FLOW RELATIONSHIPS IN REVERSE OSMOSIS U L R I C H MERTEN John Jay Hofkins Laboratory for Pure and Applied Scipnce, General ..ltomic Division, General Dynamics Gorp., San Diego, Calif.
Reverse osmo’sishas been receiving increasing attention in the past two years as a means of water desalination. The emphasis has been almost entirely on membrane properties, with the result that osmotic membranes have now been developed which have promising throughput characteristics. Little attention has been given to the mass transfer problem which arises in the brine. The present work presents an over-all phenomenological description of reverse osmosis, with special emphasis on boundary-layer effects. The treatment permits a qucintitative calculation of the effects of the interface salt concentration gradient on membrane performance.
EVERSE
osmosis has, been examined as a method of \rater
OSMOTIC BARRIER /-h7
R desalination by se\.eral groups in recent years, particularl>by Locb and others ( 8 . 9 ) a t the University of California a t Los .4ngeles and by Reid and his students (2. 7 7 : 72) a t the University of Florida. Both groups have demonstrated that from certain organic polymers: notably cellulose acetate. membranes can be prepared Lrhich are reasonably permeable to w’ater and yet sufficiently impermeable to dissolved salts to permit their use as semipermeable membranes in a reverse osmotic process for converting sea-\rater brines to potable \rater in a single pass. In the present paper, a phenomenological description of the process is given Xvhich should prove useful in evaluating its potential and interpreting experimental results.
MIXED
I82 LAYER
REGION DIFFUSION REGION
P
I
Reverse Osmotic Process
Figure 1 gi\zes a schematic representation of a reverseosmosis cell and a diagram shoiring the qualitative behavior of several important variables during stead>--state flow of component 1 of a binary solution from left to right through the membrane in Figure 1. We assume a n essentially infinite supply of the mixture containing p l and p? mass units of components 1 and 2. respectively. per unit volume and make the approximation that p , = ( p l p?) is independent of concentration over the range encountered in the high-pressure chamber. (The densit)- of the most concentrated brines of interest in the practical problem \rill not differ by more than about IOy6from that of \rater at the same temperature. so the density approximation is reasonable.’) The membrane is assumed to be ideally semipermeable -i.e.. p ? = 0 everyLshere to the right of the membrane. The system is assumed to be isothermal, and mixing is provided in the high-pressure chamber to assure a uniform concentration except in a boundary layer next to the membrane. The membrane may be thought of either as a diffusion barrier or as a filter. ‘4similar distinction has been made by others. For instance, ‘Ticknor ( 7 3 ) finds that the flow of water and methanol through cellophane films is viscous, while that of higher alcohols is primarily diffusive. It can be shorvn that the same general relationship describes the flow in the two cases. In Figure 1 the barrier is depicted as consisting of trro regions in series-region b. in which the flow is diffusive, and region c. a porous structure in which the flow is viscous. If the membrane acts as a diffusion barrier which is permeable to component 1 but not to component 2, the chemical potential of
+
DIRECTION OF FLOW, x
Figure 1. Reverse-osmosis cell, showing qualitative variation of several parameters across cell
P. Total pressure p1, p?. Concentrations of components 1 and 2 PI,p?. Chemical potentials of components 1 and 2
component 1. P I . \vi11 vary roughly as indicated in Figure 1 and its flo\r. F I . in the r direction through the membrane, region b. can be Xrritten FI
= - a ’ ~grad P I
(1)
as has been pointed out by Clark (3) in a recent paper. 1L.e \rill take the coefficient CY'^ independent of p1 and Y . Then
rl
where is the partial molar volume of component 1, Ax* is the thickness of the diffusion barrier, APb is the pressure drop across this barrier, and R ’ is the osmotic pressure of the solution at the left-hand membrane surface. The diffusion membrane will generally be backed by a porous structure, region c. through which pure component 1 flows by viscous flow. so that
R
=
PAP,
VOL, 2
=
P(AP
NO. 3
- APb) AUGUST
(3) 1963
229
3.0 I
27 2.5 f
i
7
I
MEMBRANE CONSTANT, A = 1.0 x WCM? -SEC-ATM.
+
N
9 1 2.0
BULK BRINE CONCENTRATION, = 5 . 0 WT.-%
pg
TEMPERATURE.25OC.
v
I-w
U
l
1.5
3
9
: t 0
1.0
13
-
a
0
E
0.5
1
0 0
2
I
3
4
5
6
0
PRODUCT FLOW RATE (MG./CM?-SEC)
Figure 2. a t 25°C.
Boundary-layer-limited flow rates for sea water
if Poiseuille flow obtains, lvhere i P c is the pressure drop across region c and p is a constant involving the viscosity of pure component 1 and the physical dimensions and pore configuration of region c. Solving Equations 2 and 3, we obtain
(4) where A is the membrane constant. If the membrane does not act as a diffusion barrier but as a filter which permits Poiseuille flow of component 1 but excludes component 2, the membrane itself may be thought of as region c and Equation 3 describes flow through the membrane. Region b in this case still exists, but is now simply the interface between the membrane and the solution. In the case of the diffusion membrane, the barrier itself is essentially a third solution component lvhich appears at the interface to replace component 2 and thus makes possible a continuous variation in the chemical potential of component 1 across the interface despite the abrupt change in pz which takes place there. In the case of the filter, p i increases abruptly to the density of pure component 1 at the interface. Since component 1 can be transmitted across the interface, its chemical potential must be continuous, a condition which is compatible with the composition change only if the pressure change across the interface, AP,, is the osmotic pressure, T ' . (This pressure drop is transmitted to the filter structure by the forces which exclude component 2. The case of a finite solution-filter interface thickness is treated in the Appendix.) Thus, Equation 3 becomes F1
p(AP
- x')
(5
which is a special case of Equation 4 with a1 >> &Le., negligible resistance to flow in the diffusion layer-just as Equation 2 is a special case of Equation 4 for p >> a1 and APb = AP-i.e., negligible resistance to flow in the viscous-flow region. Equation 4, then, should give a satisfactory representation of the flow-rate-pressure relationships for both cases. provided that a1 and p are independent of p2 and AP. 230
l&EC
FUNDAMENTALS
50
100
200
150
250
300
PRESSURE DIFFERENCE ACROSS MEMBRANE ( A T M
Figure 3. Effect of boundary-layer thickness on flow ratepressure relationship
The Boundary layer
The flow rate through a membrane of either type could. in principle, be made arbitrarily high by increasing iP if the boundary layer, region a in Figure 1. did not present a series resistance to flow of component 1. In the high-pressure chamber the pressure is uniform, assuming negligible pressure drop due to the bulk flow, and the flow, F,, of either component past a plane parallel to the membrane may be described as the sum of two terms: Fa
=
J,
+
(6)
P ~ U
J irepresenting diffusion of component i relative to the center of mass of the system and p,v representing motion with the center-of-mass velocity. u , measured relative to the membrane surface. At a mass flow rate, F I , per unit area through the membrane, & = -
R
(7)
Pt
throughout the high-pressure region with the assumptions already stated. Also,
J:
=
- D I ? grad
(8)
p2
so
\vhere D12 is the diffusion coefficient in center-of-mass coordinates .4t steady state. F2 = 0 everywhere, and therefore grad In p 2 =
~
R-
D12pt
If we assume D I ?to be independent of
pz,
then
2.5
I
I
I
3
$ 2.0 I
P 0 0.5
a a
:: 0.003 CM. TE:MPERATURE = 25OC.
0 0
50
1
I
I
IO0
I50
200
I
PRESSURE DIFFERENCE ACROSS MEMBRANE (ATM)
Figure 4. Effect of membrane constant on flow rate-pressure relationship
where p 2 ’ and pso are the values of p 2 in the bulk and a t the right-hand surface of th.e boundary layer of thickness Axa> respectively. Since a’ can be related to p2‘, it is possible, in principle, to eliminate these quantities by solving Equations 4 and 11 simultaneously and thus to obtain F1 as a function of p 2 O and AP. An analytic solution is :not readily obtained, however, and for our purposes it Mill suffice to solve Equation 11 numerically for In p 2 ‘ and use the result to calculate a ’for use in Equation 4. As a result of this boundary-layer effect, the flow rate through any membrane \vi11 reach a limiting value with increasing A P that corresponds to sorne limiting value of p 2 ’ . I n real systems, precipitation of component 2 in the interface, or a maximay impose a limit mum in J 2 with p 2 due i:o variations in D12, on attainable values of p2’ (or excessive transmission of component 2 at high p2‘ may place a practical limit on permissible values). Precipitation is apt to be limiting in the case of seaFvater desalination by reverse osmosis. because under most conditions of temperature and feed-brine concentration. a relatively small increase in )02 will cause precipitation of some components of the solution. The boundary layer is mechanically produced by the stirring of the solution in the high-pressure chamber. I n the absence of stirring, Ava takes on -the dimensions of the chamber, while under turbulent-flow conditions the boundary-layer thickness is just that of the laminar sublayer. Thus: the circulation rate directly affects Axa and thus p2’ and F1. An effect of circulation condition on flow through a n osmotic barrier has been observed by Loeb and Sourira.jan ( 9 ) . Sea-Water Desalination
Calcium carbonate, magnesium hydroxide, and calcium sulfate are the solids commonly precipitated from sea-water brines as they are concentrated. Considerable supersaturation with respect to calcium carbonate is commonly possible, and, in any case. carbonate and hydroxide precipitation is relatively
easily prevented through p H control. Calcium sulfate precipitation is a serious problem in evaporative processes, ho\l-ever. and this substance will almost certainly set the limiting surface concentration, p z * , in reverse-osmosis apparatus using sea-water brines. Assuming this to be the case. we may use the theoretical treatment just given to construct Aow-rate relationships for hypothetical osmotic membranes and sea-water brines of varying concentration. A difficulty arises in these calculations because the diffusion problem is a complex one, involving several different solutes with somewhat different diffusion constants. As a result, the relative concentrations of the dissolved solids \vi11 be somewhat different at the membrane interface from those in the bulk solution, and the gross sa!t concentration at the interface, p ? ’ , which is needed to reach a n osmotic pressure a’ or to reach saturation \l-ith respect to a particular solute will be some\vhat different than for sea \vater of normal composition. We will not try to account for this difficulty quantitatively. but recognize that actual va!ues for limiting floxl- rates and the like may be somewhat different from those calculated here because of this simplification. Sea Ivater contains about 3.5 \\.eight % solids. According to Posnjak (70), concentration of sea water by a factor of -3.3i.e.. to -11.5 weight % solids-is sufficient to cause calcium sulfate precipitation at room temperature. From data for similar salts given by Harned and Owen (6). we may estimate 0.75 X sq. cm. per second as the diffusion constant for C a s 0 4 near its saturation concentration in water a t 25OC. LVe will adopt this value for the sea-ivater case. These data and Equation 11 with p2’ = p2* allo\l- us to construct Figure 2, which shows the relationship betjl-een the boundary-layerlimited flow rate and bulk brine concentration, p z 0 > for three boundary-layer thicknesses, A x a , The limitations imposed by the boundary layer, quite independently of membrane properties or operating pressure. are apparent from these curves. The flow-rate-pressure relationships a t flow rates below that leading to boundary-layer limitation are readily calculated for a given brine concentration from Equations 4 and 11 if the membrane constant, A , the boundary layer thickness, Ax,, and the relationship between T ’ and p2‘ are knoll-n. For the latter, we will use the approximate expression 7’
= 6.8~2’
(12)
with a ’in atmospheres and p2’ in weight per cent. [The data 4, 7) indicate significant deviacollected by Gastaldo (I? tions a t the highest concentrations of interest, but uncertainties about the salt composition a t the interface may negate the advantages of using more precise data. ] An analysis of results given in Loeb and Sourirajan’s Figure 13 ( 9 ) indicates that their best membranes have membrane constants of approximately 1.0 X 10-5 gram per sq. cm.-sec.-atm. Figure 3 shows the effect of boundary-layer thickness on the flow rates which may be obtained with a membrane of this kind. Here we have used D, = 1.5 X IO-Ssq. cm. per second, the diffusion constant for NaCl in water at room temperature ( 6 ) , in making the osmotic pressure correction due to the boundary layer, but we have retained the C a s 0 4 value stated earlier in arriving a t the precipitation limitation. Figure 4,in which a boundary-layer thickness of 0.001 5 cm. is assumed, indicates the improvement which may be realized by increasing the membrane constant. Finally, Figure 5 indicates the effect of brine concentration on the flow-rate-pressure relationship for a particular membrane constant and boundary-layer thickness. It is apparent from these examples that boundary-layer thickness and membrane constant are of similar importance in the ranges examined here, and that flow conditions on the highVOL. 2
NO. 3
AUGUST 1963
231
I
3
w
I I I I I I I 1 MEMBRANE CONSTANT, A = 1.5 X WCM? - SEC.-ATM. BOUNDARY-LAYER THICKNESS, Axa = 0.003 CM. TEMPERATURE = 25% BULK BRINE CONCENTRATION, ,Dg IN WT.-%
3.5
1
2.0
/
p,g
= 6.0
= 0, a n d f i becomes a force acting between the membrane and component 2. -4s in Equations 6 and -. >\.emay write
fl
FI
i
1
z
LL
+ 0
L
3
n
0
0 R - OS5 -
0
cz
so F1 = L E:
[ -C?f:
- C:(&
- 62) grad P
-
(*’) dC!
grad CI] r.p
and. integrating across the interface thickness, - I t b 8
I
4
s
1 J1
50 100 150 200 250 300 350 400 450
PRESSURE DIFFERENCE ACROSS MEMBRANE (ATMJ
T h e first of the integrals on the right is just the force per unit area on the membrane,
and by setting F I = 0 we can see that the second integral is LPb at zero flow. which is, by definition. the osmotic pressure, n’. Thus.
Figure 5. Effect of brine concentration on flow rate-pressure relationship
pressure side of a reverse-osmosis apparatus \vi11 be a very important consideration in determining attainable product flows Tvith the best existing membranes. Minority constituent precipitation a t the membrane surface will seriously limit attainable product flow rates under certain conditions. and control of these constituents may well become as important in reverse osmosis as it is in evaporation. Appendix
Flow across Diffusion Layer with Finite Solution-Filter Interface Thickness. I n discussing the case of membrane filters and arriving at Equation 5, we assumed a n abrupt decrease in p2 at the solution-filter interface. If the solutionfilter interface thickness is taken as finite, there may be significant resistance to flow across it. I n this case the interface acts as a diffusion region much as the membrane does in the case of Equation 2. but not\. the thickness of this region, Ax*, is determined by the range of the forces excluding component 2, rather than by the dimensions of the membrane. D e Groot (5) Xvrites the complete isothermal diffusion expression in the form
This is of the same form as Equation 2 if the value of the integral is independent of pz a t the interface. and Equation 4 is found to hold for this case to a similar degree of approximation as for the others. Acknowledgment
The author is indebted to H. K. Lonsdale. J. IVestmoreland. and D. T. Bray of this laboratory for helpful discussions. Literature Cited
(1) Arons, ’4.B., Kientzler, C. F., Trans. .4m. Geqbhu. Union 35, No. 5, 722 (1954). (2) Breton, E. J., Jr., Office of Saline Lt’ater Research and Development, Progr. Rept. 16 (April 1957). (3) Clark, IY. E., Science 138, 148 (1962). (4) Gastaldo, C., UCLA Department of Engineering, Rept. 61-80 (February 1962). (5) Groot, S. R., de, “Thermodynamics of Irreversible Processes,” North Holland Publishing Co., Amsterdam, 1951. (6) Harned, H. S.,Owen, B. B., “Physical Chemistry of Electrolytic Solutions,” 3rd ed., p. 255, Reinhold, New York, 1958. (7) Higashi, K., Nakamura, K., Hara, R., J . Soc. Ch~ni.2nd. Japan 34, 166 (1931). (8) Loeb, S., UCLA Department of Engineering. Rept. 61-42 (August 1961). ( 9 ) -Loeb. - - ~S.. -, > Souriraian. S.. Zbid.. 60-60 iJulv 19601. (10) Posnjak, E., Am.dJ. Sci.238, 559 (1940). (11) Reid, C. E., Final Report, Office of Saline LYater Contract No. 14-01-001-71, January 1959. (12) Reid, C. E., Breton, E. J., J . Appl. PolymerScz. 1, 133 (1959). 13) Tieknor, L. B., J . Phys. Chem. 62, 1483 (1958). \ - I
~~
’
where J 1 is the diffusive flux of component 1 ; J 1 a n d j z are external forces per unit volume acting on components 1 and 2, respectivel).; fjl and 62 are the partial specific volumes; L11 is a phenomenological coefficient; and C1 = p 1 / p t , CP = ~ 2 , ’ p t . T h e membrane is selective against component 2 only, so we set
232
l&EC FUNDAMENTALS
RECEIVED for review November 13, 1962 ACCEPTED hlarch 5, 1963 It-ork supported by the Office of Saline Water, U. S. Department of the Interior, under Contract 14-01-0001-250.
