diffusivities of H+to Na+ in the liquid phase, y = D A / D B , was taken to be 6.71 and the ratio in the resin-phase, 8 = D A / & , was assumed equal to y. The selectivity coefficient, KAB,was 1.5 as described in the previous section. The values of e were calculated for the cases of W = 2,5,7, and 10 g and were taken as 8.82,3.53, 2.52, and 1.77, respectively. The value of XA was varied from 2.0 to 12.0. Figures 5 and 6 show typical plots of ( k ~ ) a / ( VS. k ~F )~ for e = 2.52, XA = 10.0 and 6.0. The fraction of sodium ion
374
Ind. Eng. Chem. Fundam., Voi. 20, No. 4, 1981 \ I
08
P 05
-*
04
1 I
0
900 rprn
v
4r NO
,
V
0
01
A
rpm
9
200 r p m
02 02
03
04
05
06
07
08
09
IO
F16
\
Figure 8. (kA)&vs. P.'plots of experimental data for = 2.52).
w = 7 g (e
Table I. The Vdues of (kA)*and EA
%
(hA)s
103, kg
x 104,
900
2.0 5.0 7.0 10.0 2.0 5.0 7.0 10.0 2.0 5.0 7.0 10.0 2.0 5.0 7.0 10.0
13.58
600
It is seen from these figures that the curves for n = 1 are not straight and that the extrapolation to the limit, F = 0, is too inaccurate to determine the values of (k,& from experiments as described in the previous paper (Goto et al., 1981). Although the exponent n was introduced to make the lines straighter through the point ( k ~ ) & / ( k=~ ) ~ 1 a t Fn = 0, we did not find straight lines for any values of n. If straight lines are sought without any limitations, however, the curves for n = 1.6 in Figures 5 and 6 are approximately straight in the region 0.3 < F" < 0.95. For different values o f t and XA within our experimental conditions, the same results were obtained; that is, straight lines could be found at n = 1.6. The value of n in this work (n = 1.6) was different from that in the previous paper (Goto et al., 1981) for linear isotherms (n = 0.6). This may be caused by the extent of nonlinearity of isotherms. The Values of [(kA)&/(kA)a]+Oand [(kA)&/(kA)sIF=I Were determined by extrapolating to F = 0 and F = 1 along straight lines for n = 1.6, as indicated in Figures 5 and 6. The values of [(kA)&]F=l/[(kA)sO]F=owere calculated from the slopes of the lines because was constant. Figure 7 shows the dependence of [ (kA)sO]&l/ [(kA)sO]F=Oand [ ( k ~ ) & / ( k ~ )O ~n] XA ~ =for o = 1.77, 2.52, 3.53, and 8.82. Evaluation of (k~), and DA.The experimental values of were calculated by eq 22 from experimental C A vs. t data. Figure 8 shows (kA)& vs. P.6for the data in Figure 4. Since the linearity of this plot is expected in the region of 0.3 < P6< 0.95, the straight lines can be drawn for the data at 200,300,600, and 900 r m. The evaluation method of (kA), and A from this plot was as follows. (i) Extrapolate to F = 0 and F = 1 with the straight line to obtain values of [(kA)sO]F=O and [ ( k ~ ) & l ~ =(ii) l . Divide .by [(kA)&IF=Oto obtain the slope, [ (kA)&l~=~/ [(k&lF=,,. (iii) Read the value of XA from the plot of XA vs. [ ( ~ A ) , ~ ] F ~ ~ / [ ( & Ain) & Figure ] F = ~7. (iv) Read the value of [ ( k ~ ) ~ / ( k ~ ) from , ] ~ . ;the ~ plot of [ ( k ~ ) s ~ / ( k ~vs.) ~XA] in + ~Figure 7. (v) Divide [(kA)&]F=O in step (i) by [(kA)dO/(kA)s]min step (iv) to obtain the ylue of &A),. (vi) Calculate D A from the relation of DA = (~A)$CO/(XACO).Steps (iii) and (iv) are illustrated by dotted lines in Figure 7.
wx
N,, rpm
300
200
mZ/s
m/s
11.31 9.49 7.17
mean
7.84 6.271 6.30 mean 5.49 6.16 6.57
8.53 8.74
W 7 : ) 8.23 9.50
and D A were found by this method, The values of as indicated in Table I. Solid lines in Figure 4 indicate the numerical solutions of eq 12 through 16 in terms of the values of parameters in Table I. They are in good agreement with the experimental results. The values of ( k ~ ) ~ and DBfor sodium ion may be calculated from the equations (k& = ( k A ) J y = (k~),/6.71and DB = DA/@= DA/ 6.71, respectively. The values of (kA), in Table I are almost independent of the resin amount in the stirred tank (V = 5.0 X lo4 m3) for each stirring speed, which implies the uniform suspension of the resin in the solution. As the stirring speed, Nm,is descreased, the mean value of @A), decreases. To compare our data with published correlations in terms of the energy dissipation rate, the power consumption of the impeller, PI,was measured by the torque meter. The data show that the power number, IVY, was equal to 3.71, independent of the stirring speed in the range 200 to 900 rpm. The energy dissipation rate per unit liquid mass, e,, may be calculated from PI Np~!I5(Nm/60)~ (26) €e=-= V VP Figure 9 shows the plots of S ~ / S Cvs. ' / ~( E , Q ! , ~ / Vfor ~)~/~ the experimental results together with published correlations (Harriot, 1962; Brian et al., 1969; Sano et al., 1974). The results in this work are within the scatter of the three correlations. The values of DAin Table I seem to be independent of both the resin amount and the stirring speed, although there is some scatter in the range of (6.7-10.1) X m2/s.
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 376 Id
-
, Pr = 13.8
I Brian 2 Sam
-
Sc = 4 4 4 3 Harriot , S c = 518 I
( E. dp?,,g)n
Comparison of liquid-teparticlemaea transfer coefficients between our data and published correlations. Figure 9.
The mean value of D Ais 8.67 X lo-’* m2/s. Kataoka et ai. (1974) measured intraparticle dflusivities in isotopic ion exchange for various kinds of resins and presented the estimating equation for intraparticle diffusivity. According to this equation DAfor H+ion in Dowex 5OW-X12 was 6.39 X m2/s, which was somewhat smaller than the mean value in this work. Conclusion The evaluation method proposed in our companion paper (Goto et al., 1981) was extended to the case of nonlinear isotherms. The values of &A), and DAwere evaluated from experimental results for the exchange of H+and Na+ with Dowex 5OW-X12. They were in good agreement with those in other investigations. The proposed method may be applied to other cases such as an adsorption with a nonlinea isotherm of the Langmuir or Freundlich types. Nomenclature a = outer surface area per unit volume of swollen resin, 6 / d p , I/? Ci = ionic concentration mo€/m3 Di = ionic diffusivity, mb/s Dm = interdiffusion coefficient, m2/s dI = diameter of impeller, m d = diameter of particle, m #=fraction of ions uptaken by particles Ji = ionic flux,mol/m2 s Km = equilibrium constant (kJl = liquid-to-particle mass transfer coefficient, m/s (ki)& = apparent mass transfer coefficient, m/s
N, = stirring speed, rpm (l/min) Np= power number, P1/(pd?(N,/60)~) n = exponent on F PI = power consumption of impeller, kg m2/s3 R = radius of particles, d p / 2 ,m r = distance from particle center, m Sc = Schmidt number, v / D Sh = Sherwood number, k d p / D Ti = dimensionless time, Dit/R2 t = time, s V = volume of solution, m3 V = volume of swollen resin, m3 W = weight of swollen resin, kg x = dimensionless radial coordinate, r / R yi = dimensionless concentration in the liquid phase, CJCo zi = dimensionless concentration in the resin phase, Ci/Co Greek Letters @ = diffusivity ratio in the resin phase, DA/DB 7 = diffusivity ratio in the liquid phase, DJDS 6 = thickness of liquid film, m e = VC~/WlS, c,
= energy dissi ation rate per unit mass, Px/Vp, m2/s3
8
hi = (ki)JiCq/Di. o v = kinematic wscosity, m2/s p = density of swden resin, kg/m3
= density of solution, kg/m3 Subscripts A = H+ion B = Na+ ion i = ion species, A or B 0 = initial m = final Superscripts * = equilibrium ’ = liquid film - = resin phase Literature Cited
p
Brian, P. L. T.; Hales, H. B.; shemood,T. K. AI= J. 196% 15, 727. Wden, C. E.; Kay, W. 8. Ind. Eng. Chem. 1854, 46,2294. Qto, S.;Qto, M.; Teshima, H. I d . €ng. Chem. fundam. 1911, prewdhg article In this Issue. Harrbt, P. AIJ. 1912, 8 , 93. Hashimoto, K.; Mkre, K.; Nageta, S. J. Cbm. €ng. &n. 1976, 8, 867. Hemerlch, F. “Ion Exchangs”; McQraw HUI: New Yo&, 1962 Chaptw 6. Kamemoto, T.; Yano, M.; Harano, Y. Kagaku K-ku Ronbunshu 1977, 9 , 289. Kataoka, T.; YosMda, H.; Sanada, H. J. Chem. €ng. gn 1974, 7, 105. Mdson,R. L.; OHem,H. A.,Jr. Chem. €ng. Rug. @m. Serl959, 55, 71. Sano, Y.; Yamaguchl, N.; Adachl, T. J. Chem. €ng. gn. l9T4, 7, 255. Srnlth, T. 0.;Dranoff, J. S. Ind. €ng. Chem. fwrdem. 1984, 3, 195.
Receiued for review January 26, 1981 Accepted August 4,1981
