Journal of Membrane Science, 1(1976) 49-63 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

PRODUCTION OF ENERGY FROM CONCENTRATED BRINES BY PRESSURE-RETARDED OSMOSIS

I. PRELIMINARY TECHNICAL AND ECONOMIC CORRELATIONS

SIDNEY LOEB

Research and Development Authority, Ben-Gurion University of the Negev, Beer Sheva (Israel)

(Received July 18, 1975; in revised form August 30, 1975)

Summary

In this, the first of two papers, preliminary technical and economic correlations have been made on the production of energy from concentrated brines by Pressure-Retarded Osmosis (PRO).

In PRO, the hydraulic pressure is less than the osmotic pressure, so that water flux is against the hydraulic pressure gradient, this fact being the basis for energy production. It is visualized that a PRO power plant, operating continuously, would pump a concentrated brine such as Dead Sea brine on the outside of the fibers in hollow fiber modules at a high hydraulic pressure. Simultaneously, Jordan River water or brackish springs would be pumped through the inside of the fibers, and would permeate through the fiber wall into the pressurized brine. The permeate-enhanced brine would be depressurized through a hydroturbine, delivering net power equal to the high hydraulic pressure x the permeate rate, in the idealized case.

A transport analysis indicated that an asymmetric membrane, with the brine on the skin side, is as desirable in PRO as it is in reverse osmosis from the standpoint of high water permeation flux.

The mechanical efficiency of a PRO power plant was examined. The efficiency is very sensitive to V/A V, the ratio of concentrated brine rate to permeate rate, a low value of this ratio (less than 2) being desirable.

A preliminary analysis was made of the unit cost of energy production from a PRO plant. The results are expressed as a function of V/A V, the water flux, and the hydraulic pressure used. This analysis indicated the desired course of experimentation to obtain data for calculation of the minimum unit energy cost.

In the second and concluding paper, experimental results will be given from which the validity of correlations given here can be estimated, and unit energy costs will also be given.

I. Introduction

It is known from thermodynamics that, in principle, useful energy can be made available when a dilute and a concentrated solution are mixed. The problem is to convert this free energy of mixing to useful energy by economic means. It is believed that this can be accomplished by appropriate application

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of an osmotic process, Pressure-Retarded Osmosis (PRO), to the mixing of a concentrated brine, such as exists at the Dead Sea or Great Salt Lake, with fresh or saline water from a river, sea or underground source. In this service PRO is essentially a solar energy conversion process since it depends only on the sun for the production and continuous replenishment of the concentrated brine. Other sources of concentrated brine exist in the numerous salt moun- tains scattered throughout the world.

It is the purpose of this project to determine whether energy can be pro- duced economically by pressure-retarded osmosis. The results obtained thus far will be published in two parts, of which this is the first. These preliminary technical and economic correlations can be used to obtain unit costs of energy, providing certain indicated experiments are performed. The experimental work and the energy costs obtained will be given in the second part: Exper- imental results and projected energy costs.

II. Principles of pressure-retarded osmosis

A. Pressure-retarded osmosis explained In Fig. 1A is shown a solution separated from its solvent by a semipermeable

membrane. Under such conditions the solvent naturally flows from the solvent side of the membrane to the solution side in the process of osmosis. The os- motic flow process can be stopped if a sufficiently high hydraulic pressure is applied to the solution side (Fig. 1C). The hydraulic pressure required to maintain this osmotic equilibrium is called the osmotic pressure, ?I, and is determined completely by the concentration and composition of the solution, i.e. not by the membrane.

OSMOSIS

PRESSURE - RETARDED OSMOTIC OSMOSIS EQUILIBRIUM

J SEMlPERMEb

t MEMBRANE

I O<P<S , I

iBLE

(A) (B1 (Cl

Fig. 1. The place of pressure-retarded osmosis (PRO) in osmotic processes.

Solvent permeation rate through the membrane is a function of both n and the hydraulic pressure, P, applied to the solution, where P can have any value from zero to P > A. The solvent permeation rate (always involving water) is most conveniently expressed as flux, JI, the volumetric permeation rate per unit area, and the relation between the solvent permeation flux and the applied forces in this ideal case is:

Ji = A (n -P), (1)

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where A is the water flux constant and is characteristic of the membrane*. It is clear from eqn. (1) that if P = 0 we have osmosis, for which J1 = An,

and that if P = a we have osmotic equilibrium, i.e. J1 = 0. However, if P is between zero and n we have pressure-retarded osmosis (Fig. lB), so defined because the direction of water flux is still the same as in osmosis but the flux is decreased as the hydraulic pressure on the solution is increased**. It is im- portant to understand that in pressure-retarded osmosis, solvent flows against the hydraulic pressure gradient, i.e. the flow is uphill from a hydraulic pres- sure standpoint. By this means the free energy of mixing can be converted to potential mechanical energy, and therefore PRO can be used in power produc- tion, as discussed in the next section.

B. Application of pressure-retarded osmosis to power production It is believed that PRO will be the most economical osmotic power produc-

tion process. However, as shown in Fig. lB, PRO suffers from the fact that it is an intermittent or batch-type operation due to the dilution of the solution by the solvent. This would necessitate frequent solution replacement, with all that this entails in terms of periodic reduction of power, complicated valving equipment for changing solutions, etc.

For this reason large-scale osmotic power production is best carried out in a continuous and steady-state fashion, and these objectives are best attained as shown schematically and ideally in Fig. 2. A concentrated brine such as Dead Sea brine (DSB) having an osmotic pressure, :nsB, acquires a hydraulic pressure, Pbp , by passage through an appropriate brine pump. The power input to the pump is P,,,, V, where V is the volumetric flow rate of DSB through the pump. The pressurized brine then enters the PRO permeator on one side of a train of membranes. Simultaneously, Jordan River water, brackish spring water, or some other dilute solution enters the permeator on the other side of the membrane train at osmotic and hydraulic pressures that are very low in comparison to these quantities on the DSB side.

Permeate passes from the low to the high hydraulic pressure side of the membranes, at a volumetric flow rate A e, acquires a hydraulic pressure of Pbp atmospheres, and mixes with the DSB. The mixed brine, having an osmo- tic pressure, rMB, intermediate between that of DSB and the dilute solution, emerges from the high pressure side of the membrane train at an enhanced volumetric rate of (v + A e) and enters a hydroturbine in which the hydraulic pressure, still PM, is reduced to zero in the course of delivering power of Pbp (v + A fi. The net power delivery in this ideal case is the difference between this quantity and the power input to the brine pump:

netpower(ideal) = Pbp(e+Afi-Pp,,v = P,Ao.

*A list of symbols is given in the Nomenclature section on p. 63. **This is in contrast to reverse osmosis, a well-developed technology for which P is greater than n, and in which an increase in P increases the flux.

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BRINE PUMP

Dead P=O sea

nDSB Brine (DSB)

P ’ bp ”

PRO PERMEATOR

- Semipermeable membranes

Jordan River water or

brackish Springs

Fig. 2. Continuous pressure-retarded osmosis, idealized. 1. Efficiency of rotating compo- nents, 100%. 2. No friction losses in plant streams. 3. Membranes perfectly semipermeable.

The now depressurized mixed brine is returned to the Dead Sea. Thus, finally, the same condition vis-a-vis the Dead Sea is achieved as with the present irreversible mixing of DSB and Jordan River estuary water or brackish springs. The important difference is that, by use of PRO, a large fraction of the free energy of mixing has been converted to useful energy. The above explanations of PRO and its application to power production are deliberately simplified for clarity. More detailed and realistic expositions are given later. These expositions are largely based on the use of the asymmetric hollow fiber membrane, discussed in the next section.

III. Pressure-retarded osmosis with hollow fiber permeators

In PRO, liquid streams must flow freely on both sides of and in close inter- facial contact with a membrane across which a large hydraulic pressure dif- ference exists. These dual requirements are best met by means of the hollow fiber membrane, which is fabricated to be essentially a thick-walled tube in terms of the ratio of outside to inside diameter. In addition, hollow fibers are usually made very fine. By this means a large surface-to-volume ratio is obtained, which is important if a containing pressure vessel is used. Further- more, the wall thickness of such fine fibers can be reasonably thin, even though the fiber is classified as a thick-walled tube. A thin fiber wall is desir- able to minimize resistance to permeate flow across it.

A hollow fiber meeting many of these requirements already exists, the du Pont Permasep hollow fiber. The Permasep B-10 fiber, made at present for

53

sea water desalination by reverse osmosis (RO), is fabricated from an aromatic polyamide polymer to have outer and inner diameters of 98 and 42 microns, respectively. The fiber is asymmetric, the skin being on the outside. The asymmetry is of paramount importance in RO, and, as shown by test results, in PRO also. The reasons for this are explained in the transport analysis of the next section.

IV. Water transport during pressure-retarded osmosis through an asymmetric membrane

The discussion in Section II-A on the principle of PRO was presented in an idealized manner. In the actual case, the liquid flowing on the water-donor side of the membrane will usually possess both hydraulic and osmotic pressure. The latter can come from salt in the dilute solution entering the apparatus, or by diffusion through the membrane, which will not be perfectly semipermeable.

The external driving forces and the directions of fluxes in PRO are shown in Fig. 3, for a short section of an asymmetric hollow fiber membrane. A con- centrated brine flows axially on the outside (shell side, skin side, brine side) of the fiber at osmotic and hydraulic pressures of R,~ and P,, , respectively. Simultaneously, a dilute solution flows axially through the inside (bore side, porous substructure side, water side) of the fiber at osmotic and hydraulic pressures of n,_, and Pbo, respectively. This dilute solution provides water for permeation and also water for flushing. The flushing solution emerges from

Brine supply on shell side at ‘l(,,, and Psh

I7 sh p,h , ksh = k$ ' + v,(psh-nsh)

Fiber WOll

-- water supplj for permeation and solt flushing, plus

Porous substructure

skin Brine supply

asymmetric hollow fiber

on shell Side ot 77Sh and PSh

Fig. 3. Fluxes and forces in pressure-retarded osmosis.

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J,, water flux Transition interface

X=0 X=Xtr”AX X=AX

I,(diffusion) = -D_ dCzps/dX

k AX -I

Fig. 4. Transport conditions in pressure-retarded osmosis through an asymmetric membrane.

the exit of the bore side of the fibers, carrying with it any salt that entered the bore side either at the bore side inlet or by permeation through the fibers.

As shown in Fig. 3, the osmotic pressure (concentration) on the shell side, rsh, will be greater than that on the bore side. Second, the hydraulic pressure on the shell side, Psh, will be greater than that on the bore side, Pbo, i.e. the direction of decrease of hydraulic pressure gradient is from the shell side to the bore side. Finally, the chemical potential of water on the bore side, pIb, must be greater than that on the shell side, PI&, so that water permeates from the bore side to the shell side. In short, salt and water permeate in opposite directions, and water permeates against the hydraulic pressure gradient.

The magnitude of the water flux will be strongly influenced by the fact that an asymmetric membrane is used, and that the concentrated brine is on the skin side of the membrane. The situation will be as shown in Fig. 4. The water flux through the skin will be described by the reverse osmosis flux equa- tion:

JI = A[‘T,h - (Psh - pbo) - ntrl , (2)

where A is the water flux constant in RO and is a function of the properties of the skin only, and 7rtr is the osmotic pressure at the transition region be- tween the skin and the porous substructure.

As stated above, water permeates against the direction of the hydraulic pressure gradient. The consequences of this are two-fold. First, the hydraulic pressure drop in the membrane is entirely across the semipermeable skin. This fact is expressed in eqn. (2) by the use of Pbo as equivalent to P,, . Second, in the porous substructure, the transport of water is by diffusion only (i.e. there

55

is no laminar flow of water)*. The porous substructure is seen to have the character of a boundary layer, in which water flux is a function of concentra- tions and of concentration gradients. Assuming for simplicity that the net salt flux, J2, is negligible, we express the equal and opposite values of water- coupled salt flux and diffusional salt flux as:

J2 (net) = 0 = J1 czps - ~ZgsdcZps /d&, , (3)

where clPS is the salt concentration in the porous substructure at a distance X,, from the bore side interface, and D,, is the salt diffusion coefficient in the porous substructure.

Equation (3) may be written:

_ dX,, = dCZps . Jl

D 2P.3 czps

(4)

This equation determines that X,, = 0 must be located at the bore side inter- face, i.e. at the lowest value of c2. Because Dzps, J,, and c2 are all positive, dX and dc2 must have the same sign, i.e. as c2 increases so must X.

Equation (4) is integrated to give:

JI Xps - = In c2Ps + K . D 2PS

When X,, = 0, ln czPS = ln c2b,, . Therefore the constant K = -In c2b0 and:

Jt Xps = lnC2pS, D as c2bo

When X,, = X,, czps = czti . This is written as:

J1A.X D

=lnCZt’ , as c2bo

(5)

(6)

(7)

where I have also expressed the fact that Xt, is approximately equal to AX, the total thickness of the membrane.

Assume that c2 is proportional to 71. Then:

J,AX =lnL!&.-. D 2PS 71bo

(8)

Equation (8) is written in exponential form and substituted in eqn. (2) to eliminate 7rtr :

JI = A[n,h - (ps~, - pbo ) - nbo =P (JI Ax/D, p)b (9)

*Even a relatively small hydraulic pressure gradient in the porous substructure would cause a laminar flow in the direction of pressure decrease greater than that of diffusional flow in the reverse direction. Hence the two consequences.

5 6

Equation (9) reveals that, in order to have a high value of the water flux, J,, it is desirable that AX/Dzps and nbO are not excessive.

Under the limiting condition that xbO + 0 we have:

JI = A[nsh - (psh - pbo)] . (10)

Equation (10) is essentially the idealized equation for PRO given as eqn. (1). It is also the equation for water flux in reverse osmosis, except for a change in sign. The equal utility of eqn. (10) in both RO and PRO is subject to experi- mental verification as described later and, if true, is very encouraging from the standpoint of achieving economic unit energy costs.

It is clear why an asymmetric membrane is desirable in PRO under condi- tions such that eqn. (10) is valid. The resistance to water permeation is virtu- ally entirely in the semipermeable part of the membrane. Thus for a thicker semipermeable skin, as for example if an all-skin membrane is used, it can be expected that the water flux in PRO will be lower than if an asymmetric membrane is used.

V. Net power delivery and mechanical efficiency of a PRO power plant

In accordance with the idealized plant of Fig. 2, the net power delivery from a PRO plant would be pbp AQ. Owing to inefficiencies of rOtating com- ponents, to friction losses in the various streams, and to the fact that the membranes will not be perfectly semipermeable, the real power output will be considerably less than this. The more realistic situation is shown in Fig. 5. The concentrated brine such as DSB first passes through the brine p U m p in which hydraulic pressure is raised to Pbp. The power input to the brine pump is given by:

Power in 1 bp= Pbp P/Ebp , (11)

where E, is the brine pump efficiency expressed as a fraction (note that the brine pump is directly coupled to the hydroturbine).

DSB now mixes with the recirculating brine before entering the PRO permeator. Recirculation will probably be necessary, in spite of the thermo- dynamic mixing loss, because straight DSB will damage the membranes. The power input to the recirculating pump is given by:

POWfIX in i ( rg = Pbp - P,,F) R @/Em , (12)

where F is the fractional friction loss ratio in the PRO permeator and is given by the ratio of the hydraulic pressure of the brine leaving the permeator to that entering, R is the ratio of the recirculation brine rate to the DSB rate, and E, is the recirculating pump efficiency.

The DSB and recirculating brine now enter the PRO permeator on the shell side of the fibers at a composition approximately equal to that of the mixed brine. Meanwhile, Jordan River water or spring water is being pumped into the bore side of the fibers at a rate, (A $’ + FS), which is the sum of the water

57

HYDROTURBINE

P CA+ + F’s) Flushing Solution (F’s)-

WP To Dead Sea

Dead sea Brine (DSB)

Mixed Brine (ME) - To Dead Se0

Jordan River water

brocki% springs

Power to water pump and recycle pump

Power to bus bar = (P,,Ai/) (Mechanical

efficiency)

Fig. 5. Power production from Dead Sea by pressure-retarded osmosis (PRO), realistic. 1. Efficiency of rotating components less than 100%. 2. Friction losses in plant streams. 3. Membranes not perfectly semipermeable. Flushing solution needed.

permeate rate and the flushing solution rate (see Section IV). The power input to the water pump is given by:

Power in 1 wp = Pwg (Ap + FS)/E,, . (13)

where Pwp is the discharge pressure of the water pump, and E,, is the water pump efficiency.

Within the PRO permeator the permeate passes into the DSB at the rate Av. The pressurized brine emerging from the permeator has a composition deter- mined by the ratio (p/A@. The greater part of this brine is recirculated as discussed above, but a quantity (v + Ab) is sent to the hydroturbine at a hydraulic pressure Pbp F. In the hydroturbine, the hydraulic pressure of this stream, known as the mixed brine, is reduced to zero in the course of delivering power given by:

Power out 1 ht = PbpF Eht(v + AP) ,

where E, is the efficiency of the hydroturbine.

(14)

The power sent to the generator is given by the difference between eqn. (14) and eqn. (11). The power delivered by the generator is the same except that generator efficiency, E,, must be taken into account. Finally, some of the

58

generator output goes to the recirculation pump and water pump (eqns. (12) and (13)) so that the net power delivery for the plant (to the bus bar) is given by:

Net power delivery = [EJ [PbgF Eht (6 + AV) -PbpVsjlEbp] -

- (PW - P,,,F)RV/E, - P,,(AV + FS)/E,, . (15)

As discussed above, the net power delivery at 100% mechanical efficiency would be P,AV. Therefore, by dividing both sides of eqn. (15) by PbpAV, we obtain for the fractional mechanical efficiency, ME:

ME= + E,FEht - -

The following efficiencies are considered attainable for rotating devices in large plants :

E, = 0.98, Eht = 0.92, Ebp =E, =E, -0.90.

Some of the other terms will be determined by experiment, and all are given tentatively here as:

F= 0.98, R = 3, P&PM = 0.1, FS/AV=0.2.

If these terms are placed in eqn. (16) we have:

ME = 0.75 - 0.27 V/AV. (17)

Equation (17) is quite revealing about the efficiency of the PRO process. The maximum value of 0.75 is determined by the multiplying effect of the inefficiencies of the several rotating components, by friction losses, and by the necessity to use some energy to pump in the dilute solution, (AV + FS), to the bore side inlet. However, what is most important is the fact that mechanical efficiency is a strong function of the ratio, V/AV, of DSB rate to permeate rate. It is clear that this ratio cannot have a value much higher than about two, if any efficiency is expected from the process. One may go to the opposite extreme and say that V/A+ should be very low but this means that the osmotic pressure on the shell side of the permeator, nTsh, will also be very low thus necessitating, according to eqn. (lo), a reduction in either water flux, J1 ) or in hydraulic pressure, P, (-- Pbp). A reduction in either of these quantities will increase the unit cost of energy, as will be shown in the next section.

VI. Unit cost of energy production from a PRO power plant

In order to determine the unit cost of energy we must add up all the con- tributions to this cost, including amortization, membrane replacement, pre- treatment and other costs. As will be seen, many of these costs depend to a

59

great extent on the energy/permeate volume ratio, i.e. on the energy delivered to the bus bar per unit volume of permeate.

The energy/permeate volume ratio, kWh/m3, may be found readily from the developments of Section V. Net power delivery per unit permeate rate is P,Av(ME)/Ap = P,,,(ME), and this is most conveniently expressed as:

kW kWh __ = m3/h

- = (0.028) (F&J (0.75 -0.27 p/Air) , m3

(18)

where 0.028 is kWh/m3 atm, and we have used eqn. (17). We now consider the various contributors to the unit cost of energy, $/kWh.

A. Amortization of fixed components in PRO permeator The fixed components in the permeator would include all components

except the fiber bundle subassemblies, which are periodically replaced. It is assumed that the cost per daily cubic meter of permeate ($day/m3) for the fixed components will be inversely proportional* to the water permeation flux, J1, such that:

$dw

m3 = KIJ, , W) fixed

where K is a constant. It is estimated that, for an existing hollow fiber RO sea water desalination

plant having a flux of 0.04 m3/m2 day, the cost of fixed components would be $75 day/m3. However, this cost would be higher, perhaps $100 day/m3, for a plant having the same flux but with fibers modified for PRO service. At the higher figure, we have, for costs of fixed components:

May 4 -&3- =J,*

fixed

The cost in dollars per kilowatt of power produced would be:

$ kW =

fixed (24) (%~ ,,,) (3 '

where 24 is hours/day. Equations (18) and (20) are substituted into eqn. (21) to give:

$ kW

= (24) (4)

fixed (J1) (0.028) (P,,) (0.75 - 0.27 Q/AI+ ’

(20)

(21)

(22)

*It is possible to argue that the cost will also be directly proportional to the hydraulic pressure used. However, experimental results are anticipated here in the further assumption that the optimum hydraulic pressure in PRO will be close to that for which present sea water RO desalination plants are designed.

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The cost in dollars per kilowatt hour of energy produced is given by assuming amortization at 8% per annum:

$ (O-08) (& 1

kWh fixed = (365) (24) ’

where 365 is days per year. Equation (22) is substituted into eqn. (23) to give:

$ 0.032

kWh fixed = (J1) (I&,) (0.75 - 0.27 v/Ai7) - (24)

B, Replacement costs for fiber bundle subassemblies We assume that the cost per daily cubic meter of permeate for the fiber

bundle subassemblies is $75 day/m3. Therefore:

$day 3 =-

m3 module JI ’ (25)

The modules, however, are replaced every three years, giving a unit energy cost of:

Equations (18) and (25) are substituted in eqn. (26) to give:

$ 0.098

= kWh module (J1) (P,& (0.75 - 0.27 @/APT) ’

C. Pretreatment costs Assume pretreatment costs are $0.01/m3 of permeate. Then:

$ kWh pretreat

= (0.01) (&) .

Substitute eqn. (18) :

$ 0.36

kWh pretreat = Pbp (0.75 - 0.27 p/A+) ’

(27)

(28)

(29)

D. Hydroturbine and generator cost Assume a total of $125/kW for the hydroturbine and the generator. Then:

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$ kWh

ht--P

= (0.03) (125) (&) (;)

$ ” 0.001 . (30)

kWh h%3

E. Labor costs A lOO-megawatt plant would, it is estimated, require a labor force of 20 men

per shift, and 3 shifts per day. It is assumed that the annual salary and over- head charge on each man is $15,000. The labor contribution would then be:

$ kWh

= (20) (3) (15,000)/(100,000) (24) (365) * 0.001 , (31) labor

where 100,000 is kilowatts in a loo-megawatt plant.

F. Costs for diversion dam and attendant piping It will be necessary to divert an appreciable fraction of the Jordan River

from the Dead Sea to the PRO plant. This will require a diversion dam and attendant piping including a DSB feed pipe in a concentrated region of the Dead Sea. It is assumed that the sum of these costs will not exceed twice the cost of the hydroturbine and generator, i.e. :

$ = 0.002

kWh diversion (32)

I dam

G. Total costs Total costs are given by the sum of eqns. (24), (27), (29), (31) and (32):

$ 0.13/J, + 0.36

kWh = (pbp) (0.75 - 0.27 p/Air) + o’oo4 133)

total

In the next section is outlined an experimental program for determining the relation between the variables on the right-hand side of eqn. (33). The results of these tests will be given in the second part of this paper.

VII. Projected experimental program

A. Experiments to establish minimum unit energy costs It is clear from eqn. (33) what experimental work must be undertaken to

obtain some idea of the unit cost of energy produced by PRO. DSB and

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water must be mixed to form a mixed brine having a V/AV (= V/AV) ratio of, say, l/l. A PRO test will then be made with this brine. In this test the shell side pressure, Psh , will be varied, that on the inlet of the bore side, (Pbo,in), being held constant. By this means we obtain J1 = f(P,,). Assuming that it is approximately a linear relation, we may write it in the form of eqn. (10):

JI = Apparent [(ash - Abo.ave) - (psb -Pbo.ave)l 9 (34)

where Aapparent is the apparent value of the water flux constant, obtained experimentally by plotting J1 against(P& - Pbo,a”e), Pb,,,a”e is the arithmetic average bore side hydraulic pressure, and (R,~ - xb,,& is the intercept of the flux-pressure plot at J1 = 0.

nbo,ave is really the exponential term in eqn. (9), but under the assumed limiting conditions of eqn. (10) ‘Ilbo,a”e will approach the arithmetic average between the osmotic pressures of the inlet and outlet bore side solutions. However, most of the PRO experiments reported herein were obtained under conditions where xbO,+ve -+ 0. Therefore it is convenient to define an apparent shell side osmotic pressure, xsh,spparent, given by (A,~ - nbO,ave) as rib,-,,,, + 0. We then rewrite eqn. (34) as:

Jl = Apparent [Ash,amarent - (psh - Pbo,ave)l .

If we place eqn. (35) in eqn. (33), we have (recalling that P&, = Pbp):

(35)

$jkWh 1 total = f(Pb,) (36)

and a plot of the curve will have a minimum value of unit energy cost (for the l/l ratio of V/AV) at some optimum value of Pbp.

The tests are repeated with mixed brines having other ratios of V/AV, until the lowest unit energy cost is determined.

B. Comparison of transport parameters in RU and PRO It was concluded in the analysis of Section IV that an asymmetric mem-

brane may be as desirable in PRO as in RO. This is equivalent to saying that the water transport constant is determined in both cases by the skin, and, as a corollary to this, the constant should have the same value in RO and PRO.

This postulated equality can be verified by an experiment on a membrane known to be asymmetric at a fixed and preferably known value of x,h (the osmotic pressure on the shell side of the membrane), for example with a known solution of sodium chloride. Under the experimental conditions that xb,,a”e + 9 and that &,& iS fixed, let Psh be varied from (Psh - Pbo,ave) < a& to (Psh -Pbo.ave) > gsh. A transition from PRO to RO should be made approx- imately at a value of P& such that (Psi, - PbO,ave) = n,h. Most important, the Slope A Of the J1 US. (P&, -Pbo,& curve should have the same value in each domain if eqn. (35) is valid for both PRO and RO.

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Acknowledgement

This work was sponsored by the United States-Israel Binational Science Foundation Grant No. 337.

Nomenclature

A

; E F FS

J K P R V P X

Ax AV AV

7r lJ

water flux constant, m3/m2 day atm concentration, g/liter solution or kg/m3 solution diffusion coefficient, m2/day efficiency, expressed as fraction defined in eqn. (12) volumetric rate of flushing solution, m3/day

permeation flux, m3/m2 day for water, kg/m2 day for salt constant, eqn. (19) hydraulic pressure, atmospheres ratio, recirculating brine rate to concentrated brine rate volume of Dead Sea brine in DSB/H20 solution used as feed brine volumetric flow rate of Dead Sea brine into PRO plant, m3/day distance into membrane from bore side interface, microns or meters

total membrane thickness, microns or meters volume of water in DSB/H20 solution used as feed brine volumetric flow rate of permeate through membranes in PRO plant, m3/day osmotic pressure, atmospheres chemical potential, m3 atm/mol

Subscripts 1 water 2 salt ave average bp brine pump bo bore side of hollow fiber (always water side) DSB Dead Sea brine g generator ht hydroturbine in inlet ps porous substructure rp recirculation pump sh shell side of hollow fiber (always brine side) tr transition interface, skin to porous substructure wp water pump