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Reducing specific energy consumption in Reverse Osmosis (RO) water desalination:An analysis from first principles

Mingheng Li ⁎Department of Chemical and Materials Engineering, California State Polytechnic University, Pomona, CA 91768, USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 10 December 2010Received in revised form 10 March 2011Accepted 11 March 2011Available online 8 April 2011

Keywords:DesalinationReverse osmosisSpecific energy consumptionNon-linear optimization

The previously derived characteristic equation of RO in Li, 2010 [8] is used to describe single- or multi-stageROs with/without an energy recovery device (ERD). Analysis is made at both the theoretical limit (withanalytical solutions provided if possible) and practical conditions (using constrained nonlinear optimization).It is shown that reducing specific energy consumption (SEC) normalized by feed osmotic pressure, or NSEC inROs can be pursued using one or more of the following three independent methods: (1) increasing adimensionless group γ=AtotalLpΔπ0/Qf, (2) increasing number of stages, and (3) using an ERD. Whenγ increases, the feed rate is adversely affected and the NSEC reduces but flattens out eventually. Using morestages not only reduces NSEC but also improves water recovery. However, The NSEC flattens out when thenumber of stages increases and ROs with more than five stages are not recommended. Close to thethermodynamic limit where γ is sufficiently large, the NSEC of ROs up to five stages approaches 4, 3.60, 3.45,3.38 and 3.33 respectively. The ERD can significantly reduce the NSEC, theoretically to 1, while thecorresponding recovery approaches zero. The NSEC becomes larger when the required water recoveryincreases. It is found that a combination of all three methods can significantly reduce the NSEC whilemaintaining a high recovery and a reasonable feed or permeate rate. An NSEC around 2.5–2.8 with an 80%water recovery may be possible using 3–5 RO stages and an ERD of 90% efficiency operated at a γ about 3–5(or Qf=0.2–0.3 AtotalLpΔπ0).

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Reverse osmosis (RO) membrane separation is an important desa-lination technology to produce drinking water. Large-scale productionfrom brackish water and seawater is possible with the currentmembrane technology. Because the applied pressure requirementsreach up to 1000–1200 psi for seawater desalting and 100–600 psi forbrackishwater desalting, the energy consumption of pumps to drive themembranemoduleaccounts for amajorportionof the total costofwaterdesalination [3,10,31].Significant research efforts, dating back to earlydevelopment of RO, have been made to reduce specific energyconsumption (SEC), or the energy cost per volume of producedpermeate, in order to make this technology more affordable to people[32]. Interested readers can refer to [20] for a critical review andperspective of energy issues in desalination processes.

The approaches to reduce SEC in RO processes can be divided intothe following categories: (i) using highly permeable membranematerial [21,23,35]. Apparently, a membranewith a high permeabilityallows the water to easily pass through the membrane, which savesenergy to pump the feed. (ii) Using an energy recovery device (ERD)

[1,6,10,13,18,27]. The ERD is to use the high pressure brine to passthrough a rotary turbine that drives an auxiliary pump pressurizingthe feed, thus reducing the duty of the primary pump [1,5,16].(iii) Using intermediate chemical demineralization (ICD) at highrecoveries [4]. The ICD step removes mineral scale precursors from aprimary RO concentrate stream and allows a further recovery of waterin the secondary RO. (iv) Using renewable energy resources tosubsidize the electricity energy demand [7,26], and (v) optimizing ROconfigurations and operating conditions using mathematical models(e.g., [11,15]). The development and implementation of efficientmethods have led to a reduction in energy consumption of ROdesalination as evidenced by many industrial examples [17,19].

Model-based analysis plays an important role in reducing SEC inthe RO processes. It has been shown that operating the ROapproaching the thermodynamic limit (where the applied pressureis slightly above the concentrate osmotic pressure) significantlyreduces the SEC [21–23,25,30]. Using simplified or first-principlesbased models, it is also possible to account for capital cost, feed intakeand pretreatment, and cleaning and maintenance cost in theoptimization framework [9,12,28]. Recent research efforts have beenfocused on a formal mathematical approach to provide a clearevaluation of minimization of the production cost by studying theeffect of applied pressure, water recovery, pump efficiency, mem-brane cost, ERD, and brine disposal cost [32–35].

Desalination 276 (2011) 128–135

⁎ Tel.: +1 909 869 3668; fax: +1 909 869 6920.E-mail address: [email protected].

0011-9164/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.desal.2011.03.031

Contents lists available at ScienceDirect

Desalination

j ourna l homepage: www.e lsev ie r.com/ locate /desa l


This work aims to provide a comprehensive analysis of single- andmulti-stage RO with/without ERD from first-principles, which wouldguide the design and operation of RO at a lower SEC.Models based on thepreviously developed characteristic equation of RO are developed in adimensionless form and relationships between RO configuration, feedconditions, membrane performance, and operating conditions are un-ambiguously revealed. Optimal results at both the theoretical thermo-dynamic limits andpractical conditions are presented. Suggestions onROconfiguration and operating conditions are made to achieve a SEC lowerthan 3 times of the feed osmotic pressure with a high recovery.

2. Dimensionless characteristic equation of a RO module

The dimensionless characteristic equation of a RO module wasderived in the author's previous work [8]. A brief summary is givenbelow to provide sufficient background information for the analysis inthe next sections. Consider a general RO module shown in Fig. 1, themass balances of water and salt in a control volume (represented bydA) can be written as follows [8]:

−dQ = dA⋅Lp⋅ ΔP−Δπð ÞQ=Qf = Δπ0=Δπ

ð1Þ

where−dQ is the flow rate of water across the membrane of area dA,Lp is the membrane hydraulic permeability, ΔP and Δπ are thedifferences in the system pressure and osmotic pressure across themembrane, respectively. Q is the flow rate in the retentate and Qf isthe feed flow rate.Δπ0 is Δπ at the entrance of the membrane channel.Eq. (1) is based on the assumptions of (1) negligible salinity in thepermeate, and (2) linear relationship between osmotic pressure andsalt concentration [14]. Moreover, the effects of concentrationpolarization and fouling are not included here.

With the assumption of negligible pressure drop in the retentate,an integration of Eq. (1) from the entrance to the end of the mem-brane channel yields a dimensionless equation as follows [8]:

γ = α Y + α ln1−α

1−Y−α

� �ð2Þ

where α=Δπ0/ΔP, Y=Qp/Qf, and γ=ALpΔπ0/Qf. Qp is the permeateflow rate. α−1 is the dimensionless applied pressure, Y is thedimensionless fractional recovery, and γ might be considered as thedimensionless membrane capacity (or, alternatively, the inverse ofthe dimensionless feed rate). β−1=Y/γ=Qp/ALpΔπ0 is anotherdimensionless variable that may be considered as the dimensionlessaverage flux or average driving force. α and β are defined in such away for a better presentation of results in equations and/or figures.

Eq. (2) reveals the coupled behavior between membrane property(area and permeability), feed conditions (feed rate and osmotic pressure)and operating conditions (applied pressure and permeate rate) andthereforemaybe referred to as the characteristic equationof a ROmodule.

As mentioned earlier, the energy cost in the context of RO waterdesalination process is typically described using SEC, or the electricalenergy demand per cubic meter of permeate [32–35]. For a single-stage RO, it is readily derived that:

SECm = ηpumpSEC =QfΔPQp

ð3Þ

where SECm is the modified SEC to reflect the pump efficiency [29].The pump efficiency is considered to be constant for simplicity in thiswork. The normalized SEC (or NSEC) is defined as SECm/Δπ0 in thiswork, which is another dimensionless number.

In the operationof RO, rejection ofwater as brine is necessary becausethe required applied pressure increases drastically when the retentatebecomes more and more concentrated. Fig. 2 shows a diagram of thedimensionless driving force along the RO channel [8]. Note that thedriving force at the exit of themembranemust be positive to guarantee anonzero flux, which implies that 1−Y−αN0, orΔPNΔπ0/(1−Y). At thetheoretical thermodynamic limit, 1−Y−α=0 or ΔP=Δπ0/(1−Y).

3. NSEC in single or multi-stage ROs without ERD

Based on the dimensionless characteristic equation introduced inthe previous section, it is convenient to describe a single or multi-stage RO module and to formulate an optimization problem tominimize the NSEC. A list of key parameters in single or multi-stageRO are provided in Table 1.

The minimization of NSEC in a single or multi-stage RO module isformulated as follows:

minαj ;Yj

SECm

Δπ0=

∑N−1

j=1

Yjαj

+1αN

1−∏N

j=11−Yj

� �s:t:

0 = γj−α Yj + αj ln1−αj

1−Yj−αj

" #j = 1;…;Nð Þ

0 = γtotal− γ1 + ∑N−1

j=2γj ∏

j−1

k=11−Ykð Þ2

" #

0 = γj−γj + 1 1−Yj� �2

j = 1;…;N−1ð Þ0 ≤ αj j = 1;…;Nð Þ0 ≤ 1−αj j = 1;…;Nð Þ0 ≤ αj + 1−αj = 1−Yj

� �j = 1;…;N−1ð Þ

0 ≤ 1−∏N

j=11−Yj

� �" #−Ymin

ð4Þ

P0,Qf,Cf

Pf,Qf,Cf

P0,Qp,Cp

Pf,Cb,Qb

Fig. 1. Schematic of a reverse osmosis water desalination process.

γ

1/β1/α-1

1/α-1/(1-Y

)

Δ /Δπ0P

Δπ/Δπ0

Fig. 2. Schematic of normalized dimensionless driving force (or dimensionless flux)along the membrane channel.

129M. Li / Desalination 276 (2011) 128–135


where γtotal=AtotalLpΔπ0/Qf is the total capacity of all ROs. Equalityconstraint 0=γj−γj+1(1−Yj)2, ( j=1,…,N−1) means the stageshave the same membrane area. 0≤αj+1−αj/(1−Yj) is used so that

ΔPj ( j=2,…,N) is nonnegative. 0≤ 1− ∏N

j=11−Yj� �" #

−Ymin is included

in the optimization problem to guarantee a minimum recovery. Thisconstraint, if not included in the optimization problem,might result invery low water recoveries in some cases. To avoid taking logarithm ofnon-positive numbers in the numerical optimization procedure, anintermediate variable z= ln[(1−α)/(1−Y−α)] might be introduced[8].

The solution to Eq. (4) is shown in Fig. 3. For both single or multi-stage ROs without an ERD, the optimal NSEC is uniquely determinedby γtotal, as shown in Fig. 3(a). As γtotal increases (e.g., using a largermembrane area, a more permeable membrane, or a reduced feedrate), NSEC reduces gradually and flattens out eventually. The cut-off value of γtotal (beyond which a reduction in NSEC is notdifferentiable) is larger if the number of stages in the RO moduleincreases. It is worth pointing out that an increase in the feedosmotic pressure also leads to an increase in γtotal and thus areduction in NSEC. However, SECm as a product of NSEC and Δπ0 stillincreases [8].

It is also shown that when γtotal is small, there is little difference inNSEC for ROs with different number of stages. However, as γtotal

becomes larger, the advantage of using more stages is apparent. ForROs with 1, 2, 3, 4, and 5 stages without an ERD, the NSEC approaches4, 3.60, 3.45, 3.38 and 3.33 when γtotal becomes sufficiently large (or itlies in the flattened-out region). Another advantage of using morestages is that the water recovery corresponding to the optimal NSEC isenhanced even though the total membrane area is the same, as shownin Fig. 3(b). For example, the fractional recovery approaches 50%when the γtotal→∞. However, it increases to 57% for a two-stage ROand 63% for a five-stage RO.

The optimal stage αj and stage recovery Yj in single-, two-, andthree-stage ROs to minimize NSEC are shown in Fig. 3(c) and (d). It isseen that αj (or Yj) at different stages are typically not the same for agiven γtotal. However, when γtotal becomes sufficiently large, αj at allstages converge to the same value in the multi-stage RO. The samebehavior is found in Yj. The more stages are used in a multi-stage RO,the larger the converged value of αj becomes (which implies that thedimensionless applied pressure is smaller).

If the membrane (ALp) and the feed osmotic pressure (Δπ0) arefixed, Qf cannot be randomly chosen in the operation of RO modulesfor the sake of NSEC. Instead, Qf should be determined based on anappropriate γ. Note that a large γ adversely affects the feed rate thatthe RO module can take. An appropriate γ might be selected beforethe NSEC curve flattens out in Fig. 3(a) to achieve an NSEC slightlyabove the theoretical minimum while maintaining a reasonably highQf. The author provided a method to experimentally determine themembrane property group (ALp) in a previous work [8]. For a single-stage RO, by recording water recoveries at a fixed applied pressureand different feed rates, a plot Qf/Δπ0 vs. α Y + αln 1−α

1−Y−α

� would

yield a straight line passing through the original point with a slope ofALp. The appropriate feed rate is then calculated as Qf=ALpΔπ0/γ. Fora RO of several identical stages, one may bypass all the interstagebooster pumps and solve AtotalLp in a similar way. Then Qf=AtotalLpΔπ0/γtotal is used for the feed rate. The optimal applied pressureat each individual stage should then be determined from Fig. 3(c)based on the chosen γtotal.

Note that the computational cost of Eq. (4) increases significantlywhen the number of stages becomes very large (e.g., 100). However,calculations can be readily done when the process approaches to thethermodynamic limit, where γj becomes sufficiently large at eachstage [8].

At the thermodynamic limit, αj+Yj=1 holds at each stage. TheNSEC has the following form:

SECm

Δπ0=

∑N

j=1

1αj

−N + 1

1− ∏N

j=1αj

ð5Þ

where N is the number of stages. The optimal solution is determinedby taking the partial derivative with respect to αj (j=1,…,N) andsolving all the resulting equations. It is shown that α1=α2=…=αN=α, and α satisfies the following equation:

1−Nð ÞαN + 1 + N + 1ð ÞαN−1 = 0 ð6Þ

Eq. (6) can be solved numerically if N is greater than one. Once α isdetermined, the optimal NSEC and the overall recovery can be solvedas follows:

SECm

Δπ0=

Nα−N + 1

1−αNð7Þ

and

Yoverall = 1−αN ð8Þ

The results of NSEC and overall recovery are shown in Fig. 4. It isclearly seen that increasing the number of stages to 1000 (which maybe challenging in practice) in a RO module cannot bring the NSECbelow 3.1. This implies that using a multi-stage RO without an ERD isnot very effective to significantly reduce the NSEC. A multi-stage ROwith a total number of stages greater than five is not recommendedbecause the gain in NSEC and overall water recovery is not significant.

Table 1Expression of parameters in single or multi-stage RO processes.

Stage 1 N (N≥2)

ΔP (for pump) ΔP1 ΔPNΔP (for RO) ΔP1 ∑

N

j=1ΔPj

Y Y1 YNA A1 AN

Δπentrance Δπ0 Δπ0

∏N−1

j=11−Yj� �

Qf Q f ∏N−1

j=11−Yj� �

Qf

Qp Y1Qf YN ∏N−1

j=11−Yj� �

Qf

α Δπ0

ΔP1Δπ0

∏N−1

j=11−Yj� �

∑N

j=1ΔPj

β A1LpΔπ0

Y1Qf

ANLpΔπ0

YN ∏N−1

j=11−Yj� �2Qf

γ A1LpΔπ0

Qf

ANLpΔπ0

∏N−1

j=11−Yj� �2Qf

Yoverall Y1 1− ∏N

j=11−Yj� �

SECm

Δπ0

1Y1α1

∑N−1

j=1

Yjαj

+ 1αN

1− ∏N

j=11−Yj� �

130 M. Li / Desalination 276 (2011) 128–135


If the number of stages is not more than five, the NSEC is at least 3.3and the corresponding fractional recovery is about 63%. The NSEC willbe even greater if a higher recovery is required.

If a minimum overall water recovery has to be satisfied (Yoverall≥Ymin), one more step should be done based on the Kuhn–Tuckertheorem [2]:

α = 1−Yminð Þ1=N ð9Þ

and the optimal solution to α will be the smaller one determined byEqs. (6) and (9).

The results are shown in Fig. 5. It is seen that Yoverall is lower-bounded by the one shown in Fig. 4(b). When Ymin is set to be 50%,there is no difference in the cases with/without the overall recoveryconstraint Yoverall≥Ymin. The actual recovery is the same as the oneshown in Fig. 4(b). When Ymin is set to be 90%, however, α is deter-mined by Eq. (9), and the actual recovery is exactly 90%.

Note that at the thermodynamic limit the flux approaches zero.Therefore, the NSEC shown in Fig. 5 only provides the lower limit thata real RO can never achieve in practice. In reality, the NSEC will behigher to allow a non-zero flux. This result further confirms that anNSEC below 3 is not possible under real conditions using a multi-stageRO without an ERD.

3.1. NSEC in single or multi-stage ROs with ERD

Because the brine from the RO module has a higher pressure thanthe initial pressure of the feed, the NSEC in the RO desalinationprocess would be lower if the brine and the feed exchanges energythrough an ERD (see Fig. 6 for a schematic of a multi-stage RO withenergy recovery). Define ηerd=(Pb−Pe)/(Pb−P0), the work done bythe ERD to the feed is:

Werd = ηerdQb Pb−P0ð Þ = ηerdQf ∏N

j=11−Yj

� �∑N

j=1ΔPj ð10Þ

0.5 1 1.5 2 2.5 3 3.5 4 4.5 53.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

(SE

Cm) o

pt/Δ

π 0

1 stage2 stages3 stages4 stages5 stages

a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

γtotal=AtotalLpΔπ0/Qfγtotal=AtotalLpΔπ0/Qf

γtotal=AtotalLpΔπ0/Qf γtotal=AtotalLpΔπ0/Qf

1 stage2 stages3 stages4 stages5 stages

b)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

3−stage RO

2−stage RO

1−stage RO

α j

1st stage2nd stage3rd stage

c)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

3−stage RO

2−stage RO

1−stage RO

Yj


d)

Yov

eral

l

Fig. 3. (a) Optimal NSEC (SECm/Δπ0), (b) overall recovery (Yoverall), (c) stage α and (d) stage Y as functions of γtotal in single- and multi-stage ROs without ERD.

100 101 102 103

100 101 102 103

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Number of stages

a)

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

Number of stages

b)

(SE

Cm)/

Δπ0

Yov

eral

l

Fig. 4. (a) NSEC and (b) Yoverall at the thermodynamic limit (γ→∞).



When a multi-stage RO is combined with an ERD, it can be derivedthat the objective function in Eq. (4) has the following form:

SECm

Δπ0=

∑N−1

j=1

Yjαj

+1αN

−ηerd 1−YNð ÞαN

1−∏N

j=11−Yj

� �" # ð11Þ

Assuming a 90% efficiency in ERD, the optimization problem issolved numerically and the results are shown in Fig. 7. A comparisonbetween Fig. 7 (90% ERD efficiency) and Fig. 3 (0% ERD) shows thatthe trends are very similar. The NSEC is much lower when an ERD isused. However, the corresponding water recovery to achieve theminimum of NSEC is also lower. The stage αj at each level is higherwhich implies that the RO with ERD is operated with a lowerdimensionless applied pressure.

Because the water recovery corresponding to the optimal NSEC forROs with ERD is typically very low (see Fig. 7(b)), it is necessary toinclude the minimum recovery constraint in the optimizationproblem. In Fig. 8, Ymin is chosen to be 80%. It is seen that αj becomessmaller (or the dimensionless applied pressure is larger) in order to

boost the overall recovery. It is shown that an NSEC around 2.5–2.8may be possible using a multi-stage RO (N=3, 4, or 5) and an ERD of90% efficiency operated at a γ around 3–5 to maintain an 80% waterrecovery. In this way, the feed rate is about 0.2–0.3 AtotalLpΔπ0.

Analytical solutions are possible when single- and multi-stage ROswith an ERD approach the thermodynamic limit, where γ→∞ andηerd→100%. It can be derived that:

SECm

Δπ0=

∑N

j=1

1αj

−N

1−∏N

j=1αj

ð12Þ

Following an approach similar to the case without ERD, theoptimal solution is derived as α1=α2=…=αN=α, and α satisfiesthe following equation:

NαN + 1− N + 1ð ÞαN + 1 = 0 ð13Þ

It is shown that the only solution for 0≤α≤1 is α=1 (because the

above equation can be converted to α−1ð Þ ∑N−1

j=0αN−αj� �

= 0 and the

sum term is negative for any 0bαb1). Therefore,

SECm

Δπ0= lim

α→1

N 1−αð Þα 1−αN� � = 1 ð14Þ

and

Yoverall = 1−αN = 0 ð15Þ

The result is not surprising if the RO separation is considered as thereverse process of the mixing of one droplet of pure water with a

1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12

Number of stages

a)

1 2 3 4 5 6 7 8 9 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Number of stages

b)

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of stages

c)

1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of stages

d)

(SE

Cm)/

Δπ0

α j Yj

Ymin = 50% Ymin= 60%

Ymin= 70%

Ymin= 80%

Ymin= 90%


Ymin= 70%

Ymin= 80%

Ymin= 90%


Ymin= 70%

Ymin= 80%

Ymin= 90%

Ymin= 50%

Ymin= 60%

Ymin= 70%

Ymin= 80%

Ymin= 90%

Yov

eral

l

Fig. 5. (a) Theoretical limit of NSEC, (b) Yoverall, (c) stage α and (d) stage Y to achieve a minimum overall recovery Yoverall in multi-stage ROs without an ERD (γ→∞).

P0,Qf,Cf

ERD

Fig. 6. Schematic of a multi-stage RO with energy recovery.



solution having an osmotic pressure of Δπ0. Also note that this isconsistentwith the ideal reversible RO process that has been solved by

Spiegler [24]: SECΔπ0

= − ln 1−Yð ÞY . Based on L'Hopital's rule, lim

Y→0

SECΔπ0

= 1.

If a minimum water recovery (Yoverall≥Ymin) is required, thethermodynamic limits of α and NSEC are determined as follows:

α = 1−Yminð Þ1=N ð16Þ

0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.31 stage2 stages3 stages4 stages5 stages

a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.18

0.2

0.22

0.24

0.26

0.28

0.3


b)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.65

0.7

0.75

0.8

0.85

0.9

3−stage RO

2−stage RO

1−stage RO


c)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

3−stage RO

2−stage RO

1−stage RO


d)

SEC

m/Δ

π 0

Yov

eral

l

α j Yj



Fig. 7. Optimal (a) NSEC, (b) Yoverall, (c) stage α and (d) stage Y as a function of γ in single- and multi-stage ROs with ERD (ηerd=90%).

0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.5

3

3.5

4

4.5

5


a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.75

0.8

0.85


b)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

3−stage RO

2−stage RO

1−stage RO 1st stage2nd stage3rd stage

c)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3−stage RO

2−stage RO

1−stage RO


d)

SEC

m/Δ

π 0

Yov

eral

l

α j Yj



Fig. 8. Optimal (a) NSEC, (b) Yoverall, (c) stage α and (d) stage Y as a function of γ in single- and multi-stage ROs with ERD (ηerd=90%) and a minimum recovery of 80%.



and

SECm

Δπ0=

N 1− 1−Yminð Þ1=Nh i1−Yminð Þ1=NYmin

ð17Þ

The results corresponding to the theoretical thermodynamic limit(γj→∞ and ηerd→100%) are shown in Fig. 9. Different from the casewithout ERD (Fig. 5), the actual Yoverall is always the same as thespecified Ymin. Again, since the NSEC shown in Fig. 5 only provides thelower limit corresponding to a zero flux, the NSEC in a real multi-stageROs with ERD would always be higher than 1.5 if the recovery is atleast 50% and the number of stages is not more than 5.

The above analysis shows that the minimal recovery (Ymin) is animportant parameter in the design and operation of RO. When an ERDis used, a lower Ymin (which can be achieved by applying a smallerpressure to the membrane module) corresponds to a smaller SEC.However, the post treatment cost will increase. A tradeoff betweenthe cost of electricity to drive the pump and the cost for watertreatment should be done to identify the best Ymin to be used by theoperator.

4. Conclusions

The previously developed characteristic equation of ROmodule [8]is used to describe single- or multi-stage ROs with/without energyrecovery. Analysis is made both at the theoretical limit (withanalytical solutions provided if possible) and at practical conditions(using constrained nonlinear optimization).

Reducing NSEC (or SECm/Δπ0) in ROs can be pursued through oneor more of the following three independent methods: (1) increasingγ=AtotalLpΔπ0/Qf, (2) increasing number of stages, and (3) using anERD. The NSEC flattens out when γ becomes sufficiently large. A verylarge γ is not recommended because it will affect the feed rate, andtherefore, the permeate rate. For ROs without ERD, the theoretical

limit of NSEC is 4, 3.60, 3.45, 3.38 and 3.33 if the number of stages is 1to 5. Using more stages not only reduces NSEC, but also improves thefractional recovery of water. However, The NSEC flattens outwhen thenumber of stages increases. It is still higher than 3.1 even if thenumber of stages becomes 1000. Therefore, multi-stage ROs withoutERD is not very efficient from a view point of energy consumption. It isnot recommended to have ROs with more than five stages. The ERDcan significantly reduce the NSEC, theoretically to 1, while thecorresponding recovery approaches zero. The NSEC becomes largerwhen the required water recovery increases.

It appears that the above methods have their own advantages andlimitations. A combination of all is able to achieve a low NSEC and tomaintain a high recovery and permeate flow, if the RO is operated atoptimal stage pressures. Based on the first-principles analysispresented in this work, it might be possible to obtain an NSEC around2.5–2.8 using a multi-stage RO and an ERD of 90% efficiency operatedat a γtotal around 3–5 (the corresponding Qf=(0.2−0.33)AtotalLpΔπ0)to maintain an 80% water recovery.

In summary, this work provides a first-principles based theoreticalanalysis to reduce NSEC in RO desalination. The focus of futureresearch will be on the experimental validation and implementationof the presented theoretical results.

Nomenclatureα Δπ0/ΔP, dimensionlessβ ALpΔπ0/Qp, dimensionlessΔ difference across the membraneΔπ0 osmotic pressure difference at the entrance of the membrane

module, barηerd ERD efficiency, dimensionlessηpump pump efficiency, dimensionlessγ ALpΔπ0/Qf, dimensionlessπ osmotic pressure, barA membrane area, m2

Lp hydraulic permeability, m ⋅sec−1 ⋅bar−1

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

11

Number of stages

Ymin = 50%Ymin = 60%Ymin = 70%Ymin = 80%Ymin = 90%

a)

1 2 3 4 5 6 7 8 9 100.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Number of stages

b)

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of stages

c)

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of stages

d)

(SE

Cm)/

Δπ0

Yov

eral

l

α j Yj




Fig. 9. (a) Theoretical thermodynamic limit of NSEC, (b) Yoverall, (c) stage α and (d) stage Y to achieve a minimum overall recovery Yoverall in multi-stage ROs with energy recovery(ηerd→100% and γtotal→∞).



N number of stages, dimensionlessP pressure, barQ flow rate, m3 ⋅sec−1

SECm specific energy consumption times thepumpefficiency, J⋅m−3

Y Qp/Qf, dimensionlessb brinef feedp permeate

Acknowledgement

This work is partly supported by the American Chemical SocietyPetroleum Research Fund (PRF# 50095-UR5).

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