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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 form 7 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 chloride 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 permeate ﬂux, has been considered as response. Both RSM and ANN models have been developed based on experimental designs. Two empirical polynomial RSM models valid for different ranges of feed salt concentrations 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 polynomial model obtained by RSM and to solve non-linear problems. Analysis of variance (ANOVA) has been employed to test the signiﬁcance of response surface polynomials and ANN model. To test the signiﬁcance of ANN model, the estimation of the degree of freedom due to residuals has been detailed. Finally, both modeling methodologies RSM and ANN were compared in terms of predictive abilities by plotting the generalization graphs. The optimum operating conditions were determined by Monte Carlo simulations considering: (i) the four input variables, (ii) for typical brackish water with a ﬁxed concentration of 6 g/L and (iii) for typical seawater with a ﬁxed concentration of 30 g/L. Under the obtained optimal conditions maximum RO performance indexes have been achieved experimentally. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Construction of mathematical models for prediction of membrane 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 fundamental 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 experiments; 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 surface methodology; RS-model, response surface model; WHO, World Health Organization. ∗ Corresponding author. Tel.: +34 91 3945185; fax: +34 91 3945191. E-mail address: [email protected]ﬁs.ucm.es (M. Khayet). 0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2010.11.030

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 optimization. Response surface methodology (RSM) and artiﬁcial neural networks (ANN) are modeling tools able to solve linear and non-linear multivariate regression problems. Both methodologies do not need explicit expressions of the physical meaning of the system or process under study. Therefore, RSM as well as ANN belong to modeling tools dealing with development of non-parametric simulative models known also as “black-box” models. Such models ascertain a relationship between design variables and response or 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, RSM has been used for modeling and optimization of heavy metals removal from wastewaters 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

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

from water and to optimize the operating conditions for membranes working with multi-ionic solution [7]. The experimental design and RSM was also applied to direct contact membrane distillation [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 concentration, 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 RSM in describing the performance of thin ﬁlm composite membrane 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 RSM to optimize the hydrolysis of whey protein isolate in a tangential ﬂow membrane reactor. Concerning ANN, this modeling method has been applied progressively 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 experimental 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 neural network modeling for prediction of permeate ﬂux, separation efﬁciency and/or permeate ﬂux decline in crossﬂow membrane ﬁltration 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 models were proposed to predict the permeate ﬂux at any time during ultraﬁltration and after backwashing for arbitrary cycles. Regarding 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 approximated response surfaces to solve the optimization problem. The question is: which approximation model is more trustable offering better accuracy in ﬁtting experimental data and giving a better optimal solution conﬁrmed by experiment? Moreover, it is important 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 techniques [26–28]. Recent technological innovations make RO systems more attractive for industry using alternative energy sources like photovoltaic solar energy [29,30] and wind energy [31]. The

203

Fig. 1. Schema of the experimental RO pilot plant: (1) RO module; (2) high pressure pump; (3) vent; (4) manometer; (5) ﬂowmeter for retentate; (6) ﬂowmeter for permeate; (7) electrical conductivity monitor; (8) thermostat; (9) temperature probe; (10) feed tank; (11) low pressure pump (<4.1 bar); (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 performance index of RO pilot plant using both RSM and ANN models; (3) to compare the optimal solutions offered by RSM and ANN. To the best of our knowledge, this is the ﬁrst report comparing RSM and ANN in membrane science and technology. 2. Experimental The RO experiments were carried out using the pilot plant schematized in Fig. 1. A stainless steel spiral wound module (S2521, Osmonics), containing polyamide thin ﬁlm composite membrane was employed in this study. The effective membrane area of the membrane module is 1.2 m2 [32]. Sodium chloride, NaCl (Sigma–Aldrich) and distilled water were used to prepare the feed salt solutions of desired concentrations. The electrical conductivity of the feed and permeate aqueous solutions was measured using a conductivimeter 712 Metrohm. The salt concentration was then determined based on the calibration correlation electrical conductivity-concentration. The salt rejection factor (RE) was determined as follows [32]: RE =

C − CP C

(1)

where C is the feed concentration and CP is the permeate concentration. All experiments were conducted at different levels of feed solute concentration, feed temperature, feed ﬂow rate and hydrostatic pressure. The operating levels of these factors are given later on following the design of experiments (DoE). The permeate ﬂux was measured for different levels of operating conditions by weighing the obtained permeate for 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 ﬂux J (kg/m2 s) by integration. The RO performance index of the pilot plant (Y) was determined as the product

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between the average permeate ﬂux and the salt rejection factor: Y = J × RE

(2)

Since higher RO performance 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 secondorder 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 x1 , x2 , . . . , xn must be scaled to coded levels. In coded scale the factors vary from (−1) that corresponds to minimum level up to (+1) that suit to maximum level. The second-order models given by RSM are often used to determine the critical points (maximum, minimum, or saddle) and can be written in a general form as [34]:

n

Yˆ = ˇ0 +

n

ˇi xi +

i=1

i=1

ˇij xi xj

(3)

i

where Yˆ denotes the predicted response, xi refers to the coded levels of the input variables, ˇ0 , ˇi , ˇii , ˇij are the regression coefﬁcients (offset term, main, quadratic and interaction effects) and n is the total number of designed variables. To determine the regression coefﬁcients, the ordinary least squares (OLS) method is used. The OLS estimator can be written as follows [34–36]: ˇOLS = (X T X)

−1

logsig(S) =

XT Y

(4)

where ˇOLS is a vector of regression coefﬁcients, X is an extended designed matrix of the coded levels of the input variables, Y is a column vector of response determined according to the arrangement points into the experimental design.

1 1 + exp(−S)

(6)

The way in which the inputs and outputs of the neurons are connected is known as 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 consisting of, hidden and output neurons, respectively. Frequently the inputs are considered as additional layer. The most common neural network architecture used for solving non-linear regression problems is the multi-layer feed-forward neural network also known as multi-layer perceptron, MLP. The most common training algorithm for feed-forward neural network is back-propagation (BP) method [40]. Training of ANN by means of BP algorithm is an iterative optimization process where the performance function is minimized by adjusting the weights appropriately. The commonly employed performance function is the mean-squared-error, MSE, that is deﬁned as [41,42]:

H M

MSE =

n

ˇii xi2 +

The transfer function takes the argument, S, and produces the scalar output of a single neuron. The most used transfer functions to solve linear and non-linear regression problems are purelin, logsig and tansig [38]. For the case of logistic output the logsig transfer function may be written as:

h=1

m=1

(Yh,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, Yh,m and Y h,m are the target (experimental response) and output (predicted response) of the hth output node, respectively. According to BP algorithm the weights and biases are iteratively updated in the direction in which the performance function MSE decreases most rapidly. Generally, a single iteration of BP algorithm can be written as [38,43,44]: W (k+1) = W (k) − (k) grad (k) (MSE) where W(k)

(8)

is a vector of current weights and biases, grad(k) (MSE) is

the current gradient of the performance function MSE and (k) is the learning rate. More detailed mathematical aspects on BP training algorithms are comprehensively described elsewhere [38,39]. 4. Results and discussion

3.2. Predictive modeling using ANN 4.1. Predictive modeling using RSM ANN offers a rich framework for modeling of non-linear phenomena and for solving the multivariate regression problems. The mode of building ANN model is totally different if we compare with the construction method of polynomial RS-model. It is worth to outline some distinction between RSM and ANN terminology. RSM operates with factors (design variables) and response. In ANN modeling 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 them that can be used for mapping input and output data [37]. An artiﬁcial neuron (node) is a single computational processor, which operates with (1) summing junction and (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

xi wi + b

(5)

The ﬁrst attempt on RSM modeling 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 brackish 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, RSM has 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 seawater salinity conditions). MATLAB software has been used for all computations and graphical analysis in RSM applications. 4.1.1. RS-model for low salt concentration of feed solution In this study, the central compositional experimental design (CCD) has been used to investigate the synergistic effects of factors upon the performance of RO process. The factors involved in multivariate experimentation dealt with the feed solute (NaCl) concentration, C(x1 ), feed temperature, T(x2 ), feed ﬂow rate, Q(x3 ) and operating pressure, P(x4 ). The experimental design including the coded and actual values of factors is shown in Table 1. Note that in

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205

Table 1 Central composite design and experimental responses for desalination of low salt concentration solutions by RO. Run

A1 A1 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 A26

Factors (controllable input variables)

Responses

Feed concentration

Feed temperature

Feed ﬂow rate

Feed pressure

Rejection

Flux

Performance index

x1

C (g/L)

x2

T (◦ C)

x3

Q (L/h)

x4

P (MPa)

RE (%)

J × 10−5 (kg/m2 s)

Y × 10−5 (kg/m2 s)

+1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1.41 −1.41 0 0 0 0 0 0 0 0

10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 11.04 3.97 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

+1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 0 0 +1.41 −1.41 0 0 0 0 0 0

37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 30 30 40.58 19.43 30 30 30 30 30 30

+1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 −1 −1 −1 0 0 0 0 +1.41 −1.41 0 0 0 0

212.5 212.5 212.5 212.5 137.5 137.5 137.5 137.5 212.5 212.5 212.5 212.5 137.5 137.5 137.5 137.5 175 175 175 175 227.9 122.1 175 175 175 175

+1 +1 +1 +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 +1.41 −1.41 0 0

1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 1.0 1.0 1.0 1.0 1.0 1.0 1.35 0.65 1.0 1.0

90.173 96.430 90.681 96.665 90.730 95.947 91.078 96.498 67.163 90.957 66.002 92.265 66.317 90.787 68.973 91.695 78.071 96.375 89.713 91.568 91.305 90.773 95.374 71.891 91.203 91.460

187.909 507.898 147.360 372.572 187.690 492.289 141.302 354.309 38.416 149.083 29.088 108.259 41.472 144.931 27.496 102.234 56.376 351.413 167.976 119.713 142.719 157.003 354.378 36.825 152.764 150.970

169.443 489.766 133.628 360.147 170.291 472.337 128.695 341.901 25.801 135.601 19.199 99.885 27.503 131.579 18.965 93.743 44.013 338.674 150.696 109.619 130.310 142.516 337.984 26.474 139.325 138.077

this experimental design the feed salt concentration was varied in the range of brackish water salinity from 3.97 g/L to 11.04 g/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 (Table 1) the salt rejection factor ranged from 66.0% to 96.7%, whereas the permeate ﬂux was in the range 27.5–507.9 kg/m2 s. Based on these responses (i.e. rejection factor and permeate ﬂux) the RO performance index has been computed by means of Eq. (2). In fact, the RO performance index is an output variable that combines the salt rejection factor and the permeate ﬂux into a single overall response. Therefore, all modeling and optimizations carried out in this study are related to the performance index. The improvement 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 low salt concentration in feed solution and was written in terms of coded variables as follows: Yˆ = (139.631 − 92.41x1 + 24.214x2 + 107.727x4 + 24.173x12 −6.42x22 + 19.616x42 − 15.066x1 x2 − 43.297x1 x4 +15.548x2 x4 ) × 10−5

(9)

Subjected to: xi ∈ ˝; ˝ = {xi |−˛ ≤ xi ≤ + ˛}; ∀i = 1, 4. where ˛ denotes the star point (a property of CCD), which delimitates the boundaries of valid region ˝ known also as region of experimentation. In the case of four factors (n = 4) and an orthogonal design CCD, the star point is ˛ = 1.44. It is worth to note that the signiﬁcance of regression coefﬁcients was tested using the statistical Student’s t-test [45]. In Eq. (9) only the signiﬁcant terms have been retained. Note that the factor x3 (feed ﬂow rate) was omitted from the ﬁnal 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 from unity, 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 low as possible. Most RS-models are validated for prediction if the P-value is less than 0.05. The ANOVA results (Table 2) summarize the sum of squares of residuals and regressions together with the corresponding degrees of freedom, F-value, P-value and ANOVA coefﬁcients (i.e. coefﬁ2 ). The cients of multiple determination R2 and adjusted statistic Radj mathematical expressions used for computation of statistical esti2 ) are extensively presented in mators (i.e. SS, MS, F-value, R2 , Radj the textbooks concerning DoE and RSM [35,45–47]. According to the ANOVA results summarized in Table 2, the F-value is quite high (304.32) and the P-value is smaller than 10−4 . Note that R2 value is about 0.994, being close to unity, which is worthwhile. Moreover, the coefﬁcient R2 is in agreement with the adjusted coefﬁcient 2 . All statistical estimators disclose that the of determination, Radj developed model is validated from statistical point of view to simulate RO process for the conditions of low salt concentration in feed 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 low salt concentration in feed solution. Source Model Residual Total a b c

DFa 9 16 25

SSb

MSc −5

4.611 × 10 2.694 × 10−7 4.638 × 10−5

Degree of freedom. Sum of squares. Mean square.

−6

5.124 × 10 1.684 × 10−8

F-value

P-value

R2

2 Radj

304.32

<0.0001

0.994

0.991

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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 low salt concentration conditions.

For the graphical representation and response surface analysis it is interesting to convert the RS-model from coded to actual variables. In this case the substitution technique has been applied and the empirical model in terms of actual variables was written as: Yˆ = (−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

Fig. 4. Response surface plot of the predicted RO performance index by RSM for low salt concentration conditions as function of the feed salt concentration and the pressure for Q = 175 L/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 Student’s t-test, this factor does not affect the response surface for the conditions of low salt concentrations in feed solution.

(10)

Subjected to: 3.96 ≤ C ≤ 11.04 g/L; 19.40 ≤ T ≤ 40.60 ◦ C; 0.65 ≤ P ≤ 1.35 MPa. 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 in this case is 4-fold higher than the main effect of feed temperature. The main effect of feed salt concentration is close to the effect of pressure, in magnitude, but with an opposite sense. This means that the increase of the feed salt concentration diminishes substantially the RO performance index. Thus, the RO system operates better at low salt concentrations and high pressures. It was observed that the highest factors interaction effects exists between the pressure and salt concentration of the feed solution. 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 predicted RO performance index by RSM for low salt concentration conditions as function of the feed salt concentration and the feed temperature for Q = 175 L/h and P = 1 MPa.

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 from 12.38 g/L to 48.63 g/L. In this range, the center point corresponds to the conditions of seawater salinity, i.e. 30 g/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 rejection factor was in the range 1.5–35.3%, whereas the permeate ﬂux ranged from 6.7 kg/m2 s to 62.6 kg/m2 s. As stated earlier, the overall response, RO performance index, has been calculated using Eq. (2) and the results are summarized in Table 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 predicted RO performance index by RSM for low salt concentration conditions as function of the feed temperature and the pressure for C = 7.5 g/L and Q = 175 L/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

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 B18 B19 B20 B21 B22 B23 B24 B25 B26

Factors (controllable input variables)

Responses

Feed concentration

Feed temperature

Feed ﬂow rate

Feed pressure

Rejection

Flux

Performance index

x1

C (g/L)

x2

T (◦ C)

x3

Q (L/h)

x4

P (MPa)

RE (%)

J × 10−5 (kg/m2 s)

Y × 10−5 (kg/m2 s)

+1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1.41 −1.41 0 0 0 0 0 0 0 0

42.5 17.5 42.5 17.5 42.5 17.5 42.5 17.5 42.5 17.5 42.5 17.5 42.5 17.5 42.5 17.5 47.63 12.38 30 30 30 30 30 30 30 30

+1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 +1 −1 −1 0 0 +1.41 −1.41 0 0 0 0 0 0

37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 37.5 37.5 22.5 22.5 30 30 40.58 19.43 30 30 30 30 30 30

+1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 −1 −1 −1 0 0 0 0 +1.41 −1.41 0 0 0 0

212.5 212.5 212.5 212.5 137.5 137.5 137.5 137.5 212.5 212.5 212.5 212.5 137.5 137.5 137.5 137.5 175 175 175 175 227.9 122.1 175 175 175 175

+1 +1 +1 +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 +1.41 −1.41 0 0

1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 1.0 1.0 1.0 1.0 1.0 1.0 1.35 0.65 1.0 1.0

5.040 35.343 4.187 25.867 5.586 31.531 4.574 24.498 1.769 8.056 1.718 7.732 1.831 8.627 1.529 6.424 2.516 32.161 8.101 4.547 5.442 4.714 10.140 2.163 6.555 6.348

21.230 62.282 12.688 43.224 23.658 56.796 13.554 40.683 13.177 21.890 7.571 17.819 12.659 23.023 6.741 15.296 15.484 62.579 23.763 13.297 16.003 14.887 23.332 9.307 19.977 17.241

5.040 35.343 4.187 25.867 5.586 31.531 4.574 24.498 1.769 8.056 1.718 7.732 1.831 8.627 1.529 6.424 2.516 32.161 8.101 4.547 5.442 4.714 10.140 2.163 6.555 6.348

the coded variables was written as: Yˆ = (6.301 − 8.189x1 + 1.314x2 + 5.511x4 + 5.393x12 − 0.737x32 −1.051x1 x2 − 4.616x1 x4 + 0.968x2 x4 ) × 10−5

(11)

Subjected to: xi ∈ ˝; ˝ = {xi |−˛ ≤ xi ≤ + ˛}; ∀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 R2 is about 0.969, which is in agree2 found to ment with the adjusted coefﬁcient of determination Radj 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 empirical model has been developed in terms of the actual variables as follows:

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

Yˆ = (−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 ≤ 47.68 g/L; 19.39 ≤ T ≤ 40.61 ◦ C; 122 ≤ Q ≤ 228 L/h; 0.65 ≤ P ≤ 1.35 MPa. Figs. 7–9 show the 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

Model 8 Residual 17 Total 25

SS

MS −7

2.607 × 10 8.237 × 10−9 2.689 × 10−7

F-value P-value −8

3.258 × 10 4.846 × 10−10

67.241

R2

2 Radj

<0.0001 0.969 0.955

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 = 175 L/h and T = 30 ◦ C.

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M. Khayet et al. / Journal of Membrane Science 368 (2011) 202–214 Table 5 Additional set of experiments used for ANN modeling.

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 = 175 L/h and P = 1 MPa.

and the pressure leads to an increase of the RO performance index. However, the main effect of pressure is evidently higher than the main effect of feed temperature. The effect of feed ﬂow rate is the smallest one and it appears only as reduced quadratic effect. The increasing of the feed salt concentration minimizes considerably the RO performance index. That is low salt rejection factor and small permeates ﬂux. Regarding the interaction effects between factors, the most important interaction effect occurs between the operating 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 concentration. 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 between the salt concentration in feed solution and feed temperature diminishes the RO performance index when both factors are increased. In contrast, the interaction between the feed temperature 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 ﬂow rate 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 concentration, all data presented in Tables 1 and 3 as well as additional

Fig. 9. Response surface plot of the predicted RO performance index by RSM for high salt concentration conditions as function of the feed ﬂow rate and the temperature for C = 30 g/L and P = 1 MPa.

Run

C (g/L)

T (◦ C)

Q (L/h)

P (MPa)

RE (%)

J × 10−5 (kg/m2 s)

Y × 10−5 (kg/m2 s)

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10

49.22 3.27 26.25 26.25 26.25 26.25 26.25 26.25 26.25 26.25

30 30 40.6 19.4 30 30 30 30 30 30

175 175 175 175 228 122 175 175 175 175

1 1 1 1 1 1 1.35 6.46 1 1

20.636 96.886 38.824 43.045 42.562 42.148 54.762 26.148 41.795 41.880

13.386 400.179 26.443 15.331 19.859 20.455 33.389 11.609 21.394 21.940

2.762 387.717 10.266 6.599 8.452 8.621 18.284 3.036 8.942 9.188

experimental results given in Table 5 have been considered. A total of 66 experimental runs have been used to develop the ANN model for RO pilot plant. The inputs for the neural network were identical to the factors considered in RSM approach, namely, feed salt concentration, feed temperature, feed ﬂow rate and operating pressure. Similar to RSM modeling, the RO performance 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, which is 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 system are not of the same order of magnitude, some variables may appear more significantly than in reality are [37]. Moreover, one of the advantages of using normalization of inputs and outputs parameters is to avoid numerical overﬂows due to very large or very small 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 x1 (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 −˛ (minimum level) up to +˛ (maximum 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 (minimum level) to +1 (maximum level). 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–B26 and C1–C10 in Tables 1, 3 and 5, respectively) have been used to ﬁgure out the optimal architecture of ANN. These original data (62 samples) were divided into training, validation and test 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 samples were used. The split of data into training, validation and test subsets was carried out to estimate the performance of the neural network for prediction of “unseen” data that were not used for training. In this way, the generalization capability of ANN model can be assessed. The Neural Network Toolbox V4.0 of MATLAB mathematical 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 capability. The obtained optimal architecture (topology) of ANN model for this problem involves a feed-forward neural network with four inputs, two hidden layers (one layer with ﬁve neurons another

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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 perceptron, MLP (4:5:3:1), referring to the number of inputs and the number of neurons in the hidden and output layers, respectively. Fig. 10 shows the optimal architecture of the developed ANN model. Note that all neurons of the hidden layers have log-sigmoid transfer 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 from different layers 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 network have the bias b(l) where the superscript l indicates the layer 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 performance function (MSE). During the training phase, each neuron receives the input signals, aggregates them using 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 ≤ E0 , where the goal is E0 = 10−4 ). In the present case training was stopped after 10 iterations. Fig. 12 illustrates the training, validation and test mean squared errors. During the training step the performance functions MSE of the training and the test data subsets were lower than the goal E0 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 E0 . Therefore, the training process has been considered successfully terminated and the obtained optimal values of weights and biases are summarized in Table 6. Note that, these optimal values of connections (weights and biases) are related to coded inputs (factors) and normalized 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 architecture shown in Fig. 10 are given as matrixes and vectors in Table 6. Consequently, the ANN model for the prediction of RO performance 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) and the other terms involved deal with the aforementioned weights, biases and inputs.

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

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Table 6 Optimal values of weights and biases obtained during network training with LMA.

Source: inputs

2.2657 4.7068 (1,1) = −2.7115 IW 2.7380 −2.4246

Bias vector destination: HL-1

b

Layer weight matrix destination: HL-2

−2.6600 LW (2,1) = 0.4533 2.2031

Input weight matrix destination: HL-1

Source: HL-1

(1)

= −5.8165

(2)

−0.0461 0.8980 1.1273 −0.2777 −0.4668

= 1.2533

Bias vector destination: HL-2

b

Layer weight vector destination: OL-3 Source: HL-2

LW (3,2) = 0.0693

Bias scalar destination: OL-3

b(3) = − 0.7088

3.4903

−0.1209 −0.2102 −0.3068 0.0894 0.2784

−3.3778

2.7221 −3.8825 −2.2707

−1.5173 −0.7377 0.3553 −0.8035 0.2060

1.4972

−1.5649 2.9029 1.6865

−5.2410

−3.1806 −3.6157 4.5581

T

4.7213 5.5758 −4.0014

T

−2.0644

0.7018

3.5697

−0.3463

T

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 H is 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 = z1 (n + z2 + 1) + z2 (H + 1) + H

Fig. 12. Training, validation and test mean squared errors for the LMA (performance is 2.251 × 10−5 and goal is E0 = 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 freedom due to residual in the case of ANN model can be written as: DFresidual = N − L

(16)

where z1 and z2 mean the number of neurons within the ﬁrst and second hidden layers, respectively. Eqs. (15) and (16) are valid for feed-forward networks with neurons having biases. In our speciﬁc case, N = 62 and L = 47 [Eq. (16)] so that the degree of freedom due to residual is DFresidual = 15. In addition, ANOVA gives a very high F-value (1593.36) and a very low P-value (<10−4 ). The coefﬁcient of multiple determination is equal to unity (R2 = 1), which is perfect and the adjusted coefﬁcient is very 2 = 0.999). All these statistical estimators indicate close to unity (Radj an adequate ANN model with optimal architecture that can be used for predictive simulations of RO process within a wide range of feed salt concentration. The goodness-of-ﬁt between the experimental and the predicted RO performance index given by ANN is shown in Fig. 13. All points are located very near to the straight line indicating that ANN model prediction is excellent inside the valid region.

(14)

where N means the total number of experiments considered to develop the predictive model and L means the total number of connections (weights and biases) in the ANN model. For a feed-forward Table 7 Analysis of variance (ANOVA) for ANN model. Source Model Residual Total

DF 46 15 61

SS

MS −5

9.375 × 10 1.919 × 10−8 9.377 × 10−5

−6

2.038 × 10 1.279 × 10−9

F-value

P-value

R2

2 Radj

1593.36

<0.0001

1

0.999 Fig. 13. Predicted RO performance index by ANN model versus experimental values.

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

Fig. 14. RO performance index predicted by ANN model as function of the feed salt concentration and the pressure for Q = 175 L/h and T = 30 ◦ C.

A value of correlation coefﬁcient close to unity (r2 = 0.9998) shows the linear relationship between the experimental and predicted RO performance index. This result can be attributed to the good generalization 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 ANN architecture that ensures a positive degree of freedom (i.e. the total number of connections is smaller than total number of experiments used for developing the ANN model); (c) splitting the experimental data into 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 show the inﬂuence of the different 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 temperature 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 pressure and the feed temperature. Such interaction effects are similar 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. RO performance index predicted by ANN model as function of the feed salt concentration and the feed temperature for Q = 175 L/h and P = 1 MPa.

211

Fig. 16. RO performance index predicted by ANN model as function of the feed temperature and the pressure for C = 26.25 g/L and Q = 175 L/h.

As can be seen in Figs. 14 and 15, the effect of the salt concentration in the feed solution is signiﬁcant. An exponential increase of the RO performance index was observed with the decrease of the salt concentration below 15 g/L. This may be the reason why RSM failed 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 RSM contains 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 limited to an approximation of linear and quadratic effects only like RSM. Therefore, ANN demonstrated 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 (∼30 g/L) by RO process the salt rejection must be higher than 99.3% in order to make possible production of potable water from seawater in a practical single-stage RO plant. Concerning brackish water, its salinity is usually between 2 and 10 g/L. The World Health Organization (WHO) recommendation for salinity of potable water is about 0.5 g/L or lower, so that up to 90% of the salt must be removed from brackish feed solutions [28].

Fig. 17. RO performance index predicted by ANN model as function of the feed ﬂow rate and the temperature for C = 26.25 g/L and P = 1 MPa.

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Table 8 Optimal solutions for RO performance index given by RSM and ANN methods considering: (I) all factors as variables, (II) brackish water of ﬁxed concentration 6 g/L and (III) seawater of ﬁxed concentration 30 g/L. Method I II III

RSM ANN RSM ANN RSM ANN

C (g/L)

T (◦ C)

Q (L/h)

P (MPa)

Ypredicted ×10−5 (kg/m2 s)

Yexperimental ×10−5 (kg/m2 s)

3.97 3.30 6.00 6.00 30.00 30.00

40.6 40.6 40.3 40.3 39.9 26.1

175 228 145 170 180 225

1.35 1.35 1.35 1.34 1.35 1.34

677.2 663.5 492.1 555.8 17.5 12.5

709.5 764.1 687.3 735.6 35.2 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 concentration of 6 g/L (i.e. average salt concentration for brackish waters) was considered. The third optimization problem was solved for a ﬁxed salt concentration value of 30 g/L (i.e. a typical value of seawater). All objective functions given by RSM and ANN have been optimized by means of Monte Carlo simulation method based on pseudo random numbers (PRNs). The stochastic simulations were carried out using a multistage zooming-in approach to localize the optimal points inside the valid region more accurately. The optimal solutions found by RSM and ANN models together with the conﬁrmation runs (experimental validation of optimum) are summarized 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 concentration), one may conclude that both methods actually converged to quite similar solutions. In fact, both models provide identical optimal feed temperature (40.6 ◦ C) and optimal pressure (1.35 MPa). The optimum salt concentration in feed solution by RSM model is 3.97 g/L, while that obtained by ANN model is 3.30 g/L. The highest difference was obtained between the optimal values of feed ﬂow rate. However, the effect of this factor on the RO performance 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 RSM model was 735.98 × 10−5 kg/m2 s and the salt rejection factor was 96.4%. Under such optimum conditions the salt concentration in the permeate was about 0.143 g/L, lower than the imposed limit by WHO, 0.5 g/L. In a similar way, the measured permeate ﬂux under the optimum operating conditions given by ANN model was found to be 786.88 × 10−5 kg/m2 s and the salt rejection factor was 97.1%. The salt concentration in permeate was about 0.09 g/L, also lower than the permitted limit. It is worth mentioning that the optimal solutions obtained following both RSM and ANN models for the second optimization problem when the feed salt concentration was ﬁxed at 6 g/L, converged to an identical temperature (40.6 ◦ C) and to almost the same operating pressure (1.35 MPa for RSM and 1.34 MPa for ANN). Concerning the feed ﬂow rate, the optimal value of 145 L/h was obtained by RSM and 170 L/h by ANN. The experimental conﬁrmation runs revealed that the optimal conditions offered by ANN are better than those given by RSM (i.e. the experimental permeate ﬂuxes are as follows 735.8 × 10−5 kg/m2 s for ANN versus 687.4 × 10−5 kg/m2 s for RSM. Note that, the salt rejection efﬁciency in this case was about 99.98%. For seawater desalination conditions (third optimization problem) it seems that RSM provides a better optimal point than ANN. For instance, ANN indicates an optimal temperature of 26.1 ◦ C and an operating pressure of 1.34 MPa, while by RSM model a higher optimal temperature of 39.9 ◦ C and a pressure of 1.35 MPa were obtained. The experimental permeate ﬂuxes are as follows 51.9 × 10−5 kg/m2 s for RSM and 27.1 × 10−5 kg/m2 s for ANN with an average rejection factor of 67.83%.

Finally, it is worth to mention that both RSM and ANN models indicate that the global optimal operation conditions of the considered RO pilot plant are in the range of desalination of brackish waters. 5. Conclusions RSM and ANN methods were applied for modeling and optimization of desalination process by reverse osmosis (RO). RSM was unable to develop a global model to predict the RO performance 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 concentrations (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 performance 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 concentrations. ANN approach provides a global model describing RO performance of the pilot plant in a wide range of feed salt concentration. This is an advantage of ANN model over RSM model. Another advantage 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 permits to add new experimental data to build a trustable ANN model. 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 values as input variables: C = 3.30 g/L, T = 40.6 ◦ C, Q = 228 L/h and P = 1.35 MPa. Under such conditions of operation, a maximal RO performance index was achieved compared to all performed experiments. This is a performance index of the RO pilot plant of 764.1 × 10−5 kg/m2 s. Optimum operating conditions were also determined for typical brackish water and seawater with ﬁxed concentrations of 6 g/L and 30 g/L, respectively. The obtained optimum operating conditions, when the concentration was 6 g/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 30 g/L, the obtained optimum temperature and ﬂow rate of each model were different, whereas the optimum pressure of both ANN and RSM models was quite similar (i.e. 1.34–1.35 MPa). In this case, a higher RO performance was found for RSM model.

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Acknowledgements The authors of this work gratefully acknowledge the ﬁnancial support of the University Complutense of Madrid for granting Dr. C. Cojocaru “Estancias de Doctores y Tecnólogos en la Universidad Complutense, Convocatoria 2008” and UCM-BSCH (Project GR58/08, UCM Group 910336). M. Essalhi is thankful to the Middle East Desalination Research Centre for the grant (MEDRC 06-AS007).

Nomenclature b bias term for a node b bias vector for a layer C concentration of salt in feed solution (g/L) CP concentration of salt in permeate (g/L) DF degree of freedom E0 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 ﬂow rate r2 correlation coefﬁcient R2 coefﬁcient of multiple determination 2 Radj adjusted statistic coefﬁcient RE salt rejection factor S net input SS sum of 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 x1 , x2 , x3 , x4 coded levels of input variables Y vector of performance index Y response/target—performance index (experimental value) Yˆ predictor of response (performance index) by RSM Y predictor of response/target (performance index) by ANN number of neurons in hidden layers z1 , z2 Greek letters ˛ axial point or “star” point in CCD ˇ0 , ˇi , ˇii , ˇij regression coefﬁcients within response surface model

ˇOLS ϕ

vector of regression coefﬁcients learning rate response function valid region (region of experimentation)

Superscirpts l integer variable indicating the layer in ANN topology m integer variable (subscript) T transpose of matrix or vector

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