Asymptotic Solute Rejection in Reverse Osmosis - American Chemical

18 Asymptotic Solute Rejection in Reverse Osmosis

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on June 2, 2015 | http://pubs.acs.org Publication Date: May 21, 1981 | doi: 10.1021/bk-1981-0153.ch018

SUN-TAK HWANG Chemical and Materials Engineering, The University of Iowa, Iowa City, IA 52242 WOLFGANG PUSCH Max-Planck-Institut für Biophysik, Kennedy-Allee 70, 6000 Frankfurt am Main 70, West Germany In the literature, there are many transport theories describing both salt and water movement across a reverse osmosis membrane. Many theories require specific models but only a few deal with phenomenological equations. Here a brief summary of various theories will be presented showing the relationships between the salt rejection and the volume flux. The solution-diffusion model (1) assumes that water and salt diffuse independently across the membrane and allows no convective salt transport. The reciprocal salt rejection, 1/r, is linearly related to the reciprocal volume flux, 1/q:

1/r=1+

KsDs/d

·1/q (

1

)

where Ks i s the salt distribution coefficient, Ds is the salt d i f f u s i v i t y and d i s the membrane thickness. The finely-porous membrane model (2,3) assumes that a substantial amount of salt i s transported by convective flow through the narrow pores of the membrane. Integrating the Nernst-Planck equation for salt transport (3) and using the appropriate boundary conditions, the following relationship i s obtained between the salt rejection and the volume flux:

~z

1-r

= ^ 1 - r

00

- 1 - r oo

exp [-q(l - r )d/P ] ^ /

0

0

/

(2)

s

where r ^ i s the asymptotic s a l t r e j e c t i o n and P i s the s a l t permeability. Katchalsky and Curran (4) formulated the nonequilibrium thermodynamic equations f o r the t r a n s p o r t of m a t e r i a l through a discontinuous membrane. However, there was a d i f f i c u l t y i n t h e i r expressions, as they r e q u i r e d the "average c o n c e n t r a t i o n " i n the membrane. In order to overcome t h i s d i f f i c u l t y of e v a l u a t i n g the g

0097-615 6/81/015 3-025 3$05.00/ 0 © 1981 American Chemical Society

In Synthetic Membranes:; Turbak, A.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

254

SYNTHETIC

MEMBRANES:

DESALINATION


average c o n c e n t r a t i o n and the concept of d i s c o n t i n u i t y , S p i e g l e r and Kedem (5) introduced a continuous membrane model. This model c o n s i s t s of a l t e r n a t i n g t h i n l a y e r s of membranes and s o l u t i o n s under l o c a l e q u i l i b r i u m . Thus, S p i e g l e r and Kedem were able to convert the d i f f e r e n c e equations derived by Katchalsky and Curran i n t o a s e t of d i f f e r e n t i a l equations. Then, upon i n t e g r a t i o n across the membrane, they a r r i v e d at the same f i n a l expression f o r the s a l t f l u x as i n the f i n e l y - p o r o u s membrane model. The s a l t r e j e c t i o n i s r e l a t e d to the volume f l u x e x a c t l y i n the same manner:

d

r ~ = i T 7 - r T 7 - p I-*"- - °> /V

( 3 )

where a i s the r e f l e c t i o n c o e f f i c i e n t . I t i s apparent that r = a when q becomes i n f i n i t e l y l a r g e , that i s , r ^ = cr. Therefore, Eq. (3) i s i d e n t i c a l to Eq. ( 2 ) . In a recent study, Pusch (6) derived a l i n e a r r e l a t i o n s h i p between the r e c i p r o c a l s a l t r e j e c t i o n and the r e c i p r o c a l volume f l u x based on the model-independent nonequilibrium thermodynamic expressions: I - -L + -1^ . I r r r d q 00

(4)

oo

This i s an e x c e l l e n t model from the viewpoint of determining the transport c o e f f i c i e n t s from experimental data. A simple l e a s t squares f i t gives r ^ from the i n t e r c e p t and P^/r^d from the s l o p e . The only weakness i s i n the approximation of the mean s a l t conc e n t r a t i o n c - c^ - c'. This w i l l be discussed again l a t e r . g

Volumetric Transport As summarized above, there are many t r a n s p o r t models and flow mechanisms d e s c r i b i n g reverse osmosis. Each r e q u i r e s some s p e c i f i c assumptions regarding membrane s t r u c t u r e . In g e n e r a l , membranes could be continuous or discontinuous and porous or nonporous and homogeneous or non-homogeneous. One must be reasonably sure about the membrane s t r u c t u r e before he analyzes a part i c u l a r s e t of experimental data based on one of the above theories. Since t h i s i s d i f f i c u l t , i n many cases, i t would be d e s i r a b l e to develop a model-independent phenomenological theory which can i n t e r p r e t the experimental data. In the f o l l o w i n g d i s c u s s i o n , a p u r e l y phenomenological theory w i l l be presented f o r the t r a n p o r t of s a l t and t o t a l volume. Hence, i t does not assume any p a r t i c u l a r membrane s t r u c t u r e or t r a n s p o r t mechanism. However, the membrane i s assumed to be continuous i n the sense that any p h y s i c a l q u a n t i t y may be d i f f e r e n t i a t e d or i n t e g r a t e d throughout the membrane. In an analogous manner to the r e g u l a r treatment of t r a n s p o r t phenomena (7), the


18.

HWANG A N D PUSCH

Asymptotic

Solute

255

Rejection

transport of s a l t and t o t a l volume across a membrane can be t r e a t e d phenomenologically without r e s o r t i n g to a s p e c i f i c t r a n s port mechanism. The molar f l u x of s a l t with respect to s t a t i o n a r y coordinates or a l a b o r a t o r y observer i s cj) = C v s ss

(5)

where C i s the s a l t concentration i n the membrane and v i s the s a l t v e l o c i t y r e l a t i v e to s t a t i o n a r y coordinates. I t should be noted that t h i s f l u x (f i s not a d i f f u s i v e f l u x , which i s r e l a t i v e motion with respect to some average v e l o c i t y . Two f r e q u e n t l y used average v e l o c i t i e s are mass average v e l o c i t y and molar average v e l o c i t y . Here, we w i l l introduce a volume average v e l o c i t y v"^* (8):


s

g

s

v

+

= £ C . V i

±

v. = L * i V i

(6)

where V-^ i s the p a r t i a l molar volume of the i t h s p e c i e s . Here i represents s f o r s a l t , w f o r water, and M f o r membrane. Using t h i s reference frame, the molar f l u x of s a l t can be d i v i d e d i n t o two terms: cj) s Y

+

= C v = C (v - v ) + C v s s s s s

+

(7)

The f i r s t term on the right-hand s i d e i s the d i f f u s i v e f l u x r e l a t i v e t o the volume average v e l o c i t y . The second term represents a c o n t r i b u t i o n due to bulk flow. I t should be emphasized here that the s e p a r a t i o n of the t o t a l f l u x i n t o two c o n t r i b u t i o n s i s always p o s s i b l e r e g a r d l e s s of the a c t u a l transport mechanism through the membrane. In other words, Eq. (7) i s purely phenomen o l o g i c a l and does not r e q u i r e any s p e c i f i c t r a n s p o r t model. If the d i f f u s i v e f l u x i s expressed according to F i c k ' s d i f f u s i o n equation: d

+ f

C (v - v ) - o

s

s

D

C

—2.

(8)

s dx

then, Eq. (7) becomes: dC c}> = -D -r- - + C v (9) s s dx s In reverse osmosis, the t o t a l volume f l u x i s f r e q u e n t l y used. 5

q = (j> V WW

T

+ cf> V s s


(10)

256

SYNTHETIC

MEMBRANES:

DESALINATION

Comparing Eqs. (6) and (10), the f o l l o w i n g i d e n t i t y i s obtained (note that the membrane f l u x cb i s zero) : M v* = q (11) 1


which means that the t o t a l volume f l u x i s a c t u a l l y the volume average v e l o c i t y . Substituting this identity relationship into Eq. (9), the Nernst-Planck type equation i s obtained: dC = -D ~ - + C q s s dx s

(12)

However, a d i s t i n c t i o n should be made i n that Eq. (12) i s p u r e l y phenomenological and does not r e q u i r e any t r a n s p o r t mechanism model w h i l e the Nermst-Planck equation used i n the previous f i n e l y - p o r o u s membrane model r e q u i r e s a s p e c i f i c pore model. Another d i f f e r e n c e i s that the s a l t c o n c e n t r a t i o n i n Eq. (12) i s that i n the membrane while the quantity appearing i n the NernstPlanck equation r e f e r s to the s a l t c o n c e n t r a t i o n i n the membrane pores. The g e n e r a l i z e d equation o f motion can be w r i t t e n as ( 9 ) : 3C +

Tr

V

•

c

i V

R

i

(13

>

where R-^ i s the molar r a t e of production of the i t h species per u n i t volume, and i s zero i n a reverse osmosis system. I f the p a r t i a l molar volume can be assumed to be constant, m u l t i p l y i n g through by and summing over a l l s p e c i e s , the f o l l o w i n g equation results: 3

c

? i *± 1 3

t

+

v

•E

c. v

±

v =o ±

(14)

At the steady s t a t e the f i r s t term drops out and s u b s t i t u t i n g Eq. (6) i n t o the above equation, the equation of c o n t i n u i t y becomes: V • v

+

= 0

(15)

When t h i s i s a p p l i e d to the one dimensional reverse osmosis system Eqs. (11) and (15) y i e l d :

£

=

0

16

Thus, i t becomes apparent that the t o t a l volume f l u x remains constant throughout the membrane at steady s t a t e . I t i s a l s o obvious


18.

HWANG AND PUSCH

Asymptotic

Solute

257

Rejection

that the molar s a l t f l u x i s constant a t steady s t a t e . Therefore, the d i f f u s i v i t y i s the only v a r i a b l e c o e f f i c i e n t i n Eq. (12), which can be i n t e g r a t e d as f o l l o w s :


(17)

Defining the average d i f f u s i v i t y , D :

(18)

Eq. (17) yields:

c

s -T

=

c

( s - T)

e

x

p

(

q

x

/

( 1

V

This equation describes p r e c i s e l y how the s a l t concentration v a r i e s i n the membrane as a f u n c t i o n of p o s i t i o n . Homogeneous Membrane When the membrane i s homogeneous, the s a l t d i s t r i b u t i o n c o e f f i c i e n t s are assumed to be i d e n t i c a l at both s i d e s of the membrane. The s a l t d i s t r i b u t i o n c o e f f i c i e n t i s denoted as K , which r e l a t e s the s a l t concentration i n the membrane to the s a l t concentration i n the bulk: g

C" s K

s

=

(20)

Here, s i n g l e prime r e f e r s to the feed s i d e (high pressure) and double prime r e f e r s to the product s i d e (low pressure) and the lower case c's r e f e r to the bulk concentrations. Even i f the membrane i s homogeneous, i t i s p o s s i b l e to have a v a r i a b l e d i f f u s i v i t y , which may be a f u n c t i o n of s a l t c o n c e n t r a t i o n . The boundary c o n d i t i o n on the low pressure s i d e of a reverse osmosis c e l l r e q u i r e s that the s a l t and water f l u x e s through the membrane determine the bulk s a l t concentration on the low pressure s i d e . Thus, the f o l l o w i n g r e l a t i o n s h i p r e s u l t s :


SYNTHETIC

258 C" #

MEMBRANES:

DESALINATION

- f

w

( 2 1 )

w

In a d d i t i o n , f o r a d i l u t e s o l u t i o n the f o l l o w i n g approximations may be used: c

V

« 1

,

q = V

WW


WW

(22)

Therefore, the boundary c o n d i t i o n given by Eq. r e w r i t t e n as:

(21) can

be

C" * — q Introducing

(23)

the s a l t r e j e c t i o n , r : C"

47

oo

E s s e n t i a l l y , the same r e s u l t s w i l l be obtained from the homogeneous double l a y e r theory i f the s u b s c r i p t s 1 and 2 are exchanged i n Eq. (40). Conclusion An exact mathematical r e l a t i o n s h i p i s obtained between the s a l t r e j e c t i o n and t o t a l volume f l u x i n reverse osmosis based on a p u r e l y phenomenological theory assuming constant s a l t permeability. This approach does not r e q u i r e a s p e c i f i c membrane model;


SYNTHETIC

264

MEMBRANES:

DESALINATION

w h i l e , the r e s u l t s s t i l l agree with the f i n e l y - p o r o u s membrane model and the Spiegler-Kedem model. I t a l s o shows that there are e s s e n t i a l l y no d i s t i n c t i o n s made between the present r e s u l t s and the ones obtained by Pusch when experimental data are c o r r e l a t e d . Furthermore, the c a l c u l a t i o n s j u s t i f y the assumption introduced by Pusch that c = £ = c a t the two extreme cases when q -> 0 and q . The present study shows that the asymptotic s a l t r e j e c t i o n , r , i s determined by the top s k i n l a y e r of a membrane. This i s a r e s u l t of the steady-state mass balance and the boundary c o n d i t i o n s . Although there are no experimental data to support t h i s , i t has been shown t h e o r e t i c a l l y that the asymptotic s a l t r e j e c t i o n i s i d e n t i c a l to the r e f l e c t i o n c o e f f i c i e n t f o r the homogeneous membrane, r = a. F i n a l l y , i t should be mentioned that i t i s very d i f f i c u l t to a c c u r a t e l y measure the values of r ^ when i t i s c l o s e to u n i t y . On the other hand, when the value of r ^ i s low, membrane compaction w i l l take p l a c e . In s p i t e of these d i f f i c u l t i e s , experiments are c u r r e n t l y being conducted to t e s t the present theory at the Max-Planck-Institut f u r Biophysik i n F r a n k f u r t , Germany. s

s


00

O T

Acknowledgment The authors are g r a t e f u l to P r o f e s s o r R. S c h l o g l f o r h i s i n t e r e s t i n t h i s work. The f i n a n c i a l support by the "Bundesm i n i s t e r f u r Forschung und Technologie," Bonn, Germany, i s greatly appreciated. Nomenclature a

= s a l t d i s t r i b u t i o n c o e f f i c i e n t between two contiguous membranes s a l t c o n c e n t r a t i o n i n the membrane s a l t c o n c e n t r a t i o n i n the bulk mean s o l u t i o n c o n c e n t r a t i o n salt

diffusivity

average s a l t

diffusivity

membrane thickness K

P

g

= s a l t d i s t r i b u t i o n c o e f f i c i e n t between the bulk and the membrane = s a l t permeability


18.

HWANG A N D PUSCH

Asymptotic

Solute Rejection

q

= t o t a l volume f l u x

R

= molar r a t e of production per u n i t volume

r

= salt rejection


= asymptotic s a l t r e j e c t i o n a t q t

= time

V

= p a r t i a l molar volume

v

= velocity

v

= volume average v e l o c i t y

x

= dimension across membrane

2(

0 0

Greek l e t t e r s a

= reflection

coefficient

Asymptotic Solute Rejection in Reverse Osmosis - American Chemical

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