Ind. Eng. Chem. Process Des. Dev. IQBS, 2 4 , 297-303
297
Predictability of Performance of Batchwise and Continuous Reverse Osmosis Operations for Separation of Mixed Uni-univalent Electrolytes in Aqueous Solutions Takeshl Matsuura and S. Sourlrajan' Division of Chemistty, National Research Council of Canada, Ottawa, Canada K1A OR9
This paper combines the prediction technique of the reverse osmosis performance data including ion separation and product rate for different aqueous feed solutions containing two uni-univalent electrolytic solutes without a common ion with that of the system performance of reverse osmosls modules. Thus, this paper describes the method of predicting the average ion separatlons in product water and the operating period required to achieve a given fraction product recovery for a batchwise RO operation by which aqueous solutions of two mixed electrolytes are treated. The validity of the method is further confirmed by the experimental RO data obtained from a static cell of laboratory scale. This analytical method also enables one to predict the average ion separation and the membrane area required to achieve a given fraction product recovery for a continuous RO operation. Such calculations are conducted with respect to spiral-wound and hollow fiber modules.
Introduction The development of suitable methods for predicting membrane performance for mixed solute systems involving several electrolytes and nonelectrolytes in aqueous solution is an area of fundamental importance in reverse osmosis transport. Some prediction techniques of the membrane performance for such systems were developed and experimental works were carried out to testify to the validity of the prediction technique. As such examples, mixtures of uni-univalent electrolytes with (Agrawal and Sourirajan, 1970) and without a common ion (Rangarajan et al., 1978a, 1984), mixtures of uni-divalent electrolytes (Rangarajan et al., 1979), and mixtures of nonelectrolytes (Matsuura et al., 1974) were studied. Furthermore, aqueous solutions of partially dissociated acids, which are regarded as mixtures of electrolyte and nonelectrolyte, were studied with (Malaiyandi et al., 1982) and without (Matsuura et al., 1976) another nonelectrolyte solute in the system. All the results obtained indicate that it is actually possible to predict the membrane performance including product rate and separations of each constituent solute species on the basis of fundamental transport equations developed by Kimura and Sourirajan when a single experimental datum is given for a reference solute such as sodium chloride (Sourirajan, 1970b). On the other hand, equations of system analysis were presented for prediction of reverse osmosis module performance as an entire system (Ohya and Sourirajan, 1971). By use of system analysis equations, reverse osmosis modules were specified on the basis of the experimental data on NaCl-H,O solutions. The prediction of membrane performance was performed for the system sucrose-water and the result was tested by experimental data from several commercial modules (Tweddle et al., 1980). The object of this work is to combine the prediction method established for the mixed solute system (for the case where the fraction product recovery is nearly equal to zero) with the method of predicting system performance (at higher fraction product recoveries) which was so far developed for single-solute systems only. Two sets of calculated data are presented in this work. The first set of data was calculated for reverse osmosis concentration of systems NaC1-KN03-H20 and NaCH,COO-KN0,0 196-4305/85/ 1124-0297$0 1.50/0
Table I. Specification of Films and Modules Used in the Calculation
membranes and modules CA membrane used for experiments RO modules Roga-4000 Du Pont B9
A x 107, ( ~ ~ ~ kg-mol K ~ c I , HzO/(m2 X lo7, s kPa) m/s In C*N.FI 0.804 2.105 -16.74 1.26 0.20
4.73 0.075'
-15.93 -18.71"
1 kNsCl
lo6. m / s 12.42 10.4 1.2
" a t XA2= 0.00128.
H 2 0 using a static reverse osmosis cell of laboratory scale. For such systems, data are presented as separations of component ionic species and fraction product recovery vs. batch operating period. The result was then tested by experimental data. Since agreement of calculated and experimental values testifies to the validity of the approach, the same calculation technique was applied to the reverse osmosis concentration of mixed solute systems by two commercial modules. The result of the calculation is presented as the second set of data. Experimental Section Mixed solute systems of uni-univalent electrolytes such as NaC1-KN03 and NaCH,COO-KNO, were treated by batch 316 (10/30) type cellulose acetate membranes (Pageau and Sourirajan, 1972) in this work. The static cell used and the general experimental details were the same as those reported earlier (Sourirajan, 1970a). The cell was filled by the initial feed solution of 205 mL, and after removal of the initial 5 mL of product solution the concentration experiment was started. The amount of product solution and the concentration of each ionic species involved in the composite product solution were determined from time to time. The total molality of the feed solutions used was in the range 0.1 to 0.2. The molal ratio of one salt to the other in the feed was maintained as 1:l. The operating pressure for the reverse osmosis experiments was 6895 kPag (1000 psig). All membranes were subjected to a pure water pressure of 11721 kPag (1700 psig) for 2 h prior to reverse osmosis experiments. The initial specifications (Sourirajan, 1970b) of membranes used are given in Table I in terms of pure water permeability constant
Published 1985 by the American Chemical Society
298
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
A (in kg-mol of HzO/(m2 s kPa)) and solute transport parameter (DAM/K6),treated as a single quantity (Sourirajan, 1970b) for sodium chloride at different operating pressures. All reverse osmosis experiments were carried out at the laboratory temperature (23-25 "C). In each experiment the fraction solute separation defined as xAl f =
-xA3
XAl
(for single solute systems)
(1)
or
x*il - x*. 13 (for mixed solute systems) f = x*il
Assuming equilibrium between solution and membrane cXA = KcMXAM
(7)
NA = (DAM/K~)(czXAZ - c3xA3)
(8)
eq 4 becomes (2)
and the time required to collect a given product recovery were determined under the specified experimental conditions. The concentrations of salt and ions in the feed and product solutions were determined as follows: sodium chloride in the single-salt system NaCl-H,O, by using a conductivity bridge; in all other systems, sodium and potassium by atomic absorption technique, chloride by potentiometric titration, nitrate by UV absorption, and acetate by difference on the basis of overall electroneutrality of solution. Theoretical Section Nomenclature. All symbols are defined at the end of the paper. A brief note here is in order. The subscripts A, B, and M refer to salt, water, and membrane phase, respectively. All quantities with an asterisk refer to ions; with respect to such quantities, the first subscript i, 1,3, 2, or 4 refers to the indicated ion, and the second subscript M, 1,2, or 3 refers to the indicated phase (M = membrane phase, 1 = bulk solution phase on the high pressure side of the membrane, 2 = concentrated boundary solution phase on the high-pressure side of the membrane, and 3 = product solution phase on the atmospheric pressure side of the membrane). Ions 1and 3 are different cations and 2 and 4 are different anions; all ions are univalent. With respect to quantities which do not refer specifically to ions, the subscript M, 1, 2, or 3 refers to the indicated phase. Numerical subscripts 12, 14, 32, and 34 in boldface type refer to single salts with the ions indicated by each number. Thus, for example, the quantities X*,, X*il, X*l, X*1M, X*13,XAM, XAM3,and x A 3 represent mole fraction of ion i, ion i in phase 1, ion 1, ion 1 in membrane phase, ion 1 in product solution phase, salt A in membrane phase, salt A in membrane phase in equilibrium with XA3, and salt A in product solution phase, respectively; the quantities c2and cM2represent molar density of solution in phase 2 and membrane phase in equilibriumwith X,, respectively; and the quantity (DAM/K6)12 represents solute transport parameter for the single salt 12. Basic Reverse Osmosis Transport Equations. The following basic equations of reverse osmosis transport have been derived and discussed extensively (Sourirajan, 1970b) with respect to single-solute aqueous solution systems and porous cellulose acetate membranes for the case where water is preferentially sorbed at the membrane-solution interface. For water flux For solute flux
The meanings of all the symbols used are given in the end of this paper. Basic Equations as Applied to a Mixed Solute Syetem Involving Several Ions in Aqueous Solution. Equations 3,5-8 can be written in analogous forms for the mixed solute system involving uni-univalent electrolyte as follows.
NB = A [ P - dCX*dmixt + a(CX*i&mixtl
X*i2 = X*i3
+ (X*il - X*i3)exp {NB y
*
j
(9)
(12)
and X*i3 = N*,/(NB
+ EN*,)
(13)
In addition, the condition for overall neutrality prevails in each phase, so that CZ*,X*,1 = 0
(14)
EZ*,X*,, = 0
(15)
CZ*1X*t3= 0
(16)
CZ*,X*,M = 0
(17)
EZ*,N*, = 0
(18)
and Referring to eq 7 and 11, the quantity K in the parameter Dm/K6 (eq 8) is a true equilibrium constant independent of salt concentration; for a mixed solute system under analysis, the corresponding quantity is K*, (eq 11) which is the distribution coefficient for the ion in the particular solution system under consideration. Unlike K, K*, is composition dependent, and this dependency arises because of the competing effects of different ions on their distribution between the aqueous phase and the membrane-solution interface. Assumptions. For simplicity of analysis leading to the required equations, the following assumptions are made: (i) the molar density of the solution is constant, i.e. c1 = c2 = c3 = c (19) (ii) the flux of solvent water is very high compared to that of all ions through the membrane, i.e. NB
(4) and for concentration polarization
where
>> CN*,
(20)
(iii) the osmotic pressure of solution is proportional to the sum of the mole fraction of all ions, i.e. 7r(CX*,) = B,,C:X*,
(21)
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 299
(iv) a single average value can be used for mass transfer coefficient on the high pressure side of the membrane, Le. k = k,, = constant (22) for an operating condition and; (v) for any salt or ion, the ratio of diffusivity through the membrane to that in water is a constant (Hoffer and Kedem, 1972), i.e. DAM -=--
D*iM
DAB
D*i Assumptions i and ii are practically valid over a fairly wide range of solute concentrations. In the absence of precise data on osmotic pressure of mixed solute systems, assumption iii is a reasonable one for engineering calculations. Since the diffusivities of different uni-univalent ions are not too different from each other, an average value of k for a given operating condition should be adequate for purposes of predicting membrane performance. The assumption that the mobility of a salt or ion through the membrane phase is proportional to that in free solution can also be accepted for a practical purpose. The validity of assumptions i to v was examined in our earlier work (Rangarajan et al., 1978a, 1979). On the basis of assumptions i to v above, and defining a = exp(NB/k,,c) (24) eq 9, 10, 12, and 13 can be written as NB = A P - AB,,(X*11 - X*13 X*31 - X*33)~r(25)
+
+ (X*il - X*i3)a
(27)
X*13
= N*i/NB
(28)
N*~/NB
(29)
= N*~/NB
(30)
x*33= X*23
Equations Necessary for Solutions of Basic Transport Equations. For the solution of basic transport equations described above, some additional equations are required. Proportionality constant B,, in eq 25 can be expressed as + B32 + B14 + B34 (31) Bav = 8 The average of proportionality constant B with respect to 4 salts involved in the system was taken on the basis of 8 constituent ions. Data on osmotic pressures as a function of concentration for different salts are available in the literature (Sourirajan, 1970~).The average mass transfer coefficient used in eq 24 can be obtained by B12
k,, = (~N,cI/~DN,c~~/~)[(D*~)~/~ + (D*3)2/3 ( D * z ) ~+/ ~ (D*4)2/31(32) The data on diffusivities of different ions in water are available in the literature (Parsons, 1959). Ion fluxes given by eq 26 can be expressed for individual ions as (Rangarajan et al., 1978a) = c(D*l/Dlz)(DAM/K6)12[(X+ *1(x*ll 3 - x*13).1 (((1- 0)(X*21~~ - X*23a + X*23) p(X*lla - X*13(~ +
N*l
X*13
+ + X*31c~- X*33a + X*33))/((X*11(~ - X * ~ ~+C Y X*13) + y(X*31(~ - X*33a + X*33)})1/2 -
X*13(((1- fl)x*23 + P(x*13 + X*33)1/(X*13 + Yx*331)1’21
(33)
= C(D*3/D32)(D~~/Kb)32[(X*33 + (x*3i(((1- P)(X*21a- X*23a + x*23)+ P(x*lla X*13 + X*31~r- X*33(~ + X * ~ J ) / ( ( X * ~- ~ C Y
x*33)a) - X*13a
+
+
+ (1/y)(x*11a- x*13a+ x*13)))’/2 -
x*33(u x*33)
x*33(((1 - P)x*23
+ P(x*13+ x*33)1/(x*3+3 l/Yx*131)1/21 (34)
N*2
- constant
x*i2= x*i3
N*3
=
c(D*z/Diz)(D~~/Kb)i2[(X*23 + (x*21- x*23)a)x (((X*iia- X*13~r+ X*13) + y(X*31(~ - X*33(~+ x*33))/((1 - P)(x*21a- x*23a + x*23)+ P(X*lla X*13(~ + X*13 + X*31a - X*33(~+ X*3J))1/2x*23(ix*13 + 7x*33)/{(1- P)x*23 + P(x*13+ (35) Further N*4can be obtained from eq 18, expressing the electroneutrality of ion fluxes. Constants P and y involved in eq 33-35 are calculated by X*33)I)”21
P=
(D12/D14)2(DAM/
K6)142
(DAM /K6) 122
(36)
and =
(Di2/D32)2(D~/D8)322/(D~~/K6)i22 (37)
Solute transport parameters necessary for the calculations of eq 33,34,35,36, and 37 can be evaluated as in the following. From the concept of free energy parameter for ions (-AAG/RT)*, developed earlier (MatYuura et al., 1975, 1977; Rangarajan et al., 1976, 197813; Sourirajan and Matsuura, 1979), it has already been shown that the solute transport parameter DAMIK6 for a single solute can be expressed by the relation In ( D A M / K ~ = ) ~In, ~C*NaCl ~ + X(-AAG/Rn*i (38) where In C*NaC1 is a constant (representing the porous structure of the membrane surface) obtained from the experimental DAM/K6data for NaCl using the relation
The values of Dm/K6 for single salts 12, 14, and 32 are then obtained by In (DAM/K& = .... -AAG -AAG In C*NaCl + (w)* +1 ( w ) * 2 ] (40) ~~
[
In
C*NaC1
+
[
In
C*NaC1
+
[
-AAG
+
-AAG
(w)*l ( w ) * 4 ]
-AAG ( w ) * 3
+(
(41)
-AAG X ) * 2 ]
(42)
The applicable values of (-AAG/Rr)*i for different ions are already in the literature (Sourirajan and Matsuura, 1979). System Analysis. Suppose the overall fraction recovery from the system is A and the entire system is split into m sections. Assume that the fraction recovery of each section is Alm. Then, considering the material balance of ionic species i at (n - 1)th section (1- (n - 2)(A/m))[X*,,iln-1 = (1- (n- l)(A/m))[X*i,iIn + (A/m)[X*i,3ln-l (43)
300
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
where [ 1, denotes the quantity involved in the nth section. Rearranging eq 43, we obtain
By using a free-energy parameter applicable to Na+ and C1-, we can express In (DAM/K6) as In (DAM/K6) = In C*NaC1 (-AAG/R!f)*N,++ (-AAG/RT)*a- (53)
+
(44) The concentration of the composite product solution is n=m A n=m E -[X*i,3In E [X*i,Jn n=l m - n=l x*i,3 = (45) n=m A m
E-
m Furthermore, the area required for obtaining fraction recovery A/m of feed flow rate in weight, 6i (kg/s), in the continuous operation is n=l
[Plll)(A/m) sn= @/ [NB1nME / n
(47) The total area required for recovering A from the entire system is, therefore
Similarly, the total time required for the fraction recovery of A in a batchwise operation by a static cell is
where w and S denote the total weight of feed solution at the initiation of reverse osmosis run and the membrane area used in the static cell, respectively. Thus, the mole fraction of each ionic species and the total surface (in case of the continuous operation of a module) or the total operation period (in case of batchwise operation of static cell) required for obtaining a given fraction recovery A can be calculated by eq 45,48, and 49 if Xi,3and N B can be calculated a t consecutive sections. In the case of polyamide membrane such as Du Pont's B-9 module, another special consideration has to be given to the magnitude of the transport parameter, DAM/K6. With respect to sodium chloride solution, it was found that the latter quantity can be expressed as a function of boundary mole fraction of sodium chloride as (50)
(XA2)fNaCl
where t = 0.37 (Tweddle et al., 1980). This may be due to the significantly charged character of the polyamide material (Dickson et al., 1976). Denoting the value of (Dm/K8)NaC1a t a standard mole fraction (X'AZ)N,CI as (DAM/K8)ONaCI, eq 50 can be written as (DAM
/ K6)oNaCl((XA2)NaCI/
Combining eq 52-54, we obtain In
C*NaCI
= In
C*ONaCI
+ 1n (XA2/XoA2)NaCI (55)
We retain the form of eq 55 for mixed solute systems and replace XA2by the summation of mole fractions of all ionic species, Since the mole fraction of 0.00128 was used as the standard mole fraction in the previous work (Matsuura et al., 1977) eq 55 can be written as In
where p1 and p3 denote the density of feed and product solutions, respectively. As an approximation, we assume that p is constant throughout the whole system. Then, eq 46 can be written as
(DAM/K8)NaCI =
In (DAM/K6)ONaC] = In C*ONaC1 + (-AAG/RT)*N,++ (-AAG/RT)*cl- (54)
(46)
b31
(DAM/Ka)NaCl
Similarly
(XoA2)NaCIIC
(51) Taking the logarithm on both sides of eq 51
C*NaC1
= In C*ONaC1
+
t
EX*i2 In 0.00128
(56)
For the ROGA-4000 module, where cellulose acetate membrane is used, t = 0. Technique for Computing the Separation of Ionic Species and Area (or Time) Required for Obtaining a Given Product Recovery From the known composition of the feed solution, chosen reverse osmosis operating conditions of pressure and fluid turbulence on the feed side of the membrane (feed flow rate in case of continuous cell, and stirrer speed in case of static cell) and data on membrane specification, the quantities [X*1111, [X*3111,[X*21II,[X*,lll, P, A, (Dm/Kb)NaC1 (or (Dm/K6)oNaCI at standard mole fraction of 0.00128 in case of B-9 module), and lZNaclare known. To predict the system and the area or time required for obtaining a given value of A, one must know X*13,X*33,X*=, X*43,and NB a t each section. In order to calculate the latter quantities, proceed as follows. (1)From literature data (Sourirajan, 1970c) on osmotic pressure vs. mole fraction for the single salts 12, 32, 14, and 34, determine B I ZBS2, , BI4,and B34for the range of concentration of interest. Then calculate B,, from eq 31. (2) Determine the applicable value of molar density of solution from literature data (Sourirajan, 1970~). (3) Using literature data on diffusivity of ions (Parsons, 1959), calculate k,, for the system using eq 32. (4) Using membrane specification data on (DAhl/K6)NaC1 at the standard NaCl concentration and literature data on -AAG*/RT for sodium and chloride ions (= 5.79 and -4.42, respectively, for cellulose acetate (Matsuura et al., 1975), and = -1.35 and 1.35, respectively, for aromatic polyamide (Sourirajan and Matsuura, 1979)),calculate In C*ONaCI for the films using eq 54. (5) Assume In C*NaCI and using literature data on -AAG*/RT for different ions (Sourirajan and Matsuura, 1979) calculate (DAMIK6) for single salts 12, 14, and 32 from eq 40, 41, and 42, respectively. (6) Using the data on Dm/K6 calculated above and the data on diffusivity of single salts 12,14, and 32 in aqueous solutions obtained from the literature (Sourirajan, 1970c), calculate 0 and y from eq 36 and 37. (7) Now consider eq 24, 28, 29, 30, 25, 33, 34, and 35; as concentrations involved in the equations such as X*iI and X*i3,those in the first section [X*,,], and [X*,3]1have to be used. The unknown quantities in these equations are [NE],, [x*1,]1, [X*33l1,and [x*23]1.Solve for the latter four quantities by simultaneous solution of four equations, eq 25, 33, 34, and 35, using other equations indicated
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
B
SYSTEM NoCI-KN0,- WATER
0
005 MOLAL FOR EACH SOLUTE
d L"
l
b
dZ
d3
014
0'5
0'6
d7
FR4CTlON
1
5
01
02
PRODUCT RECOVERY
03
04
05
06
07
100
A
08
-
PREDICTED O O A A EXPERIMENTAL
3
8 8 07 60
0
above. This can be accomplished by a computer by solving the above nonlinear equation by a standard method involving an iterative procedure starting with an initial guess for [NBIl, [x*1311, [x*3311, and [ X * Z and ~ I approaching the true solution by the method of steepest descent. Then [X*43]is calculated by eq 16. (8) Calculate [X*1211,[X*3211,[ X * Z Zand ~ ~[X*4211 , by eq 27. (9) Calculate In C*NaC1 by eq 56 and check if In C*N~CI calculated is the same as the one assumed in step 5. If it is not, go back to step 5. If it is, adopt [X*1311, [X*3311, [X*z3]1,[X*43l1,and [NB]1 so calculated as true solutions and go to the next step. (10) Calculate [X*1112, [X*311z7 [X*zl12,and [X*411~ from eq 44 for given A and m values by setting n equal to 2. (11) Go back to step 5 and proceed to step 9. As a result [NB]~, [X*idz,[x*331z, [X*zdz,and [ x * 4 3 1 z are obtained(12) Proceed to section 3, section 4, ...,up to section m. As solutions we obtain [ N B ] 3 [NBlm, [X*& ... [X*13Im, [x*3313 [ X * 3 3 I m , [ X * Z ~ I[~X * d m ~[X*,I3 [X*,Im. (13) Apply eq 45 and obtain X*13, X*33,X*23,and X*43. (14) Apply eq 2 and obtain f*l, f*3, f*2,and f*4. (15) Apply eq 48 or 49 and obtain S or t. Thus f*l, f*3, f*2, f*4, and S or t can be obtained for a given value of A. Experimental Verification of Prediction Technique The foregoing prediction technique was tested with experimental reverse osmosis data obtained at the experimental conditions specified in the Experimental Section. In all the prediction calculations, the molar density of the solution was assumed to be that of pure water, and the osmotic pressure vs. mole fraction correlation was approximated to be a straight line in the concentration range of interest for calculating Bav. For calculating the ratio of diffusivities of ions and salts, the literature data on diffusivities at infinite dilution (Sourirajan, 1970c; Parsons, 1959) were used. Since the values of k, calculated from eq 32 were within 10% of those for kNaclin all cases, and since the correspondingvariations in predicted results are within experimental error, the values of kNaClwere used for k, for the mixed-salt systems in prediction calculations. The data on free energy parameter for ions reported earlier (Sourirajan and Matsuura, 1979) were used for calculating In C*NaC1and DAM/K6 for single salts. The results of prediction calculations along with the experimental results are given in Figures 1 and 2. A comparison of the calculated and experimental results on ion separations and time required for obtaining a given product recovery shows good agreement in most cases. The agreement between the calculated and experimentalresults shown in Figures 1 and 2 confirms the scientific validity
...
A
b
Figure 1. Solute separation and batch operation period vs. fraction product recovery. System: NaC1-KN03-water; membrane, cellulose acetate 316(10/30); operating pressure, 6895 kPag (= 1000 psig).
**a
- KNOI - WATER
01 M O L 4 L FOR E4CH
-
0'6 0
SYSTEM NaCH,COO
E
301
0.1
0.2
0.3
04
05
06
08
FRACTION PRODUCT RECOVERY, A
Figure 2. Solute separation and batch operation period vs. fraction product recovery. System: NaCH3C00-KN03-water; feed concentration, 0.05 m for each solute; membrane, cellulose acetate 316(10/30); operating pressure, 6895 kPag (= 1000 psig).
and practical utility of the prediction technique developed above.
Calculation with Commercial Modules Since equations and the computation technique described above were confirmed by reverse osmosis experimental results from laboratory scale batchwise operation by a static cell, the prediction method was applied to some commercial modules the memranes in which are specified in Table I. The calculation was carried out for three sets of experimental conditions. (1) For the mixture of sodium chloride (1.5 m) and potassium nitrate (1.5 m) the operating pressure was changed from 3447 to 10 342 kPag (500 to 1500 psig). The solute separation as well as the membrane area required were calculated at the fraction product recovery of 0.7. (2) For the mixed solute system of sodium chloride and potassium nitrate (molal ratio = 1:l) the total concentration was changed from 0.1 to 4.9 m at the constant operating pressure of 3447 kPag (500 psig). The solute separation and the membrane area required were calculated for the fraction product recovery of 0.7. (3) For the mixture of sodium acetate (1.5 M) and potassium nitrate (1.5 M) the operating pressure was maintained at 10 342 kPag (= 1500 psig). Solute separation and membrane area required were calculated for various values of fraction product recovery. For each set the feed flow rate was maintained at 1.2 X m3/min and the mass transfer coefficient of sodium chloride, listed also in Table I, was used for the computation of the module performance. The first set of data is illustrated in Figure 3. As the general trend, solute separation increases and the membrane area required diminishes as operating pressure increases. The separation of chloride anion is much higher while that of nitrate is much lower than the separation of cations at each operating pressure, as indicated in the previous work (Rangarajan et al., 1978a). Furthermore, the area required for Du Pont B-9 hollow fiber module is an order of magnitude higher than that for ROGA-4000 module, reflecting the low permeability of hollow fibers. For the Du Pont B-9 module the difference in separations of chloride and nitrate anions is so enhanced that the separation of NO3-is nearly equal to zero at the operating pressure of 6895 kPag (= 1000 psig). Figure 4 illustrates the results of the second set of calculation. Increase in feed concentration generally de-
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
302
SYSTEM, NaCH3COO- KN03- WATER FEED CONCENTRATION, 1.5 MOLAL FOR EACH SALT;OPERATING PRESSURE, I O 342 kPag (.I500 psig); FEED FLOW RATE, 1.2 a 1
~ 4 0 0 ~R O G A - 4 0 0 0
L
o~m3/~~~
MODULE
l
/
p ' " : L a
N a+ Kf I
O'
06
0 4
08 OPERATING
I O 06 0 8 PRESSURE X Id', kPop
I O
Figure 3. Effect of operating pressure on solute separation and area required for some RO modules. System: NaC1-KN03-water; feed concentration, 1.5 m for each salt; feed flow rate, 1.2 X m3/min; A = 0.7.
1
02
04
06
NO;
08
A Figure 5. Effect of fraction solute recovery on solute separation and area required for some RO modules. System: NaCH3COOKN03-water; feed concentration, 1.5 m for each salt; operating pressure, 10342 Wag (= 1500 psig); feed flow rate 1.2 X m3/min.
r
1000r
0
electrolytes with a proper computation procedure in which a single reverse osmosis process is regarded as a multiple step operation.
Acknowledgment The authors are grateful to H. B. MacPherson and S. Asselin for their valuable assistance of these investigations.
Nomenclature t w
5
w
zoc
K+ NO;
0
I O
2 0
TOTAL
30 LO SOLUTE M O L A L I T Y IN
50 0 F E E D SOLUTION
I O
2 0
Figure 4. Effect of total feed molality on the solute separation and area required for aome RO modules. System: NaC1-KN03-water; feed concentration ratio of solutes, 1:1;operatingpresaure, 3447 Wag m3/min; A = 0.7. (= 500 psig); feed flow rate, 1.2 X
creases the separation of each ionic species and increases the area required for achieving the fraction recovery of 0.7 for each module studied. In the case of Du Pont B-9 module the solute separation diminishes sharply with the increase in solute concentration, and the separation of nitrate anion becomes nearly equal to 0% at the total molality of 2.0, reflecting the high dependence of solute transport parameter on the salt concentration. A very significant increase in the area required for achieving A of 0.7 with the increase in the total concentration of solute in feed is also observed. Figure 5 illustrates the third set of calculation with respect to ROGA-4000 RO module. As A increases, solute separation of each ionic species decreases and area required increases very sharply. It is interesting to note that calculations predict a small negative separation for nitrate ion a t high fraction product recovery. Conclusion This paper confirms that the membrane performance for the mixed solute system can be predicted at any product recovery on the basis of the fundamental transport equations developed for the mixture of uni-univalent
A = pure water permeability constant, kg-mol/(m2s kPa) B = constant defined by eq 31, kPa BIZ, B14,B32,B34= slope of mole fraction (XA) vs. osmotic pressure (kPa) plot for single salts 12, 14, 32, and 34, respectively, in the concentration range of interest In C * N a ~ I= constant defined by eq 39 c = molar density of solution, mol/m3 cl, cz, c3, cM = molar density c in solution phase 1, solution phase 2, solution phase 3, and membrane phase, respectively, mol/m3 cM2, cM3 = molar density c in membrane phase in equilibrium with XA2and XA3,respectively DAB, DM = diffusivity of solute in water and membrane phase, respectively, m2/s D*L,D*,M = diffusivity of ion i in water and membrane phase, respectively, m2/s D*IM, D * ~ MD, * ~ M = diffusivity of ions 1, 3, 2, respectively, in membrane phase, m2/s D*l, D*3, D*2, D*4 = diffusivity of ions 1, 3, 2, and 4, respectively, in water, m2/s Dlz,D,,, D, = diffusivity of salts 12, 14, and 32, respectively, in water, mz/s D A M / K=~ solute transport parameter (treated as a single quantity), m/s ( D A M / K ~(DM/Kb)14, )~~, (DAM/Kb)32 = solute transport parameter for salts 12, 14, and 32, respectively, m/s f = solute or ion separation defined by eq 1 or eq 2 - A A G / R T = free energy parameter (-AAG/RT)*,, (-AAG/RT)*,, (-AAG/RT)*,, (-AAG/RT)*Z, (-AAG/RT)*4,= free-energy parameter for ions i, 1, 3, 2, and 4,respectively K = equilibrium constants for single salts defined by eq 7 K*, = equilibrium distribution coefficient for ion i defined by eq 11
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
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K*i2,K*i3= K*i solution phase 2 and solution phase 3, respectively k = mass transfer coefficient on the high-pressure side of the membrane, m/s k,, = average value of k , m/s M B = molecular weight of water m = number of sections set in the entire length (in case of continuous operation) or time (in case of batch operation) of RO process NA = solute flux through membrane, mol/m2 s NB = flux of solvent water through membrane, mol/m2 s N*i = ion flux through membrane, mol/m2 s N*l,N*3,N*2,N*4= flux of ions 1,3, 2,and 4,respectively, through membrane, mol/m2 s P = operating pressure, kPag (PR) = product rate through given area of membrane surface, kglh (PWP) = pure water permeation rate through given area of membrane surface, kg/h R = gas constant S = membrane area, m2 T = absolute temperature, K t = time, s w = total weight of feed solution, kg 6 = feed flow rate in weight, kg/s X = mole fraction x A 1 , Xm, Xm,X,, = mole fraction of solute in solution phase 1, solution phase 2, solution phase 3, and membrane phase, respectively XAM2, Xm3 = mole fraction of solute in membrane phase in equilibrium with x.42and XA3, respectively X*il,X*i2,X*a,X*,u = mole fraction of ion i in solution phase 1, solution phase 2, solution phase 3, and membrane phase, respectively X*l, X*3,X*2,X*4 = mole fraction of ions 1, 3, 2, and 4, respectively X*ll, X*12, X*+ X*1M = mole fraction of ion 1 in solution phase 1,solution phase 2, solution phase 3, and membrane phase, respectively X*31, X*32, X*33,X*3M = mole fraction of ion 3 in solution phase 1,solution phase 2, solution phase 3, and membrane phase, respectively X*21,X*22,X*23, X*,M = mole fraction of ion 2 in solution phase 1,solution phase 2, solution phase 3, and membrane phase, respectively X*41,X*42, X*43,x * 4 M = mole fraction of ion 4 in solution phase 1,solution phase 2, solution phase 3, and membrane phase, respectively x A , 3 = average mole fraction of solute in solution phase 3 Xi,3= average mole fraction of ion i in solution phase 3 Zi = valency of ion i
a = quantity defined by eq 24 ,8 = quantity defined by eq 36 y = quantity defined by eq 37 A = fraction product recovery 6 = effective thickness of membrane, m = constant defined by eq 50 p1 = density of solution phase 1, kg/m3 p3 = density of solution phase 3, kg/m3 r(XA),r(XAZ), f(XA3) = osmotic pressure of single solute
Greek Letters
Issued as NRC No. 24040.
aqueous solution corresponding to mole fraction XA, X A ~ , and XA3, respectively, kPa T ( ~ X * T~ (~C) X~ *, ~= osmotic ~ ) ~ pressure of mixed solute aqueous solution corresponding to total mole fraction of all ions in solution phase 2 and solution phase 3, respectively, kPa
Literature Cited Agrawai, J. P.; Sourirajan, S. Ind. Eng. Chem. Process Des. D e v . 1970, 9 . 12. Dickson, J. M.; Matsuura, T.; Blais, P.; Sourirajan, S. J. Appl. POlym. Sci. 1976, 20, 1491. Hoffer, E.; Kedem, 0. Ind. Eng. Chem. Process Des. D e v . 1972, 7 7 , 221. Maiaiyandi. P.; Matsuura, T.; Sourirajan, S. Ind. Eng . Chem. Process Des. D i v . 1982, 27, 277. Matsuura, T.; Bednas, M. E.; Sourirajan, S. J. Appl. Po/ym. Sci. 1974, 78, 587 . Matsuura, T.; Blals, P.; Pageau, L.; Sourirajan, S . Ind. Eng. Chem. Process Des. D e v . 1977, 76, 510. Matsuura, T.; Dlckson, J. M.; Sourirajan, S . Ind. Eng. Chem. Rocess Des. D e v . 1976, 75, 350. Matsuura, T.; Pageau, L.; Sourirajan, S. J. Appl. Po/ym. Sci. 1975. 79, 179. Ohya, H.; Sourlrajan, S. Reverse Osmosis System Specification and Performance Data for Water Treatment Applications"; The Thayer School of Engineering, Dartmouth College, Hanover, NH, 1971. Pageau, L.; Sourirajan, S. J. Appl. Po/ym. Sci. 1972, 76, 3185. Parsons, R. "Handbook of Electrochemical Constants"; Butterworth Scientific Publications: London, 1959; Table 73. Rangarajan, R.; Baxter, A. G.; Matsuura, T.;Sourlrajan, S . Ind. Eng. Chem. Process Des. Dev. 1984, 23, 367. Rangarajan, R.; Matsuura, T.; Goodhue, E. C.; Sourirajan, S . Ind. Eng. Chem. Process Des. D e v . 1976, 75, 529. Rangarajan, R.; Matsuura, T.; Goodhue. E. C.; Sourirajan, S. Ind. Eng. Chem. Process Des. D e v . 1978a, 77, 46. Rangarajan, R.; Matsuura, T.; Goodhue, E. C.; Sourirajan, S . Ind. Eng. Chem. Process Des. D e v . 1978b, 77, 71. Rangarajan. R.; Matsuura, T.; Goodhue, E. C.; Sourirajan, S. Ind. Eng. Chem. Process Des. D e v . 1979, 78, 278. Sourirajan, S. "Reverse Osmosis"; Academic: New York, 1970; (a) Chapter 1; (b) Chapter 3; (c) Appendix. Sourlrajan, S.;Matsuura, T. "A Fundamental Approach to Application of Reverse OSmOsis for Water Pollution Control"; I n Proceedings, EPA Symposlum on Textile Industry Technology, Williamsburg, Dec 5-8, 1978; EPA60012-79-104, May 1979. Tweddie, T. A.; Thayer, W. L.; Matsuura, T.; Hsieh, F. H.; Sourlrajan. S. Desalination 1980, 32, 181.
Received for review June 6, 1983 Accepted April 26, 1984
