Multi-objective optimization of reverse osmosis ...repository.ias.ac.in/14130/1/407.pdf ·...

Computers and Chemical Engineering 29 (2005) 1977–1995

Multi-objective optimization of reverse osmosis desalination unitsusing different adaptations of the non-dominated

sorting genetic algorithm (NSGA)

Chandan Guriaa, Prashant K. Bhattacharyab, Santosh K. Guptab,∗a Department of Polymer Engineering, Birla Institute of Technology, Mesra 835215, Ranchi, India

b Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

Received 5 October 2004; received in revised form 9 May 2005; accepted 10 May 2005Available online 27 June 2005

Abstract

Multi-objective optimization using genetic algorithm (GA) is carried out for the desalination of brackish and sea water using spiral woundor tubular modules. A few sample optimization problems involving two and three objective functions are solved, both for the operation of an

possibleentration.

alt and-side), arehe binaryughput,e permeatetowards

brackishckish

m recent,

ination

ut

-res

ionssmo-the

existing plant (which is almost trivial), as well as, for the design of new plants (associated with a higher degree of freedom). Theobjective functions are: maximize the permeate throughput, minimize the cost of desalination, and minimize the permeate concThe operating pressure difference,�P, across the membrane is the only important decision variable for anexisting unit. In contrast, for anew plant,�P, the active area,A, of the membrane, the membrane to be used (characterized by the permeability coefficients for swater), and the type of module to be used (spiral wound/tubular, as characterized by the mass transfer coefficient on the feedthe important decision variables. Sets of non-dominated (equally good) Pareto solutions are obtained for the problems studied. Tcoded elitist non-dominated sorting genetic algorithm (NSGA-II) is used to obtain the solutions. It is observed that for maximum throthe permeabilities of both the salt and the water should be the highest for those cases studied where there is a constraint on thconcentration. If one of the objective functions is to minimize the permeate concentration, the optimum permeability of salt is shiftedits lower limit. The membrane area is the most important decision variable in designing a spiral wound module for desalination ofwater as well as seawater, whereas�P is the most important decision variable in designing a tubular module for the desalination of brawater (where the quality of the permeate is of prime importance). The results obtained using NSGA-II are compared with those fromore efficient, algorithms, namely, NSGA-II-JG and NSGA-II-aJG. The last of these techniques appears to converge most rapidly.© 2005 Elsevier Ltd. All rights reserved.

PACS: 02.60.Pn

Keywords: Multi-objective optimization; Elitist non-dominated sorting genetic algorithm; Jumping genes; Genetic algorithm; Optimization; Desal;Brackish water; Seawater; Reverse osmosis; Membranes

1. Introduction

Desalination of seawater and brackish water is rou-tinely used nowadays for overcoming the huge scarcityof potable water in different parts of the world. Desali-nation involves the reduction of the concentration of

∗ Corresponding author. Tel.: +91 512 259 7031/127;fax: +91 512 259 0104.

E-mail address: [email protected] (S.K. Gupta).

the total dissolved solids (TDS) to less than abo200× 10−3 kg m−3 (200 mg L−1). Brackish water has amuch lower TDS (<10,000× 10−3 kg m−3) than seawater(>30,000× 10−3 kg m−3). This difference in the TDS is associated with substantial differences in the osmotic pressuassociated with these operations, leading to large variatin the operating pressure differences across the reverse osis (RO) membrane. The largest desalination plant inworld treating brackish water (Lohman, 1994) is located atYuma, AZ, USA. This has a capacity of 275,000 m3 day−1. It

0098-1354/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compchemeng.2005.05.002

1978 C. Guria et al. / Computers and Chemical Engineering 29 (2005) 1977–1995

Nomenclature

a permeability coefficient for water(m h−1 bar−1)

A active area of membrane (m2)b permeability coefficient of salt (m h−1)bπ osmotic coefficient (Eqs.(A2.4) and(A2.10))

(m3 bar kg−1)C salt concentration (kg m−3)Cost operating cost of desalination unit ($ h−1)Cele cost of electricity ($ kW−1 h−1)Cmain maintenance cost of membrane ($ m−2 h−1)Cmem capital cost of membrane ($ m−2 h−1)Cpump capital cost of the pump ($ h−1)dh hydraulic diameter of channel (m)DAB mass diffusivity of salt (A) through water (B)

(m2 h−1)fi ith objective function (m3 h−1; $ h−1; kg m−3)H penalty parameter defined in Eq.(3)Idist crowding distanceIrank rankJw volumetric flux of water (m h−1)Js mass flux of salt (kg m−2 h−1)ks mass transfer coefficient of salt in feed side

(m h−1)lchrom length of chromosomelsubstr length of substringlaJG string length of jumping genem defined in Eq.(A2.14)n exponent for the pumping cost (Eq.(A2.13))Ngen generation numberNgmax maximum number of generationsNp total number of chromosomes in the popula-

tionpc crossover probabilitypJG jumping gene probabilitypm mutation probabilityP pressure, barPen penalty parameter (Eq.(3))Qw volumetric flow rate (throughput) (m3 h−1)R observed rejectionRe Reynolds’ numberSc Schmidt numberSh Sherwood numberT temperature of the feed (◦C)v velocity of water in feed channel (m h−1)Wbase reference value of power for estimating

pumping cost (Eq.(A2.13)) (kW)

Subscript/superscriptb bulkbw brackish waterd desiredL lower bound

p permeateref reference, Yuma plant (Lohman, 1994)s saltsw seawaterU upper bound

Greek letters∆ differenceη efficiency of the pumpν kinematic viscosity of salt solution (m2 h−1)π osmotic pressure (bar)ρ density of seawater (kg m−3)

uses spiral wound cellulose acetate membranes to treat rawwater having 3100× 10−3 kg m−3 TDS and produces per-meate water having a TDS less than 200× 10−3 kg m−3. Thelargest desalination plant in the world processing seawater(Ayyash, 1994) operates in Jeddah, Saudi Arabia. This has acapacity of 56,800 m3 day−1 and treats water having a TDSof approximately 44,000× 10−3 kg m−3.

RO has several advantages over other desalination pro-cesses such as distillation, evaporation and electro-dialysis(Ho & Sirkar, 1992). The main advantages of RO over otherdesalination processes are its simple design, lower mainte-nance costs, easier de-bottlenecking, simultaneous removalof both organic and inorganic impurities, low discharge inthe purge stream, and energy savings. RO is a rate-governedpressure-driven process. The solvent flux depends upon theapplied pressure difference, trans-membrane osmotic pres-sure difference, concentration of feed, permeability coeffi-cients of salt and water, and the extent of concentration polar-ization. The flux increases (at the expense of high concentra-tion polarization) with an increase in the operating pressuredifference and permeability coefficients, and decreases withan increase in the salt concentration.

Rigorous optimal design (or operation) of RO moduleswill help in reducing their cost. Attempts have been madeto obtain optimal designs of RO units considering cost asthe single objective function.Wiley, Fell and Fane (1985)h ulesf )h ,1 am-m dbo udiesi on,w veralc s oft ersy.L sso-c tionst e of a

ave carried out the optimal design of membrane modor brackish water desalination using theRosenbrock (1960ill climb method without constraints, with Palmer’s (Palmer969) axis rotation method. Sequential quadratic progring (SQP;Gill, Murray, & Wright, 1991) has been usey Maskan, Wiley, Johnston, and Clements (2000)to findptimal networks of reverse osmosis modules. These st

nvolve the optimization of only a single objective functihich may, at times, be taken as a weighted-average of seonflicting objective functions. The assignment of valuehe weighting factors is subject to considerable controvike most problems, the design of RO modules is also aiated with several non-commensurate, objective funchat need to be optimized simultaneously in the presenc

C. Guria et al. / Computers and Chemical Engineering 29 (2005) 1977–1995 1979

few constraints. Such problems are best handled using multi-objective optimization (MOO) techniques. In such problems,a set of several equally good (non-dominated) optimal solu-tions is often obtained (instead of a single optimal point),called a Pareto set. The basic advantage of MOO formula-tions is that the decision-maker is not confined to look atonly a single mathematically optimal solution (usually thatinvolving the minimum cost), but he/she can examine a set ofefficient solutions using a judgment of the trade-offs involved,refining his/her final decision (Mavrotas & Diakoulaki, 1998;Deb, 2001). Indeed, Pareto sets are becoming an ‘increas-ingly effective way to determine the necessary trade-offsbetween conflicting objective functions’ (D.E. Goldberg inDeb, 2001). The use of a single objective function which isa weighted-average of several objectives also has the draw-back that certain optimal solutions may be lost since they maynever be explored, particularly when the non-convexity of theobjective function gives rise to a duality gap (Goicoechea,Hansen, & Duckstein, 1982). Unfortunately, there is no studyon the optimal design of RO modules in the literature usingmultiple objective functions, though a parallel study (Yuen,Aatmeeyata, Gupta, & Ray, 2000) on beer dialysis (mini-mizing the alcohol content of beer to give low-alcohol beer,while maximizing the taste chemicals in the product) hasbeen reported. Optimal RO design in desalination involvesthe selection of membrane material, module geometry (viz.,p em-b ter),o nd thet 2P od-u hput( ngt e arec arly,d nityf

ono-m rticu-l 9;H msi m,s s( -t agin’sp thea s fort n ofs per-a iredb thati anyd In thee tingt werer s (and

then mapped into real numbers for use in model equations).This is an unavoidable compromise and causes problems(Deb, 2001), e.g., it slows down the computing speed and,at times, renders convergence impossible. Modifications(e.g., real coded GAs, the jumping gene adaptation, etc.)are becoming available but each technique has its ownlimitations.

Several workers have extended SGA to solve multi-objective optimization problems. Any of these techniques,reviewed recently byDeb (2001)andCoello Coello, Veld-huizen and Lamont (2002), can be used to obtain the Paretofronts. A popular algorithm for such problems is the non-dominated sorting genetic algorithm (NSGA), developedby Deb and coworkers (Deb, 2001). Two versions of thistechnique are available, NSGA-I (Srinivas & Deb, 1995)and NSGA-II (Deb, Pratap, Agarwal, & Meyarivan, 2002).Bhaskar, Ray, and Gupta (2000)have reviewed the variety ofmulti-objective optimization problems in chemical engineer-ing that have been solved in the last decade using NSGA-I(as well as the earlier optimization studies using traditionaltechniques). NSGA-II introduces the concept of elitism (Deb,2001) and has been applied recently to solve two highlycomputationally intensive problems in chemical engineer-ing, namely, the multi-objective optimization of an industrialfluidized-bed catalytic cracker unit (FCCU;Kasat, Kunzru,Saraf, & Gupta, 2002) and the unsteady operation of a steamr tf ctedf ratedb comet iver-s2 ofj ,2 era-t iono theo ndp m,Zn GA-I enep ism.A d byGt eenf solu-t othfl .M dt hichc hasb 5)f er-a areg

late and frame, tubular, spiral wound, or hollow-fiber), mrane area, quality of product, solvent recovery (i.e., waperating pressure difference across the membrane, a

hroughput (Bhattacharyya, Williams, Ray, & McCray, 199;arekh, 1988). One should be able to select optimal mle parameters that provide the highest possible througfirst objective function) while simultaneously minimizihe cost of desalination (second objective function). Thesonflicting (and non-commensurate) requirements. Cleesalination through RO provides an excellent opportu

or multi-objective optimization studies.Over the last few years, scientists, engineers and ec

ists have used AI-based evolutionary techniques, paarly, genetic algorithms (GA;Deb, 1995; Goldberg, 198olland, 1975), extensively to solve optimization proble

nvolving single objective functions. This basic algorithimple GA or SGA (Goldberg, 1989), offers advantageDeb, 2001; Holland, 1975) over more traditional optimizaion approaches (e.g., several search techniques, Pontryrinciple, SQP, etc.), in some cases. Moreover, it hasdvantage that it does not require good initial guesse

he values of the ‘decision variables’. It uses a populatioeveral points simultaneously along with probabilistic otors, viz., reproduction, crossover and mutation, inspy natural genetics. In addition, SGA has the advantage

t uses only the values of the objective functions and noterivatives, as required by gradient search techniques.arly algorithms, binary coding was used for represen

he continuous decision variables, i.e., these variablesepresented/coded as a series (string) of binary number

eformer (Nandasana, Ray, & Gupta, 2003). An importaneature of NSGA-II is that the best members are selerom a combined pool of parents and daughters (geney crossover and mutation of the parents), and these be

he parents for the next generation. Elitism reduces the dity of the gene pool, but offers several advantages (Deb,001). Kasat and Gupta (2003), inspired by the concept

umping genes (JG or transposons;McKlintock, 1987; Stryer000) in biology, developed the jumping gene (JG) op

or for use with SGA/NSGA. This macro–macro mutatperation in the binary-coded NSGA-II-JG speeds upptimization of FCCUs by almost an eight-fold factor, arovides theglobal optimal Pareto front for the test probleDT4 (Deb, 2001; Zitzler, Deb, & Thiele, 2000), which couldot be solved satisfactorily using the binary-coded NS

I. The JG operator helps improve the diversity of the gool and, thus, counteracts the negative effect of elitfurther adaptation of NSGA-II-JG has been presenteuria, Verma, Mehrotra, and Gupta, 2005). This is referred

o as NSGA-II-mJG (modified JG). This algorithm has bound to speed up the convergence to the global optimalions for problems involving networks, as for example, frotation circuits (Guria et al., 2005) for mineral processingore recently,Bhat, Saraf, and Gupta (2005)further adapte

his concept and proposed NSGA-II-aJG (adapted JG), would solve ZDT4 even more efficiently. This adaptationeen applied successfully byKhosla, Saraf, and Gupta (200

or the multi-objective optimization of fuel oil blending options. Details of NSGA-II-JG as well as NSGA-II-aJGiven inAppendix A.


The present work involves the simulation of the desali-nation plant (Lohman, 1994) at Yuma, followed by the for-mulation and solution of a few multi-objective optimizationproblems for desalination using RO modules. The binarycoded NSGA-II (Deb, 2001; Deb et al., 2002) is used. Theresults are then compared with those obtained with NSGA-II-JG and NSGA-II-aJG so as to study the efficiency of thesealgorithms. The optimization is carried out both at theoper-ating stage (Lohman, 1994; optimization of the operatingconditions in an existing unit) as well as at the design stage(optimization of a new plant), to illustrate the variety ofoptimization problems that can be solved. The optimal solu-tions of the first problem are also compared with the actualoperating point of the existing unit (Lohman, 1994). Themethodology is quite general and can be used for other plantsas well. It may be mentioned that this is the first applicationof the multi-objective elitist NSGA-II with the jumping geneadaptations in the area of membrane separation processes.

2. Formulation

2.1. Model of the reverse osmosis (RO) process

Various mathematical models are available that describet . Thet thew ,&T iveni eq blest

]

P

T hev d int ,t yc sstjD o-dN ,&1 ,1K ord und,

hollow-fiber, and tubular) using the correlations summarizedin Appendix B. The throughput,Qw, also depends on oneoperating condition: the pressure difference,�P, across themembrane. The solute concentration,Cb, in the feed and thetemperature,T, of operation, are usually specified (constants).

Eq. (1a) is an implicit nonlinear algebraic equation thatcan easily be solved numerically to giveJw and Qw for aset of values ofCb, T, A, a, b, ks and�P, bπ can be esti-mated using Eqs.(A2.17) and(A2.18). The secant method(Ray & Gupta, 2004) is used to solve this equation. Thismethod requires lower and upper bounds (estimates) ofJw,as well as two initial guesses of this root. The values ofCpand the cost can then be evaluated using Eqs. (1b) and (1c),respectively.

2.2. Multi-objective optimization

The plant at Yuma, AZ, USA (Lohman, 1994) usesa spiral wound module and treats brackish water. Theparameters characterizing this unit first need to be esti-mated (‘tuned’). This is done using the following availableinformation (Lohman, 1994): A = 3.93072× 105 m2;Qw = 275,000 m3 day−1 = 11,458 m3 h−1; �P = 27.6 bar;Cb = 3.1 kg m−3; observed rejection = 97%;Cp = 0.2 kg m−3

andT = 25◦C. The exact value ofks depends on the geomet-ric parameters of the element (i.e., the number of leaves,t acera erties( ity)oD ivenb re-

s& eso yca eo ma( d( ari-ad is thes e-tb ionsam

he local behavior and performance of the RO processransport through RO membranes is well described byidely accepted solution diffusion model (Lonsdale, MertenRiley, 1965; Rautenbach, 1986; Soltanieh & Gill, 1981).

he detailed equations (for isothermal operation) are gn Appendix B. The permeate flux,Jw (=Qw/A), the permeatuality,Cp, and the cost, Cost, the three important varia

hat are used as objectives in this study, are given by

Jw = a

[�P − bπ

(Cb − bCb exp(Jw/ks)

Jw + b exp(Jw/ks)

)exp(Jw/ks)

Cp = bCb

b + Jw exp(−Jw/ks)

Cost= CmemA + CmainA + Cpump

(Qw�P

Wbaseη

)n

+ CeleQw�

η

he variables in Eq.(1)are defined in the Nomenclature. Tolumetric flow rate,Qw, of the permeate can be expresseerms of fourdesign variables: the area,A, of the membranehe permeability coefficient,a, of water, the permeabilitoefficient,b, of the salt, and the feed-side liquid film maransfer coefficient,ks (Brian, 1965, 1966; Kimura & Sourira-an, 1968; Sherwood, Brian, Fisher, & Dresner, 1965; Sirkar,ang, & Rao 1982). The values ofks depend upon the hydrynamics on the feed side (Ohya & Taniguchi, 1975; Ohya,akajima, Takagi, Kagawa, & Negishi, 1977; Perry, GreenMalony, 1997; Rao & Sirkar, 1978; Shock & Miquel,

987; Stanojevic, Lazarevic, & Radic, 2003; Taniguchi978; Taniguchi & Kimura, 2005; Taniguchi, Kurihara, &imura, 2001; Wiley et al., 1985), and may be estimated fifferent RO modules (e.g., plate and frame, spiral wo

(a)

(b)

(c)

(1)

he thickness of spacers, the porosity of the feed spnd the membrane thickness) and the physical propmainly, density, kinematic viscosity and mass diffusivf the salt solution, and is estimated using Eq.(A2.9).etails of different types of spiral wound modules are gy Shock and Miquel (1987). We have used values cor

ponding to a FilmTec FT 30 spiral wound module (ShockMiquel, 1987) in the present study. The ‘tuned’ valu

f the two unknown parameters,a and b, are obtained burve-fitting the operating data, as 1.80× 10−3 m bar−1 h−1

nd 5.04× 10−4 m h−1, respectively. A simple two-objectivptimization problem for the (operating) plant at Yureferred to asoperating-stage optimization) is first solveProblem 1). The optimal value of the single decision vble,�P, is to be obtained. Since the permeabilities,a andb,epend primarily on the membrane, and since the latterame for all values of�P, the tuned values of these paramers are used. Thus, for this problem, values ofCb, bπ, A, a,, T, and the module are specified. Two objective functre used: maximization of the permeate flow rate,Qw, andinimization of theCost. The permeate concentration,Cp, is


Table 1Details of the several optimization problems studied

Problem no. 1 2 3 4 5

Module Spiral wound (FilmTec FT30) Spiral wound(FilmTec FT30)

Tubular module(PCI)

Spiral wound(FilmTecFT30)

Tubular module(PCI)

ks (m h−1) Eq.(A2.9) (Shock & Miquel,1987)

Eq.(A2.9) (Shock &Miquel, 1987)

Eqs.(A2.15)and(A2.16)(Wileyet al., 1985)

Eq.(A2.9)(Shock &Miquel, 1987)

Eqs.(A2.15)and(A2.16)(Wileyet al., 1985)

Feed Brackish water Brackish water Brackish water Sea water Brackish WaterOperating/design Operating (Yuma) Design Design Design DesignCb (kg m−3) 3.1 (Lohman, 1994) 3.1 (Lohman, 1994) 3.1 (Lohman, 1994) 35.0 3.1 (Lohman, 1994)bπ (m3 bar kg−1) 0.789b 0.789b 0.789b 0.781b 0.789b

Values (existing) orbounds (new)

�P (bar) 10–50 10–50 10–50 75–250 10-5010−5A (m2)a 3.93072 (Lohman, 1994) 1.0–4.0 2.0–4.0 1.0–4.0 2.0-4.0103a (m3 m−2 bar−1 h−1) 1.8 (Shock & Miquel, 1987) 0.5–5.0 0.2–1.0 0.5–5.0 0.2-1.0104b (m3 m−2 h−1) 5.04 (Shock & Miquel, 1987) 0.1–1.0 0.08–0.3 0.1–1.0 0.08-0.3Cp,d

c (kg m−3) 0.2 0.2 0.2 – –

a 103a = 1.8 representsa = 1.8× 10−3, etc.b Calculated from Eqs.(A2.17)and(A2.18).c Cp,d is used as an objective function in Problem 5 (and not used as a constraint in Problem 4).

constrained to lie below a desired value,Cp,d. This problem,relevant to the operation of the existing plant at Yuma, canbe written mathematically, as:

Problem 1. Yuma plant; specifiedCb,bπ,A,a,b,T; FilmTecFT 30 spiral wound module

Maxf1(�P) ≡ Qw

Qw,ref(a)

Min f2(�P) ≡ Cost

Costref(b)

Subject to (s.t.) :

Model equations (Appendix 2) (c)

Bounds : �PL ≤ �P ≤ �PU (d)

Constraint : Cp ≤ Cp,d (e)

(2)

In Eq.(2),Qw,ref and Costref (=11,458 m3 h−1 and $ 2904 h−1,respectively), estimates for the currently operating Yumaplant using Eq.(1), are used to normalize the two objectivefunctions,�PL and�PU are the lower and upper bounds on�P (values given inTable 1) andCp,d is taken as 0.2 kg m−3.Eq. (A2.9) is used to estimateks (for any Qw), while theCost is given only by the second (constant for allQw, sinceA is constant) and fourth terms of the right hand side of Eq.(1c) (since the remaining two terms are already ‘sunk’ foraIe tants

nitsu ed -b le

(ks). The membrane permeability coefficients,a andb, arerelated to the thickness of the membrane and its properties,namely, the diffusivities of salt and water in the membrane,the partition coefficient of the solute in the membrane, thefeed temperature and, the nature of the concentration polar-ization, and can be considered as decision variables directly.A is the membrane area and can vary continuously.ks dependson the hydrodynamics associated with the membrane module,and can be estimated using the correlations inAppendix Bforany desired module. Two, two-objective optimization prob-lems are being studied here using two different modules (sothat the results can be compared), namely, the FilmTec FT 30spiral wound module (Shock & Miquel, 1987), and the PCItubular module (Wiley et al., 1985):

Problems 2 and 3. (design stage; specifiedCb, module,T):

Max f1(�P, A, a, b) ≡ Qw

Qw,ref(a)

Min f2(�P, A, a, b) ≡ Cost

Costref(b)

Subject to (s.t.) :

Model equations (Appendix B) (c)

Bounds : �PL ≤ �P ≤ �PU, AL ≤ A ≤ AU,

U U

(3)

Ta e arec rlier).T Eq.(

n existing/operating plant; see discussion inAppendix B).t may be noted that the normalization constant, Costref, isvaluated using all four terms in Eq. (1c) (this is unimporince Costref is a constant anyway).

We could also study the optimization of desalination under more flexibledesign conditions (for new units). Thecision variables, then, are not only�P, but also the memrane parameters, namely,A, a, b and, the membrane modu

aL ≤ a ≤ a , bL ≤ b ≤ b (d)

Constraint : Cp ≤ Cp,d (e)

he values of the normalization constants,Qw,ref and Costref,re taken to be the same as in Problem 1 (since thesonstants anyway; this does not matter, as discussed eaable 1gives the details. Problem 4 is also described by3), but corresponds to the desalination ofsea water using a


Table 2Computational parameters used for Problems 1–5

Problem no. 1 2 3 4 5 4(JG) 4(aJG)

Ngmax 500 1000 1000 1000 10000 1000 1000Np 100 100 100 100 100 100 100lsubstr 32 32 32 32 32 32 32lchrom 32 128 128 128 128 128 128pc 0.90 0.99 0.98 0.90 0.80 0.98 0.98pm 0.01 0.001 0.01 0.02 0.015 0.001 0.001pJG – – – – – 0.80 0.80laJG – – – – – – 12Random seed numbera 0.765 0.765 0.765 0.765 0.765 0.765 0.765H 105 105 105 – – – –

a Random numbers are generated using theKnuth (1997)portable subtractive pseudo-random number generator.

FilmTec FT30 spiral-wound module. Because the salt con-centration in sea water is high, the constraint onCp is omittedfor this problem. In Problems 2–4, all four terms on the righthand side of Eq. (1c) are used to estimate the Cost (see dis-cussion inAppendix B).

The constraint onCp in Problems 1–3 (Eqs. (2e) and (3e))is taken care of using a penalty function approach (Deb, 1995,2001). We add (for minimization of an objective function) orsubtract (for maximization) a penalty, Pen, given by

Pen≡ H

[1 −

(Cp

Cp,d

)](4)

to both the objective functions (in Eqs.(2)and(3)). In Eq.(4),the penalty, Pen, is taken to be a very large number (comparedto the values of the two objective functions) whenever thevalue ofCp is aboveCp,d, but Pen is zero whenCp is belowCp,d (Deb, 1995, 2001). Table 2gives the values ofH used.Its value is large enough so that the solutions do not changewith a further increase ofH.

The NSGA-II (Kasat et al., 2002), NSGA-II-JG (Kasat& Gupta, 2003) and NSGA-II-aJG (Bhat et al., 2005) codesavailable to us maximizeall the objective functions. A pop-ular transformation for an objective function,f, that has to beminimized, to one involving the fitness function,F, that hasto be maximized, is given by

F

T1

3

testp rs.P ing� reos Thisc e of

errors. Several other standard tests, described byKasat et al.(2002)were also tried. The best values of the computationalparameters were then obtained for the different problems.These are given inTable 2.

In Problem 1, there is only one degree of freedom (aselected value ofJw determines the complete solution) and sothe solution of Eq.(2)can be obtained analytically. It is foundthat as�P increases,Qw increases and the Cost goes down,a characteristic of a Pareto set. This problem is a relativelytrivial one and so detailed results are not being presentedhere (but can be supplied on request), and this problem is notpursued further.

Problem 2 is a more interesting, design-stage multi-objective optimization problem, involving more than a singledegree of freedom. Pareto sets are obtained, as shown inFig. 1a. The CPU time taken for this problem (as well asall others, using any of the adaptations of NSGA-II) for 1000generations, and using 100 chromosomes is 1 min (on a Pen-tium IV, 1.7 GHz, 256 MB RAM). The mean value of thecrowding distance,Ii,dist (see Step 3c inAppendix A), as wellas the standard deviation of these values, in any generation,can be used to get an idea of the degree of convergence ofthe Pareto set. Details of this method are described inKasatand Gupta (2003). Alternatively, an eye estimate can be usedto get an idea of when the Pareto set has stopped changing(converged) from generation to generation, and if the ‘spread’o evia-t given alm ilityc dm thatt lueo ith� bar).T thei tam ec di-t n the

= 1

1 + f(5)

his transformation does not alter the optimal solutions (Deb,995, 2001).

. Results and discussion

The NSGA-II computer code was tested on a fewroblems (Deb, 2001) to make sure that it was free of erroroblem 1 (Table 1) was run with a single chromosome usPL =�PU = 27.6 bar. The following optimal values webtained:Qw = 11458.0 m3 h−1 and Cost =$ 2904.0 h−1, theame as actually used (simulation values) in the plant.onfirms that the code finally used for optimization is fre

f the points in the set are near-uniform (the standard dion of Ii,dist can be used for this). The two approachesearly similar results.Fig. 1d and e show that the optimodule must have the maximum permissible permeab

oefficients (a and b) for the FilmTec FT30 spiral wounembrane. This is not surprising. What is interesting is

he increase inQw is first achieved by an increase in the vaf membrane area,A, to its maximum permissible value (wP being constant at an intermediate value of about 33hereafter,Qw and the Cost both increase because of

ncrease in�P (with the membrane area,A, being constant its maximum specified value). This indicates thatQw isore sensitive toA than to�P. It may be mentioned that th

onstraint onCp in Problem 2 can be replaced by an adional term in the Cost that accounts for the decrease i


Fig. 1. Optimal solutions for Problem 2 (seeTable 1for details).


value of the permeate concentration when its specificationis violated. This, and several other interesting optimizationproblems, can be solved but are not presented since the aimis to present results of only a few simple problems.

Fig. 2 shows results for the design-stage two-objectiveoptimization problem for the desalination of brackish waterusing a different module, namely the PCI tubular module.Fig. 2a shows the Pareto set.Fig. 2d and e show that theoptimal values of the permeabilities of water and salt musthave the maximum possible values. This is similar to obser-

vations fromFig. 1(for the spiral wound module). In the caseof the PCI module, the increase inQw is first achieved by anincrease of�P to its maximum possible value (with the mem-brane area being constant at its minimum specified value).Thereafter,Qw and Cost both increase with an increase inthe membrane area (with the�P being constant at its upperlimit). This indicates thatQw is more sensitive to�P than toA for this module. The contrast in the behaviors of the resultsfor the two modules is clearly brought out inFigs. 1 and 2.The value ofCp remains almost constant after�P attains

Fu

ig. 5. Optimal Pareto solutions for Problem 4 using NSGA-II, NSGA-II-JGpwards by 10,000 and 20,000 $ h−1, respectively, on the ordinate so that the p

and NSGA-II-aJG (results for NSGA-II-JG and NSGA-II-aJG are displacedlots can be easily compared).


Fig. 5. (Continued ).

its maximum value. This is consistent with intuitive expec-tations. A small amount of scatter is observed in the optimalvalues of�P, A, a and b in Figs. 1 and 2. It is clear thatdifferences in these four decision variables compensate foreach other, and do not affect the Pareto set much. This is acharacteristic of problems associated with several degrees offreedom, and such insensitivity of the Pareto set to scatterin a few decision variables has been encountered earlier inreal-life studies (Bhaskar et al., 2000; Sareen & Gupta, 1995;Tarafder, Rangaiah, & Ray, 2005). It is known that GA doesnot guarantee optimality of the final solutions (Deb, 2001)and a few sub-optimal points/solutions are almost always

encountered. However, one can easily infer the optimal Paretosolution from the results generated. One way of eliminatingthe scatter is to express the decision variables as low-orderpolynomials (Sareen & Gupta, 1995), and obtain optimal val-ues of the coefficients used. These would givenear-optimalsolutions that are more useful. It is interesting to observe fromFigs. 1 and 2that spiral wound modules give higher through-puts than tubular ones (for similar values of the operatingvariables), of course at higher costs.

Fig. 3 presents the Pareto set for the design-stage, two-objective optimization of asea water desalination unit usingthe FilmTec FT 30 spiral wound module (Problem 4,Table 1).


Fig. 5. (Continued ).

Here, the bounds of the membrane permeability coefficientsof water and salt (i.e.,a andb) of Problem 2 are used, butmuch higher ranges of�P are imposed for obvious reasons.In this case,Qw increases initially because of the increase inboth the membrane area,A, as well as�P. After the maxi-mum area of the membrane is attained, a further increase inQw is obtained due to the increase in�P. Small amounts ofscatter in the decision variables, mainly,�P, A, a andb isobserved, but the final Pareto set is insensitive to these vari-ations. The qualitative similarity of the optimal solutions for

the two cases (Problems 2 and 4, involving different ranges for�P) for treating brackish water and sea water, respectively,using the FilmTec FT 30 spiral wound membrane is to benoted, and contrasted to the results for the PCI tubular module(Problem 3).

The occurrence of a minimum inCp in Problem 3(Fig. 2) suggests that we can take the minimization ofCpas a third objective function. We, therefore, solve the fol-lowing three-objective optimization problem (at the designstage):


Problem 5. (design stage; specifiedCb, T, PCI module):

Maxf1 (�P, A, a, b) ≡ Qw

Qw,ref(a)

Min f2 (�P, A, a, b) ≡ Cost

Costref(b)

Min f3 (�P, A, a, b) ≡ Cp (c)

Subject to (s.t.) :

Model equations (Appendix B) (d)

Bounds : �PL ≤ �P ≤ �PU, AL ≤ A ≤ AU,

aL ≤ a ≤ aU, bL ≤ b ≤ bU (e)

(6)

The Cost is estimated for Problem 5 using all four terms on theright in Eq. (1c). The reference values ofQw,ref and Costref,are the same as in Problem 1.

Fig. 4shows the results of Problem 5. The optimal pointsin Fig. 4a and b, together, comprise a three-dimensionalParetosurface. A 3D plot involving the objective functionsis shown in Fig. 4g. Since the cost increases (worsens)(andCp increases (worsens), albeit slightly) asQw increases(improves) over the entire range of the latter, the optimalsolution has the characteristics of a Pareto set. In the 3D plot,some peaks are observed because of the outliers inFig. 4a andb, but a general increase is observed from the lowQw–lowC –low Cost end to the highQ –high C –high Cost end.T kerc selecta ree-o f thet ofs tivef5 auset themi t( itht 3)).O lem,v as m.

taN neo (JG)a f thec theset l-o threec -II-a -t bec aft um Ta

ble

3C

ompa

rison

ofth

ech

arac

teris

tics

ofth

eP

aret

oso

lutio

nsob

tain

edfr

omth

eth

ree

tech

niqu

es(P

robl

em4)

Gen

erat

ion

no.

NS

GA

-II

rang

eafr

omN

SG

A-I

Ira

nge

toN

SG

A-I

Im

ean

ofI i,

dist

NS

GA

-II

S.D

.ofI

i,dis

t

NS

GA

-II-

JGra

nge

from

NS

GA

-II-

JGra

nge

toN

SG

A-I

I-JG

mea

nof

I i,di

st

NS

GA

-II-

JGS

.D.o

fIi,d

ist

NS

GA

-II-

aJG

rang

efr

omN

SG

A-I

I-aJ

Gra

nge

toN

SG

A-I

I-aJ

Gm

ean

ofI i,

dist

NS

GA

-II-

aJG

S.D

.ofI

i,dis

t

198

6.71

7034

4.35

1.32

181.

4925

1012.7

975

788.

651.

3219

1.49

2571

0.5

773

049.

411.

3219

1.49

262

579.

5285

304.

780.

8682

1.13

6110

12.79

8633

4.10

1.06

111.

4814

1083.9

581

857.

830.

9461

1.21

595

481.

7787

069.

660.

6376

1.04

1214

88.75

7542

0.80

0.63

741.

0552

1094.2

288

141.

740.

6465

1.09

6910

1024

.68

8264

8.16

0.67

021.

0592

563

.21

8804

0.48

0.65

911.

0575

888

.33

9247

1.25

0.64

791.

0202

2029

9.41

9417

2.77

0.70

081.

1051

1135.6

388

375.

730.

7074

1.10

6730

03.1

795

147.

500.

6726

1.07

0830

299.2

793

857.

760.

7121

1.11

7658

3.5

382

762.

290.

7250

1.11

2049

6.7

292

996.

290.

6966

1.08

5440

284.3

793

715.

160.

7151

1.12

5427

9.9

092

911.

110.

7286

1.12

7412

13.76

9260

4.98

0.70

881.

1062

5029

1.27

9554

4.68

0.71

641.

1211

773

.29

9540

9.55

0.72

841.

1320

546

.01

9596

6.83

0.72

321.

1118

6027

6.33

9592

5.77

0.72

111.

1234

309.

1196

586.

800.

7309

1.13

1010

58.6

996

659.

940.

7280

1.13

4870

1216

.87

9405

4.49

0.72

011.

1387

952

.87

9597

0.74

0.73

271.

1321

4003.0

196

625.

170.

7301

1.13

4480

276.0

596

060.

300.

7249

1.12

5510

99.54

9544

0.73

0.73

401.

1334

417

.34

8319

3.50

0.73

261.

1377

9027

9.86

16b

9636

1.52

0.73

121.

1519

308.

8696

146.

710.

7315

1.13

3678

2.3

896

890.

850.

7301

1.14

3210

011

90.2

497

027.

840.

7321

1.13

7130

8.9

096

507.

960.

7324

1.13

3839

0.2

095

974.

300.

7298

1.13

5050

062

1.21

9440

7.30

0.73

711.

1457

641

.36

9728

9.16

0.73

871.

1444

871.

7396

562.

550.

7342

1.13

6010

0010

41.7

897

586.

340.

7380

1.14

6322

1.7

697

568.

990.

7388

1.14

8923

3.8

396

782.

560.

7348

1.13

94

aVa

lues

ofQ

w(m

3h−

1).

bE

ntrie

sin

bold

-fac

ear

esa

tisfa

ctor

y,i.e

.,va

lues

are

alm

osta

sgo

odas

for

the

1000

thge

nera

tion.

p w phis problem is, clearly, more meaningful. A decision maan be provided these results, and may be asked ton appropriate ‘preferred’ solution. The results of the thbjective Problem 5 are also compared with those o

wo-objective Problem 3 inFig. 4. It is seen that the degreecatter increases with the introduction of the third objecunction to Problem 3. The optimal values ofCp for Problemare always lower than those obtained in Problem 3 bec

his variable is being minimized (Eq. (6c)). This forcesembrane permeability coefficient of the salt,b, to take on

ts lowest value (Fig. 4e), while a shifts to its upper limiFig. 4d) (as compared with the two-objective problem whe constraint on the permeate concentration (Problemther optimal parameters for the three-objective probiz., �P, a andA, vary withQw (Fig. 4c, d and f) almost inimilar manner as compared to the two-objective proble

Two recent improvements of NSGA-II (Deb, 2001; Deb el., 2002), namely, NSGA-II-JG (Kasat & Gupta, 2003) andSGA-II-aJG (Bhat et al., 2005), have been used to solve of the problems (Problem 4) to see if the jumping genedaptations provide any advantage. The best values oomputational parameters have been obtained for allechniques, and are listed inTable 2. Fig. 5shows the devepment of the Pareto set over the generations using theodes (the values of the cost for NSGA-II-JG and NSGAJG have been increased by 10,000 and 20,000 $ h−1, respec

ively, to displace their plots vertically, so that they canompared easily), whileTable 3gives numerical values atew generations. It is observed fromFig. 5as well asTable 3hat the ‘range’ of the Pareto set (minimum and maxim


Fig. 6. Mean and standard deviation ofIi,dist for the Pareto solutions (shown inFig. 5) of Problem 4 (forNgen≥ 3).

values ofQw) increases with the generation number, with therange of NSGA-II-aJG becoming satisfactory (i.e., almost thesame as that at the 1000th generation) at the 20th generationitself, faster than for NSGA-II and NSGA-II-JG. Anothercharacteristic of the Pareto sets is the distribution/spread ofthe several points. Two parameters describe this aspect of thePareto sets: themean distance between consecutive points(note that the same number of chromosomes are used forall three techniques), and thestandard deviation of thesedistances. Kasat and Gupta (Kasat & Gupta, 2003) have sug-gested the use of the mean and the standard deviation ofIi,dist(seeAppendix A) in any generation for this purpose.Fig. 6andTable 3show these parameters for the three techniques, atdifferent generations.Fig. 6a andTable 3show that the meanvalue ofIi,dist is lower at the beginning (after some initial largevalues). This is because the range (of the Pareto set) is smallerand the same number of points is present in all generations.It is found that the mean and standard deviation ofIi,dist (and

the range) do not change much above about 80–100 gener-ations. This can be taken as an indication that convergencehas been attained (in fact, this can be used for all previousresults, even though higher values ofNgmax have actuallybeen used). Interestingly, the mean and standard deviation ofIi,dist at the 100th generation are almost the same for all thethree algorithms.Fig. 6andTable 3show that there are oscil-lations in both these parameters, even for as high a value ofNgen as 80. Similar oscillations in the behavior of the Paretoset have been observed earlier, though qualitatively. It is alsoobserved that the mean and the standard deviation ofIi,distconvergefaster to their final converged value for NSGA-II-aJG than for the other two techniques (see italicized entries inTable 3). This means that this technique is the least expensive,computationally (since the computational time to achieveconvergence is directly proportional to the number of gener-ations necessary). It may be added that one could improvethe ‘spread’ of the Pareto sets by using theε–constraint


method (Deb, 2001), in which we replace one objective func-tion (in this case,Qw) by an equality constraint and solvethe resulting optimization problem (with a single objectivefunction, in this case) several times over for several constantvalues ofQw. This methodology has its own problems (Deb,2001).

4. Conclusions

A few two-objective (maximizing the throughput whileminimizing the cost) and three-objective optimization prob-lems (maximizing throughput while minimizing the cost aswell as the permeate concentration) are studied for the desali-nation of brackish water and sea water. Pareto optimal sets ofequally good non-dominated solutions are obtained. The opti-mal solutions for spiral wound modules are compared to thosefor tubular modules. The membrane area,A (design parame-ter), is the most important decision variable in the desalinationof brackish water and seawater using spiral wound modules.In contrast, the applied pressure,�P (operating parameter),is the most important decision variable in the desalination ofbrackish water using tubular modules. Three AI-based algo-rithms, NSGA-II, NSGA-II-JG and NSGA-II-aJG, are usedto obtain the optimal solutions and it is observed that NSGA-I stedi mallc

A

ncea ughg

Ase(

o-entingariesa setodelunc-

non-

Fig. A1. Flow chart of NSGA-II and the JG adaptations.

(c) compare chromosome,i, with each member,j, in P′,one at a time;

(d) if i dominatesj (i.e., all objective functions ofi aresuperior to/better than those ofj), removej from P′and put it back inP at its place;

(e) if i is dominated byj, removei from P′ and put itback inP at its place;

(f) if i andj are non-dominated (i.e., at least one objec-tive function of i is inferior to that ofj, while allothers are superior), keep bothi andj in P′. Exploreall j in P′;

(g) repeat, sequentially, for all chromosomes inP. P′constitutes the first front or sub-box (of size≤ Np)of non-dominated chromosomes. Assign all chro-mosomes in this frontIi,rank= 1;

(h) create subsequent fronts in (lower) sub-boxes ofP′using the chromosomes remaining inP. Comparethese membersonly with members present in thecurrent sub-box. Assign all chromosomes in the indi-vidual sub-boxes,Ii,rank= 2, 3, . . . Finally, all Npchromosomes are inP′, boxed into one or morefronts.

3. Evaluate the crowding distance,Ii,dist, for the ith chro-mosome in any front using:

I-aJG is the most rapid of these algorithms if one is interen obtaining reasonable, near-optimal solutions with a somputational effort.

cknowledgement

Partial financial support from the Department of Sciend Technology, Government of India, New Delhi (throrant III-5(13)/2001-ET) is gratefully acknowledged.

ppendix A. Binary coded elitist non-dominatedorting genetic algorithm, NSGA-II (Deb, 2001; Debt al., 2002) with the jumping gene operators, JGKasat & Gupta, 2003) and aJG (Bhat et al., 2005)

1. Generate box,P, of Np binary-coded parent chromsomes (see flowchart inFig. A1), using a sequencof random numbers (e.g., a chromosome represetwo decision variables, each represented by five bincould be 11010 10110). Map each chromosome intoof real values of the decision variables. Use the mequations to compute the values of all the objective ftions (for each chromosome).

2. Classify these chromosomes into fronts based ondomination (Deb, 2001) as follows:(a) create new (empty) box,P′, of size,Np;(b) transfer theith chromosome fromP to P′, starting

with the first;


(a) rearrange all chromosomes in front,j, in ascendingorder of the values of any one of their fitness func-tions,Fk;

(b) find the largest cuboid (rectangle for two fitnessfunctions) enclosingi, that just touches its nearestneighbors in the F-space;

(c) Ii,dist ≡ 1/2 (sum of all sides of this cuboid);(d) assign large values ofIi,dist to solutions at the bound-

aries to make them important.4. Copy the better of theNp chromosomes ofP′ in a new

box,P′′ (‘better’ parents). Use:(a) select any pair,i andj, from P′ (randomly, irrespec-

tive of fronts);(b) identify the better of these two chromosomes.i is

better thanj if (for minimization of all fitness func-tions):

Ii,rank �= Ij,rank : Ii,rank < Ij,rank,

Ii,rank = Ij,rank : Ii,dist > Ij,dist;

(c) copy (without removing fromP′) the better chromo-some in a new box,P′′;

(d) repeat tillP′′ hasNp members;(e) copy all ofP′′ in a new box,D, of sizeNp;

Not all of P′ need be inP′′ or D.5. Carry out crossover and mutation (Deb, 1995) of chro-

-

esthirdlow:

-

ched

quen-

dom

san

ningry in

andtheenth

pec-af

(d) replace the set of binaries between these two loca-tions by a new set of binaries (use random numbers).For example, we may get 110|00 11|110

7. Copy allNp members ofP′′ and all theNp members ofD into box PD (elitism). Box PD has 2Np chromosomes.

8. Reclassify these 2Np chromosomes into fronts (box PD′)usingonly non-domination (see Step 2 above).

9. Take the bestNp from box PD′ and put into boxP′′′.10. This completes one generation. Stop if criteria are met.11. CopyP′′′ into starting box,P. Go to Step 2 above.

Appendix B. Model equations

The volumetric flux, Jw (Lonsdale et al., 1965;Rautenbach, 1986; Sherwood, Brian, & Fischer, 1967;Soltanieh & Gill, 1981) of the solvent is represented phe-nomenologically by

Jw = a(�P − �π) (A2.1)

while the mass flux,Js, of the solute is given by

Js = b(Cwall − Cp) (A2.2)

In the presence of concentration polarization (Sherwoodet al., 1967), Jw, at steady state, is also given by

J

W

�

t e. Wec

J

Co

J

a

C

T

R

E

S

S

mosomes inD. This gives a box ofNp daughter chromosomes:(a) Crossover: randomly select two chromosom

and a random crossover site (say, after theposition) and exchange the binaries as shown be

(b) Mutation: for each binary ineach chromosome, generate a random number and check (usingpm) if itneeds to be changed by this operator. If yes, switit over (from 0 to 1 or vice versa).

6. Do JG or aJG operation: select a chromosome (setially) from D, say 110|10 10|110.

Check if JG/aJG operation is needed, using a rannumber andpJG. If yes:(a) generate a random number between 0 and 1;(b) multiply this bylchrom, the total number of binarie

in the chromosome. Round off to convert intointeger. This represents the position of the beginof a transposon (say, at the end of the third binathe above chromosome);

(c) JG or aJG:• JG: generate another similar random number

identify a second location (end of the JG) inselected chromosome (say, the after the sevbinary);

• aJG: fix the second end of the JG using the sified string length,laJG (say laJG= 4; so placemarker at the end of the3 + 4 = seventh binary) othe jumping gene (Bhat et al., 2005);

w = ks lnCwall − Cp

Cb − Cp(A2.3)

e use

π = bπ(Cwall − Cp) (A2.4)

o estimate the osmotic pressure across the membranan also write the solute flux as

s = JwCp (A2.5)

ombining Eqs. (A2.1)–(A2.3) (eliminating Cwall), webtain, finally (Rautenbach, 1986):

w = a

[�P − bπ

(Cb − bCb exp(Jw/ks)

Jw + b exp(Jw/ks)

)exp(Jw/ks)

]

(A2.6)

nd

p = bCb

b + Jw exp(−Jw/ks)(A2.7)

he observed rejection is given by

= 1 − Cp

Cb(A2.8)

stimation of mass transfer coefficient, ks

piral wound module (Shock & Miquel, 1987)

h = 0.065Re0.865Sc0.25 (A2.9)


where, Sh = ksdh

DAB, Re = dhv

νand Sc = ν

DAB

The hydraulic diameter of a spiral wound module depends onthe channel height, the specific surface area of the spacer andthe void fraction. Details for various membranes are given byShock and Miquel (1987).

For brackish water, the kinematic viscosity,ν, can beestimated from the data given bySourirajan (1970)for theNaCl–H2O system at 25◦C:

ν = 0.0032+ 3.0 × 10−6C + 4.0 × 10−9C2 (A2.10)

The mass diffusivity,DAB (NaCl–H2O; T = 25◦C), is esti-mated as 5.5× 10−6 m2 h−1 at C = 3.1 kg m−3 (Sourirajan,1970).

For seawater,DAB, µ andρ (Sekino, 1994; Taniguchi &Kimura, 2005; Taniguchi et al., 2001) can be estimated fromthe following equations:

DAB =6.725× 10−6 exp

(0.1546× 10−3C − 2513

273.15+ T

)

(A2.11)

µ = 1.234× 10−6 exp

(0.00212C − 1965

273.15+ T

)

(A2.12)

a

ρ

w

B

eL

S

a ;W

S

S

S )

B1

enbc

π

Therefore, the osmotic coefficient,bπ, can be obtained as

bπ = π

C(A2.18)

B.3. Estimation of the cost

The cost of production of desalinated water is given bythe following equation (Maskan et al., 2000):

Cost= CmemA + CmainA + Cpump

(Qw�P

Wbaseη

)n

+CeleQw�P

η(A2.19)

Substituting the appropriate cost coefficients andη = 0.6, oneobtains (Maskan et al., 2000; Perry et al., 1997)

Cost= 1.946× 10−3A + 3.57× 10−3A

+ 0.0943

(Qw �P

1611.36

)0.67

+ 2.315× 10−3Qw �P

(A2.20)

The first and third terms on the right hand side of Eq.(A2.20)are not used in evaluating the ‘Cost’ for the operating-stageoptimization Problem 1, since they represent ‘sunken’ capitalt

R

A eddah

B lti-

B angene

B 92).s.),

B osis

B erten.

C 2).

D

D

D and

G

G

nd

= 498.4m +√

248000m2 + 752.4mC (A2.13)

here, m = 1.0069− 2.757× 10−4T (A2.14)

.1. Tubular module (Wiley et al., 1985)

For laminar flow, i.e., forRe ≤ 2100 in a circular tube, theveque relationship (Perry et al., 1997):

h = 1.62

(Re Sc

d

l

)0.33

(A2.15)

nd for turbulent flow, i.e., forRe ≥ 2100 (Perry et al., 1997iley et al., 1985):

h = 0.023Re0.8 Sc0.33 for Sc < 1,

h = 0.023Re0.875Sc0.25 for 1 ≤ Sc ≤ 1000,

h = 0.0096Re0.91Sc0.35 for Sc > 1000 (A2.16

.2. Estimation of the osmotic coefficient (Sourirajan,970)

The osmotic pressure,π, is obtained from the data givySourirajan (1970)for the NaCl–H2O system at 25◦C (con-entration range: 0–49.95 kg m−3) and is correlated as:

= 0.7949C − 0.0021C2+ 7.0 × 10−5C3 − 6.0 × 10−7C4

(A2.17)

hat is already invested in an existing unit.

eferences

yyash, Y. (1994). Performance of reverse osmosis membrane in JI plant. Desalination, 96, 215.

haskar, V., Ray, A. K., & Gupta, S. K. (2000). Applications of muobjective optimization in chemical engineering.Reviews in ChemicalEngineering, 16, 1.

hat, S. A., Saraf, D. N., & Gupta, S. K. (2005). Optimization ofindustrial semi-batch nylon 6 reactor using the adapted jumpingoperator with genetic algorithm, in preparation.

hattacharyya, D., Williams, M. E., Ray, R. J., & McCray, S. B. (19Reverse osmosis: design. In W. S. W. Ho & K. K. Sirkar (EdMembrane handbook. New York: Van Nostrand Reinhold.

rian, P. L. T. (1965). Concentration polarization in reverse osmdesalination with variable flux and incomplete salt rejection.Industrialand Engineering Chemistry Fundamentals, 4, 439.

rian, P. L. T. (1966). Mass transport in reverse osmosis. In U. M(Ed.), Desalination by reverse osmosis. Cambridge, MA: MIT Press

oello Coello, C. A., Van Veldhuizen, D. A., & Lamont, G. B. (200Evolutionary algorithms for solving multi-objective problems. NewYork: Kluwer.

eb, K. (1995).Optimization for engineering design: algorithms andexamples. New Delhi: Prentice Hall of India.

eb, K. (2001).Multi-objective optimization using evolutionary algo-rithms. Chichester, UK: Wiley.

eb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fastelitist multiobjective genetic algorithm: NSGA-II.IEEE TransactionsEvolutionary Computation, 6, 182.

ill, P. E., Murray, W., & Wright, M. H. (1991).Numerical linear algebraand optimization. Reading, MA: Addison-Wesley.

oicoechea, A., Hansen, D. R., & Duckstein, L. (1982).Multiobjectivedecision analysis with engineering and business applications. NewYork: Wiley.


Goldberg, D. E. (1989).Genetic algorithms in search, optimization andmachine learning. Reading, MA: Addison-Wesley.

Guria, C., Verma, M., Mehrotra, S. P., & Gupta, S. K. (2005).Multi-objective optimal synthesis and design of froth flotation cir-cuits for mineral processing using the jumping gene adaptation ofgenetic algorithm.Industrial and Engineering Chemistry Research, 44,2621.

Ho, W. S. W., & Sirkar, K. K. (Eds.). (1992).Membrane handbook. NewYork: Van Nostrand Reinhold.

Holland, J. H. (1975).Adaptation in natural and artificial systems. AnnArbor, MI: University of Michigan Press.

Kasat, R., Kunzru, D., Saraf, D. N., & Gupta, S. K. (2002). Multi-objective optimization of industrial FCC units using elitist non-dominated sorting genetic algorithm.Industrial and EngineeringChemistry Research, 41, 4765.

Kasat, R. B., & Gupta, S. K. (2003). Multi-objective optimization of anindustrial fluidized-bed catalytic cracking unit (FCCU) using geneticalgorithm (GA) with the jumping genes operator.Computers andChemical Engineering, 27, 1785.

Khosla, D., Saraf, D. N., & Gupta, S. K. (2005). Multi-objective optimiza-tion of fuel oil blending using the jumping gene adaptation of geneticalgorithm.Fuel Processing Technology, submitted for publication.

Kimura, S., & Sourirajan, S. (1968). Concentration polarization effects inreverse osmosis using porous cellulose acetate membrane.Industrialand Engineering Chemistry Fundamentals, 7, 41.

Knuth, D. E. (1997). (3rd ed.).The art of computer programming,seminumerical algorithms: 2. Reading, MA: Addison Wesley.

Lohman, E. M. (1994). Operating report of the largest reverse osmosisdesalting plant.Desalination, 96, 349.

Lonsdale, H. K., Merten, U., & Riley, R. L. (1965). Transport propertiesof cellulose acetate membranes.Journal of Applied Polymer Science,

M 00).

M thm

M pos-

N odelpti-

O 77).od-

O rac-

P thesation

P l

P

Rao, G. H., & Sirkar, K. K. (1978). Explicit flux expressions in tubularreverse osmosis desalination.Desalination, 27, 99.

Rautenbach, R. (1986). Process design and optimization. In P. M. Bungay,H. K. Lonsdale, & M. N. de Pinho (Eds.),Synthetic membranes:science, engineering and applications. New York: Kluwer.

Ray, A. K., & Gupta, S. K. (2004).Mathematical methods in chemicaland environmental engineering. Singapore: Thomson Learning.

Rosenbrock, H. H. (1960). An automatic method for finding the greatestor least value of a function.Computer Journal, 3, 175.

Sareen, R., & Gupta, S. K. (1995). Multiobjective optimization of anindustrial semibatch nylon 6 reactor.Journal of Applied Polymer Sci-ence, 58, 2357.

Sekino, M. (1994). Performance data analysis for hollow-fiber reverse-osmosis modules in seawater desalination plant.Kagaku KougakuRonbunsyu, 20, 574.

Sherwood, T. K., Brian, P. L. T., & Fischer, R. E. (1967). Desalination byreverse osmosis.Industrial and Engineering Chemistry Fundamentals,6, 2.

Sherwood, T. K., Brian, P. L. T., Fischer, R. E., & Dresner, L. (1965). Saltconcentration at phase boundaries in desalination by reverse osmosis.Industrial and Engineering Chemistry Fundamentals, 4, 113.

Shock, G., & Miquel, A. (1987). Mass transfer and pressure loss in spiralwound modules.Desalination, 64, 339.

Sirkar, K. K., Dang, P. T., & Rao, G. P. (1982). Approximate designequations for reverse osmosis desalination by spiral-wound modules.Industrial and Engineering Chemistry Fundamentals, 21, 517.

Soltanieh, M., & Gill, W. (1981). Review of reverse osmosis membranesand transport models.Chemical Engineering Communication, 12, 279.

Sourirajan, S. (1970).Reverse osmosis. New York: Academic.Srinivas, N., & Deb, K. (1995). Multi-objective function optimization

using nondominated sorting genetic algorithm.Evolutionary Computu-

S raneustry.

ST cs of

T ters

T rsend its

T pti-cess.

W m-

Y lti-netic

Z ive

9, 1341.askan, F., Wiley, D. E., Johnston, L. P. M., & Clements, D. J. (20

Optimal design of reverse osmosis module networks.American Insti-tute of Chemical Engineers Journal, 46, 946.

avrotas, G., & Diakoulaki, D. (1998). A branch and bound algorifor mixed zero-one multiple objective linear programming.Eur. J. Op.Res., 107, 530.

cKlintock, B. (1987). The discovery and characterization of transable elements. InThe collected papers of Barbara McKlintock. NewYork: Garland.

andasana, A. D., Ray, A. K., & Gupta, S. K. (2003). Dynamic mof an industrial steam reformer and its use for multiobjective omization. Industrial and Engineering Chemistry Research, 42, 4028.

hya, H., Nakajima, H., Takagi, K., Kagawa, S., & Negishi, Y. (19Analysis of reverse osmotic characteristics of B-9 hollow fiber mule. Desalination, 21, 257.

hya, H., & Taniguchi, Y. (1975). An analysis of reverse osmotic chateristics of ROGA-4000 spiral-wound module.Desalination, 16, 359.

almer, J. R. (1969). An improved procedure for orthogonalisingsearch vectors in Rosenbrock’s and Swann’s direct search optimimethods.Computer Journal, 12, 69.

arekh, B. (Ed.). (1988).Reverse osmosis technology. New York: MarceDekker.

erry, R. H., Green, D. W., & Malony, J. O. (Eds.). (1997).Perry’schemical engineers’ handbook (7th ed.). New York: McGraw-Hill.

tation, 2, 221.tanojevic, M., Lazarevic, B., & Radic, D. (2003). Review of memb

contactors: designs and applications of different modules in indFME Transactions, 31, 91.

tryer, L. (2000).Biochemistry (4th ed.). New York: W.H. Freeman.aniguchi, M. (1978). An analysis of reverse osmosis characteristi

ROGA spiral wound modules.Desalination, 25, 71.aniguchi, M., & Kimura, S. (2005). Estimation of transport parame

of RO membranes for seawater desalination.American Institute ofChemical Engineers Journal, 46, 1967.

aniguchi, M., Kurihara, M., & Kimura, S. (2001). Behavior of a reveosmosis plant adopting a brine conversion two-stage process acomputer simulation.Journal of Membrane Science, 183, 249.

arafder, A., Rangaiah, G. P., & Ray, A. K. (2005). Multi-objective omization of an industrial styrene monomer manufacturing proChemical Engineering Science, 60, 347.

iley, D. E., Fell, C. J. D., & Fane, A. G. (1985). Optimization of mebrane module design for brackish water desalination.Desalination,52, 249.

uen, C. C., Aatmeeyata, Gupta, S. K., & Ray, A. K. (2000). Muobjective optimization of membrane separation modules using gealgorithm.Journal of Membrane Science, 176, 177.

itzler, E., Deb, K., & Thiele, L. (2000). Comparison of multiobjectevolutionary algorithms: empirical results.Evolutionary Computation,8, 173.

Multi-objective optimization of reverse osmosis ...repository.ias.ac.in/14130/1/407.pdf ·...

Documents

Transcript of Multi-objective optimization of reverse osmosis ...repository.ias.ac.in/14130/1/407.pdf ·...