Thermodynamic limits of extractable energy by pressure retarded osmosis

Energy &EnvironmentalScience

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Department of Chemical and Environmental

Connecticut 06520-8286, USA. E-mail: men

432-2789

† Electronic supplementary informa10.1039/c4ee01020e

Cite this: Energy Environ. Sci., 2014, 7,2706

Received 1st April 2014Accepted 21st May 2014

DOI: 10.1039/c4ee01020e

www.rsc.org/ees

2706 | Energy Environ. Sci., 2014, 7, 27

Thermodynamic limits of extractable energy bypressure retarded osmosis†

Shihong Lin, Anthony P. Straub and Menachem Elimelech*

Salinity gradient energy, which is released upon mixing two solutions of different concentrations, is

considered to be a promising source of sustainable power. Of the methods available to harvest the

salinity gradient energy, pressure retarded osmosis (PRO) has been one of the most widely investigated

processes. In this study, we identify the thermodynamic limits of the PRO process by evaluating the

obtainable specific energy, or extractable energy per total volume of the mixed solutions. Three distinct

operation modes are analyzed: an ideal case for a reversible process, and constant-pressure operations

with either co-current or counter-current flow in a membrane module. For module-scale operation,

counter-current flow mode is shown to be more efficient than co-current flow mode. Additionally, two

distinct thermodynamically limiting operation regimes are identified in counter-current flow mode—the

draw limiting regime and the feed limiting regime. We derive analytical expressions to quantify the

maximum specific energy extractable and the corresponding optimal feed flow rate fraction and applied

pressure for each operation mode. Using the analytical expressions, we determine that maximum

extractable energy in constant-pressure PRO with seawater (0.6 M NaCl) as a draw solution and river

water (0.015 M NaCl) as a feed solution is 0.192 kW h per cubic meter of mixed solution (75% of the

maximum specific Gibbs free energy of mixing). Considering that this is the theoretical upper bound of

extractable energy by the PRO process, we discuss further efficiency losses and energy requirements

(e.g., pretreatment and pumping) that may render it difficult to extract a sizable net specific energy from

a seawater and river water solution pairing. We analyze alternative source waters that provide a higher

salinity difference and hence greater extractable specific energy, such as reverse osmosis brine paired

with treated wastewater effluent, which allow for a more immediately viable PRO process.

Broader context

The concept of harnessing the energy released when two solutions of different salinities are mixed has been proposed as a promising source of clean energy.Pressure retarded osmosis (PRO) has been identied as one of the furthest developed methods to extract this energy of mixing and many studies have looked atthe performance of small-scale membrane coupons in PRO. However, these coupon-sized experiments cannot give insight into the amount of energy extractablein the full-scale PRO process since, in practice, it is operated with membrane modules at a constant applied hydraulic pressure. We theoretically evaluate thePRO process to understand ow and concentration behavior in constant-pressure modules. Using this analysis, we establish the thermodynamic limit ofextractable energy at the module scale. For river water mixing with the sea, one of the most widely considered source water pairings, we identify the maximumextractable energy in the process to be 0.192 kW h per cubic meter of mixed total solution. Even though this is only 25% less than the Gibbs free energy of mixing,we discuss additional efficiency losses and energy costs that may substantially reduce the net specic energy available from the river water and seawater. Wehighlight other source water pairings for the PRO process which, in some cases, can circumvent the need for pretreatment and still offer relatively high effi-ciencies in a constant-pressure module.

Introduction

The quest for renewable energy has been accelerated in recentyears by the urgent need to reduce greenhouse gas emissionsand mitigate global warming.1 In consequence, a variety of

Engineering, Yale University, New Haven,

[email protected]; Tel: +1 (203)

tion (ESI) available. See DOI:

06–2714

engineered systems have been conceived and developed toharness the energy constantly provided by nature. Salinitygradient energy, which is released upon mixing two solutions ofdifferent salt concentrations,2 is considered a promisingrenewable energy source with an estimated global powerpotential of 1.4–2.6 TW.3 In addition to its large potential,salinity gradient energy also has the theoretical advantage ofrelatively high energy density. For instance, in a reversiblethermodynamic process, the energy released from mixing acubic meter of fresh water into the sea is estimated to be about

This journal is © The Royal Society of Chemistry 2014

http://dx.doi.org/10.1039/C4EE01020E

http://pubs.rsc.org/en/journals/journal/EE

http://pubs.rsc.org/en/journals/journal/EE?issueid=EE007008

Paper Energy & Environmental Science

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0.8 kW h,4 which is equivalent to a hydraulic water head of�290meters.

Pressure retarded osmosis (PRO), reverse electrodialysis, andcapacitive mixing are emerging processes to harvest salinitygradient energy.3,5–10 Of these, PRO is the most widely investi-gated process. In the PRO process, the chemical potentialdifference drives water from a low-salinity feed solution across asemipermeable membrane into a high-salinity draw solution. Ahydraulic pressure lower than the osmotic pressure difference isapplied to the draw solution and the incremental increase involume (or ow rate) of the pressurized draw solution is used todrive a hydro-turbine and generate power.11–14

Most existing PRO studies were carried out at the scale of amembrane coupon with the goal of understanding the localmass transfer kinetics and membrane power density.15–19

However, information from coupon level analysis cannot beused to predict the performance of a full-scale PRO processbecause, in practice, a PRO system comprises membranemodules and is operated at a constant applied hydraulic pres-sure.20 In this operation mode, the salt buildup in the feedsolution and dilution of salts in the draw solution impose athermodynamic limit on the extractable energy that is consid-erably lower than the efficiency of a reversible process. Similarthermodynamic limitations have been demonstrated for sepa-ration processes involving high-salinity feed solutions, such asreverse osmosis21,22 and direct contact membrane distillation,23

but PRO modeling at the module scale to establish the ther-modynamic limits of the process has not been conducted yet.This type of analysis is critical to understand the efficiency ofthe process as well as to determine the optimal moduleconguration and operation conditions.

When quantifying the energy efficiency of a PRO process, it isnecessary to consider the method used to normalize theextractable energy. Previous analyses on salinity gradient energymainly focused on the extractable energy per volume of feed(low-salinity) solution, with the goal of evaluating the limit ofavailable energy assuming fresh water as the scarceresource.4,24,25 A straightforward conclusion with this approachis that the energy extractable per volume of feed solution will bemaximized if the feed volume fraction (i.e. the initial volume offeed divided by the total volume of mixed solution) approacheszero. However, the feed volume fraction approaching zero formaximizing the energy per volume of feed solution is animpractical operation condition in PRO, which provides norealistic rationale for process optimization with respect to thefeed volume fraction. In addition, while fresh water is indeedthe limiting resource compared to the virtually inexhaustibleseawater, Post et al. identied that there are economic costsassociated with pretreating and transporting both the feed anddraw solutions.26 Therefore, from a practical point of view, amore reasonable metric is the extractable energy normalized bythe total volume of the mixed solutions, which is hereby denedas specic energy.

In this paper, we evaluate the thermodynamic limits of themaximum specic energy extractable in a PRO process anddetermine the corresponding optimal operating conditions.Three different operation modes are analyzed: an ideal case for


a reversible process, and constant-pressure operations witheither co-current ow or counter-current ow in a membranemodule. Our module-scale analysis identies two distinctthermodynamically limiting operation regimes in counter-current ow operation: the draw limiting regime and the feedlimiting regime. We also derive analytical expressions toquantify the optimal operating conditions and the corre-sponding maximum specic energy extractable in the differentoperation modes. Practical implications for potential applica-tions of PRO are discussed based on the theoretical limits of themaximum specic energy.

Thermodynamically reversible PROprocess

The Gibbs free energy of mixing is the thermodynamic upperbound of the energy extractable by mixing two solutions ofdifferent salinities and can be attained only via a thermody-namically reversible process. In PRO, a thermodynamicallyreversible process can be realized in a batch mode by keepingthe applied pressure innitesimally smaller than the osmoticpressure difference throughout the entire process. It has beenproven that the energy generated in such a reversible PROprocess exactly equals the Gibbs free energy of mixing.4 In thissection, we discuss the Gibbs free energy of mixing per volumeof mixed solution, DGVM

(i.e. the specic Gibbs free energy ofmixing), as a function of feed volume fraction, f, with the goalof identifying the optimal f that yields the maximum specicGibbs free energy of mixing.

Specic Gibbs free energy of mixing

The molar Gibbs free energy of mixing, DGnM, is dened as theenergy per mole of mixed solution generated from mixing twosolutions of different salinities in an isothermal and isobaricprocess.27,28 It can be evaluated as the difference between themolar entropy aer and before mixing:26

�DGnM

RT¼

Xi

xi;M ln�gi;Mxi;M

�� nF

nM

Xi

xi;F ln�gi;Fxi;F

�

� nD

nM

Xi

xi;D ln�gi;Dxi;D

�(1)

Here, R is the ideal gas constant; T is the absolute temperature;nM, nF, and nD are the total amounts (in moles) of mobilespecies (i.e. solvent molecules, dissociated ions from solutes,and, if present, neutral solute molecules) in the mixed, feed,and draw solutions, respectively; xi,M, xi,F, and xi,D are the molefractions of species “i” in the mixed, feed, and draw solutions,respectively; and gi,M, gi,F, and gi,D are the activity coefficients ofspecies “i” in the corresponding solutions.

For dilute solutions, the activity coefficients can beapproximated as unity.29 Further, assuming negligible contri-bution of solute to the volume of the solution, the ratio of thetotal moles in the feed and draw solutions to the total moles inthe mixed solution (i.e. nF/nM and nD/nM, respectively) can beapproximated by the corresponding volume fractions: nF/nM zf and nD/nM z 1 � f.4 Applying these simplications to eqn (1)

Energy Environ. Sci., 2014, 7, 2706–2714 | 2707


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leads to an expression for the specic Gibbs free energy ofmixing per volume of total mixed solution, DGVM

, as a functionof the molar concentrations of the feed, draw, and mixedsolutions (cF, cD, and cM) as well as the feed volume fraction,4 f:

DGVM

nRT¼ cM lnðcMÞ � fcF lnðcFÞ � ð1� fÞcD lnðcDÞ (2)

where n is the van't Hoff factor for strong electrolytes (e.g., n ¼ 2for NaCl).

Fig. 1 presents the specic Gibbs free energy of mixing as afunction of the feed volume fraction, f, as predicted by themoreprecise eqn (1) (open symbols) and the simplied eqn (2)(dashed and dotted curves). The simplied equation (eqn (2))always overestimates DGVM

as compared to eqn (1). The absolutedeviation is more signicant when DGVM

is higher, but is alwaysless than 10% of DGVM

. The optimal feed volume fractions arevery similar from the predictions by both the precise andsimplied equations. Therefore, hereaer we will only use thesimplied equation (eqn (2)) for the convenience of derivation.

Optimal feed fraction and maximum specic energy

In a thermodynamically reversible PRO process, the feedvolume fraction, f, is the only operational parameter. Themaximum DGVM

occurs when its derivative with respect to f iszero:

dðDGVMÞ

df¼ 0 (3)

Solving eqn (3) using the simplied expression for DGVM(eqn

(2)) leads to the critical feed fraction, fcr,DG, with which thespecic Gibbs free energy of mixing is maximized:

Fig. 1 Specific Gibbs free energy of mixing (i.e. the reversible energy ofmixing per volume of mixed feed and draw solutions) for waters ofdifferent sources as a function of the feed volume fraction, f, definedas the ratio of the feed solution volume to the volume of the mixedfeed and draw solutions. The following sources of waters wereconsidered: seawater (SW, 0.6 M NaCl), river water (RW, 0.015 MNaCl),brackish water (BW, 0.05 M NaCl), wastewater effluent (WW, 0.015 MNaCl), and brine (1.2 M NaCl). The dashed and dotted curves wereobtained using the simplified molarity based equation (eqn (2)),whereas the open symbols are calculated using the more preciseequation (eqn (1)).

2708 | Energy Environ. Sci., 2014, 7, 2706–2714

fcr;DG ¼exp

�cF lnðcFÞ � cD lnðcDÞ

cF � cD� 1

�� cD

cF � cD(4)

Combining eqn (2) and (4) yields the maximum specic Gibbsfree energy, DGVM,max:

DGVM ;max

nRT¼ cDcF

cD � cFðlnðcDÞ � lnðcFÞÞ

� exp

�cD lnðcDÞ � cF lnðcFÞ

cD � cF� 1

�(5)

Both fcr,DG and DGVM,max are simply functions of the feed anddraw concentrations, cF and cD, respectively. Thorough inspec-tion of calculated fcr,DG over a wide range of reasonable feedand draw concentrations reveals that fcr,DG is mostly centeredaround 0.6 and decreases slightly for higher feed concentra-tions (Fig. S1†).

Module-scale PRO analysis

While the Gibbs free energy of mixing represents the theoreticalupper bound of extractable energy, it is difficult to approach inpractice, as it requires a thermodynamically reversible processwhere the applied pressure always equals the varying osmoticpressure difference between the feed and draw solutions. A full-scale PRO process operates with membrane modules underconstant pressure. In this section, we describe the mass transferin a PROmodule that is of either a counter-current or co-currentow conguration. The goal is to understand the impacts of twomajor operation parameters — the applied hydraulic pressure,DP, and the feed ow rate fraction, f — on the performance ofPRO at the module scale.

Because we are interested in understanding the thermody-namic limits of system performance, we deliberately ignorenon-idealities in conducting our system scale analysis. Speci-cally, the idealizing assumptions include (i) the absence ofreverse draw salt ux, which implies a membrane with perfectselectivity, and (ii) the absence of internal and externalconcentration polarizations, which is equivalent to having amembrane with a negligible structural parameter and operatingthe PRO process with perfect hydrodynamics (i.e., completemixing). As will be shown in the subsequent sections, ignoringnon-idealities enables the derivation of simple analyticalexpressions for the upper bound of extractable energy and thecorresponding operating conditions. With these underlyingassumptions, the ideal trans-membrane water ux, JW,i, can berelated to the bulk concentration difference by the followingequation:

JW,i(cD,cF) ¼ A[p(cD) � p(cF) � DP] (6)

where A is the pure water permeability of the PRO membraneand p(c) is the osmotic pressure of a solution of molarconcentration c. To further simplify our analysis, we assume theosmotic pressure as a function of molar concentration followingthe van't Hoff equation:

p(c) ¼ nRTc (7)




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We note that predictions based on this equation deviate slightlyfrom the actual osmotic pressure when the solution is highlyconcentrated.30

Mass transfer in counter-current ow

The mass balance of water and draw solute in a counter-currentow module is given by:

dQD

ds¼ JW;iðcDðsÞ; cFðsÞÞ (8A)

dQF

ds¼ JW;iðcDðsÞ; cFðsÞÞ (8B)

cDðsÞ ¼ cD;0QD;0

QDðsÞ (8C)

cFðsÞ ¼ cF;0QF;0

QFðsÞ (8D)

Specically, eqn (8A) and (8B) quantify the mass transfer ofwater across the semi-permeable membrane, and eqn (8C) and(8D) describe the mass balance of solute in the draw and feedsolutions, with the assumption that the membrane perfectlyrejects the draw solute. The position in the module is repre-sented by the normalized membrane area, s, which is denedas the membrane area between the entrance of the drawsolution stream (as the convention in this paper) and theposition being described, normalized by the total membrane

Fig. 2 Distribution of flow rate fractions along a membrane module for fand co-current flowmode (dashed curves) with (A) f ¼ 0.3 and (B) f¼ 0.(black) and draw (red) in co-current flow mode (dashed curves) and couThe feed solution is river water (0.015 M) and the draw solution is seawatemembrane total area, s, and pure water permeability, A, satisfy the conditmass transfer.


area in the module. Using s to quantify the relative positionin the module eliminates the need to specify the congurationof the cross-section and the membrane area per unit length ofthe module, and thus renders the analysis generallyapplicable.

For module-scale operation in the counter-current congu-ration with a total membrane area, s, the boundary conditionsare cD(0) ¼ cD,0 and QD(0) ¼ QD,0 (i.e. s ¼ 0 corresponds to thedraw solution entrance), and cF(s) ¼ cF,0 and QF(s) ¼ QF,0 (i.e.s ¼ s corresponds to the feed solution entrance).

Mass transfer in co-current ow

The mass balance of water and draw solute in a co-current owmodule is given by

dQDðsÞds

¼ JW;iðcDðsÞ; cFðsÞÞ (9A)

dQFðsÞds

¼ �JW;iðcDðsÞ; cFðsÞÞ (9B)

cDðsÞ ¼ cD;0QD;0

QDðsÞ (9C)

cFðsÞ ¼ cF;0QF;0

QFðsÞ (9D)

Note that these equations that describe the mass balance in aco-current ow module are almost the same as those for

eed (black) and draw (red) in counter-current flow mode (solid curves)8, and the corresponding distributions of solute concentrations of feednter-current flow mode (solid curves) with (C) f ¼ 0.3 and (D) f ¼ 0.8.r (0.6 M). The applied pressure, DP, is 14.5 bar. The feed flow rate,QF,0,ion that c¼ As/QF,0 ¼ 0.2 bar�1. Note that c characterizes the extent of




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the counter-current ow module (eqn (8)), except for eqn (9B)due to the different ow direction. The boundary conditionsfor the co-current ow operation are cD(0) ¼ cD,0 and QD(0) ¼QD,0, and cF(0) ¼ cF,0 and QF(0) ¼ QF,0 (i.e. s ¼ 0 corresponds tothe entrances for both the feed and draw solutions).

Operation regimes in the PRO module

Solving the mass balance equations (eqn (8) and (9)) yields theow rate and solute concentration distributions along themodules. The solute concentration distributions for both co-current (dashed curves) and counter-current (solid curves)congurations are presented in Fig. 2C and D for differentinitial feed ow rate fractions, f. To be more general, insteadof showing the ow rate distributions, Fig. 2A and B show thedistribution of ow rate fractions which are independent of theabsolute ow rates of the system. In Fig. 2, the membrane area,s, pure water permeability, A, and feed ow rate, QF,0, aredeliberately chosen so that mass transfer can proceed to thethermodynamic limit. The extent of mass transfer is charac-terized by the parameter c ¼ As/QF,0. Note that for a given feedconcentration, draw concentration, and applied hydraulicpressure, the ow rate fraction and solute concentrationdistribution proles will remain identical as long as the c valueis unchanged. Completion of mass transfer in Fig. 2 is indi-cated by the existence of a portion of the membrane modulewhere both the ow rate fractions and solute concentrationsremain constant. In such portions of the membrane module,the driving force for mass transfer (i.e., the difference betweenthe osmotic pressure difference and the applied hydraulicpressure) vanishes so that the trans-membrane water ux givenby eqn (6) becomes zero.

For co-current ow operation, the draw solute concentrationdecreases and the feed concentration increases considerablyalong the module with a small feed ow rate fraction f ¼ 0.3(Fig. 2C). However, with f¼ 0.8, a relatively large value, the feedconcentration barely changes (Fig. 2D) because only a smallfraction of the water from the feed solution transfers across themembrane before the draw concentration decreases to the levelthat the condition for zero driving force is reached. For counter-current ow operation, there are two distinct operation regimes,depending on the initial feed ow rate fraction, f. When f issmall (e.g., f ¼ 0.3), the feed concentration increases to a crit-ical feed concentration, c*F (Fig. 2C), which is dependent on theinitial draw concentration, cD,0, and the applied hydraulicpressure, DP:

c*FhcD;0 � DP

nRT(10)

The driving force vanishes when cF reaches c*F, as the osmoticpressure difference is equal to the applied hydraulic pressure. Acounter-current ow PRO operation with its effluent feedconcentration, cF,f, reaching the critical feed concentration, c*F,is considered to be in the feed limiting regime (FLR). In contrast,when f is large (e.g., f ¼ 0.8), the draw concentration, cD,decreases to a critical draw concentration, c*D (Fig. 2D), which isa function of the initial feed concentration, cF,0, and the appliedpressure, DP:


c*DhcF;0 þ DP

nRT(11)

Eqn (11) corresponds to a thermodynamic equilibrium in whichthe osmotic pressure difference between the solutions of c*D andcF,0 is equal to the applied hydraulic pressure. Analogously, weidentify a PRO operation in which the effluent draw concen-tration, cD,f, reaches the critical draw concentration, c*D, as anoperation in the draw limiting regime (DLR). We note thatsimilar concepts of thermodynamically limiting regimes havebeen identied in the module scale operation of reverseosmosis21 and direct contact membrane distillation,23 in whichthe driving force for mass transfer decreases as the processproceeds.

Regardless of the ow conguration and operation regime,the specic energy of the process, w, is always dened as thepower extractable from the system, _W , normalized by the totalow rate, QTot:

w ¼c

W

QTot

¼ DPDQ

QD;0 þQF;0

¼ DPfgðDP;fÞ (12)

where DQ is the trans-membrane ow rate (i.e. the integral ofwater ux with respect to the membrane area for the entiremodule) and g is the feed recovery rate, dened as the ratio ofDQ over the initial feed ow rate QF,0. For a given PRO system,the feed recovery rate, g, is a function of both the appliedpressure, DP, and the initial feed ow rate fraction, f. It is worthnoting that in both cases presented in Fig. 2 (f ¼ 0.3 and f ¼0.8), the feed recovery rates are very similar for co-current andcounter-current operations, which, as we will show later, is notnecessarily the case with an intermediate f.

The specic energy normalization shown above assignsequal value to feed and draw solutions, an assumption that hasbeen made previously25 and allows for the derivation of simpleanalytical equations. However, in reality, the normalizationshould take into account the difference between the feed anddraw solutions in their relative energetic cost of procurement,pretreatment, and pumping. This type of extensive analysis,though useful, is beyond the scope of this work.

Thermodynamic analysis of limitingregimes

From the numerical solution of eqn (8), as presented inFig. 2, we have identied two distinct regimes (FLR and DLR)for PRO operation with the counter-current ow congura-tion. It appears that as long as the membrane area is suffi-ciently large so that these thermodynamic limiting regimesare reached, the performance of the system can be analyti-cally determined based on simple mass balance with eqn(10)–(12).

When the PRO operation is in FLR, mass balance and thecondition cF,f ¼ c*F dictate the feed recovery rate, gFLR:

gFLR ¼ 1� cF;0

c*F(13)




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The specic energy in FLR is then given by

wFLR ¼ DPfgFLR ¼ DPf

�1� cF;0

c*F

�(14)

In FLR, both the boundary conditions cF,f ¼ c*F and cD,f $ c*Dshould be satised, which leads to the necessary operationcondition:

f#fFLR ¼ c*FcD;0 þ cF;0

(15)

We note that c*F and c*D are both functions of DP, and therefore,fFLR is a function of cF,0, cD,0 and DP.

Analogously, when the PRO operation is in DLR, the feedrecovery rate, gDLR, can be readily determined by mass balanceand the condition cD,f ¼ c*D:

gDLR ¼ 1� f

f

�cD;0

c*D� 1

�(16)

The specic energy in DLR is then given by

wDLR ¼ DPfgDLR ¼ DPð1� fÞ�cD;0

c*D� 1

�(17)

The necessary conditions for a PRO to operate in DLR can bedetermined by the boundary conditions that cD,f¼ c*D and cF,f# c*F:

f$fDLR ¼ c*FcD;0 þ cF;0

(18)

Comparing eqn (15) and (18) reveals that under the sameapplied pressure as well as feed and draw concentrations, fDLR

is exactly equal to fFLR, which means that such a critical initialfeed ow rate fraction, f* (f* ¼ fFLR ¼ fDLR), demarcatesoperation in both FLR and DLR as long as there is sufficientmembrane area for the operation to reach the thermodynamiclimiting regimes. If the membrane area is insufficient, thecondition of zero driving force will not be reached anywhere inthe module. Consequentially, the effluent feed concentrationwill always be lower than c*F, and the effluent draw concentra-tion will always be higher than c*D.

Finally, it should be noted that the above analysis on ther-modynamic limiting regimes is only applicable for counter-current ow operation, in which the equilibrium condition canoccur on either end of the module. In the special case of f¼ f*,the equilibrium occurs on both ends of the module simulta-neously. In a co-current ow operation, although the concen-tration distributions seem to be very different between the casewhen f is small and that when f is large, there is no distinctiveoperation regime that can be dened in a way similar to that ofthe counter-current ow operation, because the equilibriumcondition can only occur at the exit of the module.

Optimal conditions for moduleoperation

The analysis of thermodynamic limiting operation regimespresented above is important for understanding the behavior ofa counter-current ow PRO process at a module scale. Inaddition, eqn (14) and (17) allow us to evaluate the specic


energy of the process, which, for given sources of feed and drawsolutions, is dependent on the applied pressure, DP, and theinitial feed ow rate fraction, f. In this section, we will identifythe optimal operation conditions and the correspondingmaximum specic energy for both counter-current and co-current ow operation.

Counter-current ow operation

Because the specic energy in FLR, wFLR(f) (eqn (14)), is amonotonically increasing function, and the upper bound of f inFLR is fFLR (eqn (15)), the optimal f in FLR is fFLR, and themaximum specic energy is wFLR(fFLR). Analogously, since thespecic energy in DLR, wDLR(f) (eqn (17)), is a monotonicallydecreasing function, and the lower bound of f in DLR is fDLR

(eqn (18)), the optimal f in DLR is fDLR, and the maximumspecic energy is wDLR(fDLR). However, it has been shown thatfFLR is exactly equal to fDLR for a given applied pressure and agiven set of working concentrations (cD,0 and cF,0). Therefore,the optimal initial feed ow rate fraction in a counter-currentow operation (fopt,CT) is the critical feed ow rate (f*) thatdivides the FLR and DLR:

fopt;CTðDPÞ ¼c*F

cD;0 þ cF;0¼ cD;0 � DP=ðnRTÞ

cD;0 þ cF;0(19)

The corresponding maximum specic energy is then given byeither wDLR(f*) or wFLR(f*), with both yielding the sameanalytical equation:

wmax;CTðDPÞ ¼ DPcD;0 � cF;0 � DP=ðnRTÞ

cD;0 þ cF;0(20)

The optimal initial feed ow rate fraction and the correspond-ing maximum specic energy are plotted in Fig. 3 as functionsof the applied pressure, DP.

The blue dashed curve shown in Fig. 3 gives the maximumspecic energy extractable for a given applied hydraulic pres-sure, which can be considered as the local maximum attainableif the initial feed ow rate fraction is optimally tailored to agiven pressure. However, there also exists an optimal appliedpressure that leads to a global maximum of the specic energy.From observing the maximum specic energy curve (Fig. 3) andalso inspecting the structure of eqn (20), it is evident that theglobal maximum of specic energy occurs when the appliedpressure is half of the osmotic pressure difference (i.e. DP ¼nRT(cD,0 � cF,0)/2). This is also well known as the condition forachieving a maximum power density for a small membranecoupon (i.e. not at the module scale).15 However, it should beemphasized that the underlying principles behind these twooptimal conditions are totally different.

When the applied pressure is half of the osmotic pressuredifference, the corresponding optimal initial feed ow ratefraction is fopt,CT(Dp/2) ¼ 0.5. In other words, the conditionsleading to a global maximum of specic energy are DP ¼ Dp/2and f ¼ 0.5. Applying these conditions to either eqn (14) or (17)yield the expression for the maximum specic energy of PRO inthe counter-current ow conguration, w*

max,CT:



Fig. 3 Optimal feed flow rate fraction, f, (red solid line, left axis) andthe corresponding maximum specific energy (blue dashed curve, rightaxis) at different applied hydraulic pressures, DP, for PRO in counter-current flow mode under constant pressure. The black dotted lineindicates the global optimal specific energy and operating conditions(f ¼ 0.5, DP ¼ p/2). The feed solution is river water (0.015 M) and thedraw solution is seawater (0.6 M). It is assumed that the membranearea is sufficiently large so that the mixing proceeds to completion.

Fig. 4 Specific energy as a function of applied pressure, DP, and feedflow rate fraction, f, for PRO in co-current flow mode under constantpressure. The feed solution is river water (0.015 M) and the drawsolution is seawater (0.6 M). It is assumed that the membrane area issufficient so that the mixing proceeds to completion. The optimalapplied pressure for a given feed flow rate fraction, f, is shown with adotted line. The white star indicates the global optimal condition thatleads to the highest specific energy.


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w*max;CT ¼ wmax;CT

�Dp

2

�¼ nRT

4

ðcD;0 � cF;0Þ2ðcD;0 þ cF;0Þ (21)

As a global maximum, w*max,CT is simply a function of the initial

feed and draw concentrations as well as the working temperature.

Co-current ow operation

For co-current ow operation, no distinct operation regimes canbe dened as in counter-current ow operation. The specicenergy for a module scale constant pressure PRO operation,regardless of operation mode, is always given by eqn (12). In aco-current ow operation, the feed recovery, gCO, can be deter-mined from

DP ¼ nRT

�1=f� 1

1=f� 1þ gCO

cD;0 � 1

1� gCO

cF;0

�(22)

The analytical expression of gCO as a function DP and f basedon eqn (22) is too complicated to derive a simple analyticalexpression for an optimal initial feed ow rate fraction, fopt,CO,as a function of the applied pressure, DP. However, the specicenergy as dened by eqn (12) can be readily solved numericallywith different DP and f. From the numerical results shown inFig. 4, the global optimal operation conditions for co-currentow PRO operation are identied as DP ¼ Dp/2 and f ¼ 0.5,which are exactly the same as those for counter-current owoperation. The global maximum of specic energy, w*

max,CO,under these optimal operation conditions is given by

w*max;CO ¼ wCO

�DP ¼ Dp

2;f ¼ 1

2

�¼ nRT

4

� ffiffiffiffiffiffiffifficD;0

p � ffiffiffiffiffiffifficF;0

p �2(23)

It can be readily proven, by comparing eqn (21) and (23), thatunder optimal conditions, operation in counter-current owmode always yields a higher specic energy than that in co-current ow mode:

w*max,CT $ w*

max,CO (24)


The only condition for the equality to hold is that cD,0 ¼ cF,0,which is a trivial condition as no mixing energy is extractablefrom a system with two streams of equal salinity.

Implications

Based on our preceding analysis, the analytical expressions forthe optimal operating conditions and the correspondingmaximum specic energy in the three different operationmodes are summarized in Table 1. The optimal operatingconditions and themaximum specic energy are both functionsof the initial feed and draw concentrations as well as theworking temperatures, and are independent of any properties ofthe module, except for the assumption of having sufficientlylarge membrane area. The expressions summarized in Table 1can be used for facile evaluation of the maximum specicenergy in different operation modes for given sources of water.

Comparisons between the maximum specic energy atdifferent operation modes using the expressions in Table 1 arepresented in Fig. 5A for different combinations of feed and drawsolutions. Depending on the specic combinations, a counter-current ow PRO process can yield a maximum specic energythat is about 70–90% of the specic Gibbs free energy of mixing,DGVM,max, and a co-current ow PRO process can, at best,harvest 50–60% of DGVM,max.

For constant pressure PRO with seawater (SW) as drawsolution and river water (RW) as feed solution (Fig. S2†), thetheoretical maximum specic energy is 0.192 kW h m�3 (i.e.75% of themaximum specic Gibbs free energy of mixing, 0.256kW hm�3) with the chosen concentrations (0.015 M for RW and0.6 M for SW). This theoretical maximum is predicted byassuming the absence of detrimental effects, such as reversedraw ux as well as internal and external concentration polar-izations, which would further reduce the maximum specicenergy. In addition, a PRO system will require pretreatment to



Table 1 Summary of analytical expressions for optimal operation conditions and the corresponding maximum specific energies in differentoperation modesa

Optimal operation conditions Maximum specic energy (nRT)

Reversible process

f ¼exp

�cF lnðcFÞ � cD lnðcDÞ

cF � cD� 1

�� cD

cF � cD

exp

�cD lnðcDÞ � cF lnðcFÞ

cD � cF� 1

�� cDcF

cD � cFðlnðcDÞ � lnðcFÞÞ

Constant pressure Counter-current f ¼ 1/2, DP ¼ Dp/2 1

4

ðcD � cFÞ2ðcD þ cFÞ

Co-current f ¼ 1/2, DP ¼ Dp/2 ð ffiffiffiffifficD

p � ffiffiffiffifficF

p Þ24

a Note: for simplicity, the initial draw and feed concentrations in this table are expressed as cD and cF, respectively (instead of cD,0 and cF,0 as in themain text).


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mitigate fouling that would, in the long run, underminemembrane performance. While the extent of pretreatment isdependent on a variety of factors, the energetic cost may be ashigh as conventional water treatment31 or seawater ROpretreatment,32 making it comparable to, if not more than, themaximum specic energy of the PRO process. From thisperspective, the energy loss from constant-pressure PRO oper-ation may have a signicant impact on the net specic energyobtainable in the process. This energetic loss in conjunctionwith the pretreatment requirements, pumping energy costs tocirculate the feed and draw solutions, inefficiencies in thehydroturbine and pressure exchanger, and the aforementionedlosses due to non-ideal membranes, may render it very chal-lenging to harvest a sizable amount of energy in river water and

Fig. 5 (A) Maximum specific energy obtainable in PRO for three differepressure PRO in counter-current flow mode (blue), and constant-presmaximum specific energy obtained in counter-current (blue) and co-reversible PRO process, red bars in kW h m�3) are indicated for each paiflow mode as a function of draw and feed solution concentrations. Thseawater reverse osmosis brine, Dead Sea water, and Great Salt Lake waThe temperature used in modeling was 25 �C.


seawater PRO. However, with signicant advances in theunderstanding of pretreatment requirements and innovativesystems designed to minimize efficiency losses, a viable netenergy output may still be achieved.

While harvesting the energy of mixing between SW and RWusing PRO may be difficult with current technologies, alterna-tive salinity gradients may offer greater immediate potential inPRO (Fig. 5A). For example, Great Salt Lake water (�27% salinityin its highest concentration region, approximately equal to 4.6M NaCl) has been proposed as draw solution to work with riverwater (�0.015 M NaCl) as feed solution,33 which can yield atheoretical maximum specic energy of �1.6 kW h m�3 withcounter-current ow PRO (eqn (21), with f ¼ 0.5). Anotherpotential combination entails using Dead Sea water (�33.7%

nt cases: thermodynamically reversible PRO process (red), constant-sure PRO in co-current flow mode (green). The percentages of thecurrent (red) modes (with reference to the ideal thermodynamicallyring of source waters. (B) Maximum specific energy in counter-currente salinities of seawater (SW), river water or wastewater (RW or WW),ter are specified to be 0.6, 0.015, 1.2, 5.7, and 4.6 M NaCl, respectively.




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salinity, approximately equivalent to �5.7 M NaCl) as drawsolution and RO brine as feed solution (�1.2 M NaCl),34 whichcan yield a maximum ideal specic energy of about 1.0 kW hm�3 using counter-current ow PRO (eqn (21), with f ¼ 0.5).Note that pretreatment is not necessary in the second case forthe RO brine feed solution, as it has been pretreated prior to theRO process. In both cases, sizable net energy output per volumeof mixed solution can be attained even aer considering thenon-idealities that signicantly reduce the specic energy.Fig. 5B shows the maximum specic energy in counter-currentow mode as a function of draw and feed solution concentra-tions and can be used to evaluate additional solution pairings.Tapping into energy sources of high salinity requires PROoperation under high pressure that has been experimentallydemonstrated to be feasible.35

Beyond harnessing energy from natural salinity gradients,PRO shows promise as an energy recovery component in hybridengineered systems. For example, the Mega-ton project in Japanuses PRO to recover energy from mixing RO brine (draw solu-tion) and treated wastewater effluent (feed solution) to supple-ment the energy input for the RO process (a maximum specicenergy of �0.4 kW h m�3 with counter-current ow as shown inFig. 5).36 Extensive pretreatment is unnecessary in this case,because the draw and feed solutions have been treated in thepreceding processes. Furthermore, the system has the addedbenet of abating the environmental impact of the dischargedRO brine.

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Thermodynamic limits of extractable energy by pressure retarded osmosis

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