Predictability of Membrane Performance in Reverse-Osmosis Systems

13 Predictability of Membrane Performance in Reverse-Osmosis Systems Involving Mixed Ionized Solutes in Aqueous Solutions A General Approach RAMAMURTI RANGARAJAN, TAKESHI MATSUURA, and S. SOURIRAJAN

Downloaded by CORNELL UNIV on July 1, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch013

Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R9

This paper derives generalized transport equations for the RO system involving solvent water and different completely ionized solutes with different ionic valencies. The analysis imposes no limit to the number of ions or their valencies. The transport equations enable one to predict the separation of individual ions and the permeation rate of product solution at given operating conditions from only a single set of RO primary data for an aqueous sodium chloride reference feed solution.

It was shown in the earlier papers on reverse osmosis (RO) separation of electrolyte mixtures (J--^) that prediction of RO performance data such as the separation of individual ions involved in the mixture, and the membrane permeated product rate, could be successfully accomplished by transport equations based on Kimura-Sourirajan analysis incorporating the necessary modifications arising from ionic equilibria at the solution-membrane interface. It was also shown that the system performance of RO modules by which electrolyte mixtures are separated could also be predicted on the basis of the same fundamental principles (5_). The earlier papers cited above have illustrated in no uncertain terms that (1) the basis of KimuraSourirajan analysis is sound (2) the concept of free energy parameters which was found useful in predicting RO performance data of single solute systems (67JO is also relevant in mixed electrolyte solute systems, and (3) the assumptions introduced in the process of the numerical calculations using the transport equations are valid. In this paper, general transport equations are developed, which enable one to predict the performance of RO separations of mixed electrolytes involving any number of ions and ionic valences provided no ionic association takes place in the solution. The system treated in earlier papers are shown to be the special cases of the general expressions derived in this paper.

0097-6156/85/0281-0167$06.00/0 Published 1985, American Chemical Society

In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

168

REVERSE OSMOSIS AND ULTRAFILTRATION


Theoretical Analysis and Discussion Nomenclature* The symbols used (same as those in Part 3 (3) of this series unless otherwise stated) are listed at the end of the paper. To facilitate the reader, a brief note is in order. The subscripts A, B and M refer to salt, water and membrane respectively. All quantities with asterisks (*) refer to ions; with respect to such quantities the first subscript, i, 1, 3, 5... refers to the indicated cations and j, 2, 4, 6... refers to the indicated anions; and, the second subscript M, 1, 2, 3 refers to the indicated phase, namely, membrane, feed solution, concentrated boundary solution on the high pressure side or product solution, respectively. Cations are represented by odd numbers, while anions are represented by even numbers. With respect to the quantities which do not refer to ions specifically, the subscript M, 1, 2, or 3 refers to the indicated phase. Numerical subscripts 4J, XA> &Z> XZ refer to single salts with the ions indicated by each number. Thus, for example the quantities X£, X*, XJ lf X*, XJM, X } 3 , X ^ , X A M 3 , X A 3 represent mole fractions of cation i, anion j, cation i in phase 1, cation 1, cation 1 in membrane phase, cation 1 in product solution phase, salt A in membrane phase, salt A in membrane phase in equilibrium with X A 3 and salt A in product solution phase, respectively; the quantities c 2 and Cj|2represent molar densities of solutions in phase 2 and in membrane phase in equilibrium with Xô* respectively; and the quantity (D^/Kfi),« represents solute transport parameter for the single salt

12. Reverse Osmosis Transport Equations and the General Description of the System. For a RO system involving several ions, under steady state isothermal operating conditions at the gauge pressure P, the flux equations for solvent and ion transport can be written as follows, using the form of the Kimura-Sourirajan analysis (90. Water transport:

The cation transport:

The anion transport:

where

and Both Equation 4 and Equation 5 are applicable to concentrated boundary solution on the high pressure side or the dilute product


13. RANGARAJAN ET AL.

RO Systems Involving Mixed Ionized Solutes

169

solution on the atmospheric pressure side of the membrane with respect to the second subscript. In addition to the above equations, the total number of which is equal to (n + 1) where n is the total number of all the ions present, we have the following relations to represent the boundary concentrations of ions: For cations


and for anions

Furthermore, the product mole fractions of cations and anions are given by, for cations

and for anions

Equations 1 to 9 describe completely the equilibria and transport for all the ionic species. It should, however, be noted that all flux equations have to be considered by suitably subscribing the Equations 2 to 9. The system is also subject to the electroneutrality conditions that prevail in the feed, boundary and the membrane phases: for the feed solution,

for the concentrated boundary solution,

for the product solution,

for the membrane phase,

and also for the product water flux,


170


Thus, Equations steady state reverse of several ions, and terms of N B , N*, N*,

1 to 14 are a complete general description of a osmosis system with a feed solution consisting our object is to solve the above equations in X£ 3 and X* 3 .

Mathematical Analysis


In order to simplify the solution of the above transport equations, the following assumptions are necessary (1)

iii) For evaluating N« using Equation 1, we must have either experimental osmotic pressure data for electrolyte mixtures, or a method to estimate osmotic pressure data for such solutions. In view of the infinite number of solution compositions possible for concentrated boundary solution and product solution phases for a given feed composition, depending upon the operating conditions and the inherent properties of membranes, it is not possible to obtain osmotic pressure data for all possible compositions experimentally. Also there is no theoretical approach to calculate osmotic pressure of solutions containing several ions. For the purpose of our calculations, the osmotic pressure of an electrolyte solution involving several ions is approximated by the equation:

Equation 17 implies that the osmotic pressure of an electrolyte solution is the result of combination of the contribution from each ion to the total osmotic pressure. Secondly, the osmotic pressure contribution of each ion is proportional to its mole fraction. With these assumptions, sets of B*(X*) and B*(X*) could be evaluated at different mole fractions from the literature data on osmotic pressure of several salts by regression analysis by the method described earlier (3^. From BJ(XJ) and B*(X*) values, for any known composition of a solution, the osmotic pressure of the solution of mixed electrolytes can be calculated.

where k£ and k ^ ™ are mass transfer coefficients for ion i (calcd) and for NaC£ respectively. The quantity kflaC£ i s obtained from Kimura-Sourirajan analysis of the experimental data with NaC£ feed. The quantities D£ and % a C £ are the self-diffusion coefficients of ion i and NaC£ in water, both at infinite dilution. v) The ratio of diffusivity through the membrane to that in water for all species is constant, so that


13.

RANGARAJAN ET AL.


171

With the above assumptions the transport equations 1 to 14 can be simplified as follows: Corresponding to Equation 1,


Corresponding to Equation 2 and Equation 3,

Corresponding to Equation 6

where

Corresponding to Equation 7

where Corresponding to Equation 8 and Equation 9

Expressions for Interfacial Equilibrium Constants and Corresponding Ionic Solute Transport Parameters* It has been shown in the earlier paper (40 that the following equilibrium constants for the salt and ions comprising it are applicable and they are related to each other as follows:


172


and


Expressions for the Ionic Equilibrium Constants, Kjf and D-fa/K^S * It can be seen from Equation 2 that (D*M/K*6) has to be known for evaluating NJ. In this section an attempt is made to evaluate K* first, and then to evaluate (D*^/K#6) by using assumption (v). In developing expressions for K*, we nave to keep in mind that: (1) when an ion is in equilibrium between the aqueous phase and the membrane, we have the composition dependent, K|, given by Equation 4; once (c M X* M ) is obtained in terms of the composition (c X*) with which membranes are in contact K£ can be evaluated; and (2) electroneutrality should be maintained in the membrane phase (Equation 13). After expanding and rearranging, Equation 13 becomes

or

Further, using the general equation given by Equation 30, general expressions for the concentration ratios of cations and anions can be obtained as:

for cations, and

for anions. obtain

Inserting Equation 33 and Equation 34 in Equation 32 we




173


Also, using Equation 30 in terms of i = 1 and j = 2, Equation 35 becomes,

Equation 36 is a general expression, which is a power series in terms of (cM X f M ) . It has to be noted that special cases of Equation 36 appear as Equation 50 in ( O , Equations 52-54 in G O , Equation 44 in (30 and Equation A-21 in (U) with numerical values of z,, z^, z. and z. appropriate for the respective cases. Such approach to the evaluation of the roots (in terms of Cj4 of Equation 36 is straight forward, provided values of and

are available.

If they are not available, they are

obtainable using the assumption number (v) in the following way: From Equation 19,

Then,

therefore,

Replacing

in Equation 36 by

using Equation 31 and further

using the relation given by Equation 39, and rearranging we obtain



174


Equation 40 is a polynomial function in terms of ( c M x ^ ) which is readily soluble when all required numerical values of (D^/KS ). . , J),. , and the solution compositions are given. The method of obtaining numerical values for (DAM^K^îi a n d D ii w i l 1 b e s n o w n lat^r. Suppose such a solution is i^, then, since = DJM/D* respectively. Furthermore, using Equations 42, 30 and J7, we can deduce,

Similarly, we can obtain


13.

RANGARAJAN ET AL.


175


Both Equations 43 and 44 are applicable for the respective phases, namely

By using Equation 45 to Equation 48, the flux equations given by Equation 2 and Equation 3 can be written as

for cations (i = 1, 3, 5...), and

for anions (j = 2,4,6...)• Method of Prediction of Separation of Ions and Product Rate for a Given Membrane for Which RO Data with NaC& Feed Solution are Available. From the RO data with NaC£ feed solution, by KimuraSourirajan analysis (9), A, ( D A M / K < $ ) N c o an and S) and the following equations,

In


The mass transfer coefficients of all ions are evaluated using the following relationship (10),

The diffusivities of different ions can be obtained from the literature (11), In addition to ( D AM/K$ )•« .«> k?> etc., the other necessary information to be used in the transport analysis are values of B* versus X* for different ions. This is obtainable from the literature values of osmotic pressure for different salts and listed in the literature 0 } 4 ) for all ions involved in this study. The procedure to compute the pertinent BJ values is illustrated in our previous work (3^). Thus, from the values of A, in Cg a C ^, (-AAG/RT)f, k£ and BJ versus X£, the separation of all the ions and the product rate, [PR], can be computed by the standard procedure of solving the simultaneous equations including Equation 20 and Equations 49 and 50 (3). Experimental Verification of the Prediction Technique. The prediction technique described above was tested in our previous work for systems including 1:1 and 1:1 electrolyte mixtures, 1:1 and 2:1 electrolyte mixtures, 1:1:1 and 1:1:1 electrolyte mixtures, and electrolyte mixtures simulating the sea water. The test also included the cases where common anions or cations were involved in the mixed electrolytes. Some examples of the feed solutions tested are given in Table I. All numerical values for -in Cj$aC£, A> k, Bf(Xf), (-AAG/jlT^, Df and D.., which are necessary for the calculation, are given in the literature Oj"^)* A comparison of the calculated and experimental results on ion separations and product rates are shown in Table II. The agreement is excellent, indicating the validity and practical utility of the prediction technique developed above. Conclusion The usefulness of the prediction technique based on membrane equilibria, free energy parameters, and Kimura-Sourirajan analysis, has been experimentally substantiated in Part 1-4 of this series and the nethod has been extended to the most general case where no limit to the number of ions and valences is imposed. Thus, this approach presents a simple prediction technique of practical value and also lends support to the preferential sorption-capillary flow mechanism for RO, which forms the basis for this mathematical treatment.


13.

RANGARAJAN ET AL.


111

Table I. Description of Experimental Conditions Used in the RO Runs with Mixed Electrolytes

Run No.


1 2 3 4

5 6

7 8 9 10

11 12 13 14

15 16

17 18

19 20

Operating Pressure, kPag

10342 10342 10342 6895

10342 10342

6895 6895 6895 6895

6895 6895 6895 6895

6895 6895

6895 6895

6895 10342

Composition of Feed Solution, Molality

NaCl

KNO3

1.38 1.38 0.49 0.31

1.31 1.31 2.73 0.21

NaN0 3

NaC*

0.82 1.014

0.26 1.014

NaCA

Ca(N0 3 ) 2

0.076 0.250 0.763 0.500

0.035 0.310 0.326 0.615

NaBr

MgCA 2

0.254 0.113 0.113 0.073

0.094 0.202 0.202 0.400

NaC£

MgC* 2

0.42 0.20

0.11 0.32

NaC£

KI

LiBr

0.07 0.15

0.07 0.14

0.34 0.69

NaCA

KBr

LINO 3

0.32 1.30

0.67 0.32

0.70 0.20

mole fraction x 10^

Na+ 21 22

10342 10342

10.045 10.045

K+

Mg"1"1"

Ca"1"*

0.222 0.222

1.259 1.259

0.223 0.223

Continued on next page.


178


Table I. Continued Run No.

Operating Pressure, kPag

Mole Fraction x 103 Cl~

Sr^


21 22

10342 10342

11.795 11.795

0.005 0.005

Br"

HCO3

SO4

0.015 0.015

0.043 0.043

0.695 0.695

Table II. Comparison of Experimental and Predicted RO Performance of Cellulose Acetate Membranes with Different Mixed Electrolyte Feeds Run No.

(PR) e x 103 kg/h

Solute Separation, % 1-1, 1-1 electro lyte mixtures Na+

K+

CJT

NO3

1

10.3 (10.3)

68.3 (65.5)

60.6 (61.4)

75.0 (78.0)

52.6 (48.3)

2

60.0 (61.3)

35.5 (30.2)

24.6 (26.8)

44.3 (41.3)

15.3 (15.2)

3

96.1 (97.8)

17.2 (16.7)

12.7 (14.4)

36.2 (36.6)

9.2 (10.8)

4

34.9 (34.3)

80.2 (78.4)

74.3 (75.0)

83.5 (86.0)

69.5 (64.0)

NaN03

NaCA

5

28.9 (26.4)

94.3 (95.3)

99.6 (98.5)

6

97.7 (105.8)

21.4 (16.2)

32.3 (35.2)

3 1-1, 2- 1 electrolyte mixtures'

Na +

Ca ++

CJT

N0 3

7

48.6 (48.7)

81.7 (85.8)

97.2 (98.6)

93.9 (95.2)

82.3 (87.1)

8

138.1 (152.3)

0.5 (-2.0)

11.9 (18.5)

15.3 (16.1)

5.9 (11.2)


13.

RANGARAJAN ET AL.


Table II. Continued Run No.

(PR) 6 x 10^ kg/h

Solute Separation, % 1-1, 2-1 electrolyte mixtures*3


Na+

Ca ++

CI"

N0^

9

18.0 (16.5)

51.3 (52.7)

91.5 (91.6)

77.3 (84.9)

61.1 (54.0)

10

14.3 (13.8)

29.3 (29.1)

88.1 (88.0)

80.9 (85.2)

67.0 (65.0)

Na+

Kg**

C£"

Br"

11

107.1 (102.6)

66.9 (65.7)

91.5 (85.2)

78.4 (80.2)

68.9 (69.3)

12

49.8 (48.7)

82.3 (83.0)

97.9 (97.1)

95.4 (94.7)

90.3 (91.6)

13

91.8 (93.0)

55.3 (56.0)

90.8 (93.9)

86.3 (88.0)

73.9 (77.2)

14

66.7 (70.5)

34.7 (42.0)

88.9 (89.8)

83.9 (86.6)

80.0 (77.4)

NaC I

MgC &2

15

32.4 (29.8)

82.4 (83.6)

96.9 (99.4)

16

45.0 (42.8)

50.7 (54.4)

86.7 (91.2)

1-1-1, 1-1-1 electrolyte mixtures Na+

K+

Li +

Cf

17

22.6 (23.3)

93.5 (93.9)

92.3 (94.7)

94.7 (92.8)

96.2 (95.1)

18

47.9 (51.5)

30.0 (31.8)

28.2 (30.9)

31.9 (28.7)

42.1 (33.0)

Continued on next page.


179

180


Table I I .

Run No.

(PR) e x 10 J kg/h

Continued

Solut e Separation, 70


1-1-1, 1-1- 1 electrolyte mixtures c I"

Br""

17

22.6 (23.3)

89.6 (91.6)

95.0 (93.3)

18

47.9 (51.5)

14.3 (13.9)

29.9 (31.8)

Na +

K+

Li +

CT

34.9 (36.4)

36.8 (37.0)

35.0 (33.6)

50.0 (56.9)

55.8 (52.7)

19

27.7 (28.8)

32.3 (33.3)

20

50.6 (52.0)

52.5 (51.9)

56.3 (56.1)

Br"

NO3

19

27.7 (28.8)

34.3 (37.3)

33.3 (36.0)

20

50.6 (52.0)

50.0 (56.2)

52.6 (51.3)

€electrolyte mixture of sea water desalination

Na +

K+

Mg ++

Ca4

99.8 (99.9)

99.2 (99.9)

21

60.2 (61.7)

96.4 (96.9)

95.2 (96.6)

22

37.0 (38.7)

97.4 (98.1)

97.0 (97.9)

99.4 (-100)

99.0 (-100)


13.

RANGARAJAN ET AL.



Table I I . Run No.

(PR) x 10 kg/h

Continued

e

Solute Separation, %

electrolyte mixture of sea water desalination Sr

CJT

Br"

HCO3

S0 4

21

60.2 (61.7)

99.8 (-100)

96.5 (95.9)

(94.3)

99.7 99.8 (99.2) (-100)

22

37.0 (38.7)

99.4 (-100)

97.7 (97.6)

(96.6)

99.7 99.4 (99.5) (-100)

Note: Numbers without parentheses are experimental values. Numbers inside the parentheses are calculated values. From literature (1) . From literature (2^). From literature

(3).

From literature (4). Effective film area = 13.2 x 10~ 4 m2.


181

182



Nomenclature A

= pure water permeability constant, kmol/m 2 «s # kPa

B*

= B*(X*), osmotic pressure coefficient, kPa (per unit mole fraction of cation i)

B*

= B*(X$), osmotic pressure coefficient, kPa (per unit mole fraction of anion j)

BfojB*^

= osmotic pressure coefficient of cation i in the boundary of high pressure side of the solution and the product solution, respectively

B*2>B^

- osmotic pressure coefficient of anion j in the boundary of high pressure side of the solution and the product solution, respectively

^ftaCA

~ constant characterizing the porous structure of the membrane defined by Equation 51

c

= molar density of the solutions, mol/m3

cj,C2>C3,cM = molar density, c, in solution phase 1, phase 2, phase 3, and the membrane phase, respectively C

M2 , C M3

D

AB , D AM

D

£,D?M

~ m ° l a r density, c, in membrane phase in equilibrium with X ^ > and X ^ , respectively =

diffusion coefficient of solute in water and membrane phase, respectively, m 2 /s

" diffusion coefficient of cation i, in water and in membrane phase respectively, m 2 /s

D*,D* M

• diffusion coefficient of anion j in water and in membrane phase respectively, nr/s

Dj.

= diffusion coefficient of salts in water, m 2 /s

(D^/K6)

• solute transport parameter, m/s

(D A M /K6).. = solute transport parameter for salt, m/s f

= fraction solute separation

(-AAG/£T)£ = free energy parameter for cation i (-AAG/&T)* • free energy parameter for anion j K.,

= interfacial equilibrium constants for single solute, ij, defined by Equation 29




183

= interfacial equilibrium constants for single solute, ij, defined bv Eauation 30 = equilibrium distribution coefficients for cation i, cation i in solution phase 2, and cation i in solution phase 3, respectively = equilibrium distribution coefficients for anion j, anion j in solution phase 2, and anion j in solution phase 3, respectively


= mass transfer coefficients on the high pressure side of the membrane; k for cation i and anion j, respectively, m/s = mass transfer coefficient of NaCi as determined by experiment, m/s = solute flux through the membrane, kraol/m2»s = solvent flux through the membrane, kmol/m 2# s = ionic flux of cation i through the membrane, kmol/m2*s = ionic flux of anion j through the membrane, kmol/m^s = number of moles of anions and cations, respectively, arising from dissociation of one mole of solute P

= operating pressure, kPa

(PR)

= product rate through a given area of the membrane surface, kg/h

(PWP)

= pure water permeation rate through a given area of the membrane surface, kg/h

£

- gas constant

T

= absolute temperature

X

= mole fraction

Â1 , Â2»Â3 ,X AM " mo^-e fraction of solute A, in solution phase 1, solution phase 2, solution phase 3, and membrane phase, respectively X

AM2 ,X AM3 ~ m ° l e fraction of solute in membrane phase in equilibrium with XA2> and X ^ , respectively

X J , X J p X ^ 2 > X ^ , X ^ = mole fraction of cation i, cation i in solution phase 1, cation i in solution phase 2, cation i in solution phase 3 and cation i in membrane phase, respectively


184 x


?> x ?l> x ?2 , X t3 , X iM = m o l e fraction of anion j, anion j in solution phase 1, anion j in solution phase 2, anion j in solution phase 3 and anion j in membrane phase, respectively

îi*

z

^JjpM

i,zj

~

= mo

^-e fraction of salt i j, X. . in membrane phase respectively

va

l e n c e °f cation i and anion j, respectively


Greek Letters aâ,

= quantities defined by Equation 24 and Equation 26, respectively

6

= effective thickness of the membrane, m

TT(ZXJ + EX|), ir(EXJ2 + Z X * 2 ) , TT(EXJ 3 + EX* 3 ) = osmotic pressure of the mixed electrolyte system, that in the boundary phase, and that in the product phase, respectively, kPa

= quantity defined by Equation 37

ij>l

= quantity defined by Equation 41

Acknowledgment This paper was issued as NRC No. 24034.

Literature Cited 1.

Rangarajan, R.; Matsuura, T.; Goodhue, E.C.; Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 46.

2.


3.

Rangarajan, R.; Baxter, A.G.; Matsuura, T.; Sourirajan, S. Eng. Chem. Process Des. Dev. 1984, 23, 367.

4.

Rangarajan, R.; Mazid, M.A.; Matsuura, T.; Sourirajan, S. Eng. Chem. Process Des. Dev. 1985, in press.

5.

Matsuura, T.; Sourirajan, S. 1985, in press.

6.

Matsuura, T.; Pageau, L.; Sourirajan, S. 1975, 19, 179.

7.


8.


Ind.

Ind.

Ind. Eng. Chem. Process Des. Dev.

J. Appl. Polym.


Sci.

13. 9.

RANGARAJAN ET AL. Sourirajan, S. Chap. 3.

RO Systems Involving Mixed Ionized Solutes "Reverse Osmosis", Academic:

New York, 1970;

10.

Reid, R.C.; Sherwood, T.K. "The Properties of Gases and Liquids", McGraw-Hill: New York, N.Y., 1958, p. 295.

11.

Parsons, R. "Handbook of Electrochemical Constants", Butterworths: London, 1959; Table 73.


RECEIVED February 22, 1985


185

Predictability of Membrane Performance in Reverse-Osmosis Systems

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