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Journal of Membrane Science 368 (2011) 202–214
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Journal of Membrane Science
j ournal homepage:www.el sevi ocat e/memsci
Artificial neural network modeling and response surface methodology of
desalination by reverse osmosis

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
Reverse osmosis
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.
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

Corresponding author.Tel.:+34 91 3945185;fax:+34 91 3945191.
E-mail (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-
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.
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.
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
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
where C is the feed concentration and C
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
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
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]:
Y = ˇ
Y denotes the predicted response,x
refers to the coded
levels of the input variables,ˇ



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]:
= (X
Y (4)
where ˇ
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
S =
+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 +exp(−S)
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]:

H · M
where H denotes the number of output nodes,M is the number
of patterns used in the training set,Y

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
(MSE) (8)
is avector of current weights andbiases,grad
(MSE) is
thecurrent gradient of theperformancefunctionMSEand
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-
),feed temperature,T(x
),feed flowrate,Q(x
) and
operating pressure,P(x
).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
C (g/L) x
T (

C) x
Q (L/h) x
P (MPa) RE (%) J ×10
s) Y×10
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
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:
Y = (139.631 −92.41x
) ×10
Subjected to:x

i =
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
(feedflowrate) was omittedfromthefinal model since
it is an insignificant variable according to the results of Student’s
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
and adjusted statistic R
mathematical expressions used for computation of statistical esti-
mators (i.e.SS,MS,F-value,R
) 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
.Note that R
is about 0.994,being close to unity,which is worthwhile.More-
over,thecoefficient R
is inagreement withtheadjustedcoefficient
of determination,R
.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
F-value P-value R
Model 9 4.611×10
304.32 <0.0001 0.994 0.991
Residual 16 2.694×10
Total 25 4.638×10
Degree of freedom.
Sumof squares.
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:
Y = (−133.72 −1.58C +7.811T +73.76P +3.867C
−0.804CT −69.28CP
+8.293TP) ×10
Subjected to:3.96≤C≤11.04g/L;19.40≤T≤40.60

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

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
s to 62.6kg/m
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
C (g/L) x
T (

C) x
Q (L/h) x
P (MPa) RE (%) J ×10
s) Y×10
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:
Y = (6.301 −8.189x
) ×10
Subjected to:x

i =
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
.Furthermore,the value R
is about 0.969,which is in agree-
ment with the adjusted coefficient of determination R
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
Y = (−25.243 −0.913C −4.96 ×10
T +0.183Q +50.863P
−5.242 ×10
−1.477CP +0.516TP) ×10
Subjected to:12.32≤C≤47.68g/L;19.39≤T≤40.61

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
Model 8 2.607×10
67.241 <0.0001 0.969 0.955
Residual 17 8.237×10
Total 25 2.689×10
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

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
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
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
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
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
(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
in Fig.10 indicates the
input weight matrix of size (5×4).LW
and LW
denote the
layer weight matrixes,where the superscripts indicate the source
and destination connections,respectively.All neurons from the
networkhavethebias b
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
The training has been imposed to be finished at the point where
the network error (MSE) becomes sufficiently small (MSE≤E
where the goal is E
).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
reaching a value of 2.25×10
Furthermore,as can be seen in Fig.12,the performance function
for validation data subset was very close to the goal E
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
x +b
) +b
) +b
where f
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
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
Bias vector destination:HL-1 b
−5.8165 3.4903 −3.3778 1.4972 −5.2410
Layer weight matrix destination:HL-2
−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
Bias vector destination:HL-2 b
1.2533 −2.0644 0.7018
Layer weight vector destination:OL-3
0.0693 3.5697 −0.3463
Bias scalar destination:OL-3 b
Fig.12.Training,validationandtest meansquarederrors for the LMA(performance
is 2.251×10
and goal is E
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
= 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
Model 46 9.375×10
1593.36 <0.0001 1 0.999
Residual 15 1.919×10
Total 61 9.377×10
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
(n +z
+1) +z
(H +1) +H (16)
where z
and z
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
=15.In addi-
tion,ANOVA gives a very high F-value (1593.36) and a very low
P-value (<10
).The coefficient of multiple determination is equal
=1),whichis perfect andtheadjustedcoefficient is very
= 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

A value of correlation coefficient close to unity (r
=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
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
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
s) Y
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
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
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
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
s for ANN versus
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

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
s for RSMand 27.1×10
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
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-
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
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
M.Khayet et al./Journal of Membrane Science 368 (2011) 202–214 213
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).
b bias termfor a node
b bias vector for a layer
C concentration of salt in feed solution (g/L)
concentration of salt in permeate (g/L)
DF degree of freedom
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
correlation coefficient
coefficient of multiple determination
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
coded levels of input variables
Y vector of performance index
Y response/target—performance index (experimental
Y predictor of response (performance index) by RSM

Y predictor of response/target (performanceindex) by
number of neurons in hidden layers
Greek letters
˛ axial point or “star” point in CCD



regression coefficients within response surface
vector of regression coefficients
 learning rate
ϕ response function
 valid region (region of experimentation)
l integer variable indicating the layer in ANN topol-
m integer variable (subscript)
transpose of matrix or vector
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