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Journal of Membrane Science, 8 (1981) 141-171 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands 141 MEMBRANES FOR POWER GENERATION BY PRESSURE-RETARDED OSMOSIS K.L. LEE, R.W. BAKER and H.K. LONSDALE Bend Research, Inc., 64550 Research Road, Bend, Oregon 97701 (U.S.A.) (Received July 23, 1980; accepted in revised form September 26, 1980) Summary A model has been developed for obtaining the projected performance of membranes in pressure-retarded osmosis (PRO) from direct osmosis and reverse osmosis measure- ments. The model shows that concentration polarization within the porous substrate of the membrane markedly lowers the water flux under PRO conditions. The model has been used along with experimental data obtained with a variety of reverse osmosis mem- branes to project PRO performance with several water--brine sources. Some literature data on PRO have been similarly examined. Based on these results and a simple econom- ic analysis we conclude that membranes with significantly improved performance will be needed if PRO is to become an economically feasible method for power generation using seawater-fresh water as the salinity gradient resource. However, the economics of a brine/fresh water system appear competitive with conventional power generation technologies. Introduction A potentially vast amount of energy is stored in the waters of the earth, as a result of the unequal salt concentrations of fresh water and seawater. This energy, in reality a form of solar energy, is now irreversibly dissipated when rivers mix with the sea. Much of this energy could be recovered by al- lowing the waters to mix through an osmotic membrane, thereby converting the osmotic pressure of seawater into a hydrostatic pressure, which could in turn be used to drive a turbine and generate electricity. Such a scheme has been proposed by Loeb [l] and, independently, by Jellinek [2], The way in which the process could work is illustrated in Figure 1. Loeb has coined the term “pressure-retarded osmosis” (PRO) for this process and has already demonstrated its feasibility on a very small scale in Israel [ 31. A recent sum- mary of the state-of-the-art has been presented by Wick [4]. It is uniformly agreed that the size of the salinity gradient resource is large, even within the United States, but estimates of the economics of extraction of this energy vary over an order of magnitude. The economics are so uncertain because very few PRO experiments have been performed and there is considerable uncertainty about membrane performance under 0376-7388/81/0000-0000/$ 02.50 0 1981 Elsevier Scientific Publishing Company

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Page 1: MEMBRANES FOR POWER GENERATION BY PRESSURE-RETARDED … · MEMBRANES FOR POWER GENERATION BY PRESSURE-RETARDED OSMOSIS K.L. LEE, R.W. BAKER and H.K. LONSDALE Bend …

Journal of Membrane Science, 8 (1981) 141-171 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

141

MEMBRANES FOR POWER GENERATION BY PRESSURE-RETARDED OSMOSIS

K.L. LEE, R.W. BAKER and H.K. LONSDALE

Bend Research, Inc., 64550 Research Road, Bend, Oregon 97701 (U.S.A.)

(Received July 23, 1980; accepted in revised form September 26, 1980)

Summary

A model has been developed for obtaining the projected performance of membranes in pressure-retarded osmosis (PRO) from direct osmosis and reverse osmosis measure- ments. The model shows that concentration polarization within the porous substrate of the membrane markedly lowers the water flux under PRO conditions. The model has been used along with experimental data obtained with a variety of reverse osmosis mem- branes to project PRO performance with several water--brine sources. Some literature data on PRO have been similarly examined. Based on these results and a simple econom- ic analysis we conclude that membranes with significantly improved performance will be needed if PRO is to become an economically feasible method for power generation using seawater-fresh water as the salinity gradient resource. However, the economics of a brine/fresh water system appear competitive with conventional power generation technologies.

Introduction

A potentially vast amount of energy is stored in the waters of the earth, as a result of the unequal salt concentrations of fresh water and seawater. This energy, in reality a form of solar energy, is now irreversibly dissipated when rivers mix with the sea. Much of this energy could be recovered by al- lowing the waters to mix through an osmotic membrane, thereby converting the osmotic pressure of seawater into a hydrostatic pressure, which could in turn be used to drive a turbine and generate electricity. Such a scheme has been proposed by Loeb [l] and, independently, by Jellinek [2], The way in which the process could work is illustrated in Figure 1. Loeb has coined the term “pressure-retarded osmosis” (PRO) for this process and has already demonstrated its feasibility on a very small scale in Israel [ 31. A recent sum- mary of the state-of-the-art has been presented by Wick [4].

It is uniformly agreed that the size of the salinity gradient resource is large, even within the United States, but estimates of the economics of extraction of this energy vary over an order of magnitude. The economics are so uncertain because very few PRO experiments have been performed and there is considerable uncertainty about membrane performance under

0376-7388/81/0000-0000/$ 02.50 0 1981 Elsevier Scientific Publishing Company

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Pump

Dilute

Semipermeable membrane

Fig. 1. Conceptual representation of an energy production scheme based on pressure- retarded osmosis (PRO).

PRO conditions. PRO experiments are indeed difficult to carry out and some workers have therefore performed reverse osmosis (RO) experiments and extrapolated their results to PRO. Calculated power generation costs ob- tained in this way are quite favorable. However, the limited number of PRO experiments described in the literature exhibited much lower performance than simple extrapolation from RO experiments would suggest. Power generating costs based on these PRO results are high.

In this paper, we generate a model for the performance of permselective membranes under PRO conditions. With this model, PRO performance can be predicted from RO and direct osmosis (DO) measurements. We have also tested a variety of high performance membranes under RO and DO condi- tions and used the model to estimate their performance under PRO condi- tions. The limited literature data on PRO have also been examined and rationalized according to the model. Finally, from an economic analysis we conclude that PRO would not be economically practical even with the best of present-day membranes. The principal problem is concentration polariza- tion in the porous substrate of these membranes, which markedly reduces the water flux.

II. Theory

A. Ideal membranes The relationship between RO, PRO, and DO is illustrated in Fig. 2 for an

ideal semipermeable membrane. In direct osmosis, the membrane separates two solutions of unequal concentration, typically a salt solution from pure water. A flow of water from the water side into the salt solution takes

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Water Flux

Reverse Osmosis (AP > An)

- AP

Pressure-Retarded Osmosis - (An > AP)

-- Direct Osmosis (AP = 0)

Fig. 2. The relationship between RO, PRO, and DO for an ideal semipermeable membrane.

place due to the difference in osmotic pressures between the two solutions. If an increasing pressure is applied to the salt solution, the magnitude of the water flow will decrease until no flow occurs when the applied pressure equals the osmotic pressure difference across the membrane. The regime in which there is a flow of water into a pressurized salt solution is known as pressure-retarded osmosis. Reverse osmosis occurs when the hydrostatic pres- sure applied to the salt solution is greater than the osmotic pressure differ- ence across the membrane. In this case, there is a flow of water from the salt solution to the water side of the membrane.

The water flux, J,, across such a membrane throughout this whole regime is given by

J, = A(An - AP) , VI*

where A is the water permeation coefficient of the membrane, An is the osmotic pressure difference across the membrane, and AP is the hydrostatic pressure difference across the membrane.

In PRO, the power that can be generated per unit membrane area is equal to the product of the water flux and the hydrostatic pressure of the salt solution. The maximum power per unit flux is therefore obtained at the maximum hydrostatic pressure under which PRO takes place, i.e. the

*This relationship is conventionally written J, = A (AP - A n). We have chosen our pre- sentation so that the water flux in PRO is positive.

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osmotic pressure difference, An. However, when AP is close to An the trans- membrane water flux is very small, and a very large membrane area would be required, resulting in a high capital cost for the system. It is preferable, therefore, to operate PRO systems under conditions corresponding to the maximum power per unit membrane area, in order to minimize capital costs.

The power per unit membrane area, W, that can be generated in PRO is equal to the product of the water flux across the membrane and the hydro- static pressure of the salt solution:

W = J,AP = A(An - AP)AP. (2)

By differentiating eqn. (2) with respect to AP, it can be shown that W reaches a maximum when AP = AT/~. Substituting this value for AP in eqn. (2) yields

W max = AAn2/4. (3)

Equation (3) shows that the maximum power in a PRO system is directly proportional to the water permeability coefficient, A, and thus high flux membranes are preferred. The maximum power is also proportional to the square of the osmotic pressure difference. This arises because increasing the osmotic pressure of the salt solution increases both the optimum pressure at which the system operates (i.e. An/2) and the water flux through the mem- brane at that pressure.

In calculating the maximum power using eqn. (3), the osmotic pressure difference An has been assumed to be constant at every point along the membrane. However, significant brine dilution has to be accepted in any real system. In optimized systems, the available osmotic pressure will be re- duced by a significant factor, e.g., about 25%, if a factor of two dilution is accepted over the length of the membrane.

B. Real membranes 1. Salt leakage Real membranes are not perfectly semipermeable, and a small amount of

salt will permeate the membrane in PRO, as in RO and DO. One effect of this salt permeation is to reduce the effective osmotic pressure difference across the membrane in PRO. Thus, for example, while the osmotic pressure difference between seawater and fresh water is approximately 25 atm at room temperature, a slightly “leaky” membrane might only generate a pres- sure difference between these solutions of perhaps 23 atm when used in an osmometer, even in the absence of any concentration polarization effects. Thus, An in eqns. (l)-(3) should be 23 atm and not 25 atm, in this example. In general, this is not a serious effect unless the membrane has very poor salt permselectivity. It is customary to refer to the ratio of the ef- fective osmotic pressure to the actual osmotic pressure when J, = 0 as u, the so-called Staverman reflection coefficient [ 51.

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2. Concentration polarization Concentration polarization is a much more serious problem in PRO.

Actually, there are two types of concentration polarization in PRO, which we will call “external” and “internal” concentration polarization. The first of these is illustrated in Fig. 3. In this figure, it is assumed that the mem- brane is homogeneous, and, for simplicity, the salt partition coefficient between the membrane and aqueous phases is taken to be unity. As shown in Fig. 3, the concentrations of salt at the membrane-solution interfaces are different from the bulk solution concentrations. This is caused by the flow of salt and water through the membrane in opposite directions, reduc- ing the salt concentration at the membrane interface with the salt solution and increasing it at the water-side interface. The osmotic pressure difference, An, across the membrane is then 7r2 - r4, corresponding to the salt con- centrations C2 and C4. This driving force is lower than the apparent value, 711 - x5, corresponding to the bulk solution salt concentrations C1 and C5.

This external concentration polarization effect can be reduced by stirring the solutions on both sides of the membrane to minimize the thickness of the boundary layers. With flat-sheet membranes in PRO, appropriate stirring is easily achieved on the pressurized salt side of the membrane. However, on the water side of the membrane, some sort of support layer is required to withstand the hydrostatic pressure applied to the salt solution. This support layer interferes with good stirring and prevents elimination of the boundary layer. For this reason, measurements of water flux in PRO with flat sheet membranes are difficult to perform.

SALT SOLUTION MEMBRANE WATER

F Water Flow

Fig. 3. A schematic representation of external concentration polarization across a

homogeneous membrane in PRO.

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146

The membranes that would be used in any practical PRO system would not be homogeneous, and the representation in Fig. 3 is too simplistic. In order to achieve the requisite water flux in PRO, it is clear that asymmetric membranes of the general type presently used in RO would be required. These membranes have a very thin salt-rejecting “skin” on one side of the membrane. The remainder of the membrane is a finely porous matrix that serves as a support for the skin. If such a membrane were used in PRO, salt would accumulate within this porous substrate, leading to what Mehta and Loeb [6] have referred to as “internal polarization”. This is illustrated in Fig. 4. As before, there exists some polarization at the outer surfaces of the

SALT SOLUTION MEMBRANE WATER

! c1 C2; :I c

3

Increasing Salt c4 5

Concentration

- Water Flow

Salt Flow w

SALT-REJECTING "SKIN" / \ POROUS SUBSTRATE

Fig. 4. A schematic representation of concentration polarization membrane in PRO.

across a skinned

membrane which can be controlled by stirring. However, stirring does not affect internal concentration polarization. The osmotic pressure difference is then not n2 - r4, but x2 - 7r3, corresponding to the concentrations C2 and CJ. As is shown below, internal concentration polarization can be a very serious problem, reducing the water fluxes in DO and PRO experiments to a fraction of the values predicted from RO experiments. The problem does not occur in RO because the water flux is in the same direction as the salt flux, and thus any salt that permeates the membrane is continuously swept away with the water permeating the membrane,

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3. Cal&a tion of the water flux in PRO As noted, PRO experiments are difficult to carry out. In this section, we

develop a method by which PRO performance can be reliably predicted from RO and DO experiments, which are much easier to perform. The essen- tial problem is internal concentration polarization.

As illustrated in Fig. 4, the effects of external and internal concentration polarization combine to reduce the osmotic pressure difference across the membrane to some effective value, which we will denote A7reff. For sim- plicity, let us assume that external concentration polarization has been re- duced to negligible levels by efficient stirring, i.e. C1 = C, and C, = C,. Equation (1) can then be modified to

JW = A(AR~R - AP) = A(n, - n3 ~ AP) . (4)

Equation (4) is not a useful expression for the water flux across a PRO membrane because the osmotic pressure 71 3 is not known. This quantity can be calculated, however, by considering the salt leakage across the membrane. Thus, the salt flux across the rejecting layer, J,, can be written as [7]

-JS = B(C2 - C,) , (5)

where B is the salt permeation coefficient. In our definition, the salt flux is negative because the direction of salt flow is opposite to that of the water flow. In the porous substrate, the salt flow consists of two components act- ing in opposite directions: a diffusive part due to diffusion down the salt concentration gradient, and a convective part due to the bulk flow of water through the membrane. The salt flux across the substrate can thus be written

dC(x) -Js = DsE ____ - &C(x) 9 (6)

dx

where E is the porosity of the substrate and is assumed equal to the volume fraction occupied by capillary water in the membrane, and D, is the diffu- sion coefficient of salt in the membrane substrate. The distance x is measured from the membrane-solution interface on the porous substrate side. Equations (5) and (6) can be combined to yield

dC(x) B(C2 - C,) = II& - - - - &C(x) .

dx

The appropriate boundary conditions for either PRO or DO are

C(x) = C4 at x = 0

C(x) = C3 at x = 7t ,

(7)

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148

where t is the thickness and 7 is the tortuosity of the porous substrate. With these boundary conditions, eqn. (7) can be integrated to give

cl Bbp(JwK)-- 11 + Jw

c3 7 ew(JwK)

2 -_= ?

(8)

c2 B[exp(JwK)- 11 + Jw

where K = 7t/DSe, and is a measure of the resistance to salt transport in the porous substrate. We will assume that the ratio of salt concentrations is ap- proximately equal to the ratio of osmotic pressures to arrive at

F [exp(J,K)- l] + 1 W

Equations (4) and (9) can be combined to give

1 - 2 exp(J,K)

J, = A n2 L’2

B

1-c” exp(J, K) c2

-AP I I- (10)

i

1 + __ [exp(J,K)- l] Jw I

Equation (10) relates water flux in PRO to membrane parameters, ap- plied pressure difference, and external solution concentrations. Loeb [3] ob- tained a less general form of eqn. (10) by not allowing for salt leakage across the membrane, i.e., B = 0. For the special case of C, = 0, i.e. with water on one side of the membrane, eqn. (10) reduces to

1 + B [exp(J,K)- 11 Jw

(11)

Neither eqn. (10) or (11) is explicit for J,, but each can be solved numer- ically. Equation (11) can be simplified because, for real membranes under PRO conditions, J,K is generally much less than unity. The exponential can thus be expanded and, when higher order terms are dropped, one obtains

712 J, z A l+BK

(12)

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The membrane parameters necessary to calculate the flux in PRO from eqns. (lo)-(12) can be obtained from RO and DO measurements. Thus, the water flux in RO is given by eqn. (l), and A can be obtained from measure- ments of J, vs. AP when An is known. B can be obtained from measure- ments of salt rejection in RO; rejection is denoted by R and is defined as

salt concentration in the permeate solution R=l- . (13)

salt concentration in the feed solution

B is related to R by [8]

A(1 - R)(AP - An) B= . (14)

R

The parameter K can be obtained from direct osmosis measurements where AP = 0 and pure water is used on one side of the membrane, i.e. C4 = 0. In this case, eqns. (11) and (12) can be rearranged to

K= Lln An, - J,

Jw ( +l

B ) (15)

and

An, - J, Kr

BJW (16)

respectively, and K can be obtained by measuring JW with a solution of known osmotic pressure.

4. PRO and k0 At this point, it is instructive to examine in some detail the differences

between RO and PRO for real membranes. In this case, PRO is not a simple extension of RO as shown schematically in Fig. 2. The real situation is il- lustrated in Fig. 5. Let us consider the several cases illustrated there.

Case A: ideaE membrane. This is the case illustrated in Fig. 2. There is no salt leakage (B = 0), and J, = 0 when AP = An, the osmotic pressure differ- ence across the membrane. The slope of the line is A, the water permeabil- ity coefficient.

Case B: leaky membrane with no substrate resistance (K = 0). The slope of the line is still A, but J, = 0 when AP = rr An because of salt leakage (that is, B > 0). Here, PRO is a simple extension of RO.

Case C: RO. In the linear region, the slope is again A, and the intercept of this linear portion on the J, = 0 line is again o An. However, consider what happens in normal RO operation as one lowers AP. The salt flux, as is well known, is only weakly dependent on pressure, while the water flux is strongly dependent on pressure [ 71. Thus, as AP is reduced, salt flux con-

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-4

-3

-1

J ( ) 10-4cm3 (

w 2 cm -see I-

L-

!-

/ /

/

3- 0

I I I I I

1

10 I I t I

20 30 40 50 3P (atm)

1 60

Fig. 5. The relationship between RO and PRO for ideal and real membranes. For Cases A, B, Dl and D2, C, = 0; for Case C, C, = C,.

tinues while water flux is diminished. Even when AP is only infinitesimal, there will still be a finite water flux and, because of the salt flux, the product water salinity will be almost identical to the feed water salinity.

Case D: PRO with a real membrane. In RO, the concentration of salt on the low pressure side of the membrane is always identical to the concentra- tion of salt in the permeate. In PRO, however, we create a different boundary condition in which the concentration of salt on the low con- centration side of the membrane is fixed by the available water supply. If we again begin in the RO regime and continually reduce the pressure, we

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find that Jw = 0 when AP = A~~ff, the osmotic pressure difference across the salt-rejecting layer as illustrated in Fig. 4. Alr,~ is less than aAn because of the salt concentration gradient in the porous substrate. In the PRO regime, the dependence of J, on AP is linear with slope A when J,K is small. However, in the region of high water flux or significant resistance to salt flow in the porous substrate, the J, vs. AP plot is non-linear.

In Fig. 5 we show two examples of real membrane behavior in PRO. Case Dl occurs when the membrane is highly permselective but the porous substrate offers significant resistance to salt flow, i.e. B is small but K is very large. Case D2 occurs when B is relatively large but K is small. Both of these cases have been calculated from eqn. (11) and are representative of two of the membranes studied in this work: PA-300 and CA-80, respective- ly (described in Section III below).

The key point is this: care must be exercised in estimating PRO per- formance from RO data. Because of concentration polarization in the porous substrate, the J, vs. AP line in PRO can-have a different slope than in RO. Furthermore, the flux-pressure relationship may be non-linear. Finally, the optimum operating pressure of a PRO system can be well below An/2 or even uA1r/2.

5. Projected power generation in PRO We are now in a position to calculate the projected power generation in a

PRO system utilizing real RO membranes. All that is required, in addition to membrane parameters and the salt concentrations, is the operating pres- sure of the PRO system. As in the case of the ideal membrane, this pres- sure is again about one half of the flux-reversal pressure, i.e. the pressure where Jw = 0.

We will denote the flux-reversal pressure as APO. It is the value of A n,ff at which J, = 0 and can be obtained from eqn. (9):

An =

BK + 1 (18)

The significance of internal concentration polarization to PRO is manifest in eqn. (18): it acts to reduce the flux-reversal pressure, which is the highest pressure at which a PRO system could operate.

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The maximum power is therefore

APO W max =J,’ __

2 (19)*

where J, ’ is the water flux at the optimum operating pressure. Combining eqn. (19) with eqns. (9) and (13), we have

AAI? W max=

4(BK + 1)

: -

i

2

1-c” Cz

l-5 exp (J&K) C2

B, [exp (J&K) - 11 + 1 BK + 1 (20)

W

For the special case of fresh water on one side of the membrane, C, = 0 and

AAK’ 2 1 W max = -p

4(BK+ 1) (exp (J&) - 1) + 1

BK + 1

I

(21)

Thus, eqns. (18).- (21) allow us to calculate PRO performance from the available osmotic pressure gradient and membrane parameters determined from RO and DO measurements.

III. Experimental

Several commercial and developmental RO membranes were evaluated in RO and DO tests to determine the parameters A, B, and K.

A. Membranes The following flat-sheet membranes were evaluated:

1. Cellulose acetates Asymmetric membranes were prepared at the Max Planck Institute for

Biophysics in Frankfurt, West Germany, from Cellit K700 grade 39.1%

*Strictly speaking the left-hand equality in eqn. (19) is valid only when J, is linear in AP. As we can see from eqns. (10) and (ll), this is not, in general, true in PRO. However, the error involved in this approximation is small for all membranes studied in this work, and the conclusions presented in Section V are unaffected.

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acetyl cellulose acetate (Bayer AG, Leverkusen, West Germany), follow- ing a solution-casting procedure first described by Manjikian et al. [9]. Batches of the membrane were annealed at 70” and 8O”C, and are de- signated CA-70 and CA-80, respectively.

2. Polyamide An asymmetric polyamide membrane, designated BM-05, was obtained

from Berghof GmbH of Tiibingen, West Germany.

3. Poly benzimidazolone (PBIL) PBIL is a recently developed asymmetric membrane from Teijin Corp. of

Japan and Abcor, Inc. of Wilmington, MA [lo]. Development samples were obtained from Abcor.

4. Composite membranes These membranes consist of an ultrathin salt-rejecting layer deposited on

a porous substrate. The sole function of the substrate is to provide mechan- ical support for the salt-rejecting layer. Membranes of this type can exhibit salt rejections in RO in excess of 99%.

a. PA-300. This is a polyamide thin-film composite membrane obtained from the Fluid Systems Division of UOP, Inc., San Diego, CA. The salt- rejecting barrier layer is formed by an interfacial reaction between an epi- chlorohydrin-ethylenediamine condensate and isophthaloyl chloride as cross- linking agent. This layer is supported on a finely porous polysulfone sub- strate [ll],

b. NS-101. This membrane was obtained from the North Star Division of Midwest Research Institute, Minneapolis, MN. The salt-rejecting layer con- sists of a polyamide formed by the interfacial reaction of polyethylenimine with isophthaloyl chloride. The substrate is microporous polysulfone [ 121.

c. NS-200. This composite membrane was also obtained from North Star. The active layer of NS-200 is a thin film of polymerized furfuryl alcohol, and the substrate is again microporous polysulfone [12].

d. BM-I-C. Berghof GmbH supplied this proprietary composite mem- brane. The only feature known to be in common with other composite membranes is its polysulfone substrate.

B. Reverse osmosis tests .4 conventional stainless steel RO loop was used for the tests [7]. A salt

concentration of 3.5 wt% NaCl was used throughout. Initially, the mem- brane was taken to a pressure of 1200 psi and maintained there until steady- state water flux and salt rejection were established, typically in about 12 hours. The water flux was then measured as a function of pressure, from 1200 to 200 psig. All measurements were made at room temperature.

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C. Direct osmosis tests A glass permeation cell was used in which the osmotic water flux and

salt flux could be measured concurrently with high precision. A schematic diagram of the cell is shown in Figure 6.

The test membrane was held between the ground glass flanges of the half- cells by a clamp. The lower half-cell contained the concentrated salt solu- tion in all cases, and the membranes were always introduced so that the skin was in contact with the concentrated solution. The inlet and outlet of the upper half-cell were connected through a recirculation pump to an external 0.5 or 1.0 liter reservoir of the dilute solution. An air damper was used on the pump outlet to minimize pulsation in the water circuit. The membrane was supported by a stainless steel screen. Stirring of the liquids was provided by a motor-driven glass impeller on the dilute side and a magnetic stirring bar on the concentrated side. The rate of osmotic flow of water into the concentrated solution was measured volumetrically with a pipet. The permeation of salt into the dilute solution chamber was deter- mined by periodically measuring the salt concentration in this chamber with

r-l- MOTOR

TO RESERVOIR

FROM RESERVOIR

TEST MEMBRANE

\ PIPET

FILLINk PORT

\

\ SUPPORT SCREEN

kAGNETlC STIRRING BAR

Fig. 6. Direct osmosis test cell, exploded view.

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155

a conductivity bridge. In cases where the water flow into the concentrated solution was sufficiently rapid to dilute the solution significantly during an experiment, the concentration was also measured at the time the water flux reading was taken, All tests were carried out at room temperature with NaCl solutions, over a range of concentrations from 3.5 to 25 wt%.

Two typical DO runs are shown in Fig. 7. The flux is plotted against time, showing the approach to the steady state. In general, the membranes prepared by the Loeb--Sourirajan technique, such as BM-05, PBIL, and the cellulose acetate membranes, all reached a steady-state flux within 20 to 50 minutes. However, the composite membranes, such as the PA-300, NS-101, and NS-200, typically required as long as 100 to 250 minutes.

The effects of concentration polarization on water flux in the DO tests are illustrated in Fig. 8. This was an experiment with pure water and a 25 wt% NaCl brine. The water flux at time zero denotes the steady-state flux reached under normal operating conditions. When the stirring on the fresh water side of the membrane was stopped, the water flux decreased markedly. This is the result of a higher salt concentration at the interface between the solution and the porous substrate than in the bulk solution. Creating a stagnant condition on the brine side produced an additional large flux decline. This decline is expected from the dilution of the brine at that interface by the permeating water. Restarting the water-side stirring resulted

Water Flux, .I w

. .

BN-05

l- -o* BM-1-C

OO I I 1 50 100 150 2

Time (min)

Fig. 7. Typical direct osmosis test results for two membranes.

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156

Water FlUX

( q 2.0-

0

1 .5-

0

l .O-

0

0 .5-

0

O-

) .o-

.8-

+

.6-

.4-

.2-

o- 0

-4 3 ( ), t ie

I , I

Water-Side

/ Stirring Stopped

Brine-Side

C-Side Stirring Resumed

Water-Side Stirring Resumed

100 200 300 400

Time (min) IO

Fig. 8. Effect of stirring on either side of a PA-300 membrane on the direct osmosis water flux (25 wt% NaCI).

in partial restoration of the original flux, as the salt build-up at that solution-membrane interface was eliminated. Resumption of stirring in the brine compartment led to a sharp rise of water flux, surpassing the steady- state value at the beginning of the experiment. This surge was real and re- flected the condition in which virtually the entire osmotic pressure gradient was instantaneously imposed across the salt-rejecting layer before the salt concentration profile could again develop within the porous substrate.

These effects are described for illustration only. In all of our experiments, stirring on both sides of the membrane was maintained at levels that virtual- ly eliminated external concentration polarization.

IV. Results and discussion

A. Reverse osmosis The water permeability coefficient, A, was obtained from the linear por-

tion of the flux vs. applied pressure curve for all of the eight membranes tested. The salt permeability coefficient, B, was obtained from salt rejection data and eqn. (14). The results are summarized in Table 1. Also tabulated is

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TABLE 1

Reverse osmosis results (3.5% NaCl, room temperature)

Membrane Water flux at Salt rejection DA?7 1200 psi at 1200 psi (atm)

A

(

Asymmetric CA-80 6.4 87 25.5 1.0 20 CA-70 11.6 39 15.0 3.3 870 BM-05 2.1 90 27.2 0.40 2.3 PBIL 3.7 92 24.5 0.65 3.2

Composite PA-300 6.8 99.3 25.1 1.1 1.7 N&101 6.6 94 24.6 1.2 4.4 NS-200 5.4 94 27.2 1.0 3.5 BM-1-C 6.3 89 20.5 0.82 6.1

the intercept of the linear portion of the J, vs. AP plot on the abscissa, i.e. the quantity denoted OAT in Fig. 5. For comparison, the osmotic pressure of a 3.5’wt% solution of NaCl at room temperature is 28.1 atm [13].

B. Direct osmosis The direct osmosis water flux is plotted against the concentration of salt

used in the DO tests in Figs. 9 and 10 for the asymmetric and composite membranes, respectively. When one considers that the osmotic pressure of a 10 wt% NaCl solution is on the order of 100 atm, it is immediately clear that the DO water flux through many of the membranes, particularly the composite membranes, is much lower than would be predicted on the basis of RO data.

The observed water fluxes with 3.5 wt% NaCl are presented in Table 2 for the eight membranes. Also tabulated there is the projected water flux, based on eqn. (1) and the RO values for A and OAR. We use oAn rather than An here to incorporate the leakiness of the membranes. The dis- crepancy, then, between the observed and projected water fluxes is due solely to internal concentration polarization in the DO tests. The parameter K, the measure for this polarization phenomenon, is tabulated in the fourth column of Table 2.

The final’two columns of Table 2 deal with an alternative measure of the resistance to salt flow in the porous substrate of the membranes. Because we have no method of evaluating the tortuosity and porosity of the membrane substrate, it is convenient to define an effective diffusion co- efficient for salt in the substrate De E D&T = t/K. From the membrane thickness, t, taken here to be identical to the porous substrate thickness, one can calculate this effective diffusion coefficient. The values reported in

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Water Flux.

10

5

0 0 5 10 15 20

NaCl Concentration (wt%)

25

Fig. 9. Direct osmosis water flux vs. salt concentration for asymmetric membranes.

TABLE 2

Direct osmosis results (3.5 wt% N&I, room temperature)

Membrane Observed Jw Projected Jw * x Membrane thickness De

Asymmetric CA-80 4.3 2.5 0.65 CA-IO 2.5 4.7 0.33 BM-05 0.57 1.7 19 PBIL 2.8 1.6 6.9

Composite PA-300 0.26 3.1 57 NS-101 0.038 2.8 290 NS-200 * * 0.80 2.8 13 BM-1-C 0.28 1.7 40

*Based on eqn. (1). with AP = 0 and An = uLs?T. **The NS-200 membrane degraded during DO testing.

(10e2 cm) ( lo-’ cm’

SW!

0.97 1.5 0.94 2.5 1.1 0.057 2.4 0.35

2.0 0.43 0.41 1.2

0.035 0.0015 0.033 0.030

Table 2 may be compared with the free solution diffusion coefficient of NaCl in water of about 1.5 X lo-’ cm’/sec at room temperature [14]. Clearly, for some of these membranes, the resistance to salt flow in the sub-

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159

&rate is substantial. That is particularly true of the composite membranes, and presumably reflects the way in which these membranes are made.

Some of the DO data require comment. First, two of the membranes, CA-80 and PBIL, exhibited higher fluxes in DO than were projected from RO data. We believe this to be the result of the compaction phenomenon in RO [ 151. That is, the A value obtained at 1200 psi appears to be lower than the value that would be obtained with no pressure applied to the membrane.

A second noteworthy feature of the DO results is the distinct non- linearity of the water flux vs. salt concentration plots. This, of course, is a direct result of internal concentration polarization and is predictable from our analysis. It should be noted, however, that increasing salt concentration in DO (and, therefore, in PRO) does not produce a proportional increase in water flux.

A further point concerns the NS-200 data. These membranes are known to be somewhat unstable. In the course of our DO measurements, the water flux changed with time. The value reported in the table is therefore suspect, by perhaps as much as a factor of 2.

a -

7 -

6 -

5 - Water FlUX, Jw 4 -

3 -

2 -

l-

o- 5 10 15 20 25

NaCl Concentration (wt%)

Fig, 10. Direct osmosis water flux vs. salt concentration for composite membranes.

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As a measure of overall membrane performance, we have calculated An,ff/A R as a function of J, from eqn. (9) for all eight membranes. This is a measure of the effectiveness of utilizing the available salinity gradient in DO or PRO; for an ideal semipermeable membrane, An,ff/An = 1.0. The re- sult is plotted in Fig. 11. Clearly, the most effective membrane on this basis is CA-80. The CA-70 membrane is less effective because of salt leakage through the salt-rejecting barrier. The composite membranes are ineffective because of severe internal concentration polarization.

1.0

0 .8

0.6

AT! eff ATI

0

I I I 1

CA-80

I 1 I I 1 ,

0 2 4 6 8 10

Water Flux, .Jw

Fig. 11. The quantity Ane~/An plotted against water flux for all eight membranes, from eqn. (9). The observed DO water flux with 3.5 wt% NaCl is shown by the filled circle for each membrane.

C. Projected PRO performance With A, B, and K from RO and DO measurements, we are now in a posi-

tion to calculate the PRO performance for these membranes. The results are presented in Table 3 for a PRO system of 3.5 wt% NaCl vs. fresh water. We chose 3.5 wt% NaCl as the brine because it simulates normal seawater rather well. The curves labelled Dl and D2 in Fig. 5 have been calculated from eqn. (11) from the A, B, and K data for the PA-300 and CA-80 mem- branes, respectively.

The CA-80 membrane is superior. While the CA-70 membrane has a high projected water flux in PRO, the optimum operating pressure is low and the maximum power output per unit membrane area is relatively low. Most of the composite membranes have very poor projected performance. The

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TABLE 3

Calculated PRO performance of various membranes with 3.5 wt% NaCl vs. fresh water (An = 28.1 atm)

Membrane aP,* Optimum PRO PRO fhrx at optimum IV,,

A?7 operating pressure pressure, AP, 12 lo+ cm3

(atm) cm2-set

Asymmetric CA-80 0.89 12.5 1.23 1.55 CA-70 0.23 3.3 1.05 0.35 BM-05 0.69 9.7 0.31 0.31 PBIL 0.82 11.5 0.68 0.79

Composite PA-300 0.51 7.2 0.23 0:17 NS-101 0.07 1.0 0.03 0.003 NS-200** 0.7 10 0.7 0.7 BM-1-C 0.29 4.1 0.17 0.07

*From eqn. (18). **Gradual degradation during testing. Only one significant figure is reported.

NS-200 membrane is the exception, but the data are somewhat suspect, as noted earlier.

D. Analysis of literature data on PRO Using the model developed in this paper, it is possible to analyze the

limited number of PRO dat.a available in the literature. Most of the reliable data were generated by Loeb and Mehta [16-181 who studied several types of hollow fiber RO membranes. These included the DuPont Permasep@ B-9 and B-10 polyamide fibers and composite hollow fibers of the NS-200 chemistry, prepared by FRL (formerly Fabric Research Laboratories) of Dedham, MA. In general, the water fluxes through these membranes in PRO were significantly lower than expected from the water permeation coeffi- cient A measured in RO. Furthermore, the PRO water flux increased .only marginally with substantial increases in the osmotic pressure of the brine. In the case of the FRL hollow fibers, water was found to flow from the salt side into the water side under certain PRO conditions, where the hydrostat- ic pressure was well below the osmotic pressure of the salt solution. All of these observations are explicable with our model.

We limit our analysis to the most complete set of data available, namely that for the FRL fibers [ 17,181. Loeb and Mehta measured water fluxes through these fibers under RO, PRO, and DO conditions. Water was used on the dilute side in all of the tests: thus C4 = 0. The observed and calculated water fluxes are shown in Table 4. The calculated values were ob tamed using eqn. (11) and assuming a value for the substrate salt resistance

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162

TABLE 4

Analysis of the PRO data of Loeb and Mehta, using the model presented in this paper

Data from Loeb and Mehta [1’7,18]*

Run number

RO-1 2 54 -4.5 -3.1 RO-2 2 53 -3.1 -3.1 PRO-3 100 58 -0.37 -0.40 PRO-4 60 50 -0.89 -0.91 PRO-5 60,. 3 1.3 0.92 PRO-6 20 3 0.07 0.31 PRO-7 100 19 0.81 0.81 DO-8 43 0 1.1 0.83 DO-9 153 0 0.96 1.7 RO-10 4 51 -3.0 -2.9

Observed J,

(y.;_::y

Calculated J, **

(‘:::_g

*In cases where a range of water flux values was given, the highest value is shown here. **We have used eqn. (10) and these values: A = 0.6 X lo-’ cm3/cma-set-atm, K = 1.3 x lo4 set/cm, and B = 9.6 x lo-’ cm/set.

factor, K, of 1.3 X lo4 set/cm, equal to that obtained in our own DO experiments with flat sheet NS-200 membranes*. The value of B used was 9.6 X lo-’ cm/set, obtained from Loeb and Mehta’s salt flux data at the end of their PRO experiments. (The salt flux increased during the experi- ments by almost a factor of two.) Despite the crude assumptions used, the overall agreement between the observed and calculated water fluxes is good. Except for run number PRO-6, the predicted water flux is within a factor of two of the observed value. More significantly, the direction of the water flow is correctly predicted in every case, including the case of negative flow in PRO.

E. Salinity gradient resources Before turning to a discussion of the economics of power generation by

PRO, it is useful to consider the salinity gradient resources potentially available in the world.

1. Seawater/Fresh Water The largest salinity gradient resource is the seawater/fresh water scheme

exemplified by a river terminating at the sea. The size of this potential re- source is enormous [4]. However, it is a relatively low-grade resource, because the osmotic pressure of seawater is only about 25 atmospheres. Thus, even with highly semipermeable membranes, the PRO plant would operate at a

*K may also be obtained by using eqns. (15) or (16).

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163

seawater pressure of only about 12 atm (AT/~). Water fluxes under these conditions would be small, and the capital cost of the system would be cor- respondingly high.

2. Brine/fresh water The osmotic pressure of NaCl brines can reach 350 atm. Thus, as shown

in eqn. (3), the power produced per unit membrane area using these solu- tions could in principle be much higher than with seawater. On the other hand, although this is a much higher-grade resource, the potential quantity of energy extractable is much smaller, because concentrated brines are generally only found in deserts where the available fresh water is both limited and valuable. Wick and Isaacs [19] have recently suggested using salt domes as a brine source, in which case the plants could be located closer to available fresh water sources. The diluted brine effluent could pose a significant disposal problem. Alternately, the dilute brine could be re- concentrated using thermal energy, in a modified or closed-loop PRO system, as has been suggested [20].

3. Brine/seawater or brine/brackish water One possible method of increasing the utility of brine lakes as a salinity

gradient resource is to use seawater or brackish water on the dilute side of the membrane. There are several locations in the southern and southwestern United States where this would be feasible. The diluted brine would be re- cycled to the brine lake to the reconstituted by solar evaporation. This is an attractive approach if the internal concentration polarization effects de- scribed earlier could be controlled. This implies using membranes with good salt rejection and very open porous substrates.

Experiments aimed at determining the effect on osmotic flux of the salt concentration on the dilute side of the membrane were performed using PBIL and PA-300 membranes. The high concentration side of the mem- brane was maintained at 23-24 wt% NaCl, and the dilute side salt con- centration was varied. The results are shown in Figs. 12 and 13. The flux through each membrane decreased markedly with increasing salt concentra- tion on the dilute side. The solid lines in Figures 12 and 13 represent the water fluxes calculated from eqn. (lo), based on the values of K deter- mined in our brine/fresh water experiments. Clearly, the good agreement between the observed and predicted fluxes supports the validity of our present model.

F. Economics of PRO power generation The merits of PRO as an alternative power generating process are

ultimately based on the cost of generating power. The operating costs of a PRO power plant are likely to be relatively low because the process uses a renewable resource. Capital costs, however, are likely to be high because of the auxiliary equipment required such as pressure vessels, pumps, and hydro-

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Wate

r Fl

ux,

A7

20

15

10 5 0

10 8 6

t

1

(E)

(5%

)

4-

2-

o-

0

5 10

15

20

25

5

10

15

20

Conc

entr

atio

n of

the

wea

ker

brin

e, C

4 Co

ncen

trat

ion

of t

he d

ilut

e so

luti

on

(% NaC

l)

(% NaC

l)

Fig

. 12

. E

ffec

t of

di

lute

-sid

e sa

lt c

once

ntr

atio

n

on

the

DO

wat

er

Fig

. 13

. E

ffec

t of

di

lute

-sid

e sa

lt c

once

ntr

atio

n

on

the

DO

wat

er

flu

x th

rou

gh

a P

BIL

m

embr

ane.

C

once

ntr

ated

so

luti

on

=

flu

x th

rou

gh

a P

A-3

00

mem

bran

e.

Con

cen

trat

ed

solu

tion

=

22.

7%

24.3

w

t%

NaC

l. T

he

soli

d li

ne

was

cal

cula

ted

from

eq

n.

(10)

. N

aCl.

Th

e so

lid

lin

e w

as c

alcu

late

d fr

om

eqn

. (1

0).

Th

e po

ints

T

he

poin

ts

are

expe

rim

enta

l. ar

e ex

peri

men

tal.

1 27

-

1 O_

0 8_

Wate

r Fl

ux,

Jw

o 6-

0.

0.

0.

0. 0. 0.

1 I

I I

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turbines. For simplicity, therefore, we can assume that only capital costs need be considered in this preliminary economic analysis. Specifically, we assume that a PRO system would be operated to produce the maximum power per unit area of membrane.

The capital cost can be estimated from the following relationship:

Installed Membrane Cost ($/m’) X lo3 (

watt

Capital Cost = kilowatt

($/kilowatt) Power Output of Membrane

(22)

The “installed membrane cost” in this case includes all equipment costs, and is substantially higher than the replacement cost of the membrane alone. The capital cost of the PRO process is therefore determined by three factors: (a) the salinity gradient resource used, (b) the membrane character- istics A, B, and K (factors (a) and (b) determine the operating pressure and the power output per unit area), and (c) the installed membrane cost, which is a function of the system design used. The effect of installed mem- brane cost on the capital cost based on eqn. (22), is shown in Fig. 14 for a range of membrane power outputs. The range of installed membrane costs has been chosen to bracket the cost of RO desalination installations, typical- ly about $ 100-200/m2. In conventional nuclear or hydroelectric power generating plants, as in a potential PRO power plant, operating costs are a small fraction of the total power costs [21] so that a comparison of capital

Capital Cost (Slkw)

0 1 2 3 4 5 Power output of th 5 PRO membrane (watts/m )

Fig. 14. Capital cost of a PRO plant vs. power output of the membrane, for various in- stalled membrane costs.

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166

Seawater / Fresh water Case

106

Minimum Capital cost (Slkw)

105

104

CA-86 N 50

10 5

1.

B*K

I I I I allons 10-l 1 lo1 102 (” ) ft2-day

PRO Water Flux, J,

Fig. 15. Minimum capital cost of a PRO system vs. water flux for a 3.5 wt% NaCl solution/fresh water salinity gradient resource and for various values of internal con- centration polarization. The projected PRO water fluxes (0) of the membranes studied are shown. Assumptions: Installed Membrane Cost = $ 100/m 2; An = 28.1 atm; AP = APO/2 for all membranes.

costs alone should be a valid method of comparing these systems. Nuclear and hydroelectric plants currently have capital costs on the order of $ 1000 to $ 1500 per kilowatt [ 221. The best of the membranes listed in Table 3 had a projected power output of only 1.6 watts/m*, and it is clear that PRO with a seawater/fresh water salinity gradient resource would not be economically competitive even at the lower limits of installed membrane costs.

As an alternative method of evaluating the membranes tested in this work for their potential applicability in PRO, the capital cost is shown as a function of membrane properties in Fig. 15. The parameter B *I( has been used to characterize the degree of internal concentration polarization in the

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167

membrane. If B = 0, the membrane is ideally semipermeable and there is no internal concentration polarization. Increasing B l K decreases the optimum operating pressure in PRO and increases the capital cost. The other principal factor affecting capital cost, of course, is the water permeation co- efficient, A. With all other factors equal, increased A increases the water flux and hence decreases the capital cost. The data points in Figure 15 rep- resent projected PRO water fluxes for the assumed conditions, calculated from Eqn. (11) and plotted against the known B *K values.

The CA-80 membrane had the highest projected water flux and the lowest capital cost of all the membranes tested*. Even in this best case, and assuming a plant cost of $ 100/m* of installed membrane, the minimum capital cost still exceeds $ 60,000 per kilowatt. As shown in Fig. 15, other membranes have even higher projected capital costs. It would appear that, for PRO to be economically feasible with a seawater/fresh water salinity gradient resource, a membrane is required which is relatively impermeable to salt, has an open substrate (so that B-K is small), and has a water perme- ability more than an order of magnitude higher than that of the CA-80 membrane.

Figure 16 depicts a similar economic analysis based on a concentrated brine/fresh water salinity gradient resource. The osmotic pressure of a 25% NaCl brine is 353 atm at 25°C. Asymmetric cellulose acetate and PBIL membranes are expected to give high power outputs because of their low internal concentration polarization. The capital cost estimates are favorable for these membranes.

V. Conclusions

This paper describes the results of a feasibility study of PRO as a method for generating power from salinity gradient resources. A number of flat- sheet desalination membranes were evaluated for their potential use in PRO. RO and DO experiments were used to predict the performance of the mem- branes under PRO conditions. Based on this work, we conclude:

(1) PRO power generation is technically feasible, but not economically viable with currently available RO membranes and a seawater/fresh water salinity resource.

(2) The properties of a membrane under PRO operating conditions can be calculated using a model which takes into account internal concentration polarization. The required parameters - the water permeation coefficient A, the salt permeation coefficient B, and the resistance to salt transport in the membrane substrate K - can all be obtained from RO and DO experiments.

(3) Concentration polarization is a major problem in PRO. External con-

*The CA-80 membrane used in our studies exhibited RO performance somewhat inferior to that of present commercial cellulose acetate RO membranes. This does not change our conclusions significantly.

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Brine (25% NaCl) / Fresh water Case

105 Minimum

Capital Cost ($/kw)

10cc

lo3

102

1 O-6

I I 1

lo B-K 5

10-5 1O-4 10- j 10-2 (cm3/cm2 -set)

I I I I

10-l 1 lo1 lo2 PRO Water Flux

Fig. 16. Minimum capital cost of a PRO system vs. water flux for a 25 wt% NaCl brine/ fresh water salinity gradient resource and for various values of internal concentration polarization. The projected PRO water fluxes (e) of the membranes studied are shown. Assumptions: Installed Membrane Cost = $ loo/m2 ; Arr = 353 atm; AP = APO/2 for all membranes.

centration polarization occurs in the liquid boundary layers on either side of the membrane. External concentration polarization can be minimized by stirring the solutions to reduce the thickness of these boundary layers.

(4) Internal concentration polarization occurs in the porous substrate of asymmetric membranes and is unaffected by stirring. Internal concentration polarization can only be reduced to an acceptable level by using membranes with an open substrate. Without due regard for internal concentration polar- ization, it is unsafe to project PRO performance from RO performance.

(5) Of the existing flat-sheet RO membranes, cellulose acetate membranes of the Loeb-Sourirajan type give the best results because their open micro-

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169

porous substrate minimizes internal concentration polarization. Conventional interfacial composite membranes, despite their high water permeabilities and good salt rejections, are not suitable for PRO because of severe internal con- centration polarization. Hollow fiber membranes were not evaluated in this study. They are known to exhibit lower water fluxes in RO than flat-sheet membranes but their lower cost per unit area may compensate for the lower flux.

(6) Useful PRO membranes do not require the very high permselectivity necessary in reverse osmosis, and a trade-off between flux and salt rejection in conventional RO membranes is possible. If the salt rejection is too low, however, internal concentration polarization due to excessive salt leakage can limit the water flux. PRO membranes should have the combination of low salt permeability in the skin layer and high rate of salt diffusion in the porous substrate to avoid internal concentration polarization.

(7) Because the operating pressures of PRO systems with a seawater/fresh water salinity gradient resource can be lower than those typically used in RO, compaction of the membranes due to the hydrostatic pressure gradient can be smaller in PRO. In this instance, PRO fluxes in the absence of con- centration polarization can be higher than high pressure RO data would suggest.

(8) Based on a simple economic analysis, it appears that, when seawater is used as the salt solution, a membrane with a water flux in PRO of about 1 X lo-’ cm3 /cm*-set (-200 gal/ft2-day) is required to make the process economically viable in today’s economy, even if the installed membrane cost is as low as $ 100/m2. The highest flux we projected under these PRO conditions among the RO membranes tested was only somewhat greater than 1 X lOA cm3/cm2-see (“2 gal/ft2-day).

(9) The economics of PRO systems using brines and fresh water sources and current membranes are more favorable. However, surface brines exist in deserts where there is limited fresh water, and brines that might be pro- duced from salt domes pose a difficult effluent disposal problem. Surface brines using seawater or brackish water as the dilute solution can be used only with Loeb-Sourirajan type asymmetric membranes with open porous substrates and thus small internal concentration polarization effects. If PRO systems can be produced at an installed cost of $ 100/m2 of membrane, the projected economics are competitive with other power-generating tech- niques. This appears to be the only salinity gradient resource worthy of further study.

Acknowledgement

This work was supported in part by the U.S. Department of Energy under Contract EG-77-C-05-5525.

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Symbols

A B Ci

C(x)

De

DS JS JW J&

K AP AP, t W W max

E xi A?7 A neff

iSAIr

7

Water permeation coefficient; cm3 /cm2-set-atm. Salt permeation coefficient; cm/set. Salt concentration at location i: 1 - bulk salt solution, 2 - brine side solution-membrane interface, 3 - boundary between salt- rejecting skin and porous substrate, 4 - membrane-dilute solution interface, 5 - dilute-side bulk solution; g/cm3. Salt concentration at distance x from the dilute solution-membrane substrate interface; g/cm3. Effective diffusion coefficient of salt in the porous substrate of the membrane; cm2 /sec. Diffusion coefficient of salt in the membrane substrate; cm2/sec. Salt flux; g/cmz-sec. Water flux; cm3 /cm2 -sec. Water flux at the optimum PRO operating pressure, AP,/2; cm3 /cm2-sec. Porous substrate resistance to salt diffusion, equal to ~t/D,e; set/cm. Hydrostatic pressure difference across the membrane; atm. Flux-reversal pressure; atm. Thickness of the porous substrate; cm. Power generated in PRO per unit membrane area; watts/m’. Maximum power generated in PRO per unit membrane area; watts/m2. Porosity; dimensionless. Osmotic pressure ofthe solution at location i; atm. Osmotic pressure difference across the membrane; atm. Effective osmotic pressure difference across the salt-rejecting layer of the membrane; atm. Effective osmotic pressure difference in reverse osmosis, equal to AP when Jw = 0; atm. Tortuosity; dimensionless.

References

1 S. Loeb, Osmotic power plants, Science, 189 (1974) 350. 2 H.H.G. Jeiiinek, Osmotic work I. Energy production from osmosis on fresh water/

saline water systems, Kagaku Kojo, 19 (1975) 87. 3 S. Loeb, Production of energy from concentrated brine by pressure-retarded osmosis.

I. Preliminary technical and economic correlations, J. Membr. Sci., 1 (1976) 49. 4 G.L. Wick, Energy from salinity gradients, Energy, 3 (1978) 95. 5 A.J. Staverman, Nonequilibrium thermodynamics of membrane processes, Trans.

Faraday Sot., 48 (1952) 176. 6 G.D. Mehta and S. Loeb, Internal polarization in the porous substructure of a semi-

permeable membrane under pressure-retarded osmosis, J. Membr. Sci., 4 (1978) 261. 7 H.K. Lonsdale, U. Merten, and R.L. Riley, Transport properties of cellulose acetate

osmotic membranes, J. Appl. Polym. Sci., 9 (1965) 1341.

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171

8 H.K. Lonsdale, Recent advances in reverse osmosis membranes, Desalination, 13 (1973) 317.

9 S. Manjikian, S. Loeb, and J.W. McCutcban, Proc. First international symposium on water desalination, Washington, D.C., 1965 (published by U.S. Department of the Interior, Office of Saline Water, Washington, D.C.), Vol. 2, 1965, pp. 159-173.

10 R.L. Goldsmith, B.A. Wechsler, S. Hara, K. Mori, and Y. Taketani, Development of PBIL low pressure brackish-water reverse osmosis membranes, Desalination, 22 (1977) 311.

11 R.L. Riley, R.L. Fox, C.R. Lyons, C.E. Milstead, M.W. Seroy, and M. Tagami, Spiral- wound poly(ether/amide) thin film composite membrane systems, Desalination, 19 (1976) 113.

12 J.E. Cadotte, K.E. Cobian, R.H. Forester, and R.J. Petersen, Continued evaluation of in situ formed condensation polymers for reverse osmosis membranes, Final report, Office of Water Research and Technology, Contract No. 14-30-3298, PB 253 193; April 1976.

13 S. Sourirajan, Reverse Osmosis, Academic Press, New York, 1970, Appendix II. 14 R.A. Robinson and R. Stokes, Electrolyte Solutions, 2nd edn., Butterworths,

London, 1959. 15 H.K. Lonsdale, Theory and practice of reverse osmosis and ultrafiltration, in

R.E. Lacey and S. Loeb (Eds.), Industrial Processing with Membranes, Wiley- Interscience, New York, 1972, Ch. VIII.

16 G.D. Mehta and S. Loeb, Performance of Permasep B-9 and B-10 membranes in various osmotic regions and at high osmotic pressures. J. Membr. Sci., 4 (1979) 335.

17 S. Loeb and G.D. Mehta, A two-coefficient transport equation for pressure- retarded osmosis, J. Membr. Sci., 4 (1979) 351.

18 S. Loeb, Solar electricity: Investigation of technological problems of osmo-hydro power systems which produce power from saline solutions. Final Report, U.S. Department of Energy, Contract No. EG-77-G-01-4006, March 1978.

19 G.L. Wick and J.D. Issacs, Salt domes: Is there more energy available from their salt than from their oil? Science, 199 (1978) 7436.

20 M.D. Fraser, S. Loeb, and G.D. Mehta, Assessment of the potential of generating power from aqueous saline solutions by means of osmo-hydro power systems, paper presented at the 13th.Intersociety Energy Conversion Engineering Conference, San Diego, CA, August 20-25, 1978.

21 D.M. Considine (Editor-in-Chief), Energy Technology Handbook, McGraw-Hill, New York, 1977, sec. 5, p. 10 and sec. 8, p. 8.

22 A.D. Rossin and T.A. Riech, Economics of nuclear Power, Science, 201 (1978) 582.