Multi-objective optimization of operating parameters

Desalination 381 (2016) 71–83

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Multi-objective optimization of operating parameters of a MSF-BR desalination plant using solver optimization tool of Matlab software Mongi Ben Ali ⁎, Lakhdar Kairouani Université de Tunis El Manar, Ecole Nationale d'Ingénieurs de Tunis, 05/UR/11-14 Unité de Recherche, Energétique et Environnement, Tunis Belvédère, BP 37, 1002, Tunisia

H I G H L I G H T S • • • •

An optimization of operating parameters of MSF-BR desalination plant is proposed. Objective functions of the optimization problem are obtained using the RSM. A solver tool based on genetic algorithms is used to solve the optimization problem. The optimization approach leads to obtaining a large set of Pareto optimal solutions.

a r t i c l e

i n f o

Article history: Received 1 August 2015 Received in revised form 27 November 2015 Accepted 28 November 2015 Available online 17 December 2015 Keywords: Multi-objective optimization Multistage flashing Response surface methodology Matlab optimization solver Genetic algorithms

a b s t r a c t Multi-Stage Flash (MSF) desalination process is energy intensive and it is, therefore, essential to search for operating the plant at its optimum parameters which lead to reduction of energy consumption and consequently lower water production cost. In this study, we used a solver optimization tool of Matlab software, for optimization of operating parameters of recirculation multi-stage flash (MSF-BR) desalting plant, taking in consideration the change of brine heater fouling factor and seasonal variation of seawater temperature. The solver uses genetic algorithms for solving multi-objective optimization problems. The operating variables over which optimization was carried out are the make-up flow rate, the cooling seawater flow rate, the brine recycle flow rate and the steam temperature. The optimization method and results analysis are based on actual plant data that includes 10 desalting units, each of 16 flashing stages and a nominal capacity of 26 700 m3/d. Three objectives were considered in this optimization approach. The first is to maximize the fresh water capacity of the installation. The second is to minimize the heating steam flow rate in order to reduce the thermal energy consumption, and the third is to minimize the sum of flow rates of main pumps of the unit production in order to reduce the electric energy consumption. The expressions of the first two objective functions are obtained using response surface methodology (RSM). Solving the optimization problem has led to obtaining a set of Pareto optimal solutions, which defining various combinations of the optimal operating parameters of each MSF-BR desalination plant unit, and thus leading to optimal plant operation policy for the whole year. © 2015 Elsevier B.V. All rights reserved.

1. Introduction About 40% of the world's population suffers from a shortage of fresh water and this trend is expected to increase in the future [1]. As more than 94% of the world's water is saline [2], desalination of seawater is becoming, in many parts of the world, the main source of freshwater. Multistage flash (MSF) desalination process (Fig. 1) is the largest sector in the thermic desalination industry. It accounts for more than 40% of the entire desalination market [3], and therefore, in some countries, it became the main source of freshwater for domestic, industrial, and agriculture consumption. ⁎ Corresponding author. E-mail address: [email protected] (M.B. Ali).

http://dx.doi.org/10.1016/j.desal.2015.11.030 0011-9164/© 2015 Elsevier B.V. All rights reserved.

The energy consumption in water desalination processes is one of the important parameters that define the unit cost of desalted water. Unfortunately, MSF desalination plant is energy intensive and it is, therefore, essential to search for operating the plant at its optimum parameters which lead to reduction of energy consumption and consequently lower water production cost. In this work, the optimization of operating variables of a real brine recycle multi-stage flash (MSF-BR) desalting plant (AL KHOBAR II, Saudi Arabia), taking in consideration the change of brine heater fouling factor and seasonal variation of seawater temperature, were considered. A rigorous process model, a Design of experiments (DoE) approach, and an efficient solver optimization tool (gamultiobj) of Matlab software using genetic algorithms were used to optimize the operation of the plant. Optimization involves finding the optimum values of the operating

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M.B. Ali, L. Kairouani / Desalination 381 (2016) 71–83

Fig. 1. Schema of brine recycle multistage flash desalination process (MSF-BR) [5].

variables in the steady-state phase and when conducting performance calculations. This operating variables, defined by Helal et al. in previous study [4], are the make-up flow rate, the cooling seawater flow rate, the brine recycle flow rate and the steam temperature. The optimality objectives for this study include the increase in capacity production of the plant by maximization of the total distillate flow rate, the reduction of thermic energy consumption by minimization of the heating steam flow rate, and the reduction of electric energy consumption by minimization of the sum of flow rates of plant's main pumps. The following sections include a brief description of the MSF-BR process, model outline, optimization results and conclusion. 2. The MSF-BR process A schematic diagram of the MSF-BR process is shown in Fig. 1. It consists essentially of a water/steam circuit forming the brine heater (also called heat input section), heat recovery section and heat rejection section. The recovery and rejection sections each consist of a series of stages connected in series. The seawater enters the plant and flows through the condenser tubes from the end of the rejection section to the left side of this section. Before entering the recovery section, most of the seawater is discharged into the sea to remove the surplus thermal energy from the plant. The other part, forming the make-up feed, is deaerated, chemically treated with a mixture of anti-scaling and then mixed with recycled brine and fed in the last stage of the recovery section. In this section, the combined stream is preheated in the condenser units by absorbing the latent heat of the distillate vapor. The preheated seawater is further heated

in the brine heater by the steam where it reaches its top brine temperature. At this point the brine enters the first flash chamber through an orifice, becomes superheated, then a part of the brine is flashed into vapor and the remaining brine flows into the second stage. This flashing process continues from one stage to the next. Vapor released in each stage passes through a wire demister to remove any entrained brine droplets, and on the tube bundle of the condenser where it is condensed and drips into the distillate tray where it is collected. At the end of the last rejection stage the concentrated brine is partly discharged to the sea and the remaining is recycled as mentioned before. Each stage of an MSF evaporator (Fig. 2) consists of: • The tube bundles of the condenser to condense the vapor released in the stage. • The demister to reject brine droplets and prevent them from contaminating the condensate. • The distillate tray to collect the distillate water. • Inlet/outlet brine orifices and a weir box to control flashing brine level. • An extraction pipe leading to ejectors to remove non-condensable gases in order to maintain vacuum. • A large brine pool. 3. Plant description The plant under consideration in this study is located at Al-Khobar in Saudi Arabia. It uses a cross-tube arrangement with recirculating brine, and consists of 10 MSF-BR desalting units and their common facilities.

Fig. 2. Schema of the j-th stage of a MSF-BR plant.

M.B. Ali, L. Kairouani / Desalination 381 (2016) 71–83

Each of the 10 units has a nominal capacity of 26,700 m3/day of fresh water. The configuration investigated in this work refers to the case study reported by Rosso et al. [5], Helal et al. [4], and Hawaidi and Mujtaba [1]. The specifications and constant parameters, which are used in this work, are shown in Table 1. Seasonal variation of seawater temperature (Tcw) and brine heater fouling factor (fbh) was given by Hawaidi and Mujtaba [1] and is shown in Table 2. Note, December is assumed to be a overhauling period. 4. Optimization of operating conditions 4.1. Statement of the problem The optimization problem to be addressed in this paper is described as follows: For each month, Given:

The design data of the MSF-BR plant, Table 1, the temperature of the feed seawater (Tcw), and the brine heater fooling factor (fbh), Table 2. Optimize: Heating steam temperature (Ths), recycled brine flow rate (MR), rejected seawater flow rate (Mcw), and make-up seawater flow rate (Mf). So as to

Maximize: Capacity production of the plant in fresh water ( f1). Minimize: Thermal energy consumption ( f2), electric energy consumption ( f3). Subject to: Inequality constraints such as bounds on optimization variables. The optimization problem consists of three objectives ( f1, f2, f3). Operationally, it is desired to maximize the capacity production of the plant in fresh water ( f1) and minimize, steam supplies energy to the brine heater of the plant (f2), and electrical energy used by pumps ( f3). The pumping power requirements for the brine recirculating pump and the seawater pump are assumed to constitute the bulk of the pumping power requirement [6]. The power requirement for these two pumps depends on the rejected seawater flow rate (Mcw), the recycle flow rate (MR) and the make-up seawater flow rate (Mf). Table 1 Design information for Al-Khobar II distillation unit [4]. Heat input section (brine heater) - Number of tubes 3800 - Tube size 22.0 mm (inside diameter) × 1219 mm (thickness) × 12.2 m (length) - Heat transfer area 3530 m2 - Tube material Cu–Ni (70–30) - Fouling factor 0.160 m2K/kW Heat recovery section - Number of stages - Stage width - Number of tubes - Tube size -

Heat transfer area Tube material Height of brine level Fouling factor

Heat rejection section - Number of stages - Stage width - Number of tubes - Tube size -

Heat transfer area Tube material Height of brine level Fouling factor

13, stages 1–13 12.2 m 4300 22.0 mm (inside diameter) × 1219 mm (thickness) × 12.2 m (length) 3995 m2 Cu–Ni (90–10) 0.457 m 0.120 m2K/kW

3, stages 14–16 10.7 m 3800 24.0 mm (inside diameter) × 0769 mm (thickness) × 10.7 m (length) 3530 m2 Titanium 0.457 m 0.020 m2K/kW

73

Table 2 Seawater temperature and brine heater fouling factor. Month

Tcw (°C)

fbh (m2K/kW)

January February March April May June July August September October November

15 17 20 25 28 30 32 35 33 30 25

0.065 0.078 0.093 0.108 0.121 0.135 0.150 0.164 0.178 0.192 0.206

The optimal solution space is reduced by adding inequality constraints. Upper and lower bounds are imposed on the operating variables according to the operational considerations. In this regard, Ths U cannot be raised above an upper value (Ths ) due to scaling problems essentially in tubes of the brine heater. This upper value depends on the type of chemical products used for make-up seawater treatment. L ) should also be imposed on Ths, because too much A lower bound (Ths reduction of Ths causes necessarily a reduction of the top brine temperature (Tb0) which can cause the pressure difference between the ejector and the vapor zone in the stage to become insufficient, which in turn causes an incomplete extraction of non-condensable gases, followed by instability due to the impossibility of maintaining the vacuum and possible vapor-side corrosion problems. Similarly, limits must be imposed on Mcw, MR and Mf. A lower limits (MLcw , MLR , MLf ) must be fixed to avoid scaling problems caused by a low velocity of brine in the tubes of the brine heater or the condensers. In addition, if (Mf + MR) is low, sealing between flash chambers may not be fulfilled. As a result, the operation of the plant will be unstable and operation will be inefficient. On the other hand, a high value of (Mf + MR) can cause distillate contamination because flooding can occur. Once the distillate is contaminated, a long time is needed to reduce salinity. The upper limits U U (MU cw , MR , Mf ) are also imposed to avoid erosion of brine heater and condensers tubes. Literature recommends that the brine velocity inside tubes in the heat recovery, heat rejection and heat input section should lie between 1.5 and 3 m/s [6]. Table 3 shows the bounds of the operating variables used in this of optimization problem and which meet the requirements we have cited. The optimization problem can thus be described mathematically by: Min Ths ; Mcw ; MR ; M f

F mois ¼

8 < f 1 ðThs ; Mcw ; MR ; M f Þ ¼ −M d ðThs ; Mcw ; MR ; M f Þ f ðT ; M ; M ; M f Þ ¼ Mhs ðThs ; Mcw ; MR ; M f Þ : 2 hs cw R f 3 ðMcw ; MR ; M f Þ ¼ Mcw þMR þM f

U ð93Þ T Lhs ≤ Ths ð CÞ ≤ T hs ð115Þ U ð3000 Þ ð1500ÞMLcw ≤ Mcw ðkg=sÞ ≤ Mcw

ð1500ÞM LR ≤ MR ðkg=sÞ ≤ M UR ð3000Þ ð1500ÞMLf ≤ M f ðkg=sÞ ≤ MUf ð3000Þ

Md is the total distillate flow rate. Mhs is the heating steam flow rate. Ths is the heating steam temperature. Mcw is the rejected seawater flow rate, MR is the recycle brine flow rate and Mf is make-up seawater flow rate. In order to solve the above optimization problem, we must previously determine the expressions of objective functions f1(Ths, Mcw, MR, Mf)

Table 3 Bounds of the operating variables. Operating variables

Upper limit

Lower limit

Ths (°C) Mcw (kg/s) MR (kg/s) Mf (kg/s)

93 1500 1500 1500

115 3000 3000 3000

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and f2(Ths, Mcw, MR, Mf). To achieve this, we used the Design of Experiments (DoE) module of StatGraphics software, and a simulation program developed by Ben Ali and Kairouani in a previous study [7] that enables the simulation of steady state behavior of MSF-BR process. 4.2. MSF-BR process model A prerequisite to solving the optimization problems is the availability of a steady-state model of the plant considered. With reference to Figs. 1–2, the steady state model Equations [7] are given in Fig. 3. The model constituted of a set of mass and energy balances plus heat transfer equations, and is based on the following assumptions: • Steady state operation, which is the industry standard. • The thermal and physical properties for feed seawater, brine and distillate product are functions of temperature and salt concentration.

• The latent heat of formed/condensed vapor depends on temperature. • The overall heat transfer coefficients in the condensers depend on the following parameters: - Flow rate of the condensing vapor. - Flow rate of the brine inside the condenser tubes. - Temperatures of the condensing vapor and the brine. - Physical proprieties of the condensing vapor and the brine, which includes thermal conductivity, viscosity, density, and specific heat. - The tube material, diameter, and wall thickness. - The fouling resistance. • Thermodynamic losses include the boiling point elevation (BPE) and the non-equilibrium allowance (NEA). • Distillate product is salt free. • No sub-cooling of condensate leaving the brine heater. • Heat losses to the surroundings are negligible. • The effect of non-condensable gases on heat transfer is negligible.

Fig. 3. MSF process model [5–8]

M.B. Ali, L. Kairouani / Desalination 381 (2016) 71–83 Table 4 The experimental/simulation factors and levels. Levels

Table 5 Central Composite Face Centered Design of 4 factors.

Factors

−1 0 1

Ths (°C): x1

Mcw (kg/s): x2

MR (kg/s): x3

Mf (kg/s): x4

93 104 115

1500 2250 3000

1500 2250 3000

1500 2250 3000

4.3. Determination expressions of objective functions f1 and f2 The determination of the objective functions f1 and f2, as a function of operating variables (Ths, Mcw, MR, Mf) of the installation, was made by applying the response surface methodology (RSM) using the Design of Experiments (DoE) module of StatGraphics software. 4.3.1. Response surface methodology Response Surface Methodology (RSM) is a powerful statistical analysis technique which is well suited to modeling complex multivariate processes. It has been employed widely in the field of engineering and manufacture [9–12] where several parameters are involved in a process. It is used to search an adequate functional relationship between a response variable, y, and a number of associated input variables denoted by x1, x2, …, xk. In general, such relationship is unknown but can be approximated by a polynomial model. In most cases a second-degree polynomial model, with the following form, is used [10]. k

k

i¼1

i¼1

k

j1

y ¼ β0 þ ∑ βi xi þ ∑ βii x2i þ ∑ ∑ βij xi x j : j¼2 i¼1

ð1Þ

This second order model includes linear terms, cross product terms and a second order term for each of the x's. The response surface methodology uses the least square method to determine the regression coefficient values β0, β1…βk, β11…βkk, β12… β(k − 1)k. An important aspect of RSM is the Design of experiments (DoE). It was developed for the model fitting of physical or numerical experiments. The objective of DoE is the choosing of the points where the response should be evaluated. The choice of the DoE can have a large influence on the accuracy of the approximation. A second-order model can be constructed efficiently with Central Composite Designs (CCD) [10]. It consists of 3 parts: factorial points, center points, and axial points. The number of experiments (N) in this case is given by the following equation N ¼ 2k þ 2k þ n0

ð2Þ

n0 is the number of repetitions in the center of the experimental domain, dedicated to the statistical analysis. Details of experimental designs for fitting response surfaces are found in Khuri and Cornell [9] and Montgomery [11]. 4.3.2. Design of RSM using StatGraphics software In our study, the different factors such as heating steam temperature (Ths), rejected seawater flow rate (Mcw), the recycle brine flow rate (MR) and make-up seawater flow rate (Mf) were chosen as the effect variables and expressed as x1, x2, x3 and x4, respectively. The total distillate flow rate (Md) and the heating steam flow rate (Mhs) were chosen as the responses and expressed as y1 and y2, respectively. As variables have different units and variations in the experimental domain, the regression analysis should not be performed. Thus we must first normalize the variables before performing a regression analysis. In general we used (Eq. 3) to calculate the coded level [12]. 

Xi ¼

 L

xi  xUi þ xi =2  U  xi  xLi =2

75

ð3Þ

No. exp

Independent variables, coded levels X1

X2

X3

X4

y1

y2

1 2 3 4 5 6 7 8 9 10 11 12 13

0 1 1 0 0 1 −1 1 −1 −1 0 1 0

−1 −1 1 0 1 −1 −1 1 −1 1 0 −1 0

0 −1 1 1 0 −1 1 −1 −1 1 0 1 0

0 1 1 0 0 −1 −1 −1 1 −1 0 1 1

…. …. …. …. …. …. …. …. …. …. …. …. ….

…. …. …. …. …. …. …. …. …. …. …. …. ….

Responses

No. exp

Independent variables, coded levels X1

X2

X3

X4

y1

y2

14 15 16 17 18 19 20 21 22 23 24 25 26

1 0 −1 1 −1 −1 0 −1 −1 1 0 −1 1

1 0 −1 1 1 1 0 −1 0 0 0 1 −1

−1 0 1 1 −1 1 −1 −1 0 0 0 −1 1

1 −1 1 −1 1 1 0 −1 0 0 0 −1 −1

…. …. …. …. …. …. …. …. …. …. …. …. ….

…. …. …. …. …. …. …. …. …. …. …. …. ….

Responses

L where xU i and xi are respectively, the upper and lower bounds of the variable xi. In this case, each of the coded variables is forced to range from −1 to 1, so that they all affect the responses more evenly, and so the units of the parameters are irrelevant. In our study, the code factors of the above four independent variables were expressed as X1, X2, X3 and X4, respectively. The code level of each variable was designated as − 1, 0 and 1, respectively (Table 4). To search a correlation for y1 and y2, simulations of 4 factors and 3 levels were performed according to a Central Composite Face Centered Design (Table 5). StatGraphics, which is a highly specified multivariate statistical analysis software, was used in this study, for experimental design of RSM, regression analysis of the data and estimation of the determination coefficient (R2adjusted) of the models. To complete the experience matrix (Table 5) by the values of the responses, we used the simulation program developed by Ben Ali and Kairouani [7] for solving the steady state model equations of Fig. 3. For each month, 26 simulations were performed. The following correlations, given by StatGraphics software, were built by regression analysis using least-squares method:

Md−January ¼ y1 ¼ 543; 13 þ 66; 6698X1 þ 8; 35183X2 þ 70; 6683X3 þ82; 9723X4 −2; 0977X22 −3; 9687X23 −2; 9127X24 þ0; 74556X1 X2 þ 9; 34456X1 X3 þ 11; 3849X1 X4 þ3; 29644X2 X3 −3; 34219X3 X4 ð4Þ MhsJanuary ¼ y2 ¼ 81; 827 þ 8; 29317X1 þ 2; 36006X2 þ 16; 3087X3 þ 17; 6169X4 þ 0; 485444X24 þ 0; 2045X1 X2 þ 1; 83513X1 X3 þ 1; 989X1 X4 þ 0; 766125X2 X3 þ 0; 44025X2 X4 þ 0; 9335125X3 X4 ð5Þ MdFebruary ¼ y1 ¼ 529; 085 þ 68; 4833X1 þ 8; 16133X2 þ 67; 7918X3 þ 79; 6161X4  2; 05741X22  4; 04541X23  3; 03341X24 þ 0; 77162X1 X2 þ 9; 2212X1 X3 þ 11; 2276X1 X4 þ 3; 18087X2 X3  3; 6891X3 X4

ð6Þ

MhsFebruary ¼ y2 ¼ 79; 4676 þ 8; 303287X1 þ 2; 25661X2 þ 15; 7678X3 þ 16; 9859X4 þ 0; 455264X24 þ 0; 199937X1 X2 þ 1; 82969X1 X3 þ 1; 95744X1 X4 þ 0; 739312X2 X3 þ 0; 44812X2 X4 þ 0; 852812X3 X4

ð7Þ

MdMarch ¼ y1 ¼ 509; 242 þ 68; 3548X1 þ 7; 95172X2 þ 64; 1205X3 þ 75; 3718X4  1; 97718X22  3; 9638X23  2; 9908X24 þ 0; 77906X1 X2 þ 9; 11469X1 X3 þ 11; 0247X1 X4 þ 3; 07069X2 X3  3; 98144X3 X4 ð8Þ

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MhsMarch ¼ y2 ¼ 76; 2051 þ 8; 3165X1 þ 2; 11111X2 þ 15; 0492X3 þ 16; 1382X4 þ 0; 390819X24 þ 0; 204563X1 X2 þ 1; 81831X1 X3 þ 1; 93231X1 X4 þ 0; 694937X2 X3 þ 0; 384188X2 X4 þ 0; 750688X3 X4 ð9Þ

MdAugust ¼ y1 ¼ 409; 378 þ 67; 7239X1 þ 6; 57389X2 þ 47; 345X3 þ 56; 0449X4  4; 53904X23  3; 72204X24 þ 0; 836875X1 X2 þ 8; 53775X1 X3 þ 10; 1156X1 X4 þ 2; 51562X2 X3  4; 66813X3 X4 ð18Þ

MdApril ¼ y1 ¼ 478; 553 þ 68; 3965X1 þ 7; 61194X2 þ 59; 1559X3 þ 69; 5943X4  1; 9612X22  3; 9257X23  3; 1272X24 þ 0; 81237X1 X2 þ 9; 03612X1 X3 þ 10; 7974X1 X4 þ 2; 9432X2 X3  4; 17825X3 X4 ð10Þ

MhsAugust ¼ y2 ¼ 61; 0392 þ 8; 23261X1 þ 1; 53744X2 þ 11; 6356X3 þ 12; 3048X4  0; 281229X22 þ 0; 204125X1 X2 þ 1; 75475X1 X3 þ 1; 79887X1 X4 þ 0; 497375X2 X3 þ 0; 26225X2 X4 þ 0; 344625X3 X4

MhsApril ¼ y2 ¼ 71; 1991 þ 8; 35217X1 þ 1; 91661X2 þ 14; 0013X3 þ 14; 9173X4 þ 0; 338431X24 þ 0; 20375X1 X2 þ 1; 80462X1 X3 þ 1; 9145X1 X4 þ 0; 627X2 X3 þ 0; 353625X2 X4 þ 0; 6415X3 X4 ð11Þ MdMay ¼ y1 ¼ 458; 871 þ 68; 3067X1 þ 7; 36089X2 þ 55; 847X3 þ 65; 7815X4  1; 80535X22  3; 93935X23  3; 15285X24 þ 0; 813812X1 X2 þ 8; 91144X1 X3 þ 10; 6237X1 X4 þ 2; 83619X2 X3  4; 31469X3 X4

ð12Þ

ð14Þ

MhsJune ¼ y2 ¼ 66; 2329 þ 8; 35489X1 þ 1; 75906X2 þ 12; 7624X3 þ 13; 5341X4 þ 0; 261563X1 X2 þ 1; 73469X1 X3 þ 1; 80931X1 X4 þ 0; 508437X2 X3 þ 0; 243562X2 X4 þ 0; 538937X X ð15Þ 3

MdSeptember ¼ y1 ¼ 416; 861 þ 67; 1209X1 þ 6; 52561X2 þ 47; 4533X3 þ 56; 2715X4  3; 97837X23  3; 19587X24 þ 0; 810625X1 X2 þ 8; 378X1 X3 þ 9; 95925X1 X4 þ 2; 4865X2 X3  4; 96538X3 X4 ð20Þ

MhsMay ¼ y2 ¼ 68; 1719 þ 8; 34339X1 þ 1; 79278X2 þ 13; 3232X3 þ 14; 1493X4 þ 0; 2855669X24 þ 0; 20925X1 X2 þ 1; 80675X1 X3 þ 1; 891X1 X4 þ 0; 588375X2 X3 þ 0; 314625X2 X4 þ 0; 556875X3 X4 ð13Þ MdJune ¼ y1 ¼ 444; 286 þ 68; 0897X1 þ 7; 1208X2 þ 53; 1737X3 þ 62; 7284X4  1; 7957X22  3; 99023X23  3; 1077X24 þ 0; 81543X1 X2 þ 8; 79219X1 X3 þ 10; 4513X1 X4 þ 2; 7308X2 X3  4; 4808X3 X4

ð19Þ

4

MdJuly ¼ y1 ¼ 429; 117 þ 67; 4268X1 þ 6; 46089X2 þ 50; 9144X3 þ 59; 231X4  4; 39358X23  3; 54608X24 þ 1; 30737X1 X2 þ 8; 2015X1 X3 þ 10; 7614X1 X4 þ 2; 162X2 X3  5; 09375X3 X4 ð16Þ MhsJuly ¼ y2 ¼ 64; 0637 þ 8; 21633X1 þ 1; 58456X2 þ 12; 3314X3 þ 12; 9705X4 þ 0; 252562X1 X2 þ 1; 71131X1 X3 þ 1; 87581X1 X4 þ 0; 479063X2 X3 þ 0; 322563X2 X4 þ 0; 361813X3 X4 ð17Þ

MhsSeptember ¼ y2 ¼ 62; 3217 þ 8; 15511X1 þ 1; 53744X2 þ 11; 7911X3 þ 12; 4815X4  0; 2861X22 þ 0; 197813X1 X2 þ 1; 72669X1 X3 þ 1; 77456X1 X4 þ 0; 502437X2 X3 þ 0; 261813X2 X4 þ 0; 314437X3 X4

ð21Þ

MdOctober ¼ y1 ¼ 429; 047 þ 66; 4334X1 þ 6; 51761X2 þ 48; 1202X3 þ 57; 1679X4  4; 1774X23  3; 4359X24 þ 0; 79475X1 X2 þ 8; 21113X1 X3 þ 9; 82237X1 X4 þ 2; 47575X2 X3  5; 30238X3 X4 ð22Þ MhsOctober ¼ y2 ¼ 64; 4428 þ 8; 06761X1 þ 1; 62544X2 þ 12; 0998X3 þ 12; 83X4  0; 294957X22 þ 0; 189188X1 X2 þ 1; 69994X1 X3 þ 1; 74669X1 X4 þ 0; 511312X2 X3 þ 0; 269563X2 X4 þ 0; 288813X3 X4

ð23Þ

MdNovember ¼ y1 ¼ 450; 861 þ 65; 661X1 þ 6; 5392X2 þ 49; 8299X3 þ 59; 288X4  1; 6093X22  4; 4638X23  3; 7833X24 þ 0; 733375X1 X2 þ 8; 0776X1 X3 þ 9; 7291X1 X4 þ 2; 49137X2 X3  5; 79088X3 X4 ð24Þ MhsNovember ¼ y2 ¼ 68; 2966 þ 7; 93456X1 þ 1; 74417X2 þ 12; 7156X3 þ 13; 5403X4  0; 312448X22 þ 0; 184938X1 X2 þ 1; 65956X1 X3 þ 1; 71019X1 X4 þ 0; 530563X2 X3 þ 0; 278437X2 X4 þ 0; 281313X3 X4

Fig. 4. Adequacy graphs of Md and Mhs in the case of January.

ð25Þ

M.B. Ali, L. Kairouani / Desalination 381 (2016) 71–83

77

Fig. 5. Principle of the algorithm used by the solver gamultiobj.

4.3.3. Results Statistical analysis of the above models were tested by the Fisher's Ftest for analysis of variance and the determination coefficient (R2ajusted). R2ajusted was between 0.9997 and 0.9999, for all models. This value reflects a very good fit between the simulation and predicted responses, and it was considered then reasonable to use the regression models to analyze trends in the responses. The F-ratio of regression model gave a low probability value (p-value b 0.0001) which means that the models obtained are well precise for predicting the Md and Mhs. Analysis using models adequacy graphs, confirmed the good quality of the obtained models. Fig. 4 shows, as example, the adequacy graphs of Md and Mhs in the case of January. The objective functions f1 and f2 were obtained by substituting in Eqs. (4)–(25) the coded variables Xi by the natural variables xi using the following equation: xi ¼

   U  X i xUi  xLi x þ xLi þ i 2 2

ð26Þ

4.4. Resolution of the optimization problem Having defined the components f1 and f2 of the objective function, it was possible to optimize this function that is to adjust the operating variables of the plant in order to ensure optimal process functioning. The optimization problem can be described mathematically as follows

Min

F mois

93 ≤ x1 ≤ 115 1500 ≤ x2 ≤ 3000 1500 ≤ x3 ≤ 3000 1500 ≤ x4 ≤ 3000

8 < f 1 ðx1 ; x2 ; x3 ; x4 Þ ¼ f ðx ; x ; x ; x Þ : 2 1 2 3 4 f 3 ðx2 ; x3 ; x4 Þ ¼ x2 þ x3 þ x4

This is a non-linear multi-objective optimization problem with inequality constraints. These constraints, in fact, define the operating limit of the decision variables. Genetic algorithms (GA) are well suited to solve multi-objective optimization problems. They are well-known and credible algorithms that have been used in many applications and their performances were tested in many studies. Several papers [13,14–22] have been published on using evolutionary algorithms for solving multi-objective optimization problems. The above problem is classified in this category, therefore, we chose a solver optimization tool of Matlab software (gamultiobj) using GA, for solving this problem. 4.4.1. Genetic algorithms Genetic algorithms (GA) belong to the larger class of evolutionary algorithms (EA), which engender solutions to optimization problems

using techniques inspired by biological evolution [23]. Genetic algorithms are used in many fields such as, pharmacology, signal and image processing, engineering and robotics, economics, manufacturing, mathematical finance, chemistry, physics, and other fields.

Table 6 Results of the resolution of the optimization problem in the case of January. Operating variables

Components of objective function

Ths (°C)

Mcw (kg/s)

MR (kg/s)

Mf (kg/s)

Md (kg/s)

Mhs (kg/s)

(Mcw + MR + Mf) (kg/s)

93 114.9 114.9 93 112.9 93 115 115 115 115 111.9 115 112.9 112.9 112.9 114.9 112.9 115 112.9 115 115 112.9 115 115 115 115 115 93 115 115 115 115 115 115 115 93 115 115 115 115 112.9 115

1500 2921.6 2921.6 2921.6 1750.6 2921.6 2921.6 2404.9 1500 1500 1500 2404.9 2404.9 1500 1500 2921.7 1500 1500 2404.9 2921.6 2921.6 1500 2404.9 1750.6 2404.9 2921.6 2404.9 1500 2921.6 2921.6 2404.9 2921.6 1750.6 2404.9 2921.6 1500 2921.6 2404.9 2404.9 2921.6 1500 2921.6

1500 2931.3 2370.4 1500 1774.2 1500 2370.4 2932.5 1500 2370.4 1500 2932.6 1500 1774.2 1500 2932.6 2370.4 1500 1774.2 1500 1500 1500 2370.4 2370.4 1774.2 2932.6 1500 1774.2 1500 2370.4 1500 1500 2932.6 1500 2932.6 1500 2932.5 2370.4 1500 2370.4 1500 1774.2

1500 2671.2 2671.2 1779.5 1500 1500 1500 2671.3 2210.1 2671.2 1500 2926.9 1500 1779.5 2671.2 2671.2 2210.1 1779.5 2210.1 2926.9 1500 1500 2926.9 2671.2 1779.5 2926.9 2671.3 1500 2671.2 2926.9 2210.1 2926.9 2926.9 1779.5 2926.9 1500 2926.9 1500 1500 2671.2 2210.0 2926.9

326.9 737.9 680.5 365.3 443.1 335.5 532.7 732.0 512.6 661.8 404.5 761.7 417.5 476.5 559.8 738.0 593.0 455.5 542.1 615.1 428.7 408.6 706.7 667 497.1 768.5 580.8 351.7 583.3 711.8 521.7 615.1 750.4 464.5 768.6 326.9 768.5 527.6 426.3 680.5 502.0 646.7

43.9 121.3 106.8 51.0 58.0 45.7 75.8 118.9 69.2 101.2 51.2 126.2 53.2 63.9 79.1 121.3 87.7 58.9 76.6 91.2 55.0 51.6 111.9 102.3 67.2 128.7 83.0 48.6 84.4 114.0 71.3 91.2 123.0 60.7 128.8 43.9 128.7 74.2 54.0 106.8 68.0 98.4

4500 8524.2 7963.2 6201.2 5024.8 5921.6 6792.1 8008.7 5210.2 6541.7 4500.0 8264.4 5404.9 5053.8 5671.2 8525.5 6080.6 4779.6 6389.2 7348.6 5921.7 4500 7702.2 6792.2 5958.6 8781.2 6576.2 4774.2 7092.9 8219.0 6115.0 7348.6 7610.1 5684.4 8781.2 4500 8781.1 6275.3 5405.0 7963.2 5210.2 7622.8

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Fig. 6. Graphical results of the resolution of the optimization problem.

GA apply an iterative, stochastic search strategy to find an optimal solution. The evolution usually starts from a population of individuals often generated randomly. An individual consists of the values of the decision variables (here: process variables) [24]. In each generation, the fitness of every individual in the population is evaluated; the fitness is usually the value of the objective function in the optimization problem being solved. The better individuals are stochastically selected from the current population. Next, recombination and mutation operators are applied to these individuals producing the offspring's, and forming a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm. In general, the algorithm terminates when a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population [25]. The first multi-objective GA, called Vector Evaluated Genetic Algorithms (VEGA), was proposed by Schaffer in 1984 [26]. Later, a number

of different multi-objective evolutionary algorithms were suggested to solve multi-objective optimization problems, such as, Niched Pareto Genetic Algorithm [27], Random Weighted Genetic Algorithm (RWGA) [28], Non-dominated Sorting Genetic Algorithm (NSGA) [29], Strength Pareto Evolutionary Algorithm (SPEA) [30], Pareto-Archived Evolution Strategy (PAES) [31], and Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) [32]. 4.4.2. Using gamultiobj to solve the multi-objective optimization problem The solver gamultiobj, used in this study to solve the multi-objective optimization problem, uses a controlled elitist genetic algorithm which a variant of NSGA II [33]. An elitist genetic algorithm always favors individuals with better fitness value. But a controlled elitist genetic algorithm also favors individuals that can help increase the diversity of the population even if they have a lower fitness value. Diversity is

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maintained by controlling the elite members of the population as the algorithm progresses. Preserving diversity is found to be important for convergence to an optimal Pareto front [34]. The solution of the multi-objective optimization problem obtained by using gamultiobj is a family of points known as the Pareto-optimal front. Each point belonging to this front is optimal in the sense that no improvement can be achieved in one component of the objective function that does not lead to degradation in at least one of the remaining components.

79

Fig. 5 shows a schematic description of the tth generation of the algorithm used by the solver gamultiobj. Firstly, a combined population Rt = Pt ∪ Qt, of size 2N, is formed. This union ensures elitism. Then, the population Rt is sorted according to the concept of Pareto dominance. Now, solutions belonging set F1 are of best solutions in Rt and must be emphasized more than any other solution in the combined population. If the size of F1 is smaller than N, we choose all individuals of the set F1 for the new population Pt + 1. The remaining members of the population Pt + 1 are chosen from the other fronts in the order of their ranking.

Fig. 7. Graphical results of the resolution of the optimization problem.

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Thus, solutions from the set F2 are chosen next, followed by solutions from the set F3, and so on. This procedure is continued until no more sets can be added. Suppose that the set Ff is the last non-dominated set beyond which no other set can be added. In general, the number of solutions in all sets from F1 to Ff would be larger than the population size N. To choose exactly N population members, we sort the solutions of the last front Ff using the crowded-comparison operator bn in descending order and choose the best solutions needed to fill all population slots. Control elitism afterward, is ensured by the use of both options Pareto fraction and Distance function. Pareto fraction limits the number of individuals to maintain on the Pareto front (elite members), and Distance function helps to maintain diversity on a Pareto front favoring individuals that are relatively far to the front. The rest of

individuals belonging to the population Pt + 1 are selected from higher fronts. The new population Pt + 1 of size N is now used for selection, crossover, and mutation to create a new population Qt + 1 of size N. The process continues, generation to generation, until a stopping criterion is validated. All optimization runs are controlled by a Matlab program. The parameters used, by the solver gamultiobj to solve the multi-objective optimization problems were set to the following values. -

Population size: 60. Using crowding distance sorting. Selection function: Tournament selection. Pareto fraction = 0.7.

Fig. 8. Variation of the plant's production capacity (Md) and the heating steam flow rate (Mhs).

M.B. Ali, L. Kairouani / Desalination 381 (2016) 71–83

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Fig. 9. Variation of the plant's production capacity as a function of the sea water temperature and the brine heating fooling factor, (Ths = 93 °C, Mcw = MR = Mf = 1500 kg/s).

- Crossover operator: scattered. - Crossover fraction = 0.8 - Criteria for stopping the resolution algorithm: • Maximum number of iteration for the genetic algorithm to perform = 800. • The algorithm stops if the average relative change in the best fitness function value is ≤10−4. 4.4.3. Results and discussion The optimization problem was solved for the 11 months (January to November) of MSF-BR plant operation, i.e., for each seawater temperature (Tcw) and brine heater fouling factor (fbh). In all cases, the stopping of the algorithm was obtained after the satisfaction of the second criteria, while the number of generations has not exceeded 250. The most important result of solving the optimization problem was the obtaining of a wide range of solutions from which we can choose the operating point which suits us. This led to a seasonal optimal operation policy for the whole year. Table 6 shows, as example, the values of the operating variables and the associated values of the objective function components, obtained during the resolution of the problem in the case of January. In Figs. 6, 7 the results for all months are represented graphically. It shows the points constituting the optimal Pareto front describing the best solutions of the problem. It is important to notice that the obtained values represent the best local optima. There is no guarantee, however, that these optimum solutions are global optimum [29]. An investigation of optimization results, reveals that the flow rate of heating steam increases as the fresh water production increases (Fig. 8). When the operating variables were constant (Ths = 93 °C, Mcw = MR = Mf = 1500 kg/s), the seawater temperature exhibits the opposite trend,

i.e., the fresh water production drops as the seawater temperature increases (Fig. 9). This figure shows also that the effect of the brine heater fouling coefficient (fbh), which is increasing from January, is less marked on the fresh water plant capacity. Indeed, despite the 90.7% increase in brine heater fouling factor (fbh) of April to November (the seawater temperature and the operating parameters are identical), the fresh water production declined only 2.2%. The same was found for the months of June and October. It was noted that for different sea water temperatures, the higher plant capacity corresponds to a higher flow rate, of the heating steam and the principal pumps (Table 7). The same trend was observed in the case of the minimum freshwater production (Table 7). There is also a possibility to maintain production capacity of the plant constant throughout the year in spite of the change of the sea water temperature and the increase in fouling factor of the brine heater tubes. This is achieved by adjusting operating parameters of the installation (Ths, Mf, Mcw, MR) to adequate and optimal values chosen in the results table. Table 8 shows 3 columns corresponding to current operating state of the plant [5] (column 1), the nearest point belonging to optimization results (column 2), and the difference between the two results (column 3). The same is shown graphically in Fig. 7(e). Two reasons could explain the difference found. The first assumes that the current operation of the plant does not correspond to the optimum operating condition. The second assumes that the current point of operation corresponds to an optimum state which has not been obtained by the resolution method. This assumption is most plausible, because, firstly Fig. 7(e) shows that the current operating point is located on the optimal Pareto front, and secondly we know that optimization method using evolutionary algorithms cannot find all optimal solutions. 5. Conclusion

Table 7 Plant parameters in the case of minimum and maximum capacity. Minimum plant capacity

Maximum plant capacity

Month

Mhs (Mcw + MR + Mf) Md Mhs (Mcw + MR + Mf) Md (kg/s) (kg/s) (kg/s) (kg/s) (kg/s) (kg/s)

January February March April May Jun. July August September October November

326.9 325.8 306.8 284.2 271.5 262.6 254.5 242.5 248.9 259.2 277.9

43.9 42.6 40.7 37.9 36.2 34.9 33.9 31.9 32.8 34.2 36.7

4500 4500 4500 4500 4500 4500 4500 4500 4500 4500 4500

768.6 757.2 730.4 674.3 654.6 625.0 602.7 583.1 570.1 600.9 633.4

128.8 125.2 121.6 110.8 108.0 102.5 98.6 95.2 92.9 99.4 106.5

8781.2 8786.0 8833.9 8666.2 8767.1 8678.0 8525.2 8708.4 8400.5 8590.3 8844.1

In this study, we used two powerful commercial tools, i.e. Statgraphics and Matlab for optimization of operating parameters of an actual Table 8 Comparison of results. Parameters

Current operating state [5]

Point belonging to optimization results

Difference %

Ths (°C) MR (kg/s) Mf (kg/s) Mcw (kg/s) Md (kg/s) Mhs (kg/s) Mcw + MR + Mf (kg/s)

97 1763.9 1577.8 1561.1 259.4 37.47 4902.8

98.6 1549.6 1591.8 1708.1 279.0 36.5 4849.5

1.62 13.83 0.88 8.60 7.02 2.66 1.10

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desalting plant using the recirculation multi-stage flash (MSF-BR) process. This plant includes 13 flashing stages in heat recovery section and 3 flashing stages in heat rejection section. The plant has a nominal capacity of 26,700 m3/d. For this optimization problem, we took in consideration the change of brine heater fouling factor and seasonal variation of seawater temperature, and we considered three objectives. The first is to maximize the fresh water capacity of the plant. The second is to minimize the flow rate of heating steam in order to reduce the thermal energy consumption, and the third is to minimize the sum of flow rates of main pumps of the plant in order to reduce the electric energy consumption. The expressions of the first two objective functions were obtained using the response surface methodology (RSM) implemented in Statgraphics software. Afterward, an optimization procedure using a solver tool (gamultiobj) based on genetic algorithms was successfully applied to solve the multi-objective optimization problem. The optimization approach leads to obtaining a large set of Pareto optimal solutions, which defining various combinations of the optimal operating parameters of a MSF-BR desalination plant. The predictions of the optimization method were compared to actual measurements. The optimization results show that the solver gamultiobj is very successful in predicting MSF plant optimal operating state, since the point defining the actual measurements is located on the optimal Pareto front. The study concluded that the optimization approach developed in this study could lead to optimal plant operation policy for the whole year. Nomenclature A Heat transfer area, m2 Bd Blow-down mass flow rate, kg/s B Flashing brine mass flow rate leaving the stage, kg/s BPE Boiling point elevation, °C Cp Specific heat at constant pressure, kJ/kg K D Distillate formed in each flashing stage, kg/s f Fouling factor, m2K/kW Components objective function fi LMTD Logarithmic mean temperature difference, °C M Mass flow rate, kg/s N Number of experiments n Number of flashing stages of the heat recovery section NEA Non-equilibrium allowance, °C Thermal resistance of the scale on the inside of the condenser rfi tubes, m2K/W Thermal resistance of the scale on the inside surface brine rbh heater tubes, m2K/W T Temperature, °C Top brine temperature, °C Tb0 Temperature of the mixture feed seawater, brine recycle, °C Tf0 U Overall heat transfer coefficient, W/m2°K X Water salinity, ppm Coded variables Xi Decision variables xi Salinity of the mixture feed seawater, brine recycle, ppm Xb0 Greek βi λ

Regression coefficient value Latent heat, kJ/kg

Subscripts b c cw d f

Brine Recovery section Cooling water Distillate product Feed stream

h hs j n r R v

Brine heater Heating steam Stage index Last stage of the heat recovery section Rejection section Brine recycle Vapor

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