Journal of Membrane Science 368 (2011) 202–214

Contents lists available at ScienceDirect

Journal of Membrane Science

j ournal homepage:www.el sevi er.com/l ocat e/memsci

Artiﬁcial neural network modeling and response surface methodology of

desalination by reverse osmosis

M.Khayet

∗

,C.Cojocaru,M.Essalhi

Department of Applied Physics I,Faculty of Physics,University Complutense of Madrid,Av.Complutense s/n,28040 Madrid,Spain

a r t i c l e i n f o

Article history:

Received 7 April 2009

Received in revised form7 October 2010

Accepted 14 November 2010

Available online 19 November 2010

Keywords:

Reverse osmosis

Desalination

Response surface methodology

Neural network

a b s t r a c t

Response surface methodology (RSM) and artiﬁcial neural network (ANN) have been used to develop

predictive models for simulation and optimization of reverse osmosis (RO) desalination process.Sodium

chloride aqueous solutions were employed as model solutions for a RO pilot plant applying polyamide

thin ﬁlm composite membrane,in spiral wound conﬁguration.The input variables were sodium chlo-

ride concentration in feed solution,C,feed temperature,T,feed ﬂow-rate,Q,and operating hydrostatic

pressure,P.The RO performance index,which is deﬁned as the salt rejection factor times the perme-

ate ﬂux,has been considered as response.Both RSM and ANN models have been developed based on

experimental designs.Two empirical polynomial RSMmodels valid for different ranges of feed salt con-

centrations were performed.In contrast,the developed ANN model was valid over the whole range of

feed salt concentration demonstrating its ability to overcome the limitation of the quadratic polyno-

mial model obtained by RSMand to solve non-linear problems.Analysis of variance (ANOVA) has been

employedtotest thesigniﬁcanceof responsesurfacepolynomials andANNmodel.Totest thesigniﬁcance

of ANN model,the estimation of the degree of freedomdue to residuals has been detailed.Finally,both

modeling methodologies RSMand ANN were compared in terms of predictive abilities by plotting the

generalization graphs.The optimumoperating conditions were determined by Monte Carlo simulations

considering:(i) the four input variables,(ii) for typical brackish water with a ﬁxed concentration of 6g/L

and (iii) for typical seawater with a ﬁxed concentration of 30g/L.Under the obtained optimal conditions

maximumRO performance indexes have been achieved experimentally.

© 2010 Elsevier B.V. All rights reserved.

1.Introduction

Construction of mathematical models for prediction of mem-

brane separation processes is a valuable tool in the ﬁeld of

membrane science and technology.The models play an important

role in simulation and optimization of membrane systems leading

to efﬁcient and economical designs of separation processes [1,2].

Mathematical modeling of any process deals with two basic

approaches:(i) theoretical (or parametric) models based on funda-

mental knowledge (mechanism) of the process,known also as the

knowledge-based approach and (ii) empirical (or non-parametric)

models,which do not involve the knowing of fundamentals prin-

Abbreviations:ANN,artiﬁcial neural network;ANOVA,analysis of variances;BP,

back-propagation method;CCD,central composite design;DoE,design of experi-

ments;HL,hidden layer;LMA,Levenberg–Marquardt algorithm;MLP,multi-layer

perceptron (feed-forward network);OL,output layer;OLS,ordinary least squares

method;PRNs,pseudo random numbers;RO,reverse osmosis;RSM,response

surfacemethodology;RS-model,responsesurfacemodel;WHO,WorldHealthOrga-

nization.

∗

Corresponding author.Tel.:+34 91 3945185;fax:+34 91 3945191.

E-mail address:khayetm@ﬁs.ucm.es (M.Khayet).

ciples governing the process [3].The advantage of empirical

modeling tools over theoretical models consists in the possibility

to develop rapidly the objective function useful for process opti-

mization.

Response surface methodology (RSM) and artiﬁcial neural net-

works (ANN) are modeling tools able to solve linear and non-linear

multivariate regressionproblems.Bothmethodologies do not need

explicit expressions of the physical meaning of the systemor pro-

cess under study.Therefore,RSMas well as ANNbelongtomodeling

tools dealing withdevelopment of non-parametric simulative mod-

els known also as “black-box” models.Such models ascertain a

relationshipbetweendesignvariablesandresponseor output of the

process using a limited number of experimental runs.Commonly,

these models are developed using the design variable settings to

optimize the response (process output) [4].

During last years,the design of experiments (DoE) and RSM

have been applied successfully in different areas of membrane

technology [5–13].In ultraﬁltration,RSMhas been used for mod-

eling and optimization of heavy metals removal fromwastewaters

using micellar-enhanced and polymer assisted methods [5,6].In

nanoﬁltration,the mixture experimental design was applied to

describe the inﬂuence of ionic composition to remove nitrate ions

0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.memsci.2010.11.030

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 203

from water and to optimize the operating conditions for mem-

branes working with multi-ionic solution [7].The experimental

design and RSMwas also applied to direct contact membrane dis-

tillation [8].In this case quadratic models between the responses

(permeate ﬂuxes) and the independent variables were built for

both commercial and various laboratory made membranes of

different characteristics.Ismail and Lai [9] employed RSM to

develop defect-free asymmetric polysulfone membranes for gas

separation and investigated the main and interaction effects of

design variables on membrane structure and performance in order

to optimize membrane formation process.RSM optimization of

polydimethylsiloxane (PDMS)/ceramic composite pervaporation

membrane preparation conditions,namely,polymer concentra-

tion,crosslinking agent concentration and dip-coating time,was

carried out by Xiangli et al.[10].Khayet et al.[11] used RSM

to optimize the operating conditions for pervaporation of binary

acetonitrile–water mixtures in order to enhance both permeate

ﬂux ratio and concentration of organic in permeate.Application

of RSMin describing the performance of thin ﬁlmcomposite mem-

brane was carried out by Idris et al.[12] in order to improve both

rejection factor and membrane permeate ﬂux.In addition,Cheison

et al.[13] applied RSMto optimize the hydrolysis of whey protein

isolate in a tangential ﬂowmembrane reactor.

Concerning ANN,this modeling method has been applied pro-

gressively during last years for simulation and optimization of

membrane separation processes.Liu and Kim [1] have evaluated

membrane fouling models based on bench-scale experiments by

comparing the constant ﬂow rate blocking laws with ANN model.

Neural networks modeling of hollow ﬁbers membrane processes

were carried out by Shahsavand and Chenar [3] using two experi-

mental sets as training data for separation of carbon dioxide from

methane.Cheng et al.[14] proposed an overlapped type of local

neural network to improve the accuracy of the permeate ﬂux

decline prediction in crossﬂow membrane ﬁltration of colloidal

solution.This type of network combined the advantages of the

multilayer feed-forward back-propagation neural network and the

radial basis function network.Other studies dealing with neu-

ral network modeling for prediction of permeate ﬂux,separation

efﬁciency and/or permeate ﬂux decline in crossﬂowmembrane ﬁl-

tration were reported more in details in Refs.[15–21].A study on

neural network modeling for ultraﬁltration and backwashing has

been reported by Teodosiu et al.[22].Two neural network mod-

els were proposed to predict the permeate ﬂux at any time during

ultraﬁltration and after backwashing for arbitrary cycles.Regard-

ing desalination using ANN,several studies have been carried out

as reported in [23–25] emphasizing the potential applicability of

ANN in desalination systems.

It is worth quoting that RSM permits to perform polynomial

empirical models for approximation of process performance and

neural networks are also known as universal tools for function

approximation of non-linear systems.Both methodologies RSM

and ANN can offer trustable approximation models to predict the

true response function (objective function) of the process.Once

the non-parametric models are developed one can use approxi-

mated response surfaces to solve the optimization problem.The

question is:which approximation model is more trustable offer-

ing better accuracy in ﬁtting experimental data and giving a better

optimal solution conﬁrmed by experiment?Moreover,it is impor-

tant to reveal the advantages of each methodology and differences

between them.

Among various desalination technologies (thermal and

membrane-based) of saline and brackish waters,the reverse

osmosis (RO) is one of the most efﬁcient and widely used tech-

niques [26–28].Recent technological innovations makeROsystems

more attractive for industry using alternative energy sources like

photovoltaic solar energy [29,30] and wind energy [31].The

Fig.1.Schema of the experimental ROpilot plant:(1) ROmodule;(2) high pressure

pump;(3) vent;(4) manometer;(5) ﬂowmeter for retentate;(6) ﬂowmeter for per-

meate;(7) electrical conductivity monitor;(8) thermostat;(9) temperature probe;

(10) feed tank;(11) lowpressure pump (<4.1bar);(12) manometer;(13) ﬂowmeter

for feed;(14) ﬁlter;(15) pressure controller;(16) switch on/off.

optimization of operating conditions for a certain RO system still

remain an issue of great concern since the optimal solution always

gives the maximal improvement of any operating system.

The present work has three objectives:(1) to obtain predictive

models based on RSM and ANN techniques for prediction of the

performance index of RO pilot plant;(2) to maximize the perfor-

mance index of ROpilot plant using bothRSMandANNmodels;(3)

to compare the optimal solutions offered by RSMand ANN.To the

best of our knowledge,this is the ﬁrst report comparing RSMand

ANN in membrane science and technology.

2.Experimental

The RO experiments were carried out using the pilot plant

schematizedinFig.1.Astainless steel spiral woundmodule(S2521,

Osmonics),containing polyamide thin ﬁlm composite membrane

was employed in this study.The effective membrane area of the

membrane module is 1.2m

2

[32].

Sodiumchloride,NaCl (Sigma–Aldrich) anddistilledwater were

used to prepare the feed salt solutions of desired concentrations.

The electrical conductivity of the feed and permeate aqueous solu-

tions was measured using a conductivimeter 712Metrohm.The

salt concentration was then determined based on the calibration

correlationelectrical conductivity-concentration.Thesalt rejection

factor (RE) was determined as follows [32]:

RE =

C −C

P

C

(1)

where C is the feed concentration and C

P

is the permeate concen-

tration.All experiments were conducted at different levels of feed

solute concentration,feed temperature,feed ﬂowrate and hydro-

static pressure.The operating levels of these factors are given later

on following the design of experiments (DoE).

Thepermeateﬂuxwas measuredfor different levels of operating

conditions byweighingtheobtainedpermeatefor a predetermined

period of time.The regression analysis was employed to ﬁt the

experimental results (permeate ﬂux versus time) and to compute

the average permeate ﬂuxJ (kg/m

2

s) byintegration.The ROperfor-

mance index of the pilot plant (Y) was determined as the product

204 M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214

between the average permeate ﬂux and the salt rejection factor:

Y = J ×RE (2)

Since higher ROperformance index involves both higher permeate

ﬂux and salt rejection factor,the response Y was used for process

modeling and optimization.

3.Theoretical background of modeling methods

3.1.Predictive modeling using RSM

RSM is an optimization approach permitting to determine the

input combination of factors that maximize or minimize a given

objective function [33].Based on the DoE and RSM the second-

order polynomial regression models can be developed to predict

the performance of any process or system.Such models are also

known as response surface models (RS-models).During response

surface modeling the input variables x

1

,x

2

,...,x

n

must be scaled

to coded levels.In coded scale the factors vary from(−1) that cor-

responds to minimumlevel up to (+1) that suit to maximumlevel.

The second-order models given by RSM are often used to deter-

mine the critical points (maximum,minimum,or saddle) and can

be written in a general formas [34]:

ˆ

Y = ˇ

0

+

n

i=1

ˇ

i

x

i

+

n

i=1

ˇ

ii

x

2

i

+

n

i<j

ˇ

ij

x

i

x

j

(3)

where

ˆ

Y denotes the predicted response,x

i

refers to the coded

levels of the input variables,ˇ

0

,ˇ

i

,ˇ

ii

,ˇ

ij

are the regression coef-

ﬁcients (offset term,main,quadratic and interaction effects) and n

is the total number of designed variables.To determine the regres-

sion coefﬁcients,the ordinary least squares (OLS) method is used.

The OLS estimator can be written as follows [34–36]:

ˇ

OLS

= (X

T

X)

−1

X

T

Y (4)

where ˇ

OLS

is a vector of regression coefﬁcients,X is an extended

designedmatrix of the codedlevels of the input variables,Yis a col-

umn vector of response determined according to the arrangement

points into the experimental design.

3.2.Predictive modeling using ANN

ANN offers a rich framework for modeling of non-linear phe-

nomena and for solving the multivariate regression problems.The

mode of building ANNmodel is totally different if we compare with

the construction method of polynomial RS-model.It is worth to

outline some distinction between RSMand ANNterminology.RSM

operates with factors (design variables) and response.In ANNmod-

eling the factors are known as inputs while the response as target

(experimental response) or output (predicted response).

ANN is a non-linear processing system composed of neurons

(nodes) and connections between themthat can be used for map-

ping input and output data [37].

An artiﬁcial neuron (node) is a single computational processor,

whichoperates with(1) summingjunctionand(2) transfer function

[38,39].The connections consist of weights w and biases b with

neurons addressing information.

The summing junction operator of a single neuron summarizes

the weights and bias into a net input,S,known as argument to be

processed:

S =

n

i=1

x

i

w

i

+b (5)

Thetransfer functiontakes theargument,S,andproduces thescalar

output of a single neuron.The most usedtransfer functions to solve

linear and non-linear regression problems are purelin,logsig and

tansig[38].For thecaseof logistic output thelogsigtransfer function

may be written as:

logsig(S) =

1

1 +exp(−S)

(6)

The way in which the inputs and outputs of the neurons are con-

nected is knownas architecture of the neural network.As usual,the

neurons of a network are divided into several groups called layers.

A multi-layer neural network has hidden and output layers con-

sisting of,hidden and output neurons,respectively.Frequently the

inputs are consideredas additional layer.The most commonneural

network architecture used for solving non-linear regression prob-

lems is the multi-layer feed-forwardneural network also knownas

multi-layer perceptron,MLP.The most commontraining algorithm

for feed-forward neural network is back-propagation (BP) method

[40].Training of ANNby means of BP algorithmis an iterative opti-

mization process where the performance function is minimized by

adjusting the weights appropriately.The commonlyemployedper-

formance function is the mean-squared-error,MSE,that is deﬁned

as [41,42]:

MSE =

H

h=1

M

m=1

(Y

h,m

−

Y

h,m

)

2

H · M

(7)

where H denotes the number of output nodes,M is the number

of patterns used in the training set,Y

h,m

and

Y

h,m

are the target

(experimental response) and output (predicted response) of the

hth output node,respectively.

According to BP algorithmthe weights andbiases are iteratively

updated in the direction in which the performance function MSE

decreases most rapidly.Generally,a singleiterationof BPalgorithm

can be written as [38,43,44]:

W

(k+1)

= W

(k)

−

(k)

grad

(k)

(MSE) (8)

whereW

(k)

is avector of current weights andbiases,grad

(k)

(MSE) is

thecurrent gradient of theperformancefunctionMSEand

(k)

is the

learning rate.More detailed mathematical aspects on BP training

algorithms are comprehensively described elsewhere [38,39].

4.Results and discussion

4.1.Predictive modeling using RSM

The ﬁrst attempt on RSMmodeling was to develop an empirical

model to describe the RO performance index over a wide range of

salt concentration of feed aqueous solution including both brack-

ish and seawater salinity conditions.However,the RSM failed in

providing a general RS-model because of inadequate prediction

of the RO performance index for simultaneous low and high salt

concentrations of feed solutions.Therefore,RSMhas been applied

to develop two separate models,one for low salt concentration

of feed solution (range of brackish water salinity conditions) and

another for high salt concentration of feed solution (including sea-

water salinity conditions).MATLAB software has been used for all

computations and graphical analysis in RSMapplications.

4.1.1.RS-model for lowsalt concentration of feed solution

In this study,the central compositional experimental design

(CCD) has been used to investigate the synergistic effects of fac-

tors upon the performance of RO process.The factors involved in

multivariateexperimentationdealt withthefeedsolute(NaCl) con-

centration,C(x

1

),feed temperature,T(x

2

),feed ﬂowrate,Q(x

3

) and

operating pressure,P(x

4

).The experimental design including the

coded and actual values of factors is shown in Table 1.Note that in

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 205

Table 1

Central composite design and experimental responses for desalination of lowsalt concentration solutions by RO.

Run Factors (controllable input variables) Responses

Feedconcentration Feedtemperature Feed ﬂowrate Feed pressure Rejection Flux Performance index

x

1

C (g/L) x

2

T (

◦

C) x

3

Q (L/h) x

4

P (MPa) RE (%) J ×10

−5

(kg/m

2

s) Y×10

−5

(kg/m

2

s)

A1 +1 10 +1 37.5 +1 212.5 +1 1.25 90.173 187.909 169.443

A1 −1 5 +1 37.5 +1 212.5 +1 1.25 96.430 507.898 489.766

A3 +1 10 −1 22.5 +1 212.5 +1 1.25 90.681 147.360 133.628

A4 −1 5 −1 22.5 +1 212.5 +1 1.25 96.665 372.572 360.147

A5 +1 10 +1 37.5 −1 137.5 +1 1.25 90.730 187.690 170.291

A6 −1 5 +1 37.5 −1 137.5 +1 1.25 95.947 492.289 472.337

A7 +1 10 −1 22.5 −1 137.5 +1 1.25 91.078 141.302 128.695

A8 −1 5 −1 22.5 −1 137.5 +1 1.25 96.498 354.309 341.901

A9 +1 10 +1 37.5 +1 212.5 −1 0.75 67.163 38.416 25.801

A10 −1 5 +1 37.5 +1 212.5 −1 0.75 90.957 149.083 135.601

A11 +1 10 −1 22.5 +1 212.5 −1 0.75 66.002 29.088 19.199

A12 −1 5 −1 22.5 +1 212.5 −1 0.75 92.265 108.259 99.885

A13 +1 10 +1 37.5 −1 137.5 −1 0.75 66.317 41.472 27.503

A14 −1 5 +1 37.5 −1 137.5 −1 0.75 90.787 144.931 131.579

A15 +1 10 −1 22.5 −1 137.5 −1 0.75 68.973 27.496 18.965

A16 −1 5 −1 22.5 −1 137.5 −1 0.75 91.695 102.234 93.743

A17 +1.41 11.04 0 30 0 175 0 1.0 78.071 56.376 44.013

A18 −1.41 3.97 0 30 0 175 0 1.0 96.375 351.413 338.674

A19 0 7.5 +1.41 40.58 0 175 0 1.0 89.713 167.976 150.696

A20 0 7.5 −1.41 19.43 0 175 0 1.0 91.568 119.713 109.619

A21 0 7.5 0 30 +1.41 227.9 0 1.0 91.305 142.719 130.310

A22 0 7.5 0 30 −1.41 122.1 0 1.0 90.773 157.003 142.516

A23 0 7.5 0 30 0 175 +1.41 1.35 95.374 354.378 337.984

A24 0 7.5 0 30 0 175 −1.41 0.65 71.891 36.825 26.474

A25 0 7.5 0 30 0 175 0 1.0 91.203 152.764 139.325

A26 0 7.5 0 30 0 175 0 1.0 91.460 150.970 138.077

this experimental design the feed salt concentration was varied in

the range of brackish water salinity from3.97g/L to 11.04g/L.Two

basic responses have been determined according to CCD,i.e.salt

rejection factor,RE,and the permeate ﬂux,J.For this experimental

design(Table1) thesalt rejectionfactor rangedfrom66.0%to96.7%,

whereas the permeate ﬂux was in the range 27.5–507.9kg/m

2

s.

Based on these responses (i.e.rejection factor and permeate ﬂux)

the ROperformance index has been computed by means of Eq.(2).

In fact,the RO performance index is an output variable that com-

bines the salt rejection factor and the permeate ﬂux into a single

overall response.Therefore,all modeling andoptimizations carried

out inthis studyarerelatedtotheperformanceindex.Theimprove-

ment of RO performance index involves the intrinsic increment of

both rejection and ﬂux.The application of DoE and RSM leads to

development of the predictive RS-model (I).The RS-model (I) can

be used for simulation of RO desalination process in the range of

lowsalt concentration in feed solution and was written in terms of

coded variables as follows:

ˆ

Y = (139.631 −92.41x

1

+24.214x

2

+107.727x

4

+24.173x

2

1

−6.42x

2

2

+19.616x

2

4

−15.066x

1

x

2

−43.297x

1

x

4

+15.548x

2

x

4

) ×10

−5

(9)

Subjected to:x

i

∈˝;˝={x

i

|−˛≤x

i

≤+˛};

∀

i =

1,4.

where˛denotes thestar point (apropertyof CCD),whichdelim-

itates the boundaries of valid region ˝ known also as region of

experimentation.In the case of four factors (n=4) and an orthog-

onal design CCD,the star point is ˛=1.44.It is worth to note that

the signiﬁcance of regression coefﬁcients was tested using the sta-

tistical Student’s t-test [45].

InEq.(9) onlythesigniﬁcant terms havebeenretained.Notethat

thefactor x

3

(feedﬂowrate) was omittedfromtheﬁnal model since

it is an insigniﬁcant variable according to the results of Student’s

t-test.

The adequacy of RS-model was tested by means of analysis of

variance (ANOVA) and the results of the statistical test are shown

in Table 2.According to ANOVA,the F-value,which is a measure of

the variance of data about the mean,was determined.If the F-value

departs signiﬁcantly fromunity,the more certain is that the input

variables adequately explain the variation in the mean of the data.

Based on F-value and the degree of freedom,the P-value is then

computed.To validate from statistical standpoint any RS-model,

the F-value must be as high as possible whereas the P-value should

be as lowas possible.Most RS-models are validated for prediction

if the P-value is less than 0.05.

The ANOVA results (Table 2) summarize the sumof squares of

residuals and regressions together with the corresponding degrees

of freedom,F-value,P-value and ANOVA coefﬁcients (i.e.coefﬁ-

cients of multiple determinationR

2

and adjusted statistic R

2

adj

).The

mathematical expressions used for computation of statistical esti-

mators (i.e.SS,MS,F-value,R

2

,R

2

adj

) are extensively presented in

the textbooks concerning DoE and RSM [35,45–47].According to

the ANOVAresults summarized in Table 2,the F-value is quite high

(304.32) and the P-value is smaller than 10

−4

.Note that R

2

value

is about 0.994,being close to unity,which is worthwhile.More-

over,thecoefﬁcient R

2

is inagreement withtheadjustedcoefﬁcient

of determination,R

2

adj

.All statistical estimators disclose that the

developed model is validated fromstatistical point of viewto sim-

ulate ROprocess for the conditions of lowsalt concentrationinfeed

solution.In addition,the goodness-of-ﬁt of RS-model is illustrated

in Fig.2.As can be seen the model shows a good prediction for the

investigated response (RO performance index).

Table 2

ANOVA table for RS-model (I) predicting the performance index for the conditions

of lowsalt concentration in feed solution.

Source DF

a

SS

b

MS

c

F-value P-value R

2

R

2

adj

Model 9 4.611×10

−5

5.124×10

−6

304.32 <0.0001 0.994 0.991

Residual 16 2.694×10

−7

1.684×10

−8

Total 25 4.638×10

−5

a

Degree of freedom.

b

Sumof squares.

c

Mean square.

206 M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214

Fig.2.RS-model (I):predicted and experimental RO performance index valid for

lowsalt concentration conditions.

For the graphical representation and response surface analysis

it is interesting to convert the RS-model fromcoded to actual vari-

ables.In this case the substitution technique has been applied and

the empirical model in terms of actual variables was written as:

ˆ

Y = (−133.72 −1.58C +7.811T +73.76P +3.867C

2

−0.114T

2

+313.92P

2

−0.804CT −69.28CP

+8.293TP) ×10

−5

(10)

Subjected to:3.96≤C≤11.04g/L;19.40≤T≤40.60

◦

C;

0.65≤P≤1.35MPa.

Figs.3–5 show the response surface plots given by RS-model

(I) as a function with different variables.The response surface

indicates that increasing both feed temperature and pressure will

enhance the RO performance index of the pilot plant.The main

effect of pressure inthis case is 4-foldhigher thanthe maineffect of

feedtemperature.Themaineffect of feedsalt concentrationis close

to the effect of pressure,in magnitude,but with an opposite sense.

This means that the increase of the feed salt concentration dimin-

ishes substantially the ROperformance index.Thus,the ROsystem

operates better at lowsalt concentrations and high pressures.

It was observedthat the highest factors interactioneffects exists

betweenthepressureandsalt concentrationof thefeedsolution.As

can be seen in Fig.4,the effect of concentration is more signiﬁcant

at higher pressure while the effect of pressure is more signiﬁcant

at lower salt concentration.Furthermore,moderate interaction

effects were observed between salt concentration in feed solution

and feed temperature as well as between feed temperature and

Fig.3.Response surface plot of the predictedROperformance index by RSMfor low

salt concentration conditions as function of the feed salt concentration and the feed

temperature for Q=175L/h and P=1MPa.

Fig.4.Response surface plot of the predicted RO performance index by RSM for

lowsalt concentration conditions as function of the feed salt concentration and the

pressure for Q=175L/h and T=30

◦

C.

pressure.According to these interactions effects the inﬂuence of

feed temperature is more signiﬁcant at lower salt concentration

in feed solution and at higher operating pressure.Since the effect

of the feed ﬂow rate was found to be insigniﬁcant based on Stu-

dent’s t-test,this factor does not affect the response surface for the

conditions of lowsalt concentrations in feed solution.

4.1.2.RS-model for high salt concentration in feed solution

To develop the empirical model for the conditions of high salt

concentrations in feed solution the central composite design (CCD)

of experiments was employed as it is shown in Table 3.In these

experiments the feed salt concentration was varied from12.38g/L

to 48.63g/L.In this range,the center point corresponds to the con-

ditions of seawater salinity,i.e.30g/L.The ranges for the other

factors (feed temperature,feed ﬂow rate and pressure) remain

the same as it was stated in the previous experimental design for

brackish waters (Table 1).For the experimental design performed

for high salt concentration feed solutions,the obtained salt rejec-

tion factor was in the range 1.5–35.3%,whereas the permeate ﬂux

ranged from6.7kg/m

2

s to 62.6kg/m

2

s.As stated earlier,the over-

all response,RO performance index,has been calculated using Eq.

(2) andthe results are summarizedinTable 3.The RS-model (II) has

been developed using OLS regression method for the prediction of

RO performance index in the conditions of high salt concentrated

solutions.By applying Student’s t-test to check the signiﬁcance of

regression coefﬁcients the ﬁnal form of RS-model (II) in terms of

Fig.5.Response surface plot of the predictedROperformance index by RSMfor low

salt concentration conditions as function of the feed temperature and the pressure

for C=7.5g/L and Q=175L/h.

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 207

Table 3

Central composite design and experimental responses for desalination of high salt concentration solutions by RO.

Run Factors (controllable input variables) Responses

Feedconcentration Feedtemperature Feed ﬂowrate Feed pressure Rejection Flux Performance index

x

1

C (g/L) x

2

T (

◦

C) x

3

Q (L/h) x

4

P (MPa) RE (%) J ×10

−5

(kg/m

2

s) Y×10

−5

(kg/m

2

s)

B1 +1 42.5 +1 37.5 +1 212.5 +1 1.35 5.040 21.230 5.040

B2 −1 17.5 +1 37.5 +1 212.5 +1 1.35 35.343 62.282 35.343

B3 +1 42.5 −1 22.5 +1 212.5 +1 1.35 4.187 12.688 4.187

B4 −1 17.5 −1 22.5 +1 212.5 +1 1.35 25.867 43.224 25.867

B5 +1 42.5 +1 37.5 −1 137.5 +1 1.35 5.586 23.658 5.586

B6 −1 17.5 +1 37.5 −1 137.5 +1 1.35 31.531 56.796 31.531

B7 +1 42.5 −1 22.5 −1 137.5 +1 1.35 4.574 13.554 4.574

B8 −1 17.5 −1 22.5 −1 137.5 +1 1.35 24.498 40.683 24.498

B9 +1 42.5 +1 37.5 +1 212.5 −1 0.75 1.769 13.177 1.769

B10 −1 17.5 +1 37.5 +1 212.5 −1 0.75 8.056 21.890 8.056

B11 +1 42.5 −1 22.5 +1 212.5 −1 0.75 1.718 7.571 1.718

B12 −1 17.5 −1 22.5 +1 212.5 −1 0.75 7.732 17.819 7.732

B13 +1 42.5 +1 37.5 −1 137.5 −1 0.75 1.831 12.659 1.831

B14 −1 17.5 +1 37.5 −1 137.5 −1 0.75 8.627 23.023 8.627

B15 +1 42.5 −1 22.5 −1 137.5 −1 0.75 1.529 6.741 1.529

B16 −1 17.5 −1 22.5 −1 137.5 −1 0.75 6.424 15.296 6.424

B17 +1.41 47.63 0 30 0 175 0 1.0 2.516 15.484 2.516

B18 −1.41 12.38 0 30 0 175 0 1.0 32.161 62.579 32.161

B19 0 30 +1.41 40.58 0 175 0 1.0 8.101 23.763 8.101

B20 0 30 −1.41 19.43 0 175 0 1.0 4.547 13.297 4.547

B21 0 30 0 30 +1.41 227.9 0 1.0 5.442 16.003 5.442

B22 0 30 0 30 −1.41 122.1 0 1.0 4.714 14.887 4.714

B23 0 30 0 30 0 175 +1.41 1.35 10.140 23.332 10.140

B24 0 30 0 30 0 175 −1.41 0.65 2.163 9.307 2.163

B25 0 30 0 30 0 175 0 1.0 6.555 19.977 6.555

B26 0 30 0 30 0 175 0 1.0 6.348 17.241 6.348

the coded variables was written as:

ˆ

Y = (6.301 −8.189x

1

+1.314x

2

+5.511x

4

+5.393x

2

1

−0.737x

2

3

−1.051x

1

x

2

−4.616x

1

x

4

+0.968x

2

x

4

) ×10

−5

(11)

Subjected to:x

i

∈˝;˝={x

i

|−˛≤x

i

≤+˛};

∀

i =

1,4.

The signiﬁcance of RS-model (II) has been checked by ANOVA

test and the results are presented in Table 4.As can be seen,

the obtained F-value is 67.241 and the P-value is smaller than

10

−4

.Furthermore,the value R

2

is about 0.969,which is in agree-

ment with the adjusted coefﬁcient of determination R

2

adj

found to

be 0.955.Therefore,all ANOVA indicators reveal that RS-model

(II) is accepted from statistical point of view for simulation of

RO process for high salt concentrations of feed solutions.The

goodness-of-ﬁt of RS-model (II) is illustrated in Fig.6.The model

gives good predictions of the response,RO performance index

of the pilot plant,for high salinity feed solutions.The empiri-

cal model has been developed in terms of the actual variables as

follows:

ˆ

Y = (−25.243 −0.913C −4.96 ×10

−3

T +0.183Q +50.863P

+0.035C

2

−5.242 ×10

−4

Q

2

−0.011CT

−1.477CP +0.516TP) ×10

−5

(12)

Subjected to:12.32≤C≤47.68g/L;19.39≤T≤40.61

◦

C;

122≤Q≤228L/h;0.65≤P≤1.35MPa.

Figs.7–9 showthe response surfaces plots of the RS-model (II)

for different temperatures,feed concentrations and pressures.It

can be observed that the increment of both the feed temperature

Table 4

ANOVA table for RS-model (II) predicting the performance index for the conditions

of high salt concentration in feed solution.

Source DF SS MS F-value P-value R

2

R

2

adj

Model 8 2.607×10

−7

3.258×10

−8

67.241 <0.0001 0.969 0.955

Residual 17 8.237×10

−9

4.846×10

−10

Total 25 2.689×10

−7

Fig.6.RS-model (II):predicted and experimental RO performance index valid for

high salt concentration conditions.

Fig.7.Response surface plot of the predicted RO performance index by RSM for

high salt concentration conditions as function of the feed salt concentration and the

pressure for Q=175L/h and T=30

◦

C.

208 M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214

Fig.8.Response surface plot of the predicted RO performance index by RSM for

high salt concentration conditions as function of the feed salt concentration and the

feed temperature for Q=175L/h and P=1MPa.

and the pressure leads to an increase of the ROperformance index.

However,the main effect of pressure is evidently higher than the

main effect of feed temperature.The effect of feed ﬂowrate is the

smallest one and it appears only as reduced quadratic effect.The

increasing of the feed salt concentration minimizes considerably

theROperformanceindex.That is lowsalt rejectionfactor andsmall

permeates ﬂux.Regarding the interaction effects between factors,

the most important interaction effect occurs between the operat-

ing pressure and the feed salt concentration.In fact,the effect of

the salt concentration is more signiﬁcant at higher pressure,while

the effect of pressure is more signiﬁcant at lower feed salt concen-

tration.Moreover,there are interaction effects between the feed

salt concentration and the feed temperature as well as between

the feed temperature and the operating pressure.The interaction

effect betweenthesalt concentrationinfeedsolutionandfeedtem-

perature diminishes the RO performance index when both factors

areincreased.Incontrast,theinteractionbetweenthefeedtemper-

ature and the pressure enhances the RO performance index when

increasing both factors.In addition,it is worth noting that there

are no interaction effects between the feed ﬂowrate and the other

factors.

4.2.Predictive modeling using ANN

For the construction of ANN model to predict RO performance

of the studied pilot plant within a wide range of feed salt concen-

tration,all data presented in Tables 1 and 3 as well as additional

Fig.9.Responsesurfaceplot of thepredictedROperformanceindexbyRSMfor high

salt concentration conditions as function of the feed ﬂowrate and the temperature

for C=30g/L and P=1MPa.

Table 5

Additional set of experiments used for ANN modeling.

Run C (g/L) T (

◦

C) Q (L/h) P (MPa) RE (%) J ×10

−5

(kg/m

2

s)

Y×10

−5

(kg/m

2

s)

C1 49.22 30 175 1 20.636 13.386 2.762

C2 3.27 30 175 1 96.886 400.179 387.717

C3 26.25 40.6 175 1 38.824 26.443 10.266

C4 26.25 19.4 175 1 43.045 15.331 6.599

C5 26.25 30 228 1 42.562 19.859 8.452

C6 26.25 30 122 1 42.148 20.455 8.621

C7 26.25 30 175 1.35 54.762 33.389 18.284

C8 26.25 30 175 6.46 26.148 11.609 3.036

C9 26.25 30 175 1 41.795 21.394 8.942

C10 26.25 30 175 1 41.880 21.940 9.188

experimental results given in Table 5 have been considered.Atotal

of 66 experimental runs have been used to develop the ANNmodel

for RO pilot plant.The inputs for the neural network were iden-

tical to the factors considered in RSMapproach,namely,feed salt

concentration,feedtemperature,feedﬂowrateandoperatingpres-

sure.Similar to RSMmodeling,the ROperformance index has been

also considered as response (target) for ANN modeling.

It is worth quoting that the development of an ANN model can

be made more effective if the pre-processing step,normalization,

is considered for both the network inputs (design variables) and

the target (output/response) [39].Normalization,whichis a simple

scaling of data set,is very important for training.It must be pointed

out that the input and output data of a given systemare not of the

same order of magnitude,some variables may appear more signif-

icantly than in reality are [37].Moreover,one of the advantages of

using normalization of inputs and outputs parameters is to avoid

numerical overﬂows duetoverylargeor verysmall weights [37,48].

In the present study the network inputs and target have been

scaled (normalized) before training.In this case,the coded levels

of the variable x

1

(feed salt concentration) were revised because

of the wide range of this factor given in all designs summarized

in Tables 1,3 and 5.Finally,the coded levels for all inputs (design

variables) were ranged from−˛ (minimumlevel) up to +˛ (max-

imum level).Therefore,the coded levels of factors were kept the

same for both RSM and ANN approaches.For the normalization

of target (RO performance index),the scaled values of response Y

were ranged from−1 (minimumlevel) to +1 (maximumlevel).The

scaled inputs and normalized target were considered in order to

avoid over-ﬁtting and to improve the training process of the model

as well as to facilitate generalization of network [39].

The data generated from all experimental designs (runs:

A1–A26,B1–B26andC1–C10inTables 1,3and5,respectively) have

beenusedtoﬁgureout theoptimal architectureof ANN.Theseorigi-

nal data(62samples) weredividedintotraining,validationandtest

subsets.As training subset a number of 41 samples,a percentage

of 66% of all available data,have been used.For validation subset

11 samples have been considered,whereas for test subset 10 sam-

ples were used.The split of data into training,validation and test

subsets was carried out to estimate the performance of the neu-

ral network for prediction of “unseen” data that were not used for

training.Inthis way,thegeneralizationcapabilityof ANNmodel can

be assessed.The Neural Network Toolbox V4.0 of MATLAB math-

ematical software has been used for scientiﬁc programming and

developing of ANN model.

In this study the number of hidden layers and neurons was

established by training different feed-forward networks of various

topologies and selecting the optimal one based on minimization

of performance function MSE and improving generalization capa-

bility.The obtained optimal architecture (topology) of ANN model

for this probleminvolves a feed-forward neural network with four

inputs,two hidden layers (one layer with ﬁve neurons another

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 209

Fig.10.Optimal architecture of ANN model for prediction of RO performance index.

with three neurons) and one output layer (including one neuron).

This feed-forward network topology is denoted as multi-layer per-

ceptron,MLP (4:5:3:1),referring to the number of inputs and the

number of neurons in the hidden and output layers,respectively.

Fig.10shows theoptimal architectureof thedevelopedANNmodel.

Notethat all neurons of thehiddenlayers havelog-sigmoidtransfer

function (logsig),while the output layer neuron has linear transfer

function (purelin).As can seen in Fig.10 the connections between

inputs and neurons as well as between neurons fromdifferent lay-

ers consist of weights and biases.IW

(1,1)

in Fig.10 indicates the

input weight matrix of size (5×4).LW

(2,1)

and LW

(3,2)

denote the

layer weight matrixes,where the superscripts indicate the source

and destination connections,respectively.All neurons from the

networkhavethebias b

(l)

wherethesuperscript l indicates thelayer

index.To ﬁgure out the optimal values of weights and biases,the

network MLP (4:5:3:1) has been trained using back-propagation

method (BP) based on Levenberg–Marquardt algorithm(LMA).The

general concept of BP method used for network training is shown

in Fig.11.The training was carried out by adjusting the weights

and biases of the entire network in order to minimize the per-

formance function (MSE).During the training phase,each neuron

receives the input signals,aggregates themusing the weights and

biases,and ﬁnally passes the result after suitable transformation as

output.

The training has been imposed to be ﬁnished at the point where

the network error (MSE) becomes sufﬁciently small (MSE≤E

0

,

where the goal is E

0

=10

−4

).In the present case training was

stopped after 10 iterations.Fig.12 illustrates the training,valida-

tion and test mean squared errors.During the training step the

performance functions MSE of the training and the test data sub-

sets were lower than the goal E

0

reaching a value of 2.25×10

−5

.

Furthermore,as can be seen in Fig.12,the performance function

for validation data subset was very close to the goal E

0

.Therefore,

the training process has been considered successfully terminated

and the obtained optimal values of weights and biases are sum-

marized in Table 6.Note that,these optimal values of connections

(weights and biases) are related to coded inputs (factors) and nor-

malized target (response).For the sake of comparison with RSM,

the weights and biases in ANN model play the role of “regression

coefﬁcients” for RS-model.The weights and biases for ANN archi-

tectureshowninFig.10aregivenas matrixes andvectors inTable6.

Consequently,theANNmodel for thepredictionof ROperformance

can be described as a composite mapping:

Y(x) = f

(3)

(LW

(3,2)

f

(2)

(LW

(2,1)

f

(1)

(IW

(1,1)

x +b

(1)

) +b

(2)

) +b

(3)

)(13)

where f

(l)

is the vector of transfer function corresponding to layer l

(l =1–3) andtheother terms involveddeal withtheaforementioned

weights,biases and inputs.

Fig.11.General scheme for network training by means of BP method.

210 M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214

Table 6

Optimal values of weights and biases obtained during network training with LMA.

Input weight matrix destination:HL-1

IW

(1,1)

=

2.2657 −0.0461 −0.1209 −1.5173

4.7068 0.8980 −0.2102 −0.7377

−2.7115 1.1273 −0.3068 0.3553

2.7380 −0.2777 0.0894 −0.8035

−2.4246 −0.4668 0.2784 0.2060

Source:inputs

Bias vector destination:HL-1 b

(1)

=

−5.8165 3.4903 −3.3778 1.4972 −5.2410

T

Layer weight matrix destination:HL-2

LW

(2,1)

=

−2.6600 2.7221 −1.5649 −3.1806 4.7213

0.4533 −3.8825 2.9029 −3.6157 5.5758

2.2031 −2.2707 1.6865 4.5581 −4.0014

Source:HL-1

Bias vector destination:HL-2 b

(2)

=

1.2533 −2.0644 0.7018

T

Layer weight vector destination:OL-3

LW

(3,2)

=

0.0693 3.5697 −0.3463

T

Source:HL-2

Bias scalar destination:OL-3 b

(3)

=−0.7088

Fig.12.Training,validationandtest meansquarederrors for the LMA(performance

is 2.251×10

−5

and goal is E

0

=1×10

−4

).

After neural network training,the developed ANN model has

been tested for its accuracy in prediction of RO performance index

using analysis of variance (ANOVA).The ANOVA results for neural

network model are given in Table 7.All ANOVA estimators have

been calculated in a similar way as RS-models.In the case of RSM,

the degree of freedom due to residuals is given by the difference

between the total number of experiments and the total number

of regression coefﬁcients from empirical model.For ANN model,

instead of the total number of coefﬁcients,the total number of

connections can be considered.The calculation of the degree of

freedomdue to residual in the case of ANN model can be written

as:

DF

residual

= N −L (14)

where N means the total number of experiments considered to

developthe predictive model andL means the total number of con-

nections (weights andbiases) intheANNmodel.For a feed-forward

Table 7

Analysis of variance (ANOVA) for ANN model.

Source DF SS MS F-value P-value R

2

R

2

adj

Model 46 9.375×10

−5

2.038×10

−6

1593.36 <0.0001 1 0.999

Residual 15 1.919×10

−8

1.279×10

−9

Total 61 9.377×10

−5

neural network with one hidden layer (HL),the total number of

connections is given by:

L = z(n +H +1) +H (15)

where n denotes the number of inputs (variables),z is the number

of neurons in HL and His the number of neurons (nodes) in output

layer (OL).In the case of a neural network with two hidden layers,

the total number of connections is estimated as:

L = z

1

(n +z

2

+1) +z

2

(H +1) +H (16)

where z

1

and z

2

mean the number of neurons within the ﬁrst and

second hidden layers,respectively.

Eqs.(15) and(16) are validfor feed-forwardnetworks withneu-

rons havingbiases.Inour speciﬁc case,N=62andL =47[Eq.(16)] so

that the degree of freedomdue to residual is DF

residual

=15.In addi-

tion,ANOVA gives a very high F-value (1593.36) and a very low

P-value (<10

−4

).The coefﬁcient of multiple determination is equal

tounity(R

2

=1),whichis perfect andtheadjustedcoefﬁcient is very

closetounity(R

2

adj

= 0.999).All thesestatistical estimators indicate

anadequate ANNmodel withoptimal architecture that canbe used

for predictive simulations of ROprocess withina wide range of feed

salt concentration.The goodness-of-ﬁt between the experimental

and the predicted ROperformance index given by ANNis shown in

Fig.13.All points are located very near to the straight line indicat-

ing that ANN model prediction is excellent inside the valid region.

Fig.13.PredictedROperformance indexby ANNmodel versus experimental values.

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 211

Fig.14.ROperformance index predicted by ANNmodel as function of the feed salt

concentration and the pressure for Q=175L/h and T=30

◦

C.

A value of correlation coefﬁcient close to unity (r

2

=0.9998) shows

the linear relationshipbetweenthe experimental andpredictedRO

performance index.This result can be attributed to the good gen-

eralization capability of the developed ANN model that has been

improved by applying different steps:(a) scaling the inputs and

normalization of target;(b) selecting the optimal ANNarchitecture

that ensures a positive degree of freedom(i.e.the total number of

connections is smaller than total number of experiments used for

developingtheANNmodel);(c) splittingtheexperimental datainto

training,validation and test subsets before starting training of the

network.

Based on the trained network MLP (4:5:3:1) the output surfaces

(3D diagrams) has been drawn to showthe inﬂuence of the differ-

ent inputs (factors) on the RO performance index.The results are

presented in Figs.14–17.

Figs.14–17 indicate that an increase of both the feed tempera-

ture and the operating pressure lead to an enhancement of the RO

performance index.However,these effects are more signiﬁcant at

lower concentration of salt in feed solution.It was also observed

that the feed ﬂow rate has the smallest non-linear effect on the

RO performance index.The interaction effects are visible between

three factors,namely,the salt concentration of feed solution,the

pressureandthefeedtemperature.Suchinteractioneffectsaresim-

ilar to those predicted by RS-models.For example,the ANN model

also predicts that the effect of pressure is more signiﬁcant at higher

feed temperatures and the effect of the feed temperature is higher

at higher operating pressures.

Fig.15.ROperformance index predicted by ANNmodel as function of the feed salt

concentration and the feed temperature for Q=175L/h and P=1MPa.

Fig.16.RO performance index predicted by ANN model as function of the feed

temperature and the pressure for C=26.25g/L and Q=175L/h.

As can be seen in Figs.14 and 15,the effect of the salt concen-

tration in the feed solution is signiﬁcant.An exponential increase

of the RO performance index was observed with the decrease of

the salt concentration below 15g/L.This may be the reason why

RSMfailed in describing the performance of the RO pilot plant for

a wide range of feed salt concentration (i.e.the RS-model I valid

for brackish water was not valid for RS-model II and vice versa).

The empirical model given by RSMcontains linear,interaction and

quadratic terms.There it cannot predict the non-linear behavior

(i.e.exponential in this case) for a large range of one of the factors.

In contrast,ANN model can predicts such exponential behavior in

similar conditions.This is an advantage of ANN modeling.ANN is

not limitedtoanapproximationof linear andquadratic effects only

like RSM.Therefore,ANNdemonstrated its ability to overcome the

limitation of the quadratic polynomial model of RSM.

4.3.Optimization of RO desalination conditions

Prior to discuss the results of optimization it is essential to

present some information about the reliable conditions of RO

desalination accepted in industry.For desalination of seawater

(∼30g/L) byROprocess thesalt rejectionmust behigher than99.3%

in order to make possible production of potable water fromseawa-

ter in a practical single-stage ROplant.Concerning brackish water,

its salinity is usually between 2 and 10g/L.The World Health Orga-

nization (WHO) recommendation for salinity of potable water is

about 0.5g/L or lower,sothat upto90%of the salt must be removed

frombrackish feed solutions [28].

Fig.17.ROperformance index predicted by ANNmodel as function of the feed ﬂow

rate and the temperature for C=26.25g/L and P=1MPa.

212 M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214

Table 8

Optimal solutions for RO performance index given by RSMand ANN methods considering:(I) all factors as variables,(II) brackish water of ﬁxed concentration 6g/L and (III)

seawater of ﬁxed concentration 30g/L.

Method C (g/L) T (

◦

C) Q (L/h) P (MPa) Y

predicted

×10

−5

(kg/m

2

s) Y

experimental

×10

−5

(kg/m

2

s)

I RSM 3.97 40.6 175 1.35 677.2 709.5

ANN 3.30 40.6 228 1.35 663.5 764.1

II RSM 6.00 40.3 145 1.35 492.1 687.3

ANN 6.00 40.3 170 1.34 555.8 735.6

III RSM 30.00 39.9 180 1.35 17.5 35.2

ANN 30.00 26.1 225 1.34 12.5 18.4

In this work the optimization part includes three issues.For the

ﬁrst optimization problem the salt concentration was considered

as variable.For the second optimization issue a ﬁxed salt concen-

tration of 6g/L (i.e.average salt concentration for brackish waters)

was considered.The third optimization problem was solved for

a ﬁxed salt concentration value of 30g/L (i.e.a typical value of

seawater).

All objective functions given by RSMand ANN have been opti-

mized by means of Monte Carlo simulation method based on

pseudo randomnumbers (PRNs).The stochastic simulations were

carried out using a multistage zooming-in approach to localize the

optimal points inside the validregionmore accurately.The optimal

solutions found by RSMand ANNmodels together with the conﬁr-

mationruns (experimental validationof optimum) aresummarized

in Table 8 for all optimization problems.

By comparing the optimal points obtained by RSM and ANN

models for the optimization I (Table 8,i.e.variable concentra-

tion),one may conclude that both methods actually converged to

quite similar solutions.In fact,both models provide identical opti-

mal feed temperature (40.6

◦

C) and optimal pressure (1.35MPa).

The optimum salt concentration in feed solution by RSM model

is 3.97g/L,while that obtained by ANN model is 3.30g/L.The

highest difference was obtained between the optimal values of

feed ﬂow rate.However,the effect of this factor on the RO per-

formance index was insigniﬁcant as stated earlier for both RSM

and ANN models.The measured permeate ﬂux of the RO system

under the optimal operating conditions given by RSMmodel was

735.98×10

−5

kg/m

2

sandthesalt rejectionfactor was96.4%.Under

such optimum conditions the salt concentration in the perme-

ate was about 0.143g/L,lower than the imposed limit by WHO,

0.5g/L.In a similar way,the measured permeate ﬂux under the

optimumoperating conditions given by ANN model was found to

be 786.88×10

−5

kg/m

2

s and the salt rejection factor was 97.1%.

The salt concentration in permeate was about 0.09g/L,also lower

than the permitted limit.

It is worth mentioning that the optimal solutions obtained fol-

lowing both RSM and ANN models for the second optimization

problemwhen the feed salt concentration was ﬁxed at 6g/L,con-

verged to an identical temperature (40.6

◦

C) and to almost the

same operating pressure (1.35MPa for RSMand1.34MPa for ANN).

Concerning the feed ﬂow rate,the optimal value of 145L/h was

obtained by RSM and 170L/h by ANN.The experimental conﬁr-

mation runs revealed that the optimal conditions offered by ANN

are better than those given by RSM (i.e.the experimental per-

meate ﬂuxes are as follows 735.8×10

−5

kg/m

2

s for ANN versus

687.4×10

−5

kg/m

2

s for RSM.Notethat,thesalt rejectionefﬁciency

in this case was about 99.98%.

For seawater desalination conditions (third optimization prob-

lem) it seems that RSMprovides a better optimal point than ANN.

For instance,ANN indicates an optimal temperature of 26.1

◦

C

and an operating pressure of 1.34MPa,while by RSM model a

higher optimal temperature of 39.9

◦

C and a pressure of 1.35MPa

were obtained.The experimental permeate ﬂuxes are as follows

51.9×10

−5

kg/m

2

s for RSMand 27.1×10

−5

kg/m

2

s for ANN with

an average rejection factor of 67.83%.

Finally,it is worth to mention that both RSMand ANN models

indicate that the global optimal operation conditions of the con-

sidered RO pilot plant are in the range of desalination of brackish

waters.

5.Conclusions

RSM and ANN methods were applied for modeling and opti-

mization of desalination process by reverse osmosis (RO).RSM

was unable to develop a global model to predict the RO perfor-

mance over a wide range of salt concentration in feed solution.

Therefore,RSM was carried out individually for low salt feed

concentrations (brackish water salinity) and high salt feed con-

centrations (seawater salinity) obtaining two empirical models.

The effects of the operating factors were investigated by response

surface analysis.The most important effects on the RO perfor-

mance index were found to be the salt concentration in feed

and the operating pressure followed by the effect of the feed

temperature.The effect of the feed ﬂow rate was negligible for

low salt concentrations and insigniﬁcant for high salt concentra-

tions.

ANN approach provides a global model describing RO perfor-

mance of the pilot plant in a wide range of feed salt concentration.

This is anadvantageof ANNmodel over RSMmodel.Another advan-

tage of ANN is that this methodology does not require a standard

experimental design to build the model.Different experimental

designs can be used.In addition,ANN model is ﬂexible and per-

mits to add newexperimental data to build a trustable ANNmodel.

In contrast,ANN methodology may require a greater number of

experiments than RSM.

When considering the concentration as variable,the optimal

solutions given by RSM and ANN models were quite similar

indicating that the optimal operating conditions of the tested

RO pilot plant are in the range of desalination of brackish

waters.However,the conﬁrmation experimental runs show that

the optimal conditions given by ANN model are the best and

represent the global optimal solution for the tested RO pilot

plant.The global optimal solution involves the following val-

ues as input variables:C=3.30g/L,T=40.6

◦

C,Q=228L/h and

P=1.35MPa.Under such conditions of operation,a maximal RO

performanceindexwas achievedcomparedtoall performedexper-

iments.This is a performance index of the RO pilot plant of

764.1×10

−5

kg/m

2

s.

Optimum operating conditions were also determined for typ-

ical brackish water and seawater with ﬁxed concentrations of

6g/L and 30g/L,respectively.The obtained optimum operating

conditions,when the concentration was 6g/L,were practically

similar for both ANN and RSM models.In this case,the response

corresponding to ANN model was found to be better than that

given by RSM with a salt rejection of about 99.98%.However,

for 30g/L,the obtained optimum temperature and ﬂow rate of

each model were different,whereas the optimum pressure of

both ANN and RSMmodels was quite similar (i.e.1.34–1.35MPa).

In this case,a higher RO performance was found for RSM

model.

M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 213

Acknowledgements

The authors of this work gratefully acknowledge the ﬁnan-

cial support of the University Complutense of Madrid for granting

Dr.C.Cojocaru “Estancias de Doctores y Tecnólogos en la Univer-

sidad Complutense,Convocatoria 2008” and UCM-BSCH (Project

GR58/08,UCMGroup 910336).M.Essalhi is thankful to the Middle

East DesalinationResearchCentrefor thegrant (MEDRC06-AS007).

Nomenclature

b bias termfor a node

b bias vector for a layer

C concentration of salt in feed solution (g/L)

C

P

concentration of salt in permeate (g/L)

DF degree of freedom

E

0

training error (goal)

f vector of transfer function

F-value ratio of variances,computed value

grad gradient of performance function

H number of neurons (nodes) in output layer

IW input weight matrix

J average permeate ﬂux

L number of connections for ANN predictive model

logsig transfer function (Matlab syntax)

LW layer weight matrix

M number of patterns used in training set

MS mean square

MSE mean-squared-error (performance function)

n number of input variables (inputs)

N number of experimental runs

P operating pressure

P-value statistical estimator

purelin transfer function (Matlab syntax)

Q feed ﬂowrate

r

2

correlation coefﬁcient

R

2

coefﬁcient of multiple determination

R

2

adj

adjusted statistic coefﬁcient

RE salt rejection factor

S net input

SS sumof squares

T temperature of feed solution

tansig transfer function (Matlab syntax)

w weight (neural network connection)

W vector of current weights and biases

X design matrix of input variables

x vector of inputs

x

1

,x

2

,x

3

,x

4

coded levels of input variables

Y vector of performance index

Y response/target—performance index (experimental

value)

ˆ

Y predictor of response (performance index) by RSM

Y predictor of response/target (performanceindex) by

ANN

z

1

,z

2

number of neurons in hidden layers

Greek letters

˛ axial point or “star” point in CCD

ˇ

0

,ˇ

i

,ˇ

ii

,ˇ

ij

regression coefﬁcients within response surface

model

ˇ

OLS

vector of regression coefﬁcients

learning rate

ϕ response function

valid region (region of experimentation)

Superscirpts

l integer variable indicating the layer in ANN topol-

ogy

m integer variable (subscript)

T

transpose of matrix or vector

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