signs of decline of its performance (see Table 11).As can be seen from the data shown in Table IIIb, the diminished rejection of the residual radioactive salts was not paralleled by a similar increase in aluminum permeation. Two possibilities were considered. (a) Rejection of Ce, Zr, and Ru is concentration dependent and a t their very low concentrations it vanishes. T o check this possibility the overall concentration of these elements in the model solution was increased by several orders of magnitude by addition of their nonradioactive isotopes. A comparison between these experiments and those in which the radioactive salts were not diluted with nonradioactive isotopes is given in Table V. No significant effect due to the increase in the overall concentration of these elements is noted. Another possibility considered to explain the observed phenomena was that, under the experimental conditions, the investigated radioactive salts were partially hydrolyzed and form semicolloidal aggregates (Orth et al., 1971) which are totally rejected, wbile the minute amounts of unaggregated salts present in solution permeate through the membrane. If the latter hypothesis is correct the efficiency of the second decontamination stage should increase
after storage of the primary product or as a result of appropriate p H adjustments. In this case, very high decontamination factors would be expected in the processing of the primary effluent. Experimental results shown in Table IV do not seem to confirm the last prediction. At the present stage of the work, this point remains, therefore, unresolved.
Literature Cited Bloch, R., Finkelstein, A,, Kedem, O., Vofsi, D., lnd. Eng. Chem., Process Des. Dev., 6, 231 (1967). Bloch. R., U S . Patents 3 450 630 (1969) and 3 450 631 (1969). Jagur-Grodzinski,J., Marian, S., Vofsi, D., Sep. Sci., 8, 33 (1973). Los Alamos Scientific Laboratory Technical Manual, LA3103, UC-4, TID-4500 (1964). Marian, S., Jagur-Grodzinski, J., Kedem, O., Vofsi, D., Biophys. J., 10, 401 (1970). Merten, U., U.S. Patent 3 386 583 (1965). Michaels, A. S., U.S. Patent 3 173 867 (1965). Vofsi, D., Jagur-Grodzinski, J., Naturwissenschaffen, 61, 25 (1974). Vogei, A. I., "Ouantititive Inorganic Analysis", p 710, 36 ed, Longmans, Green and Co., London, 1947. Orth et ai., "Proceedings of the International Solvent Extraction Conference", p 514, The Hague, 1971.
Received for reuieu November 6,1975 Accepted April 5, 1976
Free Energy Parameters for Reverse Osmosis Separations of Some Inorganic Ions and Ion Pairs in Aqueous Solutions Ramamurti Rangarajan, Takeshi Matsuura, E. C. Goodhue, and S. Sourirajan' Division of Chemistry, National Research Council of Canada, Ottawa, Canada,
K 1A OR9
Reverse osmosis separations of several inorganic salts in aqueous solutions involving polyvalent ions have been studied using porous cellulose acetate membranes. From these studies, free energy parameters for the ions Mg2+, Ca2+, Mn2+, Co2+, Ni2+, Cu2+, Zn2+, Sr2+, Cd2+, Ba2+, Pb2+, Fe3+, Cr3+, and S042-, and for the ion pairs MgS04, C0S04, ZnS04, MnS04, CuS04, CdS04, and NiS04 have been determined. These parameters offer a means of predicting reverse osmosis separations of inorganic salts in aqueous solutions involving the above ions and/or ion pairs, using porous cellulose acetate membranes from data on membrane specifications only, given in terms of a single reference solute such as sodium chloride. This prediction technique is illustrated.
Introduction Reverse osmosis separations of inorganic salts in aqueous solutions have been discussed extensively from different points of view. These include analyses based on partition coefficient of solute between membrane and water phases (Heyde e t al., 1975), or enthalpy of hydration of ions in the bulk solution phase only (Fang and Chian, 1975a,b), or partial molal free energy of hydration and entropy of ions in the bulk solution phase only (Johnston, 1975), or relative free energy parameters for ions and solutes (Matsuura e t al., 1975).This paper is concerned with the latter approach. It may be recalled (Matsuura et al., 1975) that (i) Born expression for ion-solvent interaction as applied t o the bulk separation phase and the membrane-solution interface, and (ii) the thermodynamic basis of Hammett and Taft equations representing the effect of structure on reaction rate or equilibrium, and (iii) the relationship between the parameters of Taft equation and the experimental data on solute transport parameters, together constitute the scientific basis of the concept of free energy parameters governing reverse osmosis separations. This concept was first developed with respect to inorganic ions (Matsuura et al., 1975;Dickson et al., 1975),and
later extended to organic ions and nonionized organic solutes (Matsuura et al., 1976a,b). The free energy parameters are represented by the symbol -AAG/RT for each ion or nonionized molecule, where LAG is the energy needed to bring the ion or the nonionized molecule from the bulk solution phase to the membrane-solution interface, and R and T are the gas constant and the absolute temperature, respectively. Data on -LAGIRT are functions of the chemical nature of the solute molecule and that of the membrane surface, and are independent of the porous structure of the membrane surface. These data can be built into reverse osmosis transport equations (Sourirajan, 1970), leading to a method of predicting reverse osmosis separations of different solutes in aqueous solutions with membranes of different surface porosities, only from data on membrane specifications given in terms of a single reference solute such as sodium chloride. This has been illustrated with respect to both inorganic and organic solutes (Matsuura et al., 1975, 1976a,b). This paper extends the work reported earlier (Matsuura et al., 1975) on inorganic solutes involving monovalent cations and anions. This work is concerned with inorganic solutes involving polyvalent ions in reverse osmosis separations using Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
529
Table I. List of Solutes Used with Pertinent Electrochemical Data Solute number
Solute formula
DABX lo5, cm2/sa
Dissociation constantC
1.193 3.02 6 1.99d 1.406 8 1.99d 1.5686 13 1.00 1.214 30 1.285 0.52 34 1.227 35 1.221 36 0.10d 1.085 37 1.240 38 1.218 39 1.222 40 1.211 41 1.294 0.15 42 1.277 0.39 43 0.12 1.343 44 1.406 0.065 45 1.230b 0.19 46 4.36 X 0.849 47 5.20 x 10-3 0.853 48 3.38 x 10-3 0.808 49 3.98 x 10-3 0.851 50 2.62 x 10-4 1.276 51 4.89 x 10-3 0.846 52 4.89 x 10-3 0.858 53 Calculated, unless otherwise stated. From Robinson and Stokes (1959f). From Parsons (1959). From Sillen and Martell (196410).
porous cellulose acetate membranes. The relevant free energy parameters have been generated, and their utility in predicting reverse osmosis performance of cellulose acetate membranes is illustrated with respect to several inorganic salts.
Experimental Section Twenty-four inorganic solutes listed in Table I (along with some relevant physiochemical data) were used in this work in single-solute aqueous solutions. The apparatus and experimental procedure used were the same as those reported earlier (Matsuura et al., 1975).Batch 316(10/30)-typecellulose acetate membranes (Pageau and Sourirajan, 1972)were used a t the operating pressures of 250, 500, and 1000 psig. All membranes were subjected to pure water pressure of 1700 psig for 3 h prior to use in reverse osmosis experiments. The specifications (Sourirajan, 1970)of membranes used are given in Table I1 in terms of pure water permeability constant (in g-mol of H,O/crn’ s atm) and solute transport parameter (D&K6, treated as a single quantity (Sourirajan, 1970),in cm/s) for sodium chloride a t different operating pressures. The membrane characteristics changed significantly as a result of contact with different salt solutions. Therefore the specifications of the membrane were redetermined a t frequent intervals after every few experiments, and the specifications closest to each run were used for prediction calculations on membrane performance. Data on membrane specifications were obtained using aqueous sodium chloride feed solutions containing 5000 ppm of NaCl. The concentrations of the solutions (other than the sulfates of divalent cations) used in feed solutions were in the range 10W to 10-3 m (100 to 400 ppm) so that, in each case, the osmotic pressure of the feed solution was negligible compared to the operating pressure, and consequently, the pure water permeation rate (PWP) and product rate (PR) were essentially the same in reverse osmosis experiments. With respect to sulfates of divalent cations, the solute concentrations used 530
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
Table 11. Membrane Specifications and Related Data Operating Initial Film pres- of final no. sure, psig dataa
(DAM/
A,X IO6
KB) X lo5
Solute In C * N ~ Csepn, I %d
Initial 1.625 7.71 -10.810 78.4 Final 1.416 9.32 -10.706 74.6 84.4 500 Initial 1.306 7.78 -10.831 1000 Iriitial 1.210 8.36 -10.759 88.3 Final 0.982 9.54 -10.628 87.3 250 Initial 1.803 10.16 -10.534 75.4 Final 1.574 10.86 -10.557 74.9 500 Initihl 1.518 9.29 -10.654 84.3 1000 Initial 1.419 9.98 -10.582 88.8 Final 1.209 10.98 -10.487 89.2 250 Initial 2.215 17.73 -9.978 67.8 Final 1.910 19.13 -10.014 66.8 500 Initial 1.816 15.29 -10.156 78.1 1000 Initial 1.657 14.84 -10.186 84.0 Final 1.356 17.39 -10.027 84.5 250 Initial 2.599 63.39 -8.704 43.7 Final 2.172 60.63 -8.827 45.6 44.80 -9.081 500 Initial 2.074 58.2 1000 Initial 1.878 44.78 -9.081 66.6 Final 1.570 51.46 -8.942 66.5 250 Initial 2.914 108.77 -8.164 35.1 Final 2.192 93.22 -8.291 37.0 500 Initial 2.013 74.93 -8.566 47.9 1000 Initial 1.773 76.39 -8.547 54.9 Final 1.333 91.87 -8.363 56.6 Initial = prior to all reverse osmosis experiments. Final = on completion of all reverse osmosis experiments. A in g-mol of H20/cm2 s atm. D A M / K 6 in cm/s for NaCl. Feed: 5000 ppm of NaCl-H20 with K as in Figure 2. 250
(I
were in the range 0.0006to 0.51 m so that in each case, a significant fraction of the solute existed in solution as ion pairs. The data obtained from reverse osmosis experiments involving sulfates of divalent cations were used to calculate the free energy parameter for ion pairs. All experiments were of the short run type, and they were carried out a t the laboratory temperature (23-25 “C) using a feed flow rate of 410 f 10 cm3/min. In each experiment, the fraction solute separation f, defined as
f=
(solute molality in feed) - (solute molality in product) solute molality in feed (1)
and (PR) and (PWP) in g k per given area of film surface (13.2 cm2 in this work) were determined under the specified experimental conditions. In all experiments, the terms “product” and “product rate” refer to membrane permeated solutions, and the reported permeation rates are those corrected to 25 “C using the relative viscosity and density for pure water. The concentrations of sodium chloride in feed and product solutions were determined using a conductivity bridge; the concentrations of the other solutes used were determined by the atomic absorption technique. All data presented in this paper are for 25 “C.
Results and Discussion Calculation of Degree of Dissociation, a.Inorganic solutes in aqueous solutions dissociate into ions and ion pairs (Robinson and Stokes, 1959d).For symmetrical (1-1,2-2etc) electrolytes, the dissociation equilibrium constant K , is given by the relation (Robinson and Stokes, 1959e)
K , = a2rnyk2/(1- a)
t
7 1
0
0 LL
Table 111. Data on Free Energy Parameters for Some Polyvalent Ions at 25 "C Ion
(-AAG/RT)i
Mg2+ Ca2+ Mn2+
8.72 8.88 8.58 8.76 8.47 8.41 8.76 8.76 8.71 8.50 8.40 9.82 11.28 -13.20
Co2+
W W
Ni2+
n
Cu2+ O 10-2
L 10-1
I
I
100
IO'
I
(3)
Zn2+
I
Sr2+ Cd2+ Ba2+
to3
Figure 1. Degree of dissociation as a function of Km/rny+2using eq 3.
Pb2+ Fe3+ Cr"+ SO42-
so that
All symbols used are defined in the Nomenclature section a t the end of the paper. Equation 3 is also applicable to unsymmetrical (1-2, 2-1 etc.) electrolytes if the primary equilibrium constant is used for the term K,. The data on K , and those on 7%a t different concentrations are available in the literature (Parsons, 1959; Sillen and Martell, 196413; Yatsimirskii and Vasil'ev, 1960) for many inorganic solutes. The values of the dissociation constants listed in Table I are based on molar concentrations; however, the difference in the above values based on molar and molal concentrations is insignificant when the solvent is water (Robinson and Stokes, 1959b; Sillen and Martell, 1964a). The values of y+ for MgS04 a t very low concentrations (less than 0.1 rn) are not available in the literature; these values were hence calculated using the Debye-Huckel extended equation for the mean rational activity coefficient and related quantities (Robinson and Stokes, 1959a,c). Using eq 3, one can calculate cy for different values of the quantity K,/my+2; the results of this calculation are expressed in Figure 1which shows that when K m / m y * 2is greater than 100, cy is greater than 0.99, and hence less than 1%of the solute molecules exists as ion pairs. In the case of sodium sulfate and all the nitrate, iodate, and bromate solutes studied in this work, a t the levels of feed concentrations used, the quantity K m / m y + 2had values of 140 or higher, so that ion pair formation could be considered to be practically negligible. In the case of the other sulfate solutes, a t the levels of feed concentrations used, the quantity K m / m y + 2had values in the range 0.1 to 10 so that ion-pair formation was significant. For purposes of analysis, the values of cy were calculated for the boundary concentration X A obtained ~ from basic reverse osmosis transport equations (eq 11 or 16; see discussion below). Free Energy Parameters for Some Divalent Cations and Sulfate Anion. This discussion is based on the assumption that sodium sulfate and the nitrate solutes listed in Table I were completely ionized under the conditions used for the reverse osmosis experiments. I t has been shown (Matsuura e t al., 1975) that for the case of completely ionized inorganic solutes in aqueous solutions, the solute transport parameter D A M I K is ~ related to the free energy parameter of the ions involved by the expression In ( D A M I K = ~ )In C*
+ Z(-AAG/RT)i
(4)
where In C* is a constant,,and the subscript i represents the ion involved. I t may be recalled that for a given membrane material, the quantity In C* is a function only of the porous
structure of the membrane surface. The membrane material (cellulose acetate) is the same throughout this discussion. For any given film, the quantity In C* can be expressed in terms of the experimental data on solute transport parameter for a reference solute, and free energy parameters for the ions involved in the reference solute. Considering NaCl as a suitable reference solute for this purpose, the porous structure of the membrane can be represented by the quantity In C * N ~ C ~ where
-[(-")R T
Na+
+(-")R T
CI-
]
(5)
For completely ionized inorganic solutes, eq 4 can be rewritten as
~ )In C*N,CI+ Z ( - A A G / R T ) , In ( D A M I K =
(6)
Taking ionic valencies into consideration, eq 6 can be written in the more general form
where ncand n , are the number of moles of cations and anions, respectively, arising from the dissociation of one mole of the solute. For example, for symmetrical (1-1, 2-2 etc.) electrolytes, n, = n , = 1;for a 2-1 electrolyte, n , = 1and n , = 2 etc. By using eq 6 in the form of eq 7, and using ( - A A G / R T ) , data obtained for monovalent cations and anions reported earlier (Matsuura e t al., 1975) in eq 5 and 7, a consistent set of free energy parameter data can be generated relative to those determined earlier on the basis of experimental D A M / Kdata ~ for NaCl. Using the experimental D A M I Kvalues ~ for sodium sulfate, and the nitrate solutes listed in Table I a t the operating pressure of 250 psig, the basic transport equations for reverse osmosis (Sourirajan, 1970) and the values of 5.79, -4.42, and -3.66 for ( - A A G I R T ) , for Na+, C1-, and N03- ions, respectively (Matsuura et al., 1975), the relative values of ( - L A G I R T ) , for magnesium, calcium, manganese, cobalt, nickel, copper, zinc, strontium, cadmium, barium, lead, iron, chromium, and sulfate ions were calculated using reverse osmosis data obtained with 5 membrane samples of different surface porosities. In each case, the data obtained with different membranes scattered within a narrow range, and the average value of (- AAGIRT), obtained for each ion is listed in Table 111. Free Energy Parameters for Some Ion Pairs Involving Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 4, 1976
531
Table IV. Data on Free Energy Parameters for Some Ion Pairs at 25 "C
0
40-
Ion pair
(-SAG/RT)i,
ZnSO,
0 CdSOq
o CoSO,
0
Cuso,
NiSO,
+
MnS04
LL
"m 0
MgS04 COS04 ZnS04 MnS04 CuSO4 CdSOj NiS04
' E *
3.45 3.41 2.46 2.48 2.85 3.04 2.18
0
30-
20-
x
s 10-
I
0
I
I
Divalent Cations and Sulfate Anion. When the inorganic solute is incompletely ionized in aqueous solution, part of the solute exists as ion pairs. This is so with respect t o sulfates of divalent metals particularly a t high concentrations. In this case, the ion pair may be treated as a distinct species, and the free energy parameter for such species, (-AAG/RT)lp, can be calculated as follows. For a solution system involving ions and ion pairs, eq 6 can be written as
where the subscript ip refers t o the ion pair involved. Using the experimental overall D A M / K b values (left side of eq 8) for the sulfate solutes 47 to 53 listed in Table I, at the operating pressure of 250 psig, and the data on (-AAGIRT), given in Table I11 together with the values of N calculated by the method described earlier, the relative values of (-AAG/RT),, for the ion pairs involved were calculated using the reverse osmosis data obtained with five membrane samples of different surface porosities. Again, in each case, the data obtained with different membranes scattered within a narrow range. The average value of (-SSG/RT)lp obtained for each ion pair is listed in Table IV. Utility of Free Energy Parameter Data for Predicting Membrane Performance. For predicting membrane performance (Le., solute separation and product rate), the applicable values of A , and D&K6 and h for the solute are needed (Sourirajan, 1970). Data on membrane specifications ~ NaC1. From the latter provide the values of A and D A M / Kfor data, the applicable values of D A M / Kfor ~ any other solute can be calculated using eq 3,5,7, and 8 together with the free energy parameter data given in Tables 111and IV. The values of h depend both on the porous structure of the membrane surface and the chemical nature of the solute. Figure 2 gives the correlation of k (line 1) for NaCl (represented as k N a C l ) for concentrations up to 0.1 rn,as a function of A for the particular apparatus and feed flow rate used in this work. Using Figure 2 and the relation (Matsuura and Sourirajan, 1973)
the applicable value of h for the solute under consideration can be estimated. Figure 2 also gives the correlation of k (line 2) for different sulfate solutes as a function of A , obtained from eq 9 using the diffusivities of solutes at infinite dilution calculated from data on limiting ionic conductances (Reid and Sherwood, 1958). The actual experimental values of k for the above sulfate solutes in the concentration ranges used in this work are also plotted in Figure 2 which shows that the experimental data scatter fairly along the line 2. Therefore, for practical purposes, eq 9 as represented by line 2 can be used to obtain the applicable h values. Using the values of A , and D A M I Kand ~ h for the solute thus 532
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4 , 1976
0 h 0
0
CALCULATED
EXPERIMENTAL
t IO1 - 2.0
I -1.0
I
I
1
0
10 .
2.0
E(-AAG/RT)i
Figure 3. Comparison of calculated and experimental reverse osmosis separations for solutes 6 to 46 listed in Table I. Film type, cellulose acetate (Batch 316(10/30)); operating pressure, 250 psig; feed concentration, to m; feed flow rate, 410 cm3/min; 0, A, 0 ,0 , experimental data for films 1, 2, 3, and 5, respectively.
obtained, data on solute separation and product rate can be calculated (Sourirajan, 1970) on the basis of the following transport equations and related expressions.
-
A = (PWP)/M, * S * 3600 P
(10)
NR = A [P - ~ ( X A Z n(xA3)] )
(11)
r
1
1
I
(15) For very dilute feed solutions where (PR) is essentially the same as (PWP)
X A =~X A :+~ ( X A I- X A : ~exp )
(F)
(16)
Table V. Comparison of Experimental and Calculated Solute Separation and Product Rate Data in Reverse Osmosis Experiments with Aqueous Magnesium Sulfate Feed Solutions
Film no.
a
Molal Operating Solute sepn, % concn pressure, of feed psig Exptl Calcd
0.01 500 500 0.02 1000 0.06 0.51 1000 1000 0.02 500 0.01 500 0.2 250 0.01 250 0.02 0.06 1000 0.51 1000 500 0.01 250 0.02 1000 0.02 1000 0.51 500 0.01 1000 0.02 0.06 1000 0.02 500 Film area = 13.2 cm2.
98.1 98.2 98.7 98.7 99.0 98.9 98.8 98.5 98.4 97.9 98.5 97.0 95.4 94.9 98.4 92.5 92.7 94.7 90.7
98.5 98.5 98.0 99.2 97.7 98.6 98.6 98.7 98.5 97.0 98.4 97.5 97.6 92.4 95.2 95.0 88.6 91.5 92.3
Product rate, gha Exptl
Calcd
41.7 41.0 58.1 39.2 83.9 47.7 46.9 24.4 24.0 78.6 47.8 57.1 29.2 112.1 48.0 66.6 117.0 86.8 71.5
42.3 40.6 58.4 41.1 82.0 48.7 46.8 24.9 24.0 77.2 45.0 57.9 29.0 110.5 47.9 67.1 116.0 88.5 70.3
Table VI. Comparison of Experimental and Calculated Solute Separation and Product Rate Data in Reverse Osmosis Experiments with Aqueous Feed Solutions of Different Inorganic Sulfates
solute C0S04
Operating Product rate, Molal pres- Solute sepn, % g/ha Film concn sure, no. of feed psig Exptl Calcd Exptl Calcd
1 2 3 5 1 CdS04 2 1 2 NiS04 1 2 1 MnS04 1 1 2 1 cuso4 1 3 4 1 2 ZnS04 3 4 5 a Film area =
0.242 1000 0.242 1000 0.242 1000 0.242 1000 1000 0.119 0.119 1000 0.348 1000 0.348 1000 0.052 1000 0.052 1000 0.243 1000 0.0006 250 0.138 1000 0.138 1000 250 0.001 0.108 1000 1000 0.108 1000 0.108 0.354 1000 1000 0.354 250 0.001 1000 0.333 1000 0.333 13.2 cm2.
98.9 99.6 98.6 96.7 98.9 99.6 98.9 99.7 98.6 99.2 98.4 97.8 98.9 99.5 99.3 98.7 98.5 98.0 98.8 99.8 98.2 98.4 97.1
and u,*
=
AP/c
99.4 99.4 99.1 96.0 99.6 99.6 99.6 99.6 99.3 99.3 98.9 99.8 99.8 98.9 99.8 99.5 99.1 97.6 99.1 99.1 99.5 98.8 98.2
56.9 67.6 81.3 77.6 52.1 62.5 43.0 50.6 54.5 65.7 40.8 23.2 53.5 49.2 21.6 56.4 75.0 84.6 39.6 48.1 31.2 78.6 75.3
54.9 65.8 81.2 77.6 54.2 64.5 46.2 54.5 57.4 69.5 42.5 23.3 56.5 52.1 22.2 55.8 76.2 88.6 43.6 53.3 31.9 74.3 73.1
All the above equations have been derived (Sourirajan, 1970; Matsuura and Sourirajan, 1973; Matsuura et al., 1974). The above prediction technique was applied for all the solutes studied in this work in the operating pressure range 250 to lo00 psig and feed concentration range lop4to 0.51 m using membranes of different surface porosities. The results of such prediction calculations along with the actual experimental data are given in Figure 3 and Tables V and VI. The separation data given in Figure 3 are very dilute solutions for cases where ion-pair formation could be neglected. The membrane performance data given in Tables V and VI are for solution systems involving significant ion-pair formation. The agreement between the calculated and the experimental data in Figure 3 and Tables V and VI is reasonably good. These data confirm that the data on free energy parameters are independent of operating pressure and porous structure of the membrane surface, and they can be utilized for predicting membrane performance for different solution systems just from data on membrane specifications given in terms of A and ( D A M I K ~for) NaCl only.
Conclusions The physicochemical criteria approach to reverse osmosis separations involving free energy parameters for ionized and nonionized solute molecules is further extended in this paper. This approach is applicable to all reverse osmosis membranes. Complex formation between an ion and a ligand could also be treated similar to ion-pair formation, using the appropriate equilibrium data. For cellulose acetate membranes, the general effect of ion-pair formation is to increase D&K6 for the solute and hence decrease its separation in reverse osmosis. In view of the practical utility of the above approach, generation of further data on free energy parameters is warranted.
Acknowledgment The authors are thankful to J. M. Dickson for his assistance in the computation of data. One of the authors (R.R.) thanks the National Research Council of Canada for the award of a postdoctoral fellowship. Nomenclature A = pure water permeability constant, g-mol of H20/cm2 s atm C*, C * N ~ C = ~quantities defined by eq 4 and 5, respectively, cm/s c = molar density of solution, g-mol/cm:i DAB = diffusivity of solute in water, cm2/s D A M / K ~= solute transport parameter (treated as a single quantity), cm/s f = fraction solute separation defined by eq 1 (-AAG/RTI,, ( - A A G / R T ) i p = free energy parameters for the ion and ion-pair involved, respectively K , = dissociation equilibrium constant based on molar concentrations h = mass transfer coefficient on high pressure side of the membrane, cm/s M A , M B = molecular weights of solute and water, respectively m = molality of solute N B = flux of solvent water through membrane, g-mol/cm2 S
n,, n, = number of moles of anions and cations, respectively, arising from ionization of one mole of solute P = operating pressure, atm (PR) = product rate through given area of membrane surface, g/h (PWP) = pure water permeation rate through given area of membrane surface, g/h R = gasconstant S = effective membrane area, cm2 T = absolute temperature Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
533
u,,,*
= pure water permeation velocity, cm/s
X A = mole fraction of total (dissociated and undissociated) solute
Greek Letters = degree of dissociation y+ = mean molal activity coefficient for the solute n ( X A ) = osmotic pressure of solution corresponding to (Y
X A
Subscripts 1 = bulk solution phase 2 = concentrated boundary solution on the high pressure side of the membrane 3 = membrane permeated product solution on the low pressure side of the membrane Literature Cited Dickson, J. M., Matsuura. T., Blais, P., Sourirajan, S., J. Appl. Polym. Sci., 19, 801 (1975). Fang, H. H. P., Chian, E. S. K., J. Appl. Polym. Sci., 19, 2889 (1975a). Fang, H. H.P., Chian, E. S. K., "Removal of Dissolved Solid by Reverse Osmosis," paper presented at the A.1.Ch.E. 68th Annual Meeting in Los Angeles, Calif.. Nov 16-20, 1975b. Heyde. M. E., Peters, C. R., Anderson, J. E., J. Colloidlnferface Sci., 50, 467 (1975).
Johnston, H. K., Desalination, 16, 205 (1975). Matsuura, T., Bednas, M. E., Sourirajan, S., J. Appl. Polym. Sci., 18, 567 (1974) Matsuura, T., Dickson, J. M., Sourirajan, S., hd. Eng. Chem., Process Des. Dev., 15, 149 (1976a). Matsuura, T., Dickson. J. M., Sourirajan, S., lnd Eng. Chem., Process Des. Dev., 15, 350 (1976b). Matsuura, T., Pageau, L., Sourirajan, S.. J. Appl. Polym. Sci., 19, 179 (1975). Matsuura, T., Sourirajan, S., J. Appl. Polym. Sci., 17, 1043 (1973). Pageau. L., Sourirajan, S., J. Appl. Polym. Sci. 16, 3185 (1972). Parsons, R., "Handbook of Electrochemical Constants," pp 20-27,54, Butterworths, London, 1959. Reid, R. C., Sherwood, T.K., "The Properties of Gases and Liquids," p 295, McGraw-Hill, New York. N.Y., 1958. Robinson, R. A., Stokes, R. H.,"Electrolyte Solutions," 2nd ed, (a) p 32: (b) p 39: (c) p 229; (d) p 392; (e) p 396; (f) p 513; Butterworths. London, 1959. Sillen, L. G . , Martell, A. E., "Stability Constants of Metal-Ion Complexes," (a) p x; (b) pp 166-178: Special Publication No. 17, The Chemical Society, London, 1964. Sourirajan, S., "Reverse Osmosis," Chapter 3, Academic Press, New York, N.Y., 1970. Yatsimirskii. K. B., Vasil'ev, "Instability Constants of Complex Compounds," p 120, Pergamon Press, London, 1960.
Received for review November 12, 1975 Accepted April 20, 1976
Issued as NRC No. 15428.
Optimum Operating Cycle for Systems with Deactivating Catalysts. 1. General Formulation and Method of Solution Jin Y. Park and Octave Levenspiel" Deparlment of Chemical Engineering, Oregon State University, Corvallis, Oregon, 9733 7
This paper shows how to find the best way of running a reactor whose catalyst decays with use and must consequently either be regenerated or replaced at regular intervals. Part 1 develops a one-variable search method which avoids the previously proposed multidimensional search procedures. This method is quite general and can be applied to any process or operation where one can run the unit hard or gently and where when run hard it gives a larger profit, but it also wears out more rapidly. Part 2 applies this technique specifically to catalytic reactors with both mixed flow and plug flow of fluid.
When reacting fluid flows through a batch of slowly deactivating catalyst, the activity of catalyst drops progressively to a point a t which, because of economic profitability considerations, the run is terminated, the catalyst is regenerated or replaced, and the cycle is repeated; see Figure 1. There are two important and real questions concerning the cyclic operation: (1)how to best operate the reactor during a run (the operational problem) and (2) when to stop a run and regenerate or replace the catalyst (the regeneration problem). These problems, although closely connected, can always be treated separately by means of some relay variable (or variables) which sufficiently characterizes both phases of the entire production cycle (the overall problem). In the past the operational problem has been studied by Szepe (1966) and others (Chou e t al., 1967; Ogunye and Ray, 1968; Crowe, 1970). Since temperature is the most important variable affecting reaction and deactivation of catalyst, these authors have primarily been concerned with finding the best temperature progression during the operational phase. The regeneration phase of the cycle is strongly dependent on the temperature progression during the operational phase; however, this has not yet received due attention. Nevertheless, 534
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the essential dependency of regeneration on the operational problem has recently been demonstrated in a numerical example by Miertschin and Jackson (1970). In the overall optimization where both problems are to be solved, selection of the relay variable (or variables) as the means for separation, for recombination, and for computation of the two problems is of major importance. Szepe (1966) suggested a three-variable formulation consisting of initial .catalyst activity ai, final catalyst activity af,and operational time t o pas the relay variables. He further suggested that the overall problem be solved in two steps as follows: (1)solving for the operational problem with ai, af,and t o pas parameters, and then (2) solving the regeneration problem to find the optimum set of those parameters with the corresponding previously obtained optimum operation policy. Miertschin and Jackson (1970) showed that the number of the independent variables need only be 2, with ai and t o pbeing the convenient choice for their numerical computation. The present study shows that only one independent relay variable is in fact needed. In Part 1we develop the general one-variable formulation for catalyst deactivation and other types of cyclic operation.
