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Salt rejection and flux in reverse osmosis with compactible membranes
Desalination.28 (1979) 65-85 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
SALT REJECTION AND FLUX IN REVERSE COMPACI’IBLE MEMBRANES
W.C.M. HENKENS
AND J-AM.
OSMOSIS WITH
SMIT
GorZaeus taboratonks of the State University L&den, Leiden (The Netherlands) (Received June 19.1978;
in revised form January 31,1979)
SUMMARY
The theory of reverse osmosis in bilayer membranes is extended. At high fluxes the selectivity of the layer on the high pressure side dominates the salt rejection also in case this layer has the lowest selectivity. Theoretical concentration profiles are given for both orientations. Experimentally, irreversible compaction of the membrane is controlled by a pressure treatment. For the remaining reversible compaction the transport equations can be corrected. It turns out that reversible compaction affects the flux but not the salt rejection. Correspondingly structural changes occur in the layer of low selectivity.
SYMBOLS
-
solute concentration, mol mS3. solute concentration at distance x in the membrane per volume of membrane, mol me3. - flow parameter defined by Eq. (3). - factor defined by Eq. (16). - factor defined by Eq. (22). - solute flux mol m-* s-r . - total volunk flux, m s-i. - partition coefficient defined by k/Q, (x)_ - partition coefficient defined by Eq. (27). - hydraulic permeability, N-i m3 s-i. - hydraulic permeability at AP = 0, N-i m3 s-i. - thickness of the membrane, m. - Reynold’s number. - hydrostatic pressure, Pa. - universal gas constant, N m mol-’ K-i . - salt rejection defined by Eq. (11). -
W.C.M. HENKENS
66
T
-
V
- feed velocity, m s-’ .
AND J.A.M. SMIT
absolute temperature, K.
space coordinatz perpendicular to the membrane surface, ranging from 0 to I, m. z - reciprocal characteristic diffusion length defined by Eq. (24), m-l. - operator standing for the difference in the quantity Q between (YA and @-phases, Aq = qa - qp. - compaction modulus. ;: - thickness of the layer a, m. V stoichiometric dissociation factor of the salt. X - theoretical osmotic pressure, Pa. - experimental osmotic pressure, Pa. n=u (J - reflection coefficient. w - solute permeability, m s-l. ap, (3~) - volume fraction of water at distance x in the membrane. - osmotic coefficient. @ x
-
Indices - layer u, b of the composite membrane. 0, b - the interface between layer a and b in the membrane. i -
the feed. the effluent. pressure dependent quantities.
INTRODUCTION
Anisotrope cellulose acetate membranes of the type discovered by Loeb and Sourirajan [l] are usually considered to consist of a dense skin in the top of a more porous film. Schematically they may be viewed as simple bilayer membranes of two homogeneous elements in series. Within the framework of the Spiegler-Kedem theory [ 23 this model has been applied successfully by Jagur-Grodszinsky [3]. The bilayer model has the advantage over other current models [4] of accounting more explicitly for asymmetric transport and compaction behaviour. This becomes especially apparent by comparing the membrane properties in the normal position (selective layer on the high-pressure side) with analogous properties in the reversed position (selective layer on the low-pressure side). In this paper the bilayer theory has been extended in some respects. In particular we have taken into account pressure-induced changes in the membrane structure. Furthermore we have derived and tested experimentally a flow equation for water, apart from the flow equation for salt. Finally the treatment is set up generally by assigning reflection properties to the porous support layer. A more generalized model implying a gradual change in porosity from the skin to the bottom surface
SALT RJZJECTION AND FLUX IN RO
67
of the membrane will be discussed elsewhere [5]. Non-linearity appearing in the derived flow equations at high fluxes has been clarified by discussing the concentration profiles inside the membrane. The reflection coefficient of the
skin has been obtained along different routes. Only if a correction is made
for the reversible compaction is agreement between their values found, which proves the consistency of theory and experiment.
THEORETICAL
Transport equations for ideal bilayer membranes The system to be discussed here consists of a membrane of two layers in series separating an cw-solution(the feed), in contact with layer a, from a flsolution (the effkent), in contact with layer b (Fig. 1). The composition of the external solutions is expressed in terms of the volume concentrations c, and cs of salt in water. Transport coefficients are assumed to be uniform within each layer but different in different layers. As Kedem and Katchalsky [S] have pointed out, such a composite membrane may be visualized as two homogeneous membrane elementi coupled in series via an imaginary solution which is in local equilibrium with both elements. The concentration of this virtual solution, denoted by ci, can be compared directly with the concentrations of the external solutions c, and cP. Though, as we will show elsewhere [5], the introduction of an artificial interface concentration ci is not strictly necessary, yet we shall make use of it as a convenient quantity which simplifies our expressions.
Fig. 1. Schematic representation of the system consisting of the feed solution (a) and the effluent solution (8) separated by a double-layer membrane. The arrows indicate the pcsitive directions for the fluxes.
W.C.M. HENKENS
68
AND J-AM.
SMIT
Transport of salt and water through a homogenous uncharged membrane is charac+Erizedby three transport coefficients [2] ; the reflection coefficient u, which accounts for the selectivity of the membrane 171, the solute permeability U, which indicates the diffusivity of salt, and the hydraulic permeabi%y Lp, which mainly refers to the permeation of water under the applied pressure. They are defined as *
(1)
w
A~=Q
with AP the hydrostatic pressure difference, P, - Ps, A.n the ideal osmotic pressure difference, 7ra- Xfl, between the external solutions Q and /3, Js the sait.fiux, J, the volume flow, hereafter briefly called the flux and RT having its usual meaning. For the bilayer defined above a group of 6 transport coefficients is required to describe the permeability of the membrane i.e. 0 a; ob; a,; WI,; L$; f% They appear in the J,equation and in the J, equation. For a homogenos membrane the J&quation has been derived frequently [2, 8-111. Itsderivation is usually based on a simplified NernstPlanck equation, which describes the transport process locally in the membrane. The basic local equation is linear but yields after integration across the membrane a flow equation for the solute which is non-linear and which contains o and w as concentration-independent quantities. For the two layers two separate J, equations can be written, coupled by the requirement that in a stationary state J, must remain constant across the interface. In the Spiegler-Kedem formalism the salt flux equations read J
_ s
(1 -~,)Jv(cP
-
-CiFa)
1 -F,
= (1 -Ubbk(Ct -C@b) l-F,,
(2)
in which a and b refer to the corresponding layers and F stands for F = exp [-
(3)
(1 - o)J,/w]
Eqs. (2) imply that (I--=)(I--b)C, Ci
+(I’--b)(l---a)FbCfl
(4)
= (1
-~b)(~--F,)+~a(l--b)(l--a,)
and may therefore be written as J,
=
(Cd --C&&,)(1
--Ua)(l
--b)
J
(l-~b)(l--a)+~,(l--b)(l-ua,)
o
(5)
In an analogous way the separate J, equations valid for the homogenous layers a and b can be derived [ 21 and coupled according to J,
= L",(AOP-q,A'=n) = L;(A'P-
ubAb7r)
(6)
SALT REiJ%TION
AND
FLUX
69
IN RO
with AaP equal to Pa -Pi, AbP equal to Pi - Ps and analogous expressions for A‘% and Ab x. Eqs. (6) can be condensed by introducing L,
=
L$Li
(7)
L”p -I- L::
and taking AP = AaP + AbP; An = Aan + Abn
(8)
to c& = Lp[AP-
u,An + (0, -os)AbR]
(9)
Eqs. (5) and (9) describe generally the solute and solvent flow through the composite membrane in the stationary state. In reverse osmosis they are subjected to the effluent condition [Z-4],
J, = cp J,
(10)
which states that salt is entrained out of the membrane by convective flow. Eq. (10) defines co in the reverse osmosis. If one introduces the salt rejection as r= 1 - 32 (11) CC2 the J, Eq. (5) is easily transformed with the use of Eq. (10) to r=
(1 -G)(l--06) ‘-(1--F,)(1--o,)+F,(1--o,Fb)(l--=)
(19)
a result reported earlier [3]. Remembering Eq. (3) one sees that Eq. (12) represents an analytical expression for an ideal rejection curve r us J,. The slope of this curve must correspond to (13) In the case, that the selective skin faces the feed SdUtiOn (a, >
(1
-%)ob
+ ub
(1
-(Jb)(Jb%/ab -a,
(14) I
70
W.C.M. HENKENS AND J.A.M. SMIT
which follows directly from Eq. (13) where terms on the righthand side cancel out. Maxima of this kind have been reported [11:]. These can originate from the asymmetric structure of the membrane and thus prove the existence of asymmetry. However, they can also be due to opening of pores in the skin when pressures are applied in the reverse direction. In order to separate the properties of the b-layer from those of the u-layer Eq. (12) can be rearranged with the help of Eq. (3) to
f&(1 0,
-e f = -r
exp ((1 -f-h)J~I~a~
in which f stands f
_ -
1 _(I
(15)
for
-%)tl
-Fb)ob
aa(l
-
(16)
(Jb)
If a, >> ob the factor f in Eq. (15) tends to unity and if a, < ob f may assume negative values. The former inequality is usually valid in modified cellulose acetate membranes with the selective skin faced to the feed solution [3]. Also the J, Eq. (9) may be specified to reverse osmosis. For that purpose Ab7r and An are expressed in terms of r and c, whereas the deviation of a solution from ideal behaviour is described by an osmotic coefficient $ defined by ‘IT
Q=-=-VR Tc
(17)
with v representing the number of cations and anions, when the salt is completely dissociated. Hence we have AK = vRT(&c, Abn
-+vq4
=
vRTca(@a
=
vRTc~
Gi(l -obFb)(l
= VRT(4iCi -$)oCp)
(W
-d~p(l -~1) -r)-@fi(l-r)(l-Ub) (1
-ab) (19)
where in the derivation of Lhe latter equation we have used the relationship 1 --&,Fb Ci
t1
=
-
ubFb
)(I
-
r)
C,
(l--b)
which follows from Eqs. (Z), (10) and (11). By substitution of Eq. (18) and (19) into Eq. (9) the J,, equation becomes J,
=
Lp(AP
-
in which Axe,,
An_,
)
is given by
(21)
71
SALT REJECTION AND FLUX IN RO
((70 --b) ix, Air
= a,, 1-
Axe,,
Abn
An
> ~ii(l-(JbFb)(l-r)-~8(1-r)(l-ub) (l---b)
=
II
Am
{@a
a,,gAn
-&3(l-r))
(22)
A&* has the meaning of an experimental osmotic pressure, which counteracts in reverse osmosis the applied pressure difference AP, by which it is caused. For the membrane in the normal position (cr, > ob ) the fact&r g remains smaller than unity, whereas for its reversed position (a,, < ub ) g is always larger than unity. The situation in which g equals unity is of particular interest. This happens if a, * ub the normal orientation of the bilayer membrane in reverse osmosis. h that case the ratio (1 - UbFb )/( 1 - (Jb) becomes equal to unity and consequently Ci r cs (Eq. (20)) and @i z &_ As a result Eq. (22) reduces to Are,,
= u,,A?r
(23)
At very low flow rates (Jv --f 0) r and (1 - Fb ) tend to zero and conseAre,, and AP must vanish in Eq. (21). At very high flow rates (J, * -, AP* -) r tends to o, and Fb to zero which implies that a0 becomes linear in AP (Eq. 21). It must be realized that the above limits are valid for both positions of the membrane_ quently
Concentmtion
profiles in ideal bilayer membranes
Changes in experimental quantities (r, AX=,, ) induced by varying J, will be accompanied by corresponding changes in the local solute concentration inside the membrane. It seems worthwhile to inspect the latter changes. Recently [ll] the concentration profile i.e. the function C(X) in a homogeneous membrane has been derived. It reads [12], C(X)
=
C(O)
+
cw zI
e
c(O) _
1
(ezx - 1)
in which z, now expressed in the transport quantities of this treatment, stands for (1 - u/wl)J’ and has thus the dimension of a reciprocal distance. For a bilayer membrane with an interface at x = X the analogous expressions become kxcr - 40) (25) 44 = 40) + ezaA _ 1 (e”a” - 1) for the a-layer and C(X)
=
kbCi
+
c(l) -kbCi e=b
-1
(eQJ<=-A>
-
1)
(26)
WGM,
72
HF,NKENS AND LAM.
for the b-layer, while z. = (1 - CF., /uJ) J, and zb = tl and p&ition coefficients are introduced according to lim c(x) = kaci
dX
lim c(x)
=
k&a
lim c(x) =+h
=
k&
lim c(x)
=
kacp
snllrrr
- ob [wb (1 - A)]&
(27)
xtf
XJ.0
jndicate tie direction, in which the IWits e taken, The Eqs, (27) of local equilibrium at &l i&erfaces, This seem only a sound assumption at values of Jo which are small compared to the rate tith which tie local equilibrium establishes itself. On the basis of Eqs. 125) and (26) cuncentration prof&s have been calculated, which are shown & Figs. 2 and 3, respectively referring to the n&ml position (o= > ob ) and to the reverse position (Us < a&) of the membrane during reverse osmosis. k, and hb have been simply taken equal a~ it allows a plot of (c(x)/c(O)) us x throughout the whole range 0 G x < 1. The conceMxa%ion profiles reflect how convectiun and diffusion co&ribute to the trzuxport process at different J,. Pure convectian occurs when (dc/dx) = 0 and Ja # 0, where= pure dZfuskm occurs when (d&k) # 0 and
The
amows
are based
on the concept
IO a
C(X)
c(o)
0
h
X
i
SALT
REJECTION
AND
FLUX
73
IN RO
J, = 0. From Eq. (25) and (26) the concentration gradients follow and read respectively e”G O
dCl*
= {C(l) -ktCi)
-
eZb(f_A)
At very small values of J,(z,X to dc ax=
kocf --C(O) x
dc -=
c(l) -ktci
dx
1-A
1
_
1
zt
h
4 1 and zt (1-
(29)
X) 4 1) these equations reduce
o
(30)
X
(31)
which correspond to the linear profiles I and II in Figs_ 2 and 3. Then the diffusion region is spread over the whole layer concerned. At large values of
Fig. 3. Schematic representaticn of the influence of the volume flow J, on the concentration profde inside a bilayer membrane with the porous support-layer facing the feed (ua < ot). The curves are calcukted for a, = 0.09; ob = 0.9; and In F,,:h Ft = 1O:l for differentvaluesof Jll_ The reman numerals indicate the increasing J,; viz. I-IV correspond to the respective values of In Fa - 0.01; -1; -10; -100.
W.C.M. HENKENS
74
AND J-Ah!.
SMIT
>> 1 and ezb
J,(e”a”
(32)
dc
k=Ci -C(O)
dx=
2;'
(33)
xth
This indicates that the convection occurs close behind the surface of entry of a layer, whereas near a distance of z -’ before the surface of outflow the diffusion becomes apparent (Figs. 2 and 3 profiles IV). Remarkable differences between the profiles of both figures depend on whether u, > (Tb or 0, < (Tb . In the first case the concentration profiles in the separate layers show about the same features as calculated for homogeneous membranes [ll] . As both layers retain their positive rejection in this coupling, both rejections work in the same direction. In the other case (Fig. 3) the coupling makes the salt accumulate internally, which has the effect of an apparent negative rejection of layer a contrary to a positive rejection for layer b. Here both rejections work in opposite directions. This explains that a maximal total rejection is found at finite J,. Besides the behaviour of Ci is of interest, which becomes more clear by considering the limits limit
Ci
=
limit
Ci
=
J,-0
.r,-+=
C,
=
Cp
=_A!1 -
ob
(36)
which are easily derived from Eq. (20). If a, > (76 increasing J, causes increasing salt depletion inside the membrane with ci decreasing from cq to [c,(l - 0, )/(l - ob )] _ If 0, < (Jb increasing Jv causes increasing accumulation of salt reflected by a change of Ci from c, to a maximal value [c, (l-
%)/(l
-ub)l-
Dealing with dilute solutions the local velocity of water is represented by J,/@, (x) with d?, (x) the volume fraction of water at X. On the other hand the local velocity of salt is given by J,/c(x). Hence the velocity ratio becomes
Js% (~1 = c&lJ (x) c(x) Jv
c(x)
(37)
Eq. (37) shows that the velocity ratio varies with x and more precisely that in regions with a constant +W (x), thus within each layer the velocity ratio varies inversely with the local concentration. At regions where (dc/dx) = 0 the velocity ratio is constant. At characteristic distances (Eq. (33), (34)) x=x-zz,’ and x = I -z,’ the concentration gradient dc/dx changes
SALT
REJECTION
AND FLUX IN RO
75
already owing to the jumpwise changes of the transport coefficients at the interfaces x = X and JC= 1. Consequently the velocity ratio changes in the diffusion regions implying that the salt is relatively accelerated if (dc/dx) < 0 and. retarded if (dc/dx) > 0. The latter occurs in the region around x = X in the orientation 0, < ob (see Fig. 3) giving rise to an accumulation of salt’ which even persists in the selective layer. Likewise in the orientation a, > 0s (see Fig. 2) a similar accumulation is expected around x = 0. However since the feed solution is well stirred a salt accumulation at x < 0 fails to come. Besides the velocity ratio changes at the interfaces as is indicated by Table I. For convenience another partition coefficieht has been introduced according to R = [k/a, (Jr)]. TABLE
Interface
I
Volume @w(x)
fraction
Concentration
Velocity
c(x)
4%
ratio
(XI
c(x)Jv
xto
(1 ---I
xl0
(1 -
l--(76
XtX K& X1X
r)Ki’
-
UbFb)
l--(Tb
Kb(l - a#,)
From the last column of Table I it is seen that changes at the interfaces are only due to changes in K. A realistic situation met in reverse osmosis is that O, > cb (skin turned to the feed)_ rC, < 1 (depletion of salt in the dense Skin) and & Z= 1 (neither depletion nor adsorption of salt in the porous support layer). In that case partition has the effect of increasing at x = 0, of decreasing at x = X and of letting unchanged at x = I the velocity ratio. Apart from that a variation of J, influences the velocity ratio at x = 0 and x=Xbutnotatx=Z. Transport equations for real bikzyer membranes The transport equations derived so far are strictly valid for incompressible membrane structures. However, in reality the asymmetric cellulose acetate membrane undergoes structural changes, if it is subjected to a pressure difference AP. These changes may occur as irreversible or reversible compaction of
W.C!_M.HENKENS AND J.AM. SMIT
76
the structure. If a virgin membrane is pressurized, usually its structure compacts in an irreversible way in the course of time. After a long period the membrane structure arrives at a state of weak compressibility and structural changes, measured as pressure-induced changes of Lp , have become reversible. The final compaction state depends on the magnitude and direction of the originally applied AP. The restoring property of reversible compaction is important because it enhances the establishment of a stationary state. In order to correct our expressions for reversible compaction we assume modified forms of Eqs. (15), (lS), (21) and (22) to be valid, however now with transport coefficients dependent on AP. Data in the literature [15,16] show, that compaction takes place especially in the open structure of the layer with low selectivity. Therefore we shall assume that only the open support layer in the pressurized bilayer contributes to reversible compaction. Let us indicate the pressure dependent quantities with an asterisk. Then we have for the membrane in the normal position in principle Lg, o,, a,, 02, cd: and Li* as characteristic coefficients and L$ , u,*, cd:, Li, ab , ab for the same membrane in the reversed position. Following this scheme we rewrite Eqs. (15) and (16) in case a, > a; as
-r)
%(l
f*
-r
0,
= exp KU- UJJw
(33)
I
with f
*
l_(l--a)(l--m
=
[
%x(1 - u6*)
(39) 1
In an analogous way we rewrite Eqs. (21) and (22) as J,
= L;s(AP-u,&*Ax)
(40)
with g*=
l[
Gi(l-UiFb*)(l
a, -0;: 0,
i
(l-43(&
-r)
-&(l
-r)(l
-Q&
-4)
-a:) II
(41)
Again a considerable reduction can be achieved in the Eqs. (38) to (41) if a, > (76. In that case f * and g* can be taken equal to unity. From Eq. (38) and (40) with f * = g* = 1 we may exp ect that reversible compaction influences only the flux by the way of L$. Thus if a, * ob the membrane is characterized by Go, w, and L$. In the following we shall describe the empiric relation between Lp* and AP. For the membrane in the reverse position (ua < ub ) a more complicated analysis is required, whereas an analogous reduction leading to a description with 3 transport parameters is not possible. A complete discussion of this case falls out of the scope of this paper.
SALT
REJECTION
AND
FLUX
r --------_--
IN RO -- ______
77 - _________
--,
Fig. 4. Schematic representation of the reverse osmosis eyipment. Circulation circuit: a’ stock-tank, a2 glass fiiter C 1, a three piston pump (Lewa type L3) (high pressure and circulation), a4 accumulator (Olaer, type M.P.Br ). a5 flow-meter (Fisher and Porter, 10A 3867 P.S.), a6 valve for overflow, a7 additional centrifugal pump, a* U.V. sterilizer, a9 dip cell for conductivity control of the level in stock-tank. b. Pressure-regulating sysfem: b’ back-pressure regulator (Groove, S91W), b2 gas-pressure regulator (Groove, loader 15 L.H.), b3 front-pressure regulator (Groove Mity Mite 34RR), b4 overflow, bS relief-valve, b6 on-out switch for relief valve, b7 gas manometer, bS liquid manometer, b9 capacitive pressure-sensor (Rosemo?lnt 1104 GBB l), b“ valves. c. Thermoregulating system: cl water conduit, c2 thermostat, c3 cooling and heating coils, c4 circulating pump, c5 control-thermometer, c6 thermocontrol, c7 thermoprobe. a
d. Measuring
cell.
e. Effluent.
EXPERIMENTAL A.
Characteristics of the osmotic system Salt rejection curves (r us J,) were measured on pressurized cellulose
acetate membranes (Eastman Chemical Products Inc. Kingsport U.S.A.) with NaCl (Baker Analyzed Reagent, Deventer, Holland) dissolved QI demineralized water. In addition water fluxes were determined at different “pressurecompaction” states of the membrane. These measurements were performed
W.C.M. HENKENS
78
AND J.A.M. SMIT
with an equipment schematically shown in Fig. 4. The pump in this equipment (a3 ) circulated the feed solution with a velocity of 1.25 m s-’ along the membrane surface (74 cm* ) at pressure differences up to 100 bar. With the pressure-regulating system (6) the applied AP could be maintained constant within respectively 1% and 0.2% at maximal and minimal circulation rates at minimal pressure. The fluctuations diminish considerably increasing pressure. Up to high circulation rates the thermoregulating system (c) kept the temperature in the cell at 25.0 f 0.2°C. This measuring cell (d) consisted of two stainless steel (T316) plates between which the flat membrane was clamped. In one plate a cavity was made forming the <;Ychamber, whereas in the other plate a very permeable and pressure resistant support was placed, made of sintered stainless steel (Siperm, Thijssen, Dortmund), allowing an easy drainage of the effluent on the p side. Membranes of types KP98 and KP90 were used, which differ in selectivity and permeability_ Under test conditions of AP = 41 bar; 2’ = 25°C; c, = 273 mol mm3 their specifications were given by Eastman as r Z= 0.98; Lp = (KP98) and r a 0.90; Lp % 2.7 x lo-l2 m s-’ Pa-’ 1 . 2 x lo-l2 m s-’ Pa-l (KP90). To accustom the membrane to high pressures the following procedure has been followed. After installation the membrane is exposed at the feed side water to a constant pressure of 70 or 79 bar respectively for KP98 and KP90. The membrane compacts, accompanied by a flux decline which is for the greater part irreversible. After about 1 month the hydraulic permeability becomes nearly constant. Then the membrane is exposed at the same pressure to the maximal feed concentration (O.l&f NaCl) during about 1 day resulting in a somewhat increased flux caused by the swelling of the membrane. Hereafter me;isuring the hydraulic permeability was started at pressures below the acciimatizing pressure. The membrane structure originally compacted, becomes more loose, which is reflected by an increasing Lp with decreasing AP. This phenomenon is called reversible compaction and is discussed in sub D and E below.
B. Retention
measurements
A single measuring-point required the determination of the feed concentration c,, the effluent concentration cp , the applied pressure difference A P, the flux J, and the temperature T. Then the salt rejection r follows directly from Eq. (11). First a pressure difference was settled across the membrane. After the experimental quantities have been beconie stationary, their values were obtained during a certain interval. In the experiments the feed concentration was kept constant as close as possible by tak&g the volume of the circulating salt solution rather large (40 dm3 ) and by applying a flow back of the effluent to the feed tank. Salt concentrations were determined with an accuracy of 0.3% by monitoring continuously the conductivity. Incidentally
SALT
REJECTION
AND
FLUX
79
IN RO
they were checked by a potentiometric titration of the Cl-ions. The flux was measured gravimetrically with a recording balance and corrected for small temperature variations on the basis of the known temperature-dependence of the viscosity [ 171. C. Concentration polarization Feed velocities V were chosen such that a range of turbulent mixing was obtained (Nae about 5000, see Fig. 5). At lower circulation rates boundary layers of high solute concentration c, occurred. This is clearly shown in Fig. 5 by the corresponding low values of J, (Eq. 9) and the corresponding high values of cp (Eq. 10).
10.8
10.6
10.1 I
I I
I 0
2500 I
0.5
I
NRe 5000 I 1.0
I I
I
1.5 V(mC’l
Fig. 5. Tbe dependence of the effluent-concentration cp (0) and the volume the feed velocities Valong the membrane. ~~~ is the Reynold’s number.
flux
J, (0) of
D. Reversible compaction Membranes were subjected sufficiently long to a chosen pressure, which thereafter was never exceeded. During this conditioning they lost their time-dependent compaction almost completely. However measurements of pure water fluxes as a function of AP showed that still a pressure dependence of Lp remained for pressurized membranes. Apart from a very small hysteresis also giving rise to the scattering of points in Fig. 6 a linear relationship could be established between Lp and AP [14]. This empiric relation was used for correcting the flow equations. More detailed data about the procedures A to D will be given elsewhere [X8].
W.C.M. HENKENS
80
AND J.A_M. SMIT
1.8 1
A P~lo-~lPa)
Fig. 6. Relationship between Lp and AP for pure water, showing the influence of the reversible compaction on the hydraulic permeability of pressurized membranes l KP 98 n KP90 with the active layer turned to the feed and 0 KP98 in the reversed orientation.
E. Measurements
with a
reversed membrane
Compaction could be measured along the same lines as followed in D. However a less drastic pressure treatment was chosen in order to avoid opening of pores in the selective skin or even rupture of the membrane. _ Opening of pores becomes visible as an increase of the hydraulic permeability L;1 contrary to the compaction which is reflected by a decrease of Lp*. After a sufficiently long pressure treatment of 35 bar an empirical relation between L: and AP has been determined which is shown in Fig. 6. Obviously the membrane has in this case a non-linear elastic recovery_
RESULTS
AND DISCUSSION
Salt rejecting properties of pressurized cellulose acetate membranes (in the orientation the selective shin faces the feed solution) were investigated by observing r as a function of J, (Fig. 7). These experimental findings will be compared with the theoretical behaviour of r us J= predicted by Eqs. (38) and (39). The flux J,, reflecting mainly the water transport through the membrane, was studied experimentally as a function of AP and As. These findings will be compared with the expectations suggested by Eqs. (40) and (41). On the basis of Eq. (38) and under the assumption that f * = 1 if a, > a(, a
81
SALT REJECTION AND FLUX IN RO l.O-
r
0.8-
0.6-
4
0
3
6
9
I 12
J,(prnS-‘1 Fig. 7. Salt rejection curves for KP98 and KP90 at various active layer in contact with the feed solution. OaKp90 130mol(NaCl)m~3 0 K.P98 130 mol(NaCl)m-3
feed
concentrations
l KP90
with
the
61.7mol(NaCl)m-3.
curve-fitting procedure was applied to the rejection curves. Its result is repre-+nted by the solid lines in Fig. 7 which cover the measuring points well. On the other hand the goodness of the fit is also seen in Fig. 8 where the plot of In (oa (1 - r)/(uc - r)} us J, is linear (Eq. (38)) and passes through the origin (f* = 1). Hence we may conclude that in the range of fluxes investigated the salt rejection curves can be described by two parameters a, and (1 - CJ,)/a,. Their values which do not depend on the concentration are compiled in the last two columns of Table II. Within our hypothesis these parameters are not influenced by reversible compaction. In other terms the way in which r depends on the flux is not disturbed by reversible compaction of the membrane contrary to the magnitude of the flux itself. In order to inspect the latter phenomenon more closely we recall that the hydraulic permeability for pure water decreases linearly with increasing AP (Experimental D). This can be represented by
Lp* = L$(l -
KAP)
with K a compaction modulus to be evaluated from the pure water data (Fig. 6). Dealing with salt solutions we assume that an identical K describes
W.C.M.
82
,n
HENKENS
AND
J-AM_
SMIT
dab-r1
I Uo-r
I
0.25 -
o.oI/ 0
3
6
12
9
J,(IJIYI S-‘I Fig. 8. Linearization of the salt rejection curve. KF’98 membrane in contact with a 130 mol(NaCl)m-3
feed solution.
02
03 A=/
Fig. 9. The
influence of reversible compaction
AP
correction on the hydraulic permeability
Wq. (43)). uncorrected for compaction (K = 0) 00 corrected for compaction (K # 0) 00 membrane KP98 10 membrane KPSO feed concentration in both cases: 130 mol(NaCl)mW3 _ l n
I
01
83
SALTRE!JECTIONANDFLUXINRO
which suggests a linear relationship to be verified experimentally. In fact this linearityhas been found as Fig. 9 shows. From the slopes and intercepts of these linesvalues of L$ and again values of a, have been calculated. Their values are shown in Table II. Thus the reflection coefficients o, found in this way agree reasonably with those originating from the salt rejection curves. Remembering Eq. (23) we may state that an experimental proof has been given of the relationship. limr ++-
=- A&X, AS
in which both members are equal to 0,. This result justifies the assumptions made and more generally it reflects that locally in the a-layer the Onsager Reciprocal Relation is satisfied between the transport coefficients. The necessity of the compaction correction follows from the poor results obtained when J,/AP is simply plotted uersus A?r/AP (Fig. 9). No linearity is found and moreover only erroneous values of a, are calculated (Table II, values between parentheses). We may thus conclude that the flux J,, is related to Aa and AP by the coefficients Lop, o, and K. Apparently the influence of the b-layer is reflected in the coefficient L$ which is composed from the partial hydraulic permeabilities according to Eq. (7) and from the compaction modulus ic which probably only refers to Ly _ This picture is consistent with the bilayer model which we have assumed. This basic assumption is supported by two experimental facts. First maxima in the salt rejection are measured when the dense skin is turned to the eMuent side (Fig. lo)_ These salt rejection curves were always measured with decreasing AP in order to be certain that these maxim a do not originate from opening of the pores as discussed around Eq. (14). Furthermore the membrane has in the latter orientation another compaction behaviour than in the normal orientation (Fig. 6). The values of Lj? referring to pure water are somewhat higher than the values referring to the salt solutions (Table II, column 3). This effect is probably due to an influence of viscosity. TABLEII Membrane
KP98 KF'98 KP90 KP90 KP90
=a -3 mol m
0 130 0 130 61.7
L$ x 1012 m3
pJ-1
2.079 2.026 1.738 1.711 1.716
s-’
% &.
40)
O-97(0.81) 0.90(0.61) O-89(0.22)
0,
(Eq.3'3)
l-u= ms-'
O.984
18.0
0.94O 0.944
ii.9 20.6
x106
W.C.M. HENKEXJS AND J.A.M. SMIT
r
i--0
1
I
2
L J,(pm
S-‘1
Fig. 10. Salt rejection curve for KP98 with the porous sublayer in contact with the feed solution of 130 mol(NaCl)m-3.
The effect of pressurizing the membrane emerges clearly from the strong decrease in the measured Lj! values. Even Lp of KF’90 has become smaller than Lp of KP98 after the pressure treatment which is certainly a result of the total compaction. Irreversible compaction depends on the magnitude and the duration of the original maximal hydrostatic pressure difference. It has an effect on both layers. Reversible compaction depends on the established hydrostatic pressure difference but not on the time. For a membrane placed with the skin to the high pressure side only the support layer is compressed. For both types of compaction changes in Lp are more pronounced than
changes in
r.
Though in the range of fluxes investigated by us, the layer of low selectivity p!ays a minor part compared to the highly selective skin, another situation is met at vanishing fluxes. In fact it has been shown [3,19,20] that reflection coefficients obtained at J, = 0 from osmotic measurements turn out to be smaller than the maximal retentions in reverse osmosis. The explanation of these findings must be that at J, = 0 the reflection coefficient is an overall coefficient, to which also the relatively smaIl ob contributes whereas at J,, + 00 the large a, of the selective skin predominates.
85
SALT REZJEZTIONAND FLUX IN RO ACKNOWLEDGEMENTS
The authorsthank Dr. C.P. Miming (Los Angeles, California)for technical advice, Prof. Dr. A.J. Staverman, Drs. Eijsermans and Drs. F.A.H. Pee&s (Leiden,
The Netherlands)
for helpful
discussions.
REFERENCES 1. S. Loeb and S. Sonrirajan, Sea water deminerahzation by an osmotic membrane, Advan. Chem. Ser., 38 (1963) 117. 2. K. S. Spiegier and O_ Kedem, Thermodynamics of hyperfiitration (reverse osmosis): Criteria for efficient membranes, Dessliuation, 1(1966) 311. 3. J. Jagur-Grodzinski and 0. Kedem, Transport coefficients and salt rejection in uncharged hyperfdtration membranes. Desalination. 1 (1966) 327. 4. W. Punch, Determination of transport parameters of synthetic membranes by hyperfutrstion experiments. Part 1, Ber. Buusenges. Phys. Chem., 8 (1977) 269. 5. C.W. Verslujis, forthcoming Thesis. Leiden, The Netherlands. 6. 0. Kedem and A. Katchalsky, Permeability of composite membranes. Part 1,2 and 3, Trans. Faraday Sot., 59 (1963) 1918,1931,1941. 7. A.J. Staverman, The theory of measurement of osmotic pressure, Rec. Trav. Chim., 70 (1951) 344. 8_ C.S. PatIak. D.A. Goldstein and J-F. Hoffman, The flow of solute and solvent across a two-membrane system, J. Theor. Biol., 5 (1963) 426. 9. R. SchiOgi, Non linear transport behaviour in very thin membranes, Q. Rev. Biophys., 2 (1969) 305. 10. R.J. Sha’afi, G.J. Rich, D.C. Mikulecky and A.K. Solomon, Determination of urea permeability in red cells by minimum method, J. Gen. Physiol.. 55 (1970) 427.
11. A.J. Staverman. Function and structure of membranes, Symposium on Biological and 12. 13.
14.
15_ 16. 17. 18. 19. 20. I
Artificial Membranes, Rome (1975). Pontificae Academiae Scientisrium Scripts Varia, Citta de1 Vaticano, (1976). J.A.M. Smit, J.C. Eijsermsns and A.J. Staverman, Friction and partition in membranes, J. Phys. Chem., 79 (1975) 2168. R. Griipl and W. Pusch, Asymmetric behavionr of cellulose acetate membranes in hyperfdtration experiments as a result of concentration polarization. Desalination. 8 (1970) 277. J. Bert, Membrane compaction; A theoretical and experimental explanation, J. Polym. Sci.. Part B. 7 (1969) 685. G. Jousson. The effect of pressure on the viscoelastic compaction of asymmetric cellulose acetate membranes, Proc. Seventh Int. Ccng. Rheol.. (1976) 298. L. Bssyers and S.L. Rosen, Hydrodynamic resistance and flux decline in asymmetric ceihdcse acetate reverse osmosis membranes, J. Appl. Polym. Sci., 16 (1972) 663. J. Kopecek and S. Sourirajan, Structure of porous cellulose acetate membranes and a method for improving their performan ce in reverse osmosis, J. Appl. Polym. Sci., 13 (1969) 637. W.C.M. Henkena, Thesis, Chapt. III, Leiden, 1978. W. Pusch and R. Riley, Relation between salt rejection r and reflection coefficient o of asymmetric cellulose acetate membranes. DesaIination. 14 (1974) 389. W.C.M. Henkens, J.C. Eijsermsns and J.A.M. Smit, Osmotic properties of a modified cellulose acetate membrane: the refiection coefficient and its dependence on the volume Sow history, J. Membr. Sci., in press.
SALT REJECTION AND FLUX IN REVERSE COMPACI’IBLE MEMBRANES
W.C.M. HENKENS
AND J-AM.
OSMOSIS WITH
SMIT
GorZaeus taboratonks of the State University L&den, Leiden (The Netherlands) (Received June 19.1978;
in revised form January 31,1979)
SUMMARY
The theory of reverse osmosis in bilayer membranes is extended. At high fluxes the selectivity of the layer on the high pressure side dominates the salt rejection also in case this layer has the lowest selectivity. Theoretical concentration profiles are given for both orientations. Experimentally, irreversible compaction of the membrane is controlled by a pressure treatment. For the remaining reversible compaction the transport equations can be corrected. It turns out that reversible compaction affects the flux but not the salt rejection. Correspondingly structural changes occur in the layer of low selectivity.
SYMBOLS
-
solute concentration, mol mS3. solute concentration at distance x in the membrane per volume of membrane, mol me3. - flow parameter defined by Eq. (3). - factor defined by Eq. (16). - factor defined by Eq. (22). - solute flux mol m-* s-r . - total volunk flux, m s-i. - partition coefficient defined by k/Q, (x)_ - partition coefficient defined by Eq. (27). - hydraulic permeability, N-i m3 s-i. - hydraulic permeability at AP = 0, N-i m3 s-i. - thickness of the membrane, m. - Reynold’s number. - hydrostatic pressure, Pa. - universal gas constant, N m mol-’ K-i . - salt rejection defined by Eq. (11). -
W.C.M. HENKENS
66
T
-
V
- feed velocity, m s-’ .
AND J.A.M. SMIT
absolute temperature, K.
space coordinatz perpendicular to the membrane surface, ranging from 0 to I, m. z - reciprocal characteristic diffusion length defined by Eq. (24), m-l. - operator standing for the difference in the quantity Q between (YA and @-phases, Aq = qa - qp. - compaction modulus. ;: - thickness of the layer a, m. V stoichiometric dissociation factor of the salt. X - theoretical osmotic pressure, Pa. - experimental osmotic pressure, Pa. n=u (J - reflection coefficient. w - solute permeability, m s-l. ap, (3~) - volume fraction of water at distance x in the membrane. - osmotic coefficient. @ x
-
Indices - layer u, b of the composite membrane. 0, b - the interface between layer a and b in the membrane. i -
the feed. the effluent. pressure dependent quantities.
INTRODUCTION
Anisotrope cellulose acetate membranes of the type discovered by Loeb and Sourirajan [l] are usually considered to consist of a dense skin in the top of a more porous film. Schematically they may be viewed as simple bilayer membranes of two homogeneous elements in series. Within the framework of the Spiegler-Kedem theory [ 23 this model has been applied successfully by Jagur-Grodszinsky [3]. The bilayer model has the advantage over other current models [4] of accounting more explicitly for asymmetric transport and compaction behaviour. This becomes especially apparent by comparing the membrane properties in the normal position (selective layer on the high-pressure side) with analogous properties in the reversed position (selective layer on the low-pressure side). In this paper the bilayer theory has been extended in some respects. In particular we have taken into account pressure-induced changes in the membrane structure. Furthermore we have derived and tested experimentally a flow equation for water, apart from the flow equation for salt. Finally the treatment is set up generally by assigning reflection properties to the porous support layer. A more generalized model implying a gradual change in porosity from the skin to the bottom surface
SALT RJZJECTION AND FLUX IN RO
67
of the membrane will be discussed elsewhere [5]. Non-linearity appearing in the derived flow equations at high fluxes has been clarified by discussing the concentration profiles inside the membrane. The reflection coefficient of the
skin has been obtained along different routes. Only if a correction is made
for the reversible compaction is agreement between their values found, which proves the consistency of theory and experiment.
THEORETICAL
Transport equations for ideal bilayer membranes The system to be discussed here consists of a membrane of two layers in series separating an cw-solution(the feed), in contact with layer a, from a flsolution (the effkent), in contact with layer b (Fig. 1). The composition of the external solutions is expressed in terms of the volume concentrations c, and cs of salt in water. Transport coefficients are assumed to be uniform within each layer but different in different layers. As Kedem and Katchalsky [S] have pointed out, such a composite membrane may be visualized as two homogeneous membrane elementi coupled in series via an imaginary solution which is in local equilibrium with both elements. The concentration of this virtual solution, denoted by ci, can be compared directly with the concentrations of the external solutions c, and cP. Though, as we will show elsewhere [5], the introduction of an artificial interface concentration ci is not strictly necessary, yet we shall make use of it as a convenient quantity which simplifies our expressions.
Fig. 1. Schematic representation of the system consisting of the feed solution (a) and the effluent solution (8) separated by a double-layer membrane. The arrows indicate the pcsitive directions for the fluxes.
W.C.M. HENKENS
68
AND J-AM.
SMIT
Transport of salt and water through a homogenous uncharged membrane is charac+Erizedby three transport coefficients [2] ; the reflection coefficient u, which accounts for the selectivity of the membrane 171, the solute permeability U, which indicates the diffusivity of salt, and the hydraulic permeabi%y Lp, which mainly refers to the permeation of water under the applied pressure. They are defined as *
(1)
w
A~=Q
with AP the hydrostatic pressure difference, P, - Ps, A.n the ideal osmotic pressure difference, 7ra- Xfl, between the external solutions Q and /3, Js the sait.fiux, J, the volume flow, hereafter briefly called the flux and RT having its usual meaning. For the bilayer defined above a group of 6 transport coefficients is required to describe the permeability of the membrane i.e. 0 a; ob; a,; WI,; L$; f% They appear in the J,equation and in the J, equation. For a homogenos membrane the J&quation has been derived frequently [2, 8-111. Itsderivation is usually based on a simplified NernstPlanck equation, which describes the transport process locally in the membrane. The basic local equation is linear but yields after integration across the membrane a flow equation for the solute which is non-linear and which contains o and w as concentration-independent quantities. For the two layers two separate J, equations can be written, coupled by the requirement that in a stationary state J, must remain constant across the interface. In the Spiegler-Kedem formalism the salt flux equations read J
_ s
(1 -~,)Jv(cP
-
-CiFa)
1 -F,
= (1 -Ubbk(Ct -C@b) l-F,,
(2)
in which a and b refer to the corresponding layers and F stands for F = exp [-
(3)
(1 - o)J,/w]
Eqs. (2) imply that (I--=)(I--b)C, Ci
+(I’--b)(l---a)FbCfl
(4)
= (1
-~b)(~--F,)+~a(l--b)(l--a,)
and may therefore be written as J,
=
(Cd --C&&,)(1
--Ua)(l
--b)
J
(l-~b)(l--a)+~,(l--b)(l-ua,)
o
(5)
In an analogous way the separate J, equations valid for the homogenous layers a and b can be derived [ 21 and coupled according to J,
= L",(AOP-q,A'=n) = L;(A'P-
ubAb7r)
(6)
SALT REiJ%TION
AND
FLUX
69
IN RO
with AaP equal to Pa -Pi, AbP equal to Pi - Ps and analogous expressions for A‘% and Ab x. Eqs. (6) can be condensed by introducing L,
=
L$Li
(7)
L”p -I- L::
and taking AP = AaP + AbP; An = Aan + Abn
(8)
to c& = Lp[AP-
u,An + (0, -os)AbR]
(9)
Eqs. (5) and (9) describe generally the solute and solvent flow through the composite membrane in the stationary state. In reverse osmosis they are subjected to the effluent condition [Z-4],
J, = cp J,
(10)
which states that salt is entrained out of the membrane by convective flow. Eq. (10) defines co in the reverse osmosis. If one introduces the salt rejection as r= 1 - 32 (11) CC2 the J, Eq. (5) is easily transformed with the use of Eq. (10) to r=
(1 -G)(l--06) ‘-(1--F,)(1--o,)+F,(1--o,Fb)(l--=)
(19)
a result reported earlier [3]. Remembering Eq. (3) one sees that Eq. (12) represents an analytical expression for an ideal rejection curve r us J,. The slope of this curve must correspond to (13) In the case, that the selective skin faces the feed SdUtiOn (a, >
(1
-%)ob
+ ub
(1
-(Jb)(Jb%/ab -a,
(14) I
70
W.C.M. HENKENS AND J.A.M. SMIT
which follows directly from Eq. (13) where terms on the righthand side cancel out. Maxima of this kind have been reported [11:]. These can originate from the asymmetric structure of the membrane and thus prove the existence of asymmetry. However, they can also be due to opening of pores in the skin when pressures are applied in the reverse direction. In order to separate the properties of the b-layer from those of the u-layer Eq. (12) can be rearranged with the help of Eq. (3) to
f&(1 0,
-e f = -r
exp ((1 -f-h)J~I~a~
in which f stands f
_ -
1 _(I
(15)
for
-%)tl
-Fb)ob
aa(l
-
(16)
(Jb)
If a, >> ob the factor f in Eq. (15) tends to unity and if a, < ob f may assume negative values. The former inequality is usually valid in modified cellulose acetate membranes with the selective skin faced to the feed solution [3]. Also the J, Eq. (9) may be specified to reverse osmosis. For that purpose Ab7r and An are expressed in terms of r and c, whereas the deviation of a solution from ideal behaviour is described by an osmotic coefficient $ defined by ‘IT
Q=-=-VR Tc
(17)
with v representing the number of cations and anions, when the salt is completely dissociated. Hence we have AK = vRT(&c, Abn
-+vq4
=
vRTca(@a
=
vRTc~
Gi(l -obFb)(l
= VRT(4iCi -$)oCp)
(W
-d~p(l -~1) -r)-@fi(l-r)(l-Ub) (1
-ab) (19)
where in the derivation of Lhe latter equation we have used the relationship 1 --&,Fb Ci
t1
=
-
ubFb
)(I
-
r)
C,
(l--b)
which follows from Eqs. (Z), (10) and (11). By substitution of Eq. (18) and (19) into Eq. (9) the J,, equation becomes J,
=
Lp(AP
-
in which Axe,,
An_,
)
is given by
(21)
71
SALT REJECTION AND FLUX IN RO
((70 --b) ix, Air
= a,, 1-
Axe,,
Abn
An
> ~ii(l-(JbFb)(l-r)-~8(1-r)(l-ub) (l---b)
=
II
Am
{@a
a,,gAn
-&3(l-r))
(22)
A&* has the meaning of an experimental osmotic pressure, which counteracts in reverse osmosis the applied pressure difference AP, by which it is caused. For the membrane in the normal position (cr, > ob ) the fact&r g remains smaller than unity, whereas for its reversed position (a,, < ub ) g is always larger than unity. The situation in which g equals unity is of particular interest. This happens if a, * ub the normal orientation of the bilayer membrane in reverse osmosis. h that case the ratio (1 - UbFb )/( 1 - (Jb) becomes equal to unity and consequently Ci r cs (Eq. (20)) and @i z &_ As a result Eq. (22) reduces to Are,,
= u,,A?r
(23)
At very low flow rates (Jv --f 0) r and (1 - Fb ) tend to zero and conseAre,, and AP must vanish in Eq. (21). At very high flow rates (J, * -, AP* -) r tends to o, and Fb to zero which implies that a0 becomes linear in AP (Eq. 21). It must be realized that the above limits are valid for both positions of the membrane_ quently
Concentmtion
profiles in ideal bilayer membranes
Changes in experimental quantities (r, AX=,, ) induced by varying J, will be accompanied by corresponding changes in the local solute concentration inside the membrane. It seems worthwhile to inspect the latter changes. Recently [ll] the concentration profile i.e. the function C(X) in a homogeneous membrane has been derived. It reads [12], C(X)
=
C(O)
+
cw zI
e
c(O) _
1
(ezx - 1)
in which z, now expressed in the transport quantities of this treatment, stands for (1 - u/wl)J’ and has thus the dimension of a reciprocal distance. For a bilayer membrane with an interface at x = X the analogous expressions become kxcr - 40) (25) 44 = 40) + ezaA _ 1 (e”a” - 1) for the a-layer and C(X)
=
kbCi
+
c(l) -kbCi e=b
-1
(eQJ<=-A>
-
1)
(26)
WGM,
72
HF,NKENS AND LAM.
for the b-layer, while z. = (1 - CF., /uJ) J, and zb = tl and p&ition coefficients are introduced according to lim c(x) = kaci
dX
lim c(x)
=
k&a
lim c(x) =+h
=
k&
lim c(x)
=
kacp
snllrrr
- ob [wb (1 - A)]&
(27)
xtf
XJ.0
jndicate tie direction, in which the IWits e taken, The Eqs, (27) of local equilibrium at &l i&erfaces, This seem only a sound assumption at values of Jo which are small compared to the rate tith which tie local equilibrium establishes itself. On the basis of Eqs. 125) and (26) cuncentration prof&s have been calculated, which are shown & Figs. 2 and 3, respectively referring to the n&ml position (o= > ob ) and to the reverse position (Us < a&) of the membrane during reverse osmosis. k, and hb have been simply taken equal a~ it allows a plot of (c(x)/c(O)) us x throughout the whole range 0 G x < 1. The conceMxa%ion profiles reflect how convectiun and diffusion co&ribute to the trzuxport process at different J,. Pure convectian occurs when (dc/dx) = 0 and Ja # 0, where= pure dZfuskm occurs when (d&k) # 0 and
The
amows
are based
on the concept
IO a
C(X)
c(o)
0
h
X
i
SALT
REJECTION
AND
FLUX
73
IN RO
J, = 0. From Eq. (25) and (26) the concentration gradients follow and read respectively e”G O
dCl*
= {C(l) -ktCi)
-
eZb(f_A)
At very small values of J,(z,X to dc ax=
kocf --C(O) x
dc -=
c(l) -ktci
dx
1-A
1
_
1
zt
h
4 1 and zt (1-
(29)
X) 4 1) these equations reduce
o
(30)
X
(31)
which correspond to the linear profiles I and II in Figs_ 2 and 3. Then the diffusion region is spread over the whole layer concerned. At large values of
Fig. 3. Schematic representaticn of the influence of the volume flow J, on the concentration profde inside a bilayer membrane with the porous support-layer facing the feed (ua < ot). The curves are calcukted for a, = 0.09; ob = 0.9; and In F,,:h Ft = 1O:l for differentvaluesof Jll_ The reman numerals indicate the increasing J,; viz. I-IV correspond to the respective values of In Fa - 0.01; -1; -10; -100.
W.C.M. HENKENS
74
AND J-Ah!.
SMIT
>> 1 and ezb
J,(e”a”
(32)
dc
k=Ci -C(O)
dx=
2;'
(33)
xth
This indicates that the convection occurs close behind the surface of entry of a layer, whereas near a distance of z -’ before the surface of outflow the diffusion becomes apparent (Figs. 2 and 3 profiles IV). Remarkable differences between the profiles of both figures depend on whether u, > (Tb or 0, < (Tb . In the first case the concentration profiles in the separate layers show about the same features as calculated for homogeneous membranes [ll] . As both layers retain their positive rejection in this coupling, both rejections work in the same direction. In the other case (Fig. 3) the coupling makes the salt accumulate internally, which has the effect of an apparent negative rejection of layer a contrary to a positive rejection for layer b. Here both rejections work in opposite directions. This explains that a maximal total rejection is found at finite J,. Besides the behaviour of Ci is of interest, which becomes more clear by considering the limits limit
Ci
=
limit
Ci
=
J,-0
.r,-+=
C,
=
Cp
=_A!1 -
ob
(36)
which are easily derived from Eq. (20). If a, > (76 increasing J, causes increasing salt depletion inside the membrane with ci decreasing from cq to [c,(l - 0, )/(l - ob )] _ If 0, < (Jb increasing Jv causes increasing accumulation of salt reflected by a change of Ci from c, to a maximal value [c, (l-
%)/(l
-ub)l-
Dealing with dilute solutions the local velocity of water is represented by J,/@, (x) with d?, (x) the volume fraction of water at X. On the other hand the local velocity of salt is given by J,/c(x). Hence the velocity ratio becomes
Js% (~1 = c&lJ (x) c(x) Jv
c(x)
(37)
Eq. (37) shows that the velocity ratio varies with x and more precisely that in regions with a constant +W (x), thus within each layer the velocity ratio varies inversely with the local concentration. At regions where (dc/dx) = 0 the velocity ratio is constant. At characteristic distances (Eq. (33), (34)) x=x-zz,’ and x = I -z,’ the concentration gradient dc/dx changes
SALT
REJECTION
AND FLUX IN RO
75
already owing to the jumpwise changes of the transport coefficients at the interfaces x = X and JC= 1. Consequently the velocity ratio changes in the diffusion regions implying that the salt is relatively accelerated if (dc/dx) < 0 and. retarded if (dc/dx) > 0. The latter occurs in the region around x = X in the orientation 0, < ob (see Fig. 3) giving rise to an accumulation of salt’ which even persists in the selective layer. Likewise in the orientation a, > 0s (see Fig. 2) a similar accumulation is expected around x = 0. However since the feed solution is well stirred a salt accumulation at x < 0 fails to come. Besides the velocity ratio changes at the interfaces as is indicated by Table I. For convenience another partition coefficieht has been introduced according to R = [k/a, (Jr)]. TABLE
Interface
I
Volume @w(x)
fraction
Concentration
Velocity
c(x)
4%
ratio
(XI
c(x)Jv
xto
(1 ---I
xl0
(1 -
l--(76
XtX K& X1X
r)Ki’
-
UbFb)
l--(Tb
Kb(l - a#,)
From the last column of Table I it is seen that changes at the interfaces are only due to changes in K. A realistic situation met in reverse osmosis is that O, > cb (skin turned to the feed)_ rC, < 1 (depletion of salt in the dense Skin) and & Z= 1 (neither depletion nor adsorption of salt in the porous support layer). In that case partition has the effect of increasing at x = 0, of decreasing at x = X and of letting unchanged at x = I the velocity ratio. Apart from that a variation of J, influences the velocity ratio at x = 0 and x=Xbutnotatx=Z. Transport equations for real bikzyer membranes The transport equations derived so far are strictly valid for incompressible membrane structures. However, in reality the asymmetric cellulose acetate membrane undergoes structural changes, if it is subjected to a pressure difference AP. These changes may occur as irreversible or reversible compaction of
W.C!_M.HENKENS AND J.AM. SMIT
76
the structure. If a virgin membrane is pressurized, usually its structure compacts in an irreversible way in the course of time. After a long period the membrane structure arrives at a state of weak compressibility and structural changes, measured as pressure-induced changes of Lp , have become reversible. The final compaction state depends on the magnitude and direction of the originally applied AP. The restoring property of reversible compaction is important because it enhances the establishment of a stationary state. In order to correct our expressions for reversible compaction we assume modified forms of Eqs. (15), (lS), (21) and (22) to be valid, however now with transport coefficients dependent on AP. Data in the literature [15,16] show, that compaction takes place especially in the open structure of the layer with low selectivity. Therefore we shall assume that only the open support layer in the pressurized bilayer contributes to reversible compaction. Let us indicate the pressure dependent quantities with an asterisk. Then we have for the membrane in the normal position in principle Lg, o,, a,, 02, cd: and Li* as characteristic coefficients and L$ , u,*, cd:, Li, ab , ab for the same membrane in the reversed position. Following this scheme we rewrite Eqs. (15) and (16) in case a, > a; as
-r)
%(l
f*
-r
0,
= exp KU- UJJw
(33)
I
with f
*
l_(l--a)(l--m
=
[
%x(1 - u6*)
(39) 1
In an analogous way we rewrite Eqs. (21) and (22) as J,
= L;s(AP-u,&*Ax)
(40)
with g*=
l[
Gi(l-UiFb*)(l
a, -0;: 0,
i
(l-43(&
-r)
-&(l
-r)(l
-Q&
-4)
-a:) II
(41)
Again a considerable reduction can be achieved in the Eqs. (38) to (41) if a, > (76. In that case f * and g* can be taken equal to unity. From Eq. (38) and (40) with f * = g* = 1 we may exp ect that reversible compaction influences only the flux by the way of L$. Thus if a, * ob the membrane is characterized by Go, w, and L$. In the following we shall describe the empiric relation between Lp* and AP. For the membrane in the reverse position (ua < ub ) a more complicated analysis is required, whereas an analogous reduction leading to a description with 3 transport parameters is not possible. A complete discussion of this case falls out of the scope of this paper.
SALT
REJECTION
AND
FLUX
r --------_--
IN RO -- ______
77 - _________
--,
Fig. 4. Schematic representation of the reverse osmosis eyipment. Circulation circuit: a’ stock-tank, a2 glass fiiter C 1, a three piston pump (Lewa type L3) (high pressure and circulation), a4 accumulator (Olaer, type M.P.Br ). a5 flow-meter (Fisher and Porter, 10A 3867 P.S.), a6 valve for overflow, a7 additional centrifugal pump, a* U.V. sterilizer, a9 dip cell for conductivity control of the level in stock-tank. b. Pressure-regulating sysfem: b’ back-pressure regulator (Groove, S91W), b2 gas-pressure regulator (Groove, loader 15 L.H.), b3 front-pressure regulator (Groove Mity Mite 34RR), b4 overflow, bS relief-valve, b6 on-out switch for relief valve, b7 gas manometer, bS liquid manometer, b9 capacitive pressure-sensor (Rosemo?lnt 1104 GBB l), b“ valves. c. Thermoregulating system: cl water conduit, c2 thermostat, c3 cooling and heating coils, c4 circulating pump, c5 control-thermometer, c6 thermocontrol, c7 thermoprobe. a
d. Measuring
cell.
e. Effluent.
EXPERIMENTAL A.
Characteristics of the osmotic system Salt rejection curves (r us J,) were measured on pressurized cellulose
acetate membranes (Eastman Chemical Products Inc. Kingsport U.S.A.) with NaCl (Baker Analyzed Reagent, Deventer, Holland) dissolved QI demineralized water. In addition water fluxes were determined at different “pressurecompaction” states of the membrane. These measurements were performed
W.C.M. HENKENS
78
AND J.A.M. SMIT
with an equipment schematically shown in Fig. 4. The pump in this equipment (a3 ) circulated the feed solution with a velocity of 1.25 m s-’ along the membrane surface (74 cm* ) at pressure differences up to 100 bar. With the pressure-regulating system (6) the applied AP could be maintained constant within respectively 1% and 0.2% at maximal and minimal circulation rates at minimal pressure. The fluctuations diminish considerably increasing pressure. Up to high circulation rates the thermoregulating system (c) kept the temperature in the cell at 25.0 f 0.2°C. This measuring cell (d) consisted of two stainless steel (T316) plates between which the flat membrane was clamped. In one plate a cavity was made forming the <;Ychamber, whereas in the other plate a very permeable and pressure resistant support was placed, made of sintered stainless steel (Siperm, Thijssen, Dortmund), allowing an easy drainage of the effluent on the p side. Membranes of types KP98 and KP90 were used, which differ in selectivity and permeability_ Under test conditions of AP = 41 bar; 2’ = 25°C; c, = 273 mol mm3 their specifications were given by Eastman as r Z= 0.98; Lp = (KP98) and r a 0.90; Lp % 2.7 x lo-l2 m s-’ Pa-’ 1 . 2 x lo-l2 m s-’ Pa-l (KP90). To accustom the membrane to high pressures the following procedure has been followed. After installation the membrane is exposed at the feed side water to a constant pressure of 70 or 79 bar respectively for KP98 and KP90. The membrane compacts, accompanied by a flux decline which is for the greater part irreversible. After about 1 month the hydraulic permeability becomes nearly constant. Then the membrane is exposed at the same pressure to the maximal feed concentration (O.l&f NaCl) during about 1 day resulting in a somewhat increased flux caused by the swelling of the membrane. Hereafter me;isuring the hydraulic permeability was started at pressures below the acciimatizing pressure. The membrane structure originally compacted, becomes more loose, which is reflected by an increasing Lp with decreasing AP. This phenomenon is called reversible compaction and is discussed in sub D and E below.
B. Retention
measurements
A single measuring-point required the determination of the feed concentration c,, the effluent concentration cp , the applied pressure difference A P, the flux J, and the temperature T. Then the salt rejection r follows directly from Eq. (11). First a pressure difference was settled across the membrane. After the experimental quantities have been beconie stationary, their values were obtained during a certain interval. In the experiments the feed concentration was kept constant as close as possible by tak&g the volume of the circulating salt solution rather large (40 dm3 ) and by applying a flow back of the effluent to the feed tank. Salt concentrations were determined with an accuracy of 0.3% by monitoring continuously the conductivity. Incidentally
SALT
REJECTION
AND
FLUX
79
IN RO
they were checked by a potentiometric titration of the Cl-ions. The flux was measured gravimetrically with a recording balance and corrected for small temperature variations on the basis of the known temperature-dependence of the viscosity [ 171. C. Concentration polarization Feed velocities V were chosen such that a range of turbulent mixing was obtained (Nae about 5000, see Fig. 5). At lower circulation rates boundary layers of high solute concentration c, occurred. This is clearly shown in Fig. 5 by the corresponding low values of J, (Eq. 9) and the corresponding high values of cp (Eq. 10).
10.8
10.6
10.1 I
I I
I 0
2500 I
0.5
I
NRe 5000 I 1.0
I I
I
1.5 V(mC’l
Fig. 5. Tbe dependence of the effluent-concentration cp (0) and the volume the feed velocities Valong the membrane. ~~~ is the Reynold’s number.
flux
J, (0) of
D. Reversible compaction Membranes were subjected sufficiently long to a chosen pressure, which thereafter was never exceeded. During this conditioning they lost their time-dependent compaction almost completely. However measurements of pure water fluxes as a function of AP showed that still a pressure dependence of Lp remained for pressurized membranes. Apart from a very small hysteresis also giving rise to the scattering of points in Fig. 6 a linear relationship could be established between Lp and AP [14]. This empiric relation was used for correcting the flow equations. More detailed data about the procedures A to D will be given elsewhere [X8].
W.C.M. HENKENS
80
AND J.A_M. SMIT
1.8 1
A P~lo-~lPa)
Fig. 6. Relationship between Lp and AP for pure water, showing the influence of the reversible compaction on the hydraulic permeability of pressurized membranes l KP 98 n KP90 with the active layer turned to the feed and 0 KP98 in the reversed orientation.
E. Measurements
with a
reversed membrane
Compaction could be measured along the same lines as followed in D. However a less drastic pressure treatment was chosen in order to avoid opening of pores in the selective skin or even rupture of the membrane. _ Opening of pores becomes visible as an increase of the hydraulic permeability L;1 contrary to the compaction which is reflected by a decrease of Lp*. After a sufficiently long pressure treatment of 35 bar an empirical relation between L: and AP has been determined which is shown in Fig. 6. Obviously the membrane has in this case a non-linear elastic recovery_
RESULTS
AND DISCUSSION
Salt rejecting properties of pressurized cellulose acetate membranes (in the orientation the selective shin faces the feed solution) were investigated by observing r as a function of J, (Fig. 7). These experimental findings will be compared with the theoretical behaviour of r us J= predicted by Eqs. (38) and (39). The flux J,, reflecting mainly the water transport through the membrane, was studied experimentally as a function of AP and As. These findings will be compared with the expectations suggested by Eqs. (40) and (41). On the basis of Eq. (38) and under the assumption that f * = 1 if a, > a(, a
81
SALT REJECTION AND FLUX IN RO l.O-
r
0.8-
0.6-
4
0
3
6
9
I 12
J,(prnS-‘1 Fig. 7. Salt rejection curves for KP98 and KP90 at various active layer in contact with the feed solution. OaKp90 130mol(NaCl)m~3 0 K.P98 130 mol(NaCl)m-3
feed
concentrations
l KP90
with
the
61.7mol(NaCl)m-3.
curve-fitting procedure was applied to the rejection curves. Its result is repre-+nted by the solid lines in Fig. 7 which cover the measuring points well. On the other hand the goodness of the fit is also seen in Fig. 8 where the plot of In (oa (1 - r)/(uc - r)} us J, is linear (Eq. (38)) and passes through the origin (f* = 1). Hence we may conclude that in the range of fluxes investigated the salt rejection curves can be described by two parameters a, and (1 - CJ,)/a,. Their values which do not depend on the concentration are compiled in the last two columns of Table II. Within our hypothesis these parameters are not influenced by reversible compaction. In other terms the way in which r depends on the flux is not disturbed by reversible compaction of the membrane contrary to the magnitude of the flux itself. In order to inspect the latter phenomenon more closely we recall that the hydraulic permeability for pure water decreases linearly with increasing AP (Experimental D). This can be represented by
Lp* = L$(l -
KAP)
with K a compaction modulus to be evaluated from the pure water data (Fig. 6). Dealing with salt solutions we assume that an identical K describes
W.C.M.
82
,n
HENKENS
AND
J-AM_
SMIT
dab-r1
I Uo-r
I
0.25 -
o.oI/ 0
3
6
12
9
J,(IJIYI S-‘I Fig. 8. Linearization of the salt rejection curve. KF’98 membrane in contact with a 130 mol(NaCl)m-3
feed solution.
02
03 A=/
Fig. 9. The
influence of reversible compaction
AP
correction on the hydraulic permeability
Wq. (43)). uncorrected for compaction (K = 0) 00 corrected for compaction (K # 0) 00 membrane KP98 10 membrane KPSO feed concentration in both cases: 130 mol(NaCl)mW3 _ l n
I
01
83
SALTRE!JECTIONANDFLUXINRO
which suggests a linear relationship to be verified experimentally. In fact this linearityhas been found as Fig. 9 shows. From the slopes and intercepts of these linesvalues of L$ and again values of a, have been calculated. Their values are shown in Table II. Thus the reflection coefficients o, found in this way agree reasonably with those originating from the salt rejection curves. Remembering Eq. (23) we may state that an experimental proof has been given of the relationship. limr ++-
=- A&X, AS
in which both members are equal to 0,. This result justifies the assumptions made and more generally it reflects that locally in the a-layer the Onsager Reciprocal Relation is satisfied between the transport coefficients. The necessity of the compaction correction follows from the poor results obtained when J,/AP is simply plotted uersus A?r/AP (Fig. 9). No linearity is found and moreover only erroneous values of a, are calculated (Table II, values between parentheses). We may thus conclude that the flux J,, is related to Aa and AP by the coefficients Lop, o, and K. Apparently the influence of the b-layer is reflected in the coefficient L$ which is composed from the partial hydraulic permeabilities according to Eq. (7) and from the compaction modulus ic which probably only refers to Ly _ This picture is consistent with the bilayer model which we have assumed. This basic assumption is supported by two experimental facts. First maxima in the salt rejection are measured when the dense skin is turned to the eMuent side (Fig. lo)_ These salt rejection curves were always measured with decreasing AP in order to be certain that these maxim a do not originate from opening of the pores as discussed around Eq. (14). Furthermore the membrane has in the latter orientation another compaction behaviour than in the normal orientation (Fig. 6). The values of Lj? referring to pure water are somewhat higher than the values referring to the salt solutions (Table II, column 3). This effect is probably due to an influence of viscosity. TABLEII Membrane
KP98 KF'98 KP90 KP90 KP90
=a -3 mol m
0 130 0 130 61.7
L$ x 1012 m3
pJ-1
2.079 2.026 1.738 1.711 1.716
s-’
% &.
40)
O-97(0.81) 0.90(0.61) O-89(0.22)
0,
(Eq.3'3)
l-u= ms-'
O.984
18.0
0.94O 0.944
ii.9 20.6
x106
W.C.M. HENKEXJS AND J.A.M. SMIT
r
i--0
1
I
2
L J,(pm
S-‘1
Fig. 10. Salt rejection curve for KP98 with the porous sublayer in contact with the feed solution of 130 mol(NaCl)m-3.
The effect of pressurizing the membrane emerges clearly from the strong decrease in the measured Lj! values. Even Lp of KF’90 has become smaller than Lp of KP98 after the pressure treatment which is certainly a result of the total compaction. Irreversible compaction depends on the magnitude and the duration of the original maximal hydrostatic pressure difference. It has an effect on both layers. Reversible compaction depends on the established hydrostatic pressure difference but not on the time. For a membrane placed with the skin to the high pressure side only the support layer is compressed. For both types of compaction changes in Lp are more pronounced than
changes in
r.
Though in the range of fluxes investigated by us, the layer of low selectivity p!ays a minor part compared to the highly selective skin, another situation is met at vanishing fluxes. In fact it has been shown [3,19,20] that reflection coefficients obtained at J, = 0 from osmotic measurements turn out to be smaller than the maximal retentions in reverse osmosis. The explanation of these findings must be that at J, = 0 the reflection coefficient is an overall coefficient, to which also the relatively smaIl ob contributes whereas at J,, + 00 the large a, of the selective skin predominates.
85
SALT REZJEZTIONAND FLUX IN RO ACKNOWLEDGEMENTS
The authorsthank Dr. C.P. Miming (Los Angeles, California)for technical advice, Prof. Dr. A.J. Staverman, Drs. Eijsermans and Drs. F.A.H. Pee&s (Leiden,
The Netherlands)
for helpful
discussions.
REFERENCES 1. S. Loeb and S. Sonrirajan, Sea water deminerahzation by an osmotic membrane, Advan. Chem. Ser., 38 (1963) 117. 2. K. S. Spiegier and O_ Kedem, Thermodynamics of hyperfiitration (reverse osmosis): Criteria for efficient membranes, Dessliuation, 1(1966) 311. 3. J. Jagur-Grodzinski and 0. Kedem, Transport coefficients and salt rejection in uncharged hyperfdtration membranes. Desalination. 1 (1966) 327. 4. W. Punch, Determination of transport parameters of synthetic membranes by hyperfutrstion experiments. Part 1, Ber. Buusenges. Phys. Chem., 8 (1977) 269. 5. C.W. Verslujis, forthcoming Thesis. Leiden, The Netherlands. 6. 0. Kedem and A. Katchalsky, Permeability of composite membranes. Part 1,2 and 3, Trans. Faraday Sot., 59 (1963) 1918,1931,1941. 7. A.J. Staverman, The theory of measurement of osmotic pressure, Rec. Trav. Chim., 70 (1951) 344. 8_ C.S. PatIak. D.A. Goldstein and J-F. Hoffman, The flow of solute and solvent across a two-membrane system, J. Theor. Biol., 5 (1963) 426. 9. R. SchiOgi, Non linear transport behaviour in very thin membranes, Q. Rev. Biophys., 2 (1969) 305. 10. R.J. Sha’afi, G.J. Rich, D.C. Mikulecky and A.K. Solomon, Determination of urea permeability in red cells by minimum method, J. Gen. Physiol.. 55 (1970) 427.
11. A.J. Staverman. Function and structure of membranes, Symposium on Biological and 12. 13.
14.
15_ 16. 17. 18. 19. 20. I
Artificial Membranes, Rome (1975). Pontificae Academiae Scientisrium Scripts Varia, Citta de1 Vaticano, (1976). J.A.M. Smit, J.C. Eijsermsns and A.J. Staverman, Friction and partition in membranes, J. Phys. Chem., 79 (1975) 2168. R. Griipl and W. Pusch, Asymmetric behavionr of cellulose acetate membranes in hyperfdtration experiments as a result of concentration polarization. Desalination. 8 (1970) 277. J. Bert, Membrane compaction; A theoretical and experimental explanation, J. Polym. Sci.. Part B. 7 (1969) 685. G. Jousson. The effect of pressure on the viscoelastic compaction of asymmetric cellulose acetate membranes, Proc. Seventh Int. Ccng. Rheol.. (1976) 298. L. Bssyers and S.L. Rosen, Hydrodynamic resistance and flux decline in asymmetric ceihdcse acetate reverse osmosis membranes, J. Appl. Polym. Sci., 16 (1972) 663. J. Kopecek and S. Sourirajan, Structure of porous cellulose acetate membranes and a method for improving their performan ce in reverse osmosis, J. Appl. Polym. Sci., 13 (1969) 637. W.C.M. Henkena, Thesis, Chapt. III, Leiden, 1978. W. Pusch and R. Riley, Relation between salt rejection r and reflection coefficient o of asymmetric cellulose acetate membranes. DesaIination. 14 (1974) 389. W.C.M. Henkens, J.C. Eijsermsns and J.A.M. Smit, Osmotic properties of a modified cellulose acetate membrane: the refiection coefficient and its dependence on the volume Sow history, J. Membr. Sci., in press.