1060
Ind. Eng. Chem. R e s . 1992,31, 1060-1073
Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion of Simple Penetrants in Polymers: 11. Applications-Nonvinyl Polymers. J. Polym. Sci., Polym. Phys. Ed. 1979b, 17, 453. Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion of Simple Penetrants in Polymers: 111. Applications-Vinyl and Related Polymers. J. Polym. Sci., Polym. Phys. Ed. 1979c, 17, 465. Pace, R.J.; Datyner, A. Statistical Mechanical Model for Diffusion of Complex Penetrants in Polymers: I. Theory. J. Polym. Sci., Polym. Phys. Ed. 1979d, 17, 1675. Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion of Complex Penetrants in Polymers: 11. Applications. J. Polym. .. Sci., P d y m . Phys. Ed. 1979ei 17, 1693. Paul. C. W. A Model for Predictinn Solvent Self-Diffusion Coefficients in Nonglassy Polymer/Soivent Solutions. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 425. Pauling, L. The Chemical Bond; Cornel1 University: Ithaca, NY, 1967;Chapter 7, pp 135-155. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, NY, 1977; Chapter 9,pp 391-469. Rogers, C. E.;Stannett, V.; Szwarc, M. The Sorption, Diffusion and J. Polym. Sci. Permeation of Organic Vapors in Polyethylene. 1960, 45, 61. Stern, S. A,; Fang, S.-M.; Frisch, H. L. Effect of Pressure on Gas Permeabilitv Coefficiente: A New Application of Free-Volume Theory. J. Polym. Sci., Part A-2 1972, 10, 201. Stern, S. A.; Kulkarni, S. S.; Frisch, H. L. Tests of a Free-Volume Model of Gas Permeation through Polymer Membranes: I. Pure COP,CH,, CZH4, and C3H8in Polyethylene. J.Polym. Sci., Polym. Phys. Ed. 1983, 21, 467.
Thermodynamics Research Center. TRC Thermodynamic Tables Hydrocarbon; The Texas A&M University: College Station, TX, 1987. van Krevelen, D. W.; Hoftyzer, P. J. Properties of Polymers: Their Estimation and Correlation with Chemical Structure, 2nd ed.; Elsevier: Amsterdam, 1976;Chapter 4, pp 51-79. van Velzen, D.; Cardozo, R. L.; Langenkamp, H. Liquid ViscosityTemperature-Chemical Constitution Relation for Organic Compounds. Ind. Eng. Chem. Fundam. 1972,11, 20. Vrentas, J. S.; Duda, J. L. Diffusion in Polymers-Solvent Systems: I. Reexamination of the Free-Volume Theory. J. Polym. Sci., Polym. Phys. Ed. 1977a, 15, 403. Vrentas, J. S.;Duda, J. L. Diffusion in Polymers-Solvent Systems: 11. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecular Weight. J. Polym. Sci., Polym. Phys. Ed. 1977b, 15, 417. Vrentas, J. S.;Duda, J. L.; Ling, H . 4 . Free-Volume Theories for Self-Diffusion in Polymer-Solvent Systems: I. Conceptual Differences in Theories. J. Polym. Sci., Polym. Phys. Ed. 1985a,23, 275. Vrentas, J. S.;Duda, J. L.; Ling, H.-C.; Hou, A.-C. Free-Volume Theories for Self-Diffusion in Polymer-Solvent Systems: 11. Predictive Capabilities. J. Polym. Sci., Polym. Phys. Ed. 1985b, 23, 289. Williams, M. L.; Landel, R. F.; Ferry, J. D. The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-formingLiquids. J. Am. Chem. SOC.1955, 77,3701. Receiued for review August 12, 1991 Revised manuscript received December 10, 1991 Accepted December 16, 1991
Sorption Studies on Ion Exchange Resins. 1. Sorption of Strong Acids on Weak Base Resins Vinay M. Bhandari, Vinay A. Juvekar,* and Suresh R. Patwardhan Department of Chemical Engineering, Indian Institute of Technology, Bombay 400076, India
Sorption equilibria and batch dynamics of strong acids (HC1 and HNOJ on weak base ion exchange resins, in free base form, are studied. The sorption process shows significant reversibility even a t high concentrations of acids. It is also concluded from the dynamic studies that the extent of exclusion of hydrogen ions from the resin pores is far less than that predicted on the basis of complete dissociation of counterions from the protonated ionogens. An attempt is made to correlate the sorption equilibria and sorption dynamics by considering an electric double layer a t the pore walls. A reversible sorption model, which accounts for sorption equilibrium at ionogenic sites of resin, fits the experimental dynamics satisfactorily over the entire range of the resin conversion. Values of the effective pore diffusion coefficient of the acids, regressed from the observed dynamics, have been satisfactorily correlated on the basis of the developed theory.
Introduction Weak base ion exchange resins are commonly used for removal of acids from aqueous streams. Experimental work on the uptake of acids on weak base resins has been reported by Kunin (1958), Adams et al. (19691, HOll and Sontheimer (1977), Hubner and Kadlec (1978), Rao and Gupta (1982a,b), and Helfferich and Hwang (1985). The reaction of an acid with a weak base resin involves protonation of the ionogenic sites of the resin by the acid. The mechanism of ion exchange involving ionic reactions was first postulated by Helfferich (1965). He stated that whenever there is an irreversible sorption on a resin, the dynamics of sorption can be explained on the basis of the shrinking core model. Helfferich and Hwang (1985) suggested that acid sorption by most of the commercial weak base ion exchangers can be modeled considering sorption to be irreversible under most conditions. However, the work of Kunin (1958) and also the present work on sorp-
tion equilibria on weak base resins indicate that the sorption is significantly reversible, especially at low acid concentrations and for the resins with low basicity. In such situations the shrinking core model is not expected to be valid. So far, the most general treatment of the dynamics of the sorption of acids on weak base resins has been presented by Helfferich and Hwang (1985). They have also suggested a rate-controlling-ion model, which is a simplified version of their generalized model. According to these authors, the rate-controlling step in the sorption of strong acids (such as HC1) is the diffusion of H+ ions, which are excluded to a significant extent form the resin pores as a result of Donnan exclusion. Their model assumes complete dissociation of the acid anions from the protonated ionogenic species of the resin. Concentration of H+ in the resin is computed using the ideal Donnan exclusion principle. Analytical solution of the shrinking core model
0888-5885/92/2631-1060$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1061 is used to describe the dynamics. The model has been verified on the basis of the experiments on the sorption of HC1 and H d O l on Amberlite IRA-68. The concentrations of acids in the range of 0.1-0.5 kmol/m3 have been used in the experiments. In this work we have studied equilibria and dynamics of sorption of strong acids (HC1 and “OB) on weak base resins, viz., Dowex WGR-2 and Amberlite IRA-93. These resins have much lower basicity than Amberlite IRA-68, used in the work of Helfferich and Hwang (1985). The acid concentrations in our work range from 0.001 to 0.02 kmol/m3. It was observed from the studies on equilibria that the sorption on these resins is reversible even at high acid concentrations (as high as 0.1 kmol/m3 for Dowex WGR-2). Such a high extent of reversibility is not commensurate with the basicity of these resins. Further, when the H+ concentration in the resin pores was computed on the basis of the ideal Donnan exclusion principle, absurdly high values of pore diffusivity of H+ resulted from regreasion of the experimental data. In fact, reasonably good values of diffusivity were obtained when the exclusion of H+ from the pores was neglected. These observations can be explained if we consider the existence of a charged double layer at the pore walls. The counterions (acid anions, in this case) are concentrated in this double layer. As a result, the core of a pore is relatively dilute wth respect to the counterions. The core region therefore offers only a weak Donnan potential, and hence the extent of exclusion of the co-ion (H+,in this case) from the pores is considerably reduced. In this work, we have presented a reversible sorption model which accounts for the effect of the reversibility of the sorption process on dynamics. The values of the effective pore diffusion coefficient have been regressed on the basis of the model. After taking into account the effect of a double layer, it was also possible to explain the trends in the regressed values of the diffusion coefficients.
Experimental Work Weak base resins, viz., Dowez WGR-2 (epoxyamine polymer matrix) of Dow Chemical Co. and Amberlite IRA-93 (macroreticular polystyrene polyamine resin) of Rohm and Haas Co., were used for the present study. The dry resin samples were sieved several times to obtain a fraction of narrow size range. The average particle size of the fully swollen resin beads, after equilibrating with distilled water, was measured using an image analyzing system (LEITZ TAS-PLUS), which involves digitization and analysis of the optical image of a sample. The resolution of the microscope is 1pm. Approximately 100 resin beads were used in each measurement. The average particle size was 490 pm for Dowex WGR-2 and 450 pm for Amberlite IRA-93. The deviations from the mean size were less than *5%. The resins were pretreated before use as per the procedure suggested by Helfferich (1962). Hydrochloric acid and nitric acid used for these studies were of analytical reagent grade and were obtained from companies of repute. Sorption Equilibria. Samples of known weight of the resin were equilibrated with acid solutions of known concentrations. Equilibration was carried out for more than 72 h on a continuous shaker at ambient temperature (27 1“C). Samples of the extraparticle fluid were analyzed for the acid concentrations by measuring the pH of the solution (for acid concentrations less than 0.02 kmol/m3) and by titration with standard NaOH solution (for acid concentrations greater than 0.02 kmol/m3). The acid concentrations in the extraparticle fluid after 48 and 72 h were practically the same in all the experiments, indi-
*
t.0
> EQUILIBRIA FIT -_ _ _SORPTION _ ISOTHERM FOR IRREVERSIBLE SORPIION
01
1~16~
I I”164
I
I [HI
I
I 1x16’
1x162
1.1@
, krnol/m3
Figure 1. Sorption equilibria on Amberlite IRA-93.
t
l
L
I
lr1i5
1~16‘
1.1i’ [HI
,
1”
12
1x10‘
1.0
hmollm’
Figure 2. Sorption equilibria on Dowex WGR-2.
cating attainment of equilibrium after 48 h. The equilibrium concentration of the sorbed acid was estimated from the difference between the initial and final acid concentrations in the extraparticle fluid. An appropriate correction was made for the free acid in the resin pores. Fractional pore volume required for this correction was obtained from measurements of the water content of the swollen resin. The water contents of fully swollen Dowex WGR-2 and Amberlite IRA-93 were estimated to be 0.95 and 1.18cm3/g, respectively, on the dry weight basis. The sorbed acid concentration reaches a constant value at high acid concentrations (>0.5 kmol/m3). This value of the sorbed acid concentration may be considered to represent the capacity of the resin, Q. The capacities based on the pore volume were found to be 10.1 and 4.67 kmol/m3 for Dowex WGR-2 and Amberlite IRA-93, respectively. The sorption equilibria are presented in Figures 1and 2. It is seen from these figures that the sorption isotherm deviates considerably from the rectangular form (shown by the dotted lines), which is representative of irreversible sorption. This indicates that the sorption is significantly reversible. Reversibility is more pronounced in Dowex WGR-2, which is a weaker base. Sorption Dynamics. Sorption dynamics was studied in an 85-mm-diameter, 500-cm3-capacity glass vessel provided with a 30-mm-diameter, turbine type glass sitrrer and four axial baffles of 10-mm width. A known weight of resin, partially vacuum dried to a known moisture content, was added to a 250-cm3 acid solution of known concentration. The purpose of predrying was to remove the superficial water from the resins and thereby make them free flowing so that they do not stick to the wall of the vessel during addition. The stirrer was started immediately after the resin addition. The stirring speed of 450 revfmin was used in order to ensure complete SUBpension of resin beads and also to ensure that the film diffusion resistance is negligibly small. The latter was confirmed from the fact that practically identical dynamic behavior was observed at stirrer speeds as low as 250 revfmin. A research grade digital pH meter, with a resolution of 0.001 pH, was used for measurement of the pH
1062 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 12
“2 s n 10 E *
-
0
0
0
E -
A
. I
z
-
.
w w
2.20
0
.
n
:3.39 :4.64
W
- FIT
8
AMBERLITE IRA-93 HCl INITIAL ACID CONCENTRATION.0.02 1mol 0
w i 1.01 w 2.00
rr?
i
-FIT OFTHE MODEL
E 15 0
OF THE MODEL
E
0 c 4
E
z 6 W
U
z
0
u 4 B U
4
2
0 0
2
1 -
2
0
6
4
-
* I @
a
0
a
6
4
TIME, ks
TIME, ks
Figure 3. Experimental data on sorption dynamics of Amberlite IRA-93/HCl with initial acid concentration = 0.001 kmol/m3.
Figure 6. Experimental data on sorption dynamics of Amberlite IRA-93/HCl with initial acid concentration = 0.02 kmol/m3. 12,
AMBERLITE IRA-93/HCl INITIAL ACID C O N C E N T R A T I O N ~ * 0 0 5 1 ~ 0 w .I.OL 0 W 1-96 AW ~ 3 . 0 4 FIT OF THE MOML
AMBERLITE IRA-93/HNO3 km
a
INITIAL ACID CONCENTRATK)N=O.OOI 0 W.2.36 0 w L.22 0 w. 5 . 7 9 FIT OF THE MODEL
-
0
2
0
6
4
2
Figure 4. Experimental data on sorption dynamics of Ax IRA-93/HCl with initial acid concentration = 0.0051 kmol
brlite 3.
Figure 7. Experimental data on sorption dynamics of Amberlite IRA-93/HN03 with initial acid concentration = 0.001 kmol/m3. AMBERLITE IRA- 93/HN03 k mol INITIAL ACID CONCENTRATION ~ 0 . 0 0 4 57 rn 0 W z 0.93
AMBERLITE IRA-93 HCI INITIAL ACID CONCENTRATION=O.OI 0 0
-
w 1.00 w = 2.01
A W
6
4 TIME, k s
TIME, k s
m3
0
A
- FIT
3-01 FIT OFTHE MODEL s
0 TIME, lo
Figure 5. Experimental data on sorption dynamics of Amberlite IRA-93/HCl with initial acid concentration = 0.01 kmol/m3.
of the solution. The meter was precalibrated in the required acidic range (pH 2-5). All the experiments were carried out at ambient temperature (27 f 1OC). After each experimental run, the resin beads were visually checked
2
w; 1.30 W = 1.70
OF THE MODEL
4
6
TIME, ks
Figure 8. Experimental data on sorption dynamics of Amberlite IRA-93/HN03 with initial acid concentration = 0.0045 kmol/m3.
for physical damage. No such damage was observed during any experiment. The results of sorption dynamics are presented in Figures 3-16 as plots of extraparticle acid concentration vs time.
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1063
1
10
m0
r a
m
-
1 AMBERLITE l R k 9 3 1 H N O j INITIAL ACID CONCENTRATION=O.0098 0 W: 0.80 FIT OF THE MODEL
%
INITIAL ACID CONCE 0
-
E
w =
1.1 1
5
2.00
w
0
-FIT OF THE MODEL
e *I a
0 0
2
6
4
8
TIME, Its
TIME,kr
Figure 9. Experimental data on sorption dynamics of Amberlite IRA-93/HN03 with initial acid concentration = 0.0098 kmol/m3.
Figure 12. Experimental data on sorption dynamics of Dowex WGR-2/HCl with initial acid concentration = 0.0095 km01/m3.
DOWEX WGR -2/HCl INITIAL ACID CONCENTRATION:0.0011 0
2.50 W = 3-05
A
W
O W
INITIAL ACID CONCENTRATION= 0.02
t
i
4.03
2
4
i
2.00
TIME, ks
WGR-S/HCl with initial acid concentration = 0.0011 kmol/m3.
DOWEX WGR-Z/HC I INITIAL ACID CONCENTRATION:.0051 0 w : 1.43 0 w :2 . 1 2 FIT O f THE MODEL
Figure 13. Experimental data on sorption dynamics of Dowex WGR-2/HCl with initial acid concentration = 0.02 kmol/m3.
m
0 W 0
A
-
4
1.00
z
6
Figure 10. Experimental data on sorption dynamics of Dowex
2
W . 1.50
-FIT OF THE MODEL
TIME, Its
0 0
w
0
A W
- FIT OF THE MODEL
0
0
6
i
2-51
W
:3.71 W :5.70
- FIT OF THE MODEL
8
TIME, ks
Figure 11. Experimental data on sorption dynamics of Dowex WGR-P/HCl with initial acid concentration = 0.0051 kmol/m3.
Correlation of the Equilibrium Data As a first attempt, both the sorption equilibria and dynamics are correlated under the assumptions of complete dissociation of the acid salt and ideal Donnan ex-
0
2
4 TIME, ks
6
Figure 14. Experimental data on sorption dynamics of Dowex WGR-2/HN03 with initial acid concentration = 0.001 kmol/ma.
clusion principle. A refined model based on the doublelayer theory is discussed later. For convenience the following abbreviations are used in the equations to denote the species: HA for H+A- (acid species); H for H+; A for
1064 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992
Table I. Variation of K, with Concentration for Amberlite IRA-93/HNO8 (Q = 4.67 kmol/ma)
OOWEX WGR-2/HN03 INITIAL ACID CONCENTRATION= 0.0045k mol 0 w 5 1.01 730 w = 2.20 o W i 3.02 FIT OF THE MODEL
equilibr concn of H+ [HI, kmol/m3 [HI, kmol/m3 6.82 X lo-" 9.33 X lo* 4.71 x 10-10 3.47 x 10-5 2.18 X 2.88 X 7.59 x 10-3 1.29 x 10-5 3.56 X lo-* 2.78 X lo-'
00
2
6
4
Amberlite IRA-93 Amberlite IRA-93 Dowex WGR-2 Dowex WGR-2 1 0
Figure 15. Experimental data on sorption dynamics of Dowex WGR-2/HN03 with initial acid concentration = 0.0045 kmol/m3.
456 513 7.80 28.4
-0.653 -0.709 -0.848 -0.784
m 1 Kp =
- FIT
HC1 HNOB HCl HN03
where Q is the capacity of the resin based on the porephase volume. Eliminating [R] between eqs 2 and 4 we get
DOWEX WGR -Z/HNO3 0 W ~1.04
KP' (krnol/m3)-l 5.51 x 109 2.55 x 109 2.02 x 108 1.70 X lo6 1.32 x 105
Table 11. Regressed Values of Constants KPoand m resin acid KPO m
TIME, ks
INITIAL ACID CONCENTRATION :0.0098
[=I,
kmol/m3 1.28 2.55 3.81 4.47 4.55
(Q - [=])[HI
The electroneutrality condition in the pore gives [RH] [HI = [A] (6) Since [HI is much less than (in view of strong exclusion of H+from the pore phase), we can approximately write [RH] = [A] (7) The ideal Donnan exclusion principle requires
&?!!
+
m3
w .2.00 OF THE MODEL
[m]
[H][A] = [HI2 (8) where [HI repr_esnta the extraparticle concentrationof H+. Eliminating [A] between eqs 7 and 8 and combining the resultant equation with eq 5, we get
6
0
0
I
2
I
4
1
6
1 0
TIME, k 5
Figure 16. Experimental data on sorption dynamics of Dowex WGR-2/HN03 with initial acid concentration = 0.0098 km01/m3.
A- (acid anion); RH for RH+ (protonated species); RHA for RH+A- (acid salt). The resin is assumed to consist of a solid phase and a pore phase. Concentrations of all the species inside the resin are based on the volume of the pore phase. The protonation of the ionogenic groups of the resin may be described by the following reaction: R+H*RH (1) The bars above the species refer to an intraparticle environment. The concentration-basedequilibrium constant for the above reaction is defined as
The association of the anion with the protonated ionogenic group may be written as RH+A*RHA (3) Reactions 1 and 3 are ionization reactions and are instantaneous. It is also assumed that RHA is completely dissociated, i.e., the equilibrium constant of reaction 3 is zero. The material balance on the ionogenic species yields (4) + [R] = Q
[m]
Table I lists some values of Kp for the system Amberlite IRA-93/HN03. It is seen from this table that Kpdecreases rapidly with the increase in acid concentration. The value of Kp, estimated by Adams et al. (1969) for Amberlite IRA-93,is approximately 1X 10s (kmol/m3)-'. The values of Kp of Table I range from lo9 to 105 (kmol/m3)-'. Such a large variation cannot be explained on the basis of the ionic activity coefficients alone. K may be empirically correlated with the concentration ions in the pore as follows: of Kp = KPo[H]" (10)
If+
where Kpoand m are constants. Combining eqs 10 and 9 and using eqs 7 and 8, we get
Kpo and m can now be regressed from the experimental equilibrium data of Figures 1 and 2. These values are listed in Table I1 for different resin/acid systems. These equilibrium relationships are used for correlating the data on sorption dynamics. Geometric Considerations All the resin beads are assumed to be spherical in shape and uniform in size. The pore phase is assumed to consist of cylindrical pores of uniform radius. To simplify the
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1066
analysis, it is assumed that the pores are straight and run radially from the outer surface of the particle to its center. The tortuous nature of the pores is accounted for in the tortuosity factor, 7 , which is applied to correct individual ionic diffusivities by the following equation: Di = Di0/7 (12) where Dio is the free diffusivity of ion i in solution while Di is its diffusivity in the pore phase. Ionogenic groups of resin are assumed to be located on the pore walls. Diffusion of ions occurs along the axis of the pore, which also corresponds to the radial coordinate of the spherical particle (r coordinate). The transport equations inside a pore are written in cylindrical coordinates, while those in the pore phase are written in spherical coordinates.
Correlation of Sorption Dynamics The generalized model treatment of Helfferich and Hwang (1985) is used to analyze the dynamics. Sorption reversibility is incorporated using eqs 5 and 10. Extraparticle acid balance is coupled with the intraparticle balances. The film resistance is neglected. The fluxes of ions are written in terms of the Nernst-Planck equation, and the electrical potential term is eliminated by using the no-current condition. The concentration of the co-ion at the surface of the resin is computed by applying the ideal Donnan exclusion principle between the pore solution phase and the extraparticle solution phase. It is also assumed that the acid anion is uniformly distributed in any cross section of a pore. The model equations are summarized as follows: unsteady-state continuity equation in the pore phase
H+ through electrostatic coupling effect. Sorption Equilibrium. Combining eqs 5 and 10 and rearranging the resulting expression, we get QKpo[H]m+l [RH] = 1 KpOIH]m+l
+
Modifications. Eliminating [A] and [HIbetween eqs 7,18, and 19, we get the following expression for the flux of H+ in the pore
r JH
I
=-DH 1 +
m+l (1 KpOIH]m+l)
+
+
The term in the square brackets may be considered as a factor which modifies the diffusion coefficient DH. This modification is brought about by the sorption process, as explained with reference to eq 18. Substitution of J H from eq 20 and [ m ] from eq 19 into eq 13 and subsequent rearrangement yields
(1 + KpOIH]m+l) m+l 1(QK,o[H]m+l) + K p O I H ] m + l [HI) +
tJ
(21)
boundary conditions
-m -1 - 0
at r=O ar @][A] = [HI2 at r = Rb (23) initial condition [HI = O at t=O (24) Equation 24 implies that the resin is in free base form at the start of the experiment. unsteady-state continuity equation in extraparticle fluid
Nernst-Planck equations
no-current condition JH
=J
A
Using the no-current condition, thereby eliminating the electrical potential term from the Nemt-Planck equation, we get
Since [A] >> [HI, the second term in the denominator can be neglected in comparison with the first term. Further, replacing a[A]/ar by a[RH]/ar in the resulting equation in view of eq 7, the following form results:
Inspection of eq 18 reveals that the flux of H+ in the pore phase is enhanced due to the sorption process, as indicated by the second term in the bracket. In the absence of sorption (a[RH]/ar = 0), the contribution of this term will be zero. The enhancement occurs because the sorbed acid dissociates to produce A- which enhance the diffusion of
where N is the number of particles per unit volume of the extraparticle fluid. In deriving eq 25 it is assumed that the total cross-sectional area of the pores is cp fraction of the total outer surface area of the resin. Equation 25 can be more conveniently written as
where W is the dimensionless resin loading, which is defined as (Helfferich, 1962; Adams et al., 1969) W = 4rRb3Ne,Q / 3[HIi (27) W represents the ratio of the total exchange capacity of the resin feed to the total initial acid content of the solution. J H J r = R in eq 26 can be calculated from the solution of eq 20. d e initial condition to eq 25 is [HI = [HIi at t =0 (28) Method of Solution. The partial differential equation, eq 21, along with boundary conditions, was converted into
1066 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992
Table 111. Results of the Model Based on Complete Dissociation of RHA in Resin Pores [HIi x lo3, kmol/m3 1.12
5.11 9.50 20.0 1.00 9.70
t RI iIr-Rb, W kmol/m3 Dowex WGR-PIHCl System 2.50 2.83 X 1.43 4.72 X 10” 1.11 1.51 x 10-5 LOO 6.17 x 10-5
M
IHP
I
?SCATTERED
ION
I
LAYER
regressed Dw,m2/s 1.00 X 10” 1.00 X 3.00 x 10-8 9.55 x 104
Amberlite IRA-93/HN03 System 4.22 2.49 X 6.20 X 0.80 2.13 X 2.70 X
a set of ordinary differential equations (ODES) by the orthogonal collocation technique (Villadsen and Michelsen, 1978). The stability of the solution was improved by the transformation
(r/R# (29) The resulting ODES were solved using the IMSL routine IVPAG (initial value problem Adams and Gear). The solution was found to be practically insensitive to the number of nodes on u when this number was greater than 8. Hence nine collocation nodes were used for all the simulations. Diffusivity values based on the model were regressed from the experimental data of [HA] vs time, using Simplex optimization routine UMPOL of IMSL. Comments on the Applicability of the Model. The regressed values of the pore diffusivity of H+ for some of the experimental runsare listed in Table 111. It is evident that these values are very high and unrealistic, especially at low acid concen_trations. Table I11 also lists the estimated values of [H]I,=,, at t = 0. This term represents the highest concentration of H+ in the pore. It is seen that the values of [H]Ir=Rb are extremely low relative to [HI, especially at low concentrations. It can be argued from eq 21 that the low estimates of the concentration of H+ in the pore would force the diffusivity to assume a high value so as to match the observed experimental dynamics. This observation prompts us to conclude that, for dilute acids, the estimates of the concentration of H+ in the pores on the assumption of complete salt dissociation in the pore are too low. It is also seen from Table 111that the regreased value of the diffusivity somewhat approaches a realistic value at high acid concentrations. This may be attributed to the fact that the Donnan principle predicts a lower extent of exclusion at high acid concentrations. This also explains why the value of the diffusivity of H+ (8.8 X lo* mz/s in resin phase, Le., 1.47 X 10-%n2/s based on the pore phase), regressed by Helfferich and Hwang (1985) from the experimental data on the Amberlite IRA-68/HCl system, is correct to the order of magnitude, in view of the high concentration of HCl(O.1-0.5 km01/m3) used by these authors. These observations are indicative of the inadequacy of the model in treating the sorption dynamics, especially at low acid concentrations. There is a need for a new theoretical approach which can explain the observed experimental behavior, both qualitatively and quantitatively. As an attempt, a model has been developed by considering the electrical double layer at the pore walls. Mathematical treatment has been formulated by characterizingthe effect of the double layer on diffusion. The results of this work are then discussed. u =
Existence of an Ionic Double Layer in Resin Pores We feel that the limitation of the model described in the preceding sectin lies in the assumption that the acid salt
Figure 17. Distribution of ions across a pore: (a) the Stern model; (b) concentration profile.
is completely dissociated and the counterions are uniformly distributed across the pore. It should be noted that the diameter of a typical resin pore is of the order of 10 nm. Hence, complete dissociation of the resin implies a charge separation over a distance of the order of 10 nm. It is a well-known fact that an electric double layer exists at charged interfaces. In the present context, the protonated surface of the wall of a pore forms the positively charged interface. Most of the counterions would be present in the region between the interface and “inner Helmholtz plane (I”)”, due to their “specific adsorption” at the interface. The diffuse layer of charge existe beyond IHP. These two regions are schematically shown in Figure 17a. Typical concentration profiles of H+ and A- ions across a pore are shown in Figure 17b. From the region bounded by IHP, the co-ions are practically excluded due to a high concentration of RH+. However, as we move away from the wall the disparity between the concentrations of H+ and A- decreases. If the pore diameter is sufficientlylarge, [HI would nearly become equal to [A] in the core region of the pore. This core region would therefore be freely accessible to H+ from the extraparticle solution. A major contribution to the overall diffusion process would be expected from the core region. On the other hand, the region near the wall of a pore would have a small contribution to diffusion due to exclusion of H+ from this region and probably by the drag offered by the charged interface to the diffusion of H+. The effective flux of H+ through the pores would therefore depend upon the concentration profde of H+ in the diffuse layer. These aspects have been accounted for in developing a new model for the sorption phenomenon.
“Reversible Sorption Model Based on Double-Layer Theory” Theoretical Considerations. Any cross section of a pore is viewed as consisting of three regions. The first region is between the wall of a pore and the IHP, where specific adsorption of counterions occurs. The width of this region is of the order of 1 nm (Bockris and Reddy, 1972). The second region is the “diffuse layer”, in which ions are scattered, as shown in Figure 17a. The total electrical potential in the pore is contributed by the potential of the double layer and the potential generated by diffusion of ions under the constraint of no electrical current. As suggested by Osterle and co-workers (Morrison and Osterle, 1965; Gross and Osterle, 19681,the
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1067 total electrical potential &.({,r) at any location in a pore may be split as &&,r) = $({,r) + 449 (30) where $({,r)is the contribution due to the electricaldouble layer and is a function of both { and r while $49 is the potential generated due to diffusion and is a function of r alone. We define the core region (the third region) as the region which lies outside the diffuse layer. It is assumed that in the core region the potential $ is zero. For the cases where the pore diameter is large, the core region would surround the pore axis. For small pore sizes and at low acid concentrations, the double layer may extend well up to the pore axis and the core region will have only a hypothetical existence. The concentrations of H+and A- in the core region are denoted by [HI, and [A],, respectively. Since local electroneutrality exists in the core region [HI, = [AI, [HI, and [A], will depend only on the axial location r in the pore. On the resin surface (i.e., at r = R b ) , we have [HIClr=Rb= [HI (32) Using the Boltzmann distribution, we can relate the concentrations of H+ and A- in the diffuse layer to the core concentration by the following equation:
exp(-g) [AI,, g)
[Rl, [HI, =
[AI = exp(
Defining X as = exp(
g)
we can write eqs 33 and 34 as [HI,= [RI,/X
(33)
Table IV. Regressed Values of Constants K Oand n: Reversible Sorption Model Based on Double-Layer Theory resin Amberlite IRA-93 Amberlite IRA-93 Dowex WGR-2 Dowex WGR-2
acid HCl HN03 HCl HN03
KO
600 313 2.14 6.66
n 1.107 1.320 1.540 1.464
where [A], is the concentration of the counterions in the region bounded by IHP. This 8, may be empirically related to the concentration of A- in the core region by a Freundlich type isotherm
where K and n are the isotherm constants. A more fundamental approach would be to obtain the concentration of A- by solving the Poisson-Boltzmann equation in the pore and use these results in conjunction with a Langmuir type isotherm to correlate the sorption equilibria. Difficulties underlying this approach are discussed in detail in Results and Discussion. Combining eqs 40,41, and 31, we obtain the following expression: [A], = [ m ] K (
%r
(42)
(34)
Applying eq 38 for the wall region we get [RIJAI, = [Hl,[Al, (43) [A], can now be eliminated from eq 42 using eq 43 to obtain
(35)
(44)
(36)
(37) X represents the inverse Boltzmann factor and is greater than unity in the diffuse layer. The value of X is unity in the core region. In general, X is a function of both { and r. Since the potential, $, increases with an increase in { (i.e., as we move toward the pore wall), it is expected from eq 35 that X also increases with {. Hence from eqs 36 and 37 we can deduce that the concentration of H+ decreases while that of A- increases as the poTe wall is_approached. At any point in the diffuse layer, [HI and [A] are related by the following equation: [Hl,[Al, = [Hlc[Alc (38) which follows from eqs 36 and 37. Correlation of the Sorption Equilibrium. The equilibrium constant for the protonation reaction may now be defined as (39) where [HI, is the concentration of H+ in the region bounded by IHP. Due to adsorption of counterions, the protonated sites of the resin are partially neutralized. If we denote the fraction of surface sites neutralized by A- as e,, then
Equation 44 helps us to write the equilibrium constant Kp of eq 39 as a function of [HI,
where KOis the composite equilibrium constant which combines the protonation equilibrium constant and the adsorption equilibrium constant. At equilibrium, [HI, is independent of r, and, in view of eq 32, it is also equal to [HI, the extraparticle concentration of H+. Equation 45 may therefore be written as
The values of KOand n are regressed from the observed sorption equilibria. These values for different systems are reported in Table IV. The solid lines in Figures 1and 2 represent eq 46, which incorporates the values of KOand n from Table IV.It is seen that eq 46 fits the experimental data well. Small values of KOin Table IV are indicative of high values of the adsorption constant, K. Although Dowex WGR-2 is a weaker base (KP= lo8) compared to Amberlite IRA-93 (Kp= lo9), the value of KOfor Dowex WGR-2 is about 300 times less for HC1 and 50 times less for HN03 than the corresponding values for Amberlite IRA-93. This indicates that Dowex WGR-2 adsorbs acid anions more strongly than Amberlite IRA-93. Model for Sorption Dynamics. In the derivation of the equations describing the sorption dynamics, a number of assumptions are made as and when required. These are
1068 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992
justified in Results and Discussion. The flux of H+ at location f in a pore, (JH)b is given by the followicg equation, which is obtjained from eq 17 by replacing [HI and [A] by [HI, and [A], respectively, and subsequently relating these concentrations to the core concentrations through eqs 36 and 37
OL 0
2.5-
which on simplification yields
The area average flux of H+ in a pore can be written as (49)
0 1
where A, (=(7r/4)d 2) is the cross-sectional area of the pore. Substitution of H!(r! from eq 48 into eq 49 and subsequent rearrangement yields
I
I
I
I
6
11
16
21
Figure 18. Variation of effective diffusivity with X.
m2/s for HC1 and 2.10 X m2/s for The X values corresponding to these are 2.10 and 2.20, respectively. The unsteady-state continuity eq 13 can be written as
Dm is 2.17 X
"0,. where
DHA is the effective diffusivity of H+ in the pore.
We can approximate D H A as (52)
where
In writing the first term of eq 55, we have replaced the average H+ concent_rationover the cross section at any radial location by [H],/X. We have also assumed that X is a weak function of t. Since out of the two terms on the left-hand side of eq 55 the first term has a very small contribution, this approximation is expected to have an insignificant effect on the solution of the equation. We may rewrite eq 55 as
(53)
The quantity X quantifies the effect of the double layer on the effective diffusivity of H+ in the pores. X = 1 is an asymptotic condition where there is no diffuse layer. Such a case implies that the adsorption of the counterions on the pore wall is so strong that O8 = 1 and the wall acts as a neutral surface. In such a situation, local electroneutrality exists over the entire pore cross section. Dm will then be obtained by the Nernst-Planck effective diffusivity (54)
X >> 1 indicates a severe effect of the diffuse layer on Dm. Figure 18 shows plots of DHA vs X for HC1 and HNO, which are obtained using eq 52. For calculating Dm, the following values from literature (Cussler, 1986) of individual ionic diffusion coefficients in free solution (Die) are used: DHo = 9.31 X m2/s, Dclo = 2.03 X lo4 m2/s, and DNO: = 1.90 X lo4 m2/s. The pore diffusion coefficients of these ions are then obtained by applying the tortuosity factor ( T ) of 2 (Wheeler, 1951). It is seen from these plots that if we increase X starting from X = 1, the effective diffusivity D m increases initially, passes through a maximum and then decreases. The maximum value of
where d[RH]/a[H], is obtained from eq 45 as
+ 2&@+'(2 - n)[R],'-" ((Ko8"[R]c2-y + 4K0@+'[H]:-n]1/2
(K0@)2(2 - n)[R],34"
)
(57)
The boundary conditions for eq 56 are
m - -1, ar
-0
[HI,= [HI
at
r=O
(58)
at
r=Rb
(59)
while the initial condition is
[R], = o
at t = O (60) The equations for the H+ species in the extraparticle fluid are the same as described in the previous section. However, now the flux of the species has to be computed using eq 50. For solving eq 56 we assume DHAto be a constant. This is not strictly true since we expect X to change along the
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1069
-
I. 0
z
2
a
Ambcrlitc IRA 93 I HCI Initial add conccntrotion=0401kmd/m3
0.8-
W.
regressed
t t
[H+Ii X lo3, km0i/m3 1.00
c
z
W
y
Table V. Results of the Reversible Sorption Model Based on Double-Layer Theory for the System Amberlite IRA33IHCl
0.6-
8 m m
5.10
5z 0 . 4 P m
10.0
z W
5 a
0.2-
20.0
w 2.20 3.39 4.67 1.04 1.95 2.90 1.00 2.01 3.00 1.00 2.00
D~ x 109, m2/s 0.390 0.430 0.395 1.25 1.00 0.825 1.70 1.30 1.10 1.90 1.90
x 23.7 21.4 23.4 7.11 8.75 11.2 4.45 6.45 7.80 3.62 3.62
geometric mean concn, 103 km0i/m3 0.700 0.529 0.456 2.77 1.70 1.32 4.47 2.05 1.48 8.13 2.69
0.0 0. 3
0.5
0~7
0.9
U
Figure 19. Typical concentration profile inside the resin particle of Amberlite IRA-gS/HCI: initial acid concentration = 0.001 kmol/m3; t = 5.4 ks.
pore axis as a result of variation in the concentration of H+ions. However, we have shown later that the intraparticle variation of DHAwith r has a small effect on the predicted rate of sorption and the sorption dynamics is dominated by the value of diffusivity at the surface of the particle (i.e., DHAIr=R,).The regressed values of Dm therefore may be regarded as the value at the particle surface. In order to ease the computational efforts, we have replaced l / X , the first term in the bracket on the left-hand side of eq 56, by 1. Validity of this approximation is justified later. The procedure to solve eq 56 is similar to that described in the earlier section. The only unknown parameter in these equations is DHAwhich can be regressed from the experimentally observed dynamics. The results of this analysis are summarized below.
Results and Discussion Fit of the Model to the Experimental Data. Figures 3-16 compare the experimental data of extraparticle acid concentration v8 time with the simulated profile based on the regressed values of pore diffusion coefficients. For most of the experimental runs, the fit of the “reversible sorption model“ was within i5% of the observed values over the entire range of acid concentration. For only a few runs the deviations were greater than i5% but never more than i15%. The excellent fit of the model to the dynamics of sorption over the entire range of concentrations clearly substantiates the principles behind the methodology followed in the present work. Concentration Profiles in the Pore Phase. Figure 19 shows a typical rofile of the dimensionless core concentration of H+([# ],/[HI) I and that of the RH+species in the pore phase as a function of dimensionless square radial distance (r/RbY. It is seen from this figure that there is a steep drop in the concentration of H+in the pore phase as we move inward from the surface of the min. The correaponding drop in the concentration of RH+ is relatively small. These concentration profiles resemble those found in a shrinking core type mechanism. The difference is that the reacted shell is only partially filled, and the distinction between the reacted shell and the unreacted core is not sharp. Comments on Effective Pore Diffusion Coefficients. Regressed values of the effective pore diffusion coefficient
([m]/Q)
Table VI. Results of the Reversible Sorption Model Based on Double-Layer Theory for the System Amberlite IRA-93/HNOa [H+Ii x lo3, kmol/m3 1.00 4.50 9.70
regressed
w 2.34 4.22 5.79 0.93 1.30 1.70 0.80
D~ x 109, m2/s 0.555 0.455 0.400 1.20 0.95 0.94 1.75
x 16.50 20.20 20.90 7.00 9.27 9.35 4.41
geometric mean concn, 103 kmo1/m3 0.592 0.412 0.335 2.62 2.12 1.62 5.39
Table VII. Results of the Reversible Sorption Model Based on Double-Layer Theory for the System Dowex WGRd/HCl regressed
[H+IiX lo3, kmol/m3 1.12 5.11 9.50 20.0
D~ x W 2.50 3.00 4.00 1.43 2.12 3.05 1.11 2.00 1.00 1.50 2.00
109,
mz/s 3.00 2.85 2.85 2.35 2.35 2.35 1.10 1.15 0.90 1.00 1.20
x 2.10 2.10 2.10 2.10 2.10 2.10 1.00 1.00 1.00 1.00 1.00
geometric mean concn, 103 km01/m3 0.669 0.597 0.469 3.04 2.18 1.45 6.58 4.20 13.6 9.80 6.32
Table VIII. Results of the Reversible Sorption Model Based on Double-Layer Theory for the System Dowex WGR-B/HNOI [H+Ii x lo3, kmol/m3 1.00 4.50 9.80
regressed
Dm W 2.57 3.71 5.70 1.01 2.20 3.02 1.04 2.00
X
lo’,
m2/s 3.10 3.50 3.60 2.45 2.00 2.00 1.50 1.50
X 2.20 2.20 2.20 2.20 1.60 1.60 1.00 1.00
geometric mean concn, i09kmol/m3 0.535 0.424 0.272 3.19 1.97 1.44 6.32 3.62
are listed in Tables V-VIII. The values of X are regressed from these values of DHAusing eq 52 and are also pregented in these tables. The following observations are made from these tables. Amberlite IRA-93 Resin (Tables V and VI). (1)At low acid concentrations, X is much geater than 1indicating significant effect of a double layer. X decreases with an increase in the acid concentration. This is expected since the diffuse layer shrinks with an increase in the acid concentration.
1070 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992
Table IX. Comparison of 1/x with a [ ~ ] / a [ H ] , system Dowex WGR-2/HC1 Amberlite IRAPS/HCl
[R], kmol/m3 1 x 10-7 0.01 0.10 1X 0.01 0.10
l/X
