Global optimization of MSF seawater desalination processes

Desalination 394 (2016) 30–43

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Desalination journal homepage: www.elsevier.com/locate/desal

Global optimization of MSF seawater desalination processes Chandra Sekhar Bandi, R. Uppaluri ⁎, Amit Kumar Department of Chemical Engineering, Indian Institute of Technology Guwahati, North Guwahati, Assam 781039, India

H I G H L I G H T S • For MSF-BR and MSF-M processes, DE provided global optimal solutions. • Cost based MSF process ranking is MSF-BR > MSF-M > MSF-OT* (* refers solution with penalty). • For important MSF process parameters, obtained solutions improved by 2.31%, 3.9%, 2.92%, 20.24%, 3.53% and 5.2%.

a r t i c l e

i n f o

Article history: Received 2 February 2016 Received in revised form 5 March 2016 Accepted 6 April 2016 Available online 12 May 2016 Keywords: Differential evolution algorithm Global optimization Seawater desalination Multi stage flash (MSF) MSF-BR MSF-OT MSF-M SQP Modeling Optimization

a b s t r a c t This article addresses the global optimal design of multi-stage flash desalination processes. The mathematical formulation accounts for non-linear programming (NLP) based process models that are supplemented with the non-deterministic optimization algorithm. MSF-once through, -simple mixture (MSF-M) and -brine recycle (MSF-BR) process configurations have been evaluated for their optimality. While freshwater production cost has been set as the objective function for minimization, mass, energy and enthalpy balances with relevant supplementary equations constitute the equality constraints. Differential evolution algorithm (DE/rand/bin) was adopted to evaluate the global optimal solutions. Further, obtained solutions have been compared with those obtained with MATLAB optimization toolbox solvers such as SQP and MS-SQP. The global optimal solution corresponds to a variable value set of [2794.4 m3/h, 1.0499, 7.62 m, 3.359 kW/m2 ∙K, 3.297 kW/m2 ∙K, 3.042 kW/m2 ∙K and 22] for decision variables [WM, RH, LT, UB, UR, Uj, NR] in the MSF-BR process to yield an optimal freshwater production cost of 1.0785 $/m3. Compared to the literature, the obtained global solution from DE is 2.31% better. Further, inequality constraint resolution has been excellent for DE but not other methods such as MS-SQP, SQP and DE-SQP. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Among several technologies viable for potable water production, the desalination of sea and brackish water is an established technology in several countries including the USA, Persian Gulf and European countries [15,35]. Based on the working energy principle, desalination processes are further classified primarily into two classes namely thermal processes that involve phase change due to addition of heat and membrane processes that involve pressure energy. While thermal processes are primarily classified into multi-effect evaporation (MEE), MSF and vapor-compression (VC) processes, membrane processes are primarily classified into RO and electrodialysis (ED) processes. Among various alternate technologies for sea water desalination, MSF processes have the promising features of

⁎ Corresponding author. E-mail address: [email protected] (R. Uppaluri).

http://dx.doi.org/10.1016/j.desal.2016.04.012 0011-9164/© 2016 Elsevier B.V. All rights reserved.

large scale operation and ability to deliver good quality potable water (5–50 ppm total dissolved solids). A typical MSF process involves brine heating followed with flash distillation in multiple stages and subsequent heat recovery. Thereby, a MSF process plant has three important sections namely brine heater, heat rejection and heat recovery sections. Design variations in the MSF process systems refer to either once through (OT) or simple mixer (M) or brine recirculation (BR) process configurations to yield MSF-OT or MSF-M or MSF-BR processes respectively. Among these, while MSF-OT is the simplest in design, it is not as efficient as the MSF-BR system. The design of efficient MSF processes invariably requires simulation and optimization studies. Several researchers have conducted simulation studies to obtain insights upon the process performance of MSF processes. These have been contributed by Mandil and Abdel Ghafour [19], Helal et al. [2], Al-Mutaz and Soliman [14], Rossol et al. [26], Thomas et al. [29], Abdel-Jabbar et al. [28], Hawaidi and Mujtaba [6], and Tayyebi and Alishiri [34]. Many of these literatures emphasized upon

C.S. Bandi et al. / Desalination 394 (2016) 30–43

stage-to-stage calculations and deployed Newton–Raphson method or tridiagonal matrix (TDM) formulations solved with Thomas algorithm (TA) for evaluation of MSF process performance. Further, optimization studies have also been conducted by several researchers. These include MSF-OT processes [3–5,21–23]; MSF-M processes [20] and MSF-BR processes [1,3–5,6,7,8,9,16,18,24,25,31,32,33]. Considering the minimization of water production cost as objective function, the literature refer to the deployment of either one of the following methods: genetic algorithm (GA) [24,32]; sequential quadratic programming (SQP) method [22,23], deterministic optimization methods built in gPROMS [6,25], generalized reduced gradient (GRG) [1,3–5,20] and in DICOPT++ [22,23]. Further, MATLAB programming environment has also been used in several engineering applications as a competent modeling tool for simulation and optimization studies [36,37,38,39]. A critical analysis of the available literatures in optimization studies refers to the following. Firstly, earlier research works mostly addressed either MSF-BR or MSF-M or MSF-OT processes for process optimization based insights. Only [3–5] addressed MSF-OT and MSF-BR process optimization but not the MSF-M process. The authors adopted GRG optimization method which is a local optimization tool. It is well known that GRG might provide local solutions whose quality could not be judged in conjunction with the global optimality. Further, GRG is well known to be non-rigorous and fails to solve problems with larger number of inequality constraints, as the method needs the satisfaction of all inequality constraints in each iteration. While SQP method foregoes such limitation, the SQP also could not provide insights upon the quality of generated optimal solutions. On the other hand, non-deterministic models such as GA were only investigated for the MSF-BR but not MSF-OT and MSF-M processes. Thus, it is apparent that global optimization methods have not been applied till date for the comparative assessment of MSF-BR, MSF-M and MSF-OT processes. Secondly, a critical issue with respect to alternate optimization methods such as GRG, SQP, and GA, is with respect to the satisfaction of inequality constraints. The traditional approach to couple a penalty function with cost function may or may not yield feasible solutions using GRG and SQP methods, given the fact that these algorithms may require additional fine tuning of optimization algorithm parameters such as maximum number of iterations, maximum function evaluations, and penalty parameters, to obtain feasible solutions. Thus, it might be the case that an engineer may have to spend a significant amount of time in fine tuning these parameters for the deterministic optimization methods. On the other hand, such insights may not be applicable for the non-deterministic optimization methods due to random nature of solution search. Therefore, an important issue that also needs to be addressed is the ability to fetch feasible solutions with similar penalty function parameters for both deterministic and nondeterministic optimization methods. A third and essential insight is to visualize upon the sensitivity of process and operating parameters using global optimization approaches. While such sensitivity analysis might be possible with local optimization methods, they may not provide the most stringent sensitivity analysis. Therefore, the sensitivity analysis conducted with nondeterministic methods needs to be judged with that conducted with deterministic methods. In summary, this work addresses three major objectives. The first objective refers to comparative assessment of MSF-M, MSF-OT and MSF-BR processes using non-deterministic optimization. The second objective refers to the evaluation of inequality constraint resolution ability for both deterministic and non-deterministic methods. The final objective is to conduct sensitivity analysis of all MSF processes in the light of global and local optimization. Differential evolution (DE) has been chosen as the global optimization tool as it has not been studied for MSF process optimization despite being proven effective for other engineering optimization problems. Thereby, suitable benchmarks are expected to be set for the engineering optimization of MSF processes.

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2. Process configurations A schematic representation of the MSF-OT, MSF-M and MSF-BR process configurations is presented in Fig. 1(a)–(c). Among these processes, while MSF-OT limits the temperature of the last stage to 30–40 °C for winter and summer operations, the flashing operation on several flash stages requires vacuum pressure conditions to achieve operating temperatures below 100 °C. As indicated in the figure, the common features of these process configurations are briefly summarized as follows: – The feed seawater (WMF) at temperature TSea, is de-aerated and chemically treated before being introduced into the condenser/preheater tubes of the last flashing stage in the heat recovery section. – The preheated feed seawater at temperature T2 enters the brine heater tubes, where the heating steam (WS) is condensed on the outside surface of the tubes. Eventually, the seawater reaches the maximum design temperature value also known as the top brine temperature (T3). – The feed seawater finally enters the flashing stages, where a small amount of fresh water vapor is generated by brine flashing in each stage. In each stage, the flashed off vapor condenses on the outside surface of the condenser tubes, where the feed seawater (WMF) flows inside the tubes from the cold to the hot side of the plant. Thereby, the heat recovery process enables an increase in the feed seawater temperature. The condensed fresh water vapor outside the condenser tubes accumulates across the stages and forms the distillate product stream (WMD).

Fig. 1(b) illustrates that the MSF-M process essentially consists of a brine heater, heat recovery section and brine recycle mixing tank. Hence, the MSF-M process configuration facilitates a brine recycle stream to reduce fresh seawater requirements and associated chemical pretreatment costs. This is achieved by mixing part of the blowdown brine stream (WMR) with the feed stream (WMSC), thereby generating a mixed stream (WMF) with higher salinity than that of the fresh seawater (set as 70,000 ppm for the upper bound according to El-Dessouky et al. [11]. It can be further observed in Fig. 1(c) that the MSF-BR desalination plant has heat rejection, recovery section and brine heater section. The final reject stream from the heat recovery section is being split into two streams which serve as cooling seawater stream (WMCW) and makeup stream (WM). The makeup stream is further chemically treated and mixed in the brine pool of the last flashing stage in the heat rejection section. The mixed stream is sent to blowdown splitter S2 from which the brine recycle stream (WMR) is introduced into the condenser tubes of the last stage in the heat recovery section. The stream after absorbing the latent heat of condensation from flashing vapor in several stages leaves the last stage and enters the brine heater, where its temperature is enhanced to saturation temperature (i.e., top brine temperature) at the prevalent system pressure. 3. Methodology Process optimization of alternate MSF configurations has been targeted by coding a competent simulation model that is supplemented with a non-deterministic optimization algorithm. For comparison purposes, deterministic optimization algorithms have also been considered to evaluate upon the efficacy of the non-deterministic optimization algorithm. The following sub-sections summarize the simulation and optimization models. 3.1. Simulation model The simulation models for MSF-OT and MSF-BR processes were adopted from Helal et al. [3]. For the MSF-M process, the simulation

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model was taken from Abduljawad and Ezzeghni [20]. Other than these literatures, a comprehensive account of the models for MSF processes is not apparent elsewhere in other literature. The process simulation model accounts for the mass and energy balance equations applied for each stage including the brine heater, recovery and rejection sections. The process models are non-linear in nature and consist only algebraic expressions. Further details of the process models can be obtained from the cited literature and this section briefly outlines various important features of the same. As outlined in the literatures, the following assumptions are usually applicable for the process models on a theoretical basis [3,10]:

d) The specific heat capacity of brine solution is a weak function of salt concentration. e) Distillate product is salt free; non-condensable gases have negligible effect on the heat transfer process. f) Effect of the boiling point rise and non-equilibrium losses on the stage energy balances is negligible. g) The average specific heat capacity of brine solution is equal to that of the distillate. h) The boiling point rise at the exit from the last recovery for MSF is negligible. i) Heat loss is negligible.

a) Temperature profiles of all streams flowing within the plant are linear. b) Each section has a constant value for heat transfer coefficient, heat transfer area, boiling point rise and specific heat capacity (CP) of brine solution. c) The latent heat of vaporization of water (λ) is constant, and independent of temperature.

The non-linear system of algebraic equations involving mass and energy balances has been deduced by carrying out overall mass balances across blowdown splitter, rejected sea water splitter (for the MSF-BR process), salt balances across mixer, energy balances across brine heater and condenser, enthalpy balances on the heat recovery section, heat

Fig. 1. Schematic of (a) once-through MSF (MSF-OT), (b) brine-mixing MSF (MSF-M), and (c) brine recycle MSF (MSF-BR) seawater desalination processes.

C.S. Bandi et al. / Desalination 394 (2016) 30–43

33

Fig. 1 (continued).

rejection section (MSF-BR process), brine heater, mixers and flashing brine. Supplementary expressions for the process model include expressions for distillate product temperature (equilibrium correlation), overall heat transfer coefficients in various process sections, inside film resistance, average tube side brine temperature and thermal resistance. The thermal resistance is accounted as a function of steam-side condensing film, steam-side fouling, tube metal and brine side fouling resistance components. Table 1(a) summarizes various parametric and design specifications required for the process simulation model for cross flow type MSF processes. Table 1(b) presents the process cost parameter data, which is adopted from Helal et al. [3]. For various process models, the independent simulation model variables that need to be specified are (WMF, LT, UB, UR and NR) for MSF-OT; (LT , UB , UR , WMSC , CMSMF and NR) for MSFM and (WM , LT , UB , UR , NR , RH and Uj) for the MSF-BR process. Henceforth, these variables are treated as independent (decision) variables during process optimization. 3.2. Optimization model The MSF process simulation model consisting of non-linear set of algebraic equations is formulated as a non-linear programming (NLP) optimization model defined as: Min OF ¼ ψ Subjected to f ðxÞ ¼ 0 gðxÞ≤0

where ψ refers to the objective function defined as the sum of annualized freshwater production cost and associated penalties. The process simulation model is specified as f(x) = 0 for a specific MSF processes and g(x) refers to the set of inequality constraints. The total annualized freshwater production cost function is evaluated using relevant expressions for direct capital less intake investment (CDCLIC), intake-outfall cost (CIC) (evaluated as a sum of costs of electrochemical equipment, civil work, electro chlorination, brine disposal cost and annual plant intake-outfall), direct capital investment cost (CDCC), indirect capital investment (CICC) and operating and maintenance cost (COMC) (evaluated as a sum of costs of steam, chemical treatment, power, labor and spares costs). Relevant expressions have been adopted from Helal et al. [3]. Thereby, the objective function for MSF is modified and expressed as:

ψMSF

C IC þ C DCLIC þ C ICC þC OMC ¼  3 m WY y

 
$ y

þ ðpenalty of gðxÞÞ:

ð1Þ

Inequality constraints refer to lower bound and upper bound specifications for makeup flow rate for MSF-BR (WM), feed flow rate for MSFOT (WMF) and MSF-M (WMSC), tube length (LT), number of recovery stages (NR), heat transfer coefficients in various sections, brine loading, brine velocities in various sections, rejected brine concentration values (CMBD, Clast) and absolute values of various heat transfer coefficients. The inequality constraint parameters and values have been presented in Table 2(a–b). The penalty function has been evaluated using large

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Table 1 (a) Parametric and design specifications data for cross flow type MSF processes. (b) A summary of MSF process cost parameter data. (a) Variable a CMF CPR Cpj Cp ID OD Nj T1 T3 T6 TD TS TSea WMD αR αj λ ρB ρW

Table 2 (a) Summary of lower and upper bound values for the optimal design of MSF processes. (b) A summary of inequality constraint parameters for MSF optimization model. (a)

Unit – ppm kJ/kg∙K kJ/kg∙K kJ/kg∙K m m – °C °C °C °C °C °C m3/h °C °C J/kg kg/m3 kg/m3

MSF-OT 0.88 42,000 4.18513 4.17658 4.18513 0.02199 0.024069 – – 110 – 30 114 25 1122 1.4 – 2346.3 1060 1000

MSF-M 0.88 42,000 4.18513 4.17658 4.18513 0.02199 0.024069 – – 110 – 30 114 25 1122 1.4 – 2346.3 1060 1000

MSF-BR 0.88 42,000 4.18513 4.17658 4.18513 0.02199 0.024069 3 34 110 28 30 114 25 1122 1.4 1.78 2346.3 1060 1000

(b) Unit

Value

Current work capacities MSF plant section

m3/d

26,928

Reference capacities and flow rates Ref. feed rate for intake calculation Ref. blowdown rate for outfall calculations

m3/d m3/d

1000 750

MSF plant cost data Capital recovery factor Parameter Φ in fixed capital cost term Chemicals Energy (pumping) Spares Labor

3

$/m prod $/m3 prod $/m3 prod $/m3 prod

0.0963 5500 0.024 0.03 0.082 0.1

MSF-BR Variable

Unit

WM WMF WMSC RH LT NR UB Uj UR

m3/h m3/h m3/h – m – kW/m2 ∙K kW/m2 ∙K kW/m2 ∙K

MSF-OT

MSF-M

Lower

Upper

Lower

Upper

Lower

Upper

2000 – – 0.8 7 18 2.7 2.7 2.7

3000 – – 2 10 30 3.7 3.7 3.7

– 5500 – – 7 15 2.7 – 2.7

– 10,000 – – 15 50 3.7 – 3.7

– – 5500 – 7 15 2.7 – 2.7

– – 10,500 – 15 50 3.7 – 3.7

(b) Constraints

Unit

Lower

Upper

Brine loading VB Vj VR CMBD or Clast ABS(1 − UBcal / UB) ABS(1 − Ujcal / Uj) ABS(1 − URcal / UR)

m3/h m width m/s m/s m/s ppm – – –

900 0.9144 0.9144 0.9144 – – – –

1200 1.8288 1.8288 1.8288 80,000 0.001 0.001 0.001

parametric settings that were defined for the initialization, propagation and termination strategies of various optimization methods for the MSF optimization problem. Further, it shall be noted that for several methods including SQP, MS-SQP and DE-SQP, initial vector values (independent variables) had to be defined for MSF-OT, MSF-BR and MSF-M processes. Table 3(b) summarizes the initial vector values set for the optimization studies. All simulations were conducted in MATLAB programming environment. SQP has been implemented from MATLAB optimization toolbox [27,36]. 4. Model validation

positive penalty parameters that are set to realize the satisfaction of all inequality constraints. 3.3. Optimization algorithm Differential evolution (DE) algorithm has been applied for the nondeterministic optimization of NLP process models. Introduced by Storn and Price [30], the DE optimization algorithm is a stochastic population based direct search optimization method that essentially involves the generation of new candidate solutions by combining the parent individual and several other individuals of the same population. This is facilitated by adding the weighted difference between any two population vectors to a third population vector [13]. Further, the parent vector is replaced with the mutant vector only when the mutant vector provides a better fitness value [17]. Thus, DE is an effective, fast, simple, robust, and inherently parallel technique that has few control parameters and needs less effort to tune and adopt optimization parameters. The DE has been applied as a sequence of mutation, cross-over and selection operations for all populations. The maximum number of permitted generations has been set as the termination criteria. The efficacy of the DE algorithm has been evaluated by comparing the results obtained from sequential quadratic programming (SQP) method with and without multi-start (MS) approach. Further, hybrid optimization approach involving the combination of both DE and SQP was also considered. For this case, the solution generated from DE is being provided as an initial guess value for the SQP method. Thus, the optimization methodology involves the application of either one of DE, SQP, MS-SQP and DE-SQP methods. Table 3(a) summarizes the

Model validation precisely refers to the validation of developed code for simulation and optimization models. Since appropriate data has been available only for the MSF-BR system, the simulation model code validation has been carried out using the data provided by Helal et al. [3]. Based on the input simulation variable data set of [2790 m3/h (2,790,000 kg/h), 1.0129, 18, 3.26 kW/m2 ∙ K, 3.443 kW/m2 ∙ K, 2.864 kW/m2 ∙K, 7.62 m] for [WM, RH, NR, UB, UR, Uj, and LT], the obtained results of dependent variables are [101.7 °C, 32.41 °C, 70,178 ppm, 135.5 m3/h (135,446 kg/h), 3662.8 m2, 2515.5 m2, 3355.4 m2, 9040 m3/h (9,040,000 kg/h)] for variables [T2, TBD, CMBD, WS, AB, AR, Aj, and WMF] respectively. Precisely, the same values have been reported in the literature [3] and hence the model code validation is inferred to be successful. The DE algorithm code has been tested for standard optimization model test function such as Rosenbrock Banana Function. For the algorithm parameters [F, CR, NG, and NP] specified as [0.8, 0.8, 100 and 100], the DE provided optimal solution of [1.000, 1.000] with a standard deviation of 10−6. The obtained optimal solution is in complete agreement with that available as the global optimal solution for Rosenbrook (Banana) function [12]. Hence, the DE algorithm is inferred to be effective for the optimization of alternate MSF process configurations. 5. Results and discussions 5.1. Efficacy of the DE algorithm Fig. 2 depicts the comparative performance of various optimization methods to obtain optimal solutions for the MSF-M, MSF-OT and MSF-

C.S. Bandi et al. / Desalination 394 (2016) 30–43 Table 3 (a) Parameters for various optimization techniques. (b) Initial independent variable set data for MSF processes and alternative optimization methods. (a) Parameter

DE

SQP

MS-SQP

NP/no. of starting points NG max/function evaluations DE-step-size, F CR Termination criteria

350 400 500 1000 700 1000 0.9 – – 0.9 – – NG max (DE) or 10−6 (SQP)

DE-SQP 500 1000 0.9 0.9

(b) Optimization MSF-BR method [WM, RH, NR, UB, UR, Uj, LT]

MSF-M [WMSC, NR, UB, UR, LT]

MSF-OT [WMF, NR, UB, UR, LT]

SQP

[6168, 21, 3.4599, 3.5266, 7.4, 99.21, 69,944] [6014, 23, 3.1432, 3.5221, 8.12, 103.23, 69,021] [6546, 22, 3.3565, 3.4526, 8.85, 101.45, 69,584] [6059, 25, 3.2452, 3.4411, 7.85, 102.56, 69,885] [6356, 25, 3.5232, 3.6221, 8.25, 101.23, 69,981] [5857, 25, 3.7, 3.6719, 9.74, 101.48,70,772]

[9543, 26, 3.4517, 3.6275, 8.12] [9561, 28, 3.5454, 3.5514.7.65] [9756, 27, 3.4624, 3.6454.7.69] [9628, 26, 3.3515, 3.5554, 7.18] [9643, 28, 3.3217, 3.3275, 7.98] [9500, 23, 3.4108, 3.7398, 8.62]

[2449, 1.02, 19, 3.5124, 3.1937, 2.9223, 8.51] [2767, 1.17, 18, 3.1673, 3.4619, 3.4454, 8.21] [2439, 1.16,21, 3.3351, 3.2680, 3.5171, 7.12]

MS-SQP

DE-SQP

[2566, 1.03, 23, 3. 3513, 3.4259, 3.6637.7.51] [2787, 1.07, 21, 3.4673, 3.4119, 3.4454, 7.61] [2790, 1.05, 22, 3.3624, 3.3003, 3.0448, 7.62]

BR plant configurations. It has been evaluated that the optimal fresh water production cost using DE, SQP, MS-SQP and DE-SQP methods are 1.2251, 1.2856, 1.2785 and 1.2534 $/m3 for MSF-OT, 1.198, 1.2445, 1.22 and 1.2135 $/m3 for MSF-M and 1.0785, 1.1, 1.0852 and 1.0843 for MSF-BR respectively. Hence, it can be concluded that the DE provides the lowest fresh water production cost for MSF-BR. Based on the obtained results, the optimal cost based ranking is MSF-BR b MSF-M b MSFOT*. For the MSF-BR plant, the efficacy of various methods for the optimization is in the order of DE N SQP N MS-SQP* N DE-SQP*. The solution with asterisk (*) corresponds to solution with penalty (and hence infeasible solution). An infeasible solution for the DE-SQP is due to rounding of solutions generated from DE, which were then supplied to SQP as initial guess. A summary of the important independent and dependent variable values obtained after optimization of MSF-BR, MSF-M and MSF-OT

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using various methods is presented in Tables 4, 5 and 6 respectively. In these methods, optimization methods with asterisk indicate the existence of penalties in the final solutions obtained and hence these solutions cannot be recommended as appropriate design solutions for the respective MSF process configurations. For several cases, UB inequality constraint could not be resolved without penalty. Further, the literature data for MSF-BR has also been presented in Table 4 to reflect upon the comparative optimality of obtained variable values with those presented in the literature. As outlined in Table 4, in comparison with the available literature data for the MSF-BR process, the optimal solution obtained with DE refers to marginal combinations of higher feed flow rate (2794.4 but not 2790.0 m3/h), higher specific heat ratio (1.0499 but not 1.0001), lower tube length (7.62 m but not 7.92 m), higher overall heat transfer coefficient in brine heater section (3.359 kW/m2 ∙K but not 3.260 kW/m2 ∙K), lower overall heat transfer coefficient in recovery section (3.297 kW/m2 ∙ K but not 3.443 kW/ m2 ∙ K), higher overall heat transfer coefficient in brine heater (3.042 kW/m2 ∙K but not 2.864 kW/m2 ∙K), higher number of recovery stages (22 but not 18), lower steam flow rate (130.365 m3/h but not 135.446 m3/h), lower feed flow rate, cooling water and cost. It is further interesting to note that the optimal concentration of rejected stream leaving the heat rejection section is precisely the same for this case and literature data. Compared to the literature reported optimal cost of 1.104 $/m3 , the DE generated solutions for the MSF-BR process to be 2.31% lower (1.0785 $/m3), MSF-M process to be 7.85% higher (1.198 $/m3) and MSF-OT process to be 9.88% higher (1.2251 $/m3). No relevant data was available in the same literature to compare the solutions obtained for the MSF-M and MSF-OT process configurations. It can be observed that marginal improvement in solutions can be obtained for the MSF-BR plant configuration in comparison with the literature using the DE algorithm. For the MSF-BR process configurations, solutions with penalty were obtained using the DE-SQP method. For the MSF-M process model, only DE provided solutions without penalty. Further, all optimization methods can be observed to provide solutions with penalty for the MSF-OT configurations. This conveys the efficacy of the DE algorithm to obtain solutions without penalty (feasible solutions) for both MSF-BR and MSF-M configurations. Hence, it is important to envisage that DE has potential to obtain high quality solutions even for the MSF-M process. The best set of optimal decision variable values are [WM, RH, LT, UB, UR, Uj and NR] for [2794.4 m3/h, 1.0499, 7.62 m, 3.359 kW/m2 ∙K, 3.297 kW/m2 ∙K, 3.042 kW/m2 ∙K and 22]. For the MSF-BR, the optimal variable values corresponded to production cost, thermal performance, specific heat transfer area and plant recovery of 1.0785 $/m3, 8.61, 0.009 m2/h/kg and 0.1287 respectively. These corresponded to an improvement of about 2.31, 3.90, 2.92 and 20.24% with respect to those reported in the literature [3] (1.104 $/m3, 8.283, 0.008487 m2/h/kg and 0.107 respectively). For the same case, the optimal feed flow rate and cooling water flow rate were evaluated as 8720 m3/h and 5930 m3/h respectively, which correspond to a reduction of about 3.53% and 5.2% with respect to the best known optimal values in the literature (9040 m3/h and 6250 m3/h respectively). These results indicate marginal improvement in the optimal solutions obtained and thereby convey the competence and efficacy of the DE algorithm for desalination process design and analysis.

5.2. Optimality of other dependent variables

Fig. 2. Bar chart depicting the performance of optimization methods for MSF desalination processes.

Based on thermodynamics, heat transfer and process economics based insights, several dependent variables have been defined to indicate upon the optimality of various MSF processes. These refer to specific heat transfer area, specific feed flow rate, specific cooling water rate and overall plant recovery. Definitions and formulae to evaluate the same are presented as follows:

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Table 4 Optimal independent and dependent variable values for the MSF-BR process and alternate optimization methods.

Independent variables

Dependent variables

Variable

Unit

DE

SQP

MS-SQP*

DE-SQP*

Literature [3]

WM RH LT UB UR Uj NR T2 TBD CMBD WS AB AR Aj WMF WMCW WMBD AT Cost

m3/h – m kW/m2 ∙K kW/m2 ∙K kW/m2 ∙K – °C °C ppm m3/h m2 m2 m2 m3/h m3/h m3/h m2 $/m3

2794.4 1.0499 7.62 3.359 3.297 3.042 22 102.00 32.35 70,178 130.365 3470.5 3297.1 3044.7 8721.0 5926.6 1672.4 85140.8 1.0785

2809.8 1.0001 7.92 3.362 3.161 3.114 19 102.03 32.38 67,092 139.860 3165.7 3137.9 2992.8 8732.6 6257.5 1353.1 74902.1 1.1000

2617.8 1.0339 7.98 3.7 3.294 3.687 20 102.03 32.34 67,093 129.900 3304.4 2588.7 2868.4 8856.5 6238.8 1495.8 63683.6 1.0852

2784.4 1.0149 7.61 3.254 3.436 2.858 18 101.50 32.34 70,038 135.175 3655.47 2510.47 3348.69 9021.9 6237.5 1662.4 58799.61 1.0843

2790.0 1.0129 7.62 3.260 3.443 2.864 18 101.70 32.41 70,178 135.446 3662.8 2515.5 3355.4 9040.0 6250.0 1668.0 59008.8 1.104

The specific heat transfer area is defined as the ratio of total heat transfer area to the total amount of fresh water produced i.e. Specific heat transfer area total heat transfer area A ¼ : ¼ total amount of fresh water produced W MD

ð2Þ

ð3Þ

total seawater intake fed to plant W MF ¼ : ¼ total amount of fresh water produced W MD The specific cooling water flow rate is defined as the ratio of total cooling water flow rate of the plant to the total amount of fresh water produced i.e. Specific cooling water flow rate total cooling water flow rate of plant W MCW : ¼ ¼ W MD total amount of fresh water produced

ð5Þ

Overall plant recovery

The specific flow rate is expressed as the ratio of total seawater fed to plant to the total amount of fresh water produced i.e. Specific feed flow rate

The overall plant recovery is defined as the percentage of total seawater intake that gets converted to the fresh water i.e.

ð4Þ

¼

total amount of fresh water produced W MD  100:  100 ¼ W MF total seawater intake fed to plant

Along with thermal performance, Table 7 summarizes the optimal dependent variable values for various MSF processes. The thermal performance values of MSF-OT, MSF-M, MSF-BR and MSF-BR literature data are evaluated to be 10.3457, 7.5918, 8.6066 and 8.2837 respectively. Fresh water production cost is lowest for MSF-BR plant (1.0785 $/ m3) and the corresponding water recovery is highest (12.28). For these dependent variables, the DE approach used in this work provided better results than those reported in literature [3] using the Newton– Raphson method (1.10 $/m3 and 10.7 respectively). The specific feed flow rate value is also the lowest for MSF-BR case (7.77) and this indicates lower processing cost involved with the feed pre-treatment plant. 5.3. Contributions of various cost functions Fig. 3(a), (b), (c) and (d) presents the pie charts that depict the percentage cost contributions of ICM, DCLIC, ICC, FCC and OMC to the total cost for MSF processes. For the literature data, the pie chart depicting

Table 5 Optimal independent and dependent variable values for the MSF-M process and optimization methods.

Independent variables

Dependent variables

Variable

Unit

DE

SQP*

MS-SQP*

DE-SQP*

WMSC NR LT UB UR T2 CMF TBD CMBD WS AB AR WMF WMBD AT Cost

m3/h – m kW/m2 ∙K kW/m2 ∙K °C ppm °C ppm m3/h m2 m2 m3/h m3/h m2 $/m3

5852.7 26 8.32 3.699 3.671 101.48 70,772 36.52 70,803 147.790 3661.3 1964.7 9727.1 8605.1 54743.5 1.198

5475.1 22 9.99 3.626 3.665 100.28 56,342 37.18 56,399 171.429 3907.2 1992.9 9745.3 8623.3 47751.0 1.2445

5617.8 22 8.15 3.7 3.499 101.42 75,829 36.57 75,905 148.434 3496.5 2700.0 9755.3 8633.3 62896.5 1.22

5784.4 22 9.98 3.625 3.664 100.23 56,314 37.16 56,370 171.343 3905.2 1991.9 9740.4 8618.5 47705.2 1.2135

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Table 6 Optimal independent and dependent variable values for the MSF-OT process and optimization methods.

Independent variables

Dependent variables

Variable

Unit

DE*

SQP*

MS-SQP*

DE-SQP*

WMF LT UB UR NR T2 TBD CMBD WS AB AR WMBD AT Cost

m3/h m kW/m2 ∙K kW/m2 ∙K – °C °C ppm m3/h m2 m2 m3/h m2 $/m3

10043.2 10.18 3.7 3.456 23 101.48 31.4 48,035 153.450 3053.3 3530 1672.4 84243.3 1.2251

10061.9 8.0 3.410 3.468 21 103.6 31.39 47,624 153.490 3094.4 4397.0 1353.1 95431.4 1.2856

10072.1 11.8 3.70 3.498 21 103.6 31.4 47,623 158.450 3285.1 3596.9 1495.8 78820.0 1.2785

10056.9 11.79 3.691 3.495 21 103.51 31.37 47584.9 157.363 3249.8 3594.0 1662.4 78663.9 1.2534

the cost contributions of these components is presented in Fig. 3(d). From Fig. 3(c) and (d), for the MSF-BR process configuration, it can be observed that DCLIC is 2% lower for the DE based optimal solution in comparison with that reported in the literature. For the same case, the DE based optimal solution indicated the DCLIC, ICC, OMC and optimal water production cost to be 5.44, 0.70, 22.40 M$/y and 1.0785 $/m3 respectively. Compared to the literature data [3], these values correspond to a reduction of 12.71, 7.29, 3.22 and 2.31% respectively. Further, with respect to the literature data, it can be also analyzed that the areas of heat rejection and brine heater sections are respectively marginally lower and marginally higher for the heat recovery section for the MSF-BR process configuration.

5.4. Optimality of DE algorithm parameters The effect of DE algorithm parameters (F and CR) on the solution quality was investigated. According to Storn and Price [30], among F and CR, DE is much more sensitive to F. For optimal DE algorithm performance, they further suggested that F, CR and NP can be set as F ∈ [0, 2], CR ∈ [0, 1] and NP = 10 ∗ D, where D is dimensionally of the problem. Typically, F and CR are specified as 0.9 and 0.8 respectively for engineering optimization problems. Thereby, algorithm optimality is critically investigated for the optimality of NG and NP. Significant tradeoffs exist for the optimality of NG and NP. A very low value of NG and NP may terminate the algorithm before it could reach the global optimal and a very high value of NG and NP may take significantly a long time to achieve all solutions very close to the global optimal solution. Solution clustering phenomena are typically addressed as the criteria to set optimal values of NP and NG. In this work, NP is varied from 25 to 340 and NG is varied from 50 to 2000 for the optimization of MSF processes. Fig. 4 panels (a) and (b) respectively summarize the variation of total optimal objective function value (including penalty) and total penalty function value with population size (NP = 35 to 340) and generation size. As shown, significant penalties existed for lower combinations of NG and NP for the MSF-BR process. Solutions without penalty have been achieved using a critical specification of NG and NP, over

Table 7 Optimality of thermodynamic, heat transfer and cost function variable values for MSF desalination processes. Performance model

TPR

sA, m2/(kg/h)

sF

MSF-OT* MSF-M MSF-BR MSF-BR literature [3]

10.34 7.59 8.61 8.28

0.005867 0.005014 0.008745 0.008497

8.95 8.66 7.77 8.06

sWCW – – 5.28 5.57

OPR 11.17 11.78 12.87 10.70

and above which solutions were found to exclusively cluster around the global optimal domain. For all cases of NP, a critical value of NG = 200 has been evaluated to be relevant to yield optimal solutions without penalty. Further, above an NG value of 800, almost all solutions remained fairly constant, thus indicating that the solutions generated are very close to the global optimal domain. Further, for few cases of NP and generation size where few best solutions were obtained, additional investigations were carried out to evaluate upon the solution quality in the context of the global optimal domain. For this purpose, standard deviations were evaluated for the obtained solutions. The standard deviation of the solutions obtained for a population and generation size of 270 and 800 respectively is 10−5 for the best 100 solutions. For this case, the lowest optimal solution of 1.0785 $/m3 has been obtained. However, for the case of the population and generation size of 100 and 500 respectively, a standard deviation of 10−3 was obtained for the best 100 solutions. The achievement of lower standard deviation for lower combinations of NG and NP in comparison with the higher combinations of NG and NP is expected, given the fact that higher NG and NP combinations facilitate better search of the solution space. A similar explanation could be provided for the results indicated in Fig. 4(c)–(f) for MSF-M and MSF-OT processes. Thus, the optimal DE algorithm parameter combinations for MSF-BR refer to F, CR, NG and NP values of 0.9, 0.8, 800 and 270 respectively. For the MSF-OT process configuration, Fig. 4 panels (c) and (d) respectively depict the variation of optimal total objection function value and optimal total penalty value with NP and generation size. The obtained trends are similar to those obtained for MSF-BR case and the solutions converged to optimal value for all populations (NPs) for a generation size of 800. The results reported for the MSF-OT system refer to the existence of penalties for few constraints and the reported values indicate solutions with lowest penalties. A careful analysis of the optimization results for MSF-OT indicated that brine velocity constraint was the violated beyond the specified upper bound value (6 ft/s). Eventually, with higher brine velocity, the inequality constraint presented as Þ−0:001 did not get satisfied, as UBcal is a function of brine absð1− UUBcal B density, which is in turn a function of brine velocity. The penalties for the MSF-OT have been successfully eliminated by targeting the following alterations in the parametric and design specifications: a) The brine velocity is fixed as 6 ft/s for heat recovery section and the inequality constraint has been specified to have a constraint parameter value of 10−2 but not 10−3. For such a scenario, the minimal fresh production cost has been evaluated using the DE algorithm as 1.1249 $/m3. b) The brine velocity is fixed as 6 ft/s and the brine heater width is reduced to 30 ft from 35 ft. No additional changes have been carried out for the abovementioned inequality constraint specification. For such a scenario, solutions without penalty were obtained for the

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Fig. 3. Pie-charts representing the cost contributions of various cost functions to optimal freshwater production cost; (a) MSF-OT, (b) MSF-M, (c) MSF-BR and (d) MSF-BR literature data [3].

MSF-OT system to indicate the optimal fresh water production cost of 1.1154 $/m3.

Fig. 4 panels (e) and (f) respectively illustrate the variation of total optimal objective function value (inclusive of penalties) and total optimal penalty function values for various cases of NP and generation values. The obtained trends are similar to those presented for MSF-BR case. Also, it can be observed that for MSF-M, solutions without penalty (feasible solutions) have been obtained using the DE algorithm. The MSF-OT configuration provided infeasible solutions among the three different process configurations. Using DE, the optimal water production cost for the MSF-M process is 1.1980 $/m3 respectively. 5.5. Sensitivity analysis 5.5.1. Effect of feed concentration Sensitivity analysis involves the evaluation the optimal freshwater production cost critical dependence on with various operating parameters of the MSF process. Typically, feed concentration is varied from 20,000–50,000 ppm for MSF processes [3]. Varying feed concentration in this range, the optimal freshwater production cost was evaluated using DE for MSF processes. Fig. 5(a) summarizes the results obtained for the feed concentration effect on optimal fresh water production cost. It can be observed that the optimal water production cost increased from 1.2125 to 1.2455, from 1.1876 to 1.2132, from 1.0656 to 1.1051 and from 1.09 to 1.13 $/m3 respectively for MSF-OT, MSF-M, MSF-BR and MSF-BR literature processes. From a variation in feed concentration 20,000 ppm to 40,000 ppm, the optimal water production cost varied from 1.0656 to 1.0708 for the MSF-BR process which affirms that the cost remained fairly constant. From 40,000–50,000 ppm variation in feed concentration, water product cost increased linearly from 1.0708 to 1.1051 for the MSF-BR process. The slope of the graph is about 3.433 × 10− 6. The insensitivity of the water production cost

with feed concentration up to 40,000 ppm is due to the insignificant effect of feed concentration in influencing the product flow rates, concentrations and temperature. This might not be the case for reverse osmosis process where feed concentration will have a significant effect on the water production cost. Above 40,000 ppm, the feed concentration can be observed to have a significant effect on the water production cost. The DE based optimal cost profiles lowered by 2.23–2.65% than those reported in the literature for the variation in feed seawater concentration. In comparison with the literature data, it can be observed that the lowest cost trends have been obtained for the MSF-BR process. This once again confirms the efficacy of DE to obtain high quality solutions for MSF process optimization. 5.5.2. Thermal performance The thermal performance is defined as the ratio of total amount of fresh water produced to total steam intake of the MSF process: Thermal performance ¼

ð6Þ

total amount of fresh water produced W MD : ¼ W MS total steam intake fed to plant

In general, the thermal performance ratio varies from 6 to 12 for the MSF desalination system and below 1 for the single stage flash desalination system [10]. Similarly, the top brine temperature (TBT) is varied from 90 to 100 °C for a variation in total number and thermodynamic loss range of 18–29 and 0.5–2 °C [10]. The variation of thermal performance with TBT for MSF processes is presented in Fig. 5(b). For a variation in TBT from 90 to 110 °C, the thermal performance ratio varied from 6.66 to 7.31, from 6.75 to 7.59, from 7.26 to 8.61 and from 7.02 to 8.28 for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data. It can be observed that the thermal performance is marginally sensitive with TBT for MSF-OT and MSF-M processes but not MSF-BR. Further, it can be observed that the thermal performance

C.S. Bandi et al. / Desalination 394 (2016) 30–43

39

Fig. 4. Effect of DE algorithm parameters (NG and NP) on the solution optimality for various MSF processes — (a) total optimal objective function value for MSF-BR, (b) total penalty function value for MSF-BR, (c) total optimal objective function value for MSF-M, (d) total penalty function value for MSF-M, (e) total optimal objective function value for MSF-OT and (f) total penalty function value for MSF-OT.

increased linearly with increasing TBT and the corresponding slope values are 3.2 × 10−2, 4.0 × 10−2, 6.7 × 10−2, and 6.4 × 10−2 respectively for the said process sequence. Also, in comparison with the literature data, it is apparent from the figure that the highest thermal performance values were obtained for the MSF-BR process using the DE algorithm. The thermal performance of the MSF system has been

varied from 6 to 9, where the DE based optimal costs have been evaluated to be 3.42–3.99% lower than those reported in the literature. 5.5.3. Chemical cost multiplier Typically, cost multipliers are varied from 0.25 to 2 for MSF desalination processes [3]. Fig. 5(c) presents the variation of optimal water

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Fig. 5. Sensitivity of MSF processes with respect to various process and operating parameters; (a) feed concentration, (b) TBT on TPR, (c) chemical cost multiplier, (d) steam cost multiplier. (e) Labor cost multiplier, (f) power cost multiplier, (g) spares cost multiplier, (h) TBT on cost (+[3]).

C.S. Bandi et al. / Desalination 394 (2016) 30–43

production cost as a function of chemical cost multiplier for MSF processes. It can be observed that the water production cost increased non-linearly from 1.1345 to 1.2958, from 1.0866 to 1.12798, from 1.0015 to 1.1365 and from 1.022 to 1.161 $/m3 with increasing chemical cost multiplier (0.25–2) for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data. As it is apparent in other sensitivity analyses, the lowest water production cost trends have been obtained in this work in comparison with the existing data trends of the MSF-BR process in the literature. The freshwater production cost varied linearly with chemical cost multiplier in the range of 0.25–2. For such a case, the corresponding slopes of the trend lines are 0.09, 0.11, 0.071, and 0.072. These slope values are indicative towards water production cost sensitivity with respect to chemical cost multiplier. Thus the water cost ($/m3) will increase by 9 ¢/m3, 11 ¢/m3, 7.1 ¢/m3, and 7.2 ¢/m3 respectively for a unit increase in chemical cost multiplier for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data [3]. For a variation in the cost multiplier from 0.25 to 0.5, the corresponding optimal water cost varied nonlinearly from 1.1348 to 1.1752, from 1.0866 to 1.1271, from 1.0015 to 1.0495 and from 1.022 to 1.076 $/m3. A further increase in the chemical cost multiplier from 0.5 to 2.0 enabled a linear enhancement for the optimal water cost from 1.1752 to 1.2958, from 1.1271 to 1.2798, from 1.0495 to 1.1365 and from 1.076 to 1.161 $/m3 for the said sequence of processes. Compared to the literature, the obtained optimal cost is 2–2.46% better for variation in the chemical cost multiplier. 5.5.4. Steam cost multiplier For the steam cost multiplier sensitivity analysis, the steam cost was varied from 0.00104 to 0.00832 $/kg of steam (corresponding to a variation of steam cost multiplier from 0.25 to 2.0) at a constant TBT value of 110 °C [3]. The obtained sensitivity analysis based cost trends with respect to steam cost multiplier are presented in Fig. 5(d). As shown, for a variation in steam cost multiplier from 0.25 to 2.0, the minimal water cost varied from 0.8656 to 1.7123, from 0.8202 to 1.687, from 0.7284 to 1.4876 and from 0.739 to 1.5212 $/m3 for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data [3]. The obtained cost trends with respect to steam multiplier indicate a linear variation in water cost with steam multiplier. Such a trend is expected, given the fact that the MSF is a thermal process and its performance is a strong function of steam as a heat source for flash operation. The sensitivity parameters can be obtained from the slope of the obtained data trends. These values have been evaluated correspondingly as 0.48, 0.49, 0.43 and 0.44. Thus, the water cost can be evaluated to increase by 48 ¢/m3, 49 ¢/m3, 43 ¢/m3 and 44 ¢/m3 respectively for a unit increase in steam cost for the said sequence of processes. The obtained simulation based trends indicate that MSF-BR provides the lowest water cost trends which are placed marginally below the cost trends reported in the literature [3]. For a variation in steam cost multiplier from 0.2 to 2.0, the DE based optimal cost is 1.43– 2.21% lower than that reported in the literature. 5.5.5. Labor cost multiplier Fig. 5(e) shows that the optimal water production cost evaluated for MSF processes is sensitive with respect to labor cost multiplier for all processes. As shown, for a variation in labor cost multiplier from 0.25 to 2.0, the minimal water cost varied from 1.1395 to 1.3396, from 1.209 to 1.3008, from 1.0014 to 1.1813 and from 1.0239 to 1.2001 $/ m3 for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data [3]. Hence, significantly higher cost trends can be observed for MSF-OT and MSF-M processes but not for MSF-BR and MSF-BR literature data. The obtained cost trends with respect to labor cost multiplier indicate that the variations are linear with slopes of 0.11, 0.10, 0.10 and 0.10 for MSF-OT, MSF-M, MSF-BR and MSF-BR literature [3] respectively. Thus the water cost ($/m3) will increase by 11 ¢/m3, 10.3 ¢/m3, 10 ¢/ m3, and 10 ¢/m3 respectively for a unit increase in labor cost multiplier for the said sequence of processes. The lowest data trends refer to the data obtained with DE for the MSF-BR process. For a variation in labor

41

cost multiplier from 0.2 to 2.0, the DE based optimal cost is 1.57–2.2% better than that reported in the literature. 5.5.6. Power cost multiplier For a variation in power cost multiplier from 0.25 to 2.0, Fig. 5(f) presents the variation of optimal water production cost trends for various MSF desalination processes. The observed trends are similar to those obtained for labor cost multiplier. As shown, for a variation in power cost multiplier from 0.25 to 2.0, the minimal water cost varied from 1.1286 to 1.3589, from 1.1139 to 1.3101, from 0.9915 to 1.1921 and from 1.02 to 1.213 $/m3 respectively for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data [3]. The corresponding slopes of the obtained cost trends are 0.13, 0.11, 0.11 and 0.11. Thus the water cost ($/m3) will increase by 12.9 ¢/m3, 11.2 ¢/m3, 10.9 ¢/m3, and 10.9 ¢/m3 respectively for a unit increase in labor cost multiplier for the said sequence of processes. The lowest cost trend corresponds to that obtained with MSF-BR and the DE algorithm in this work. For a variation in power cost multiplier from 0.2 to 2.0, the DE based optimal cost is 1.72– 2.79% better than that reported in the literature. 5.5.7. Spares cost multiplier Fig. 5(g) illustrates the variation of minimal water production cost as a function of spares cost multiplier for alternate MSF processes. As shown, for the variation in spares cost multiplier from 0.25 to 2.0, the minimal water cost varied from 1.1456 to 1.3102, from 1.1348 to 1.2823, from 1.0203 to 1.1628 and from 1.0375 to 1.1825 $/m3 for MSF-OT, MSF-M, MSF-BR and MSF-BR literature data [3]. The cost trends are similar to those obtained for labor and power cost multipliers. The slopes of the linearized trends are 0.09, 0.08, 0.08 and 0.08 respectively for the said sequence of processes. Thereby, the sensitivity of the spares cost multiplier has been evaluated in terms of an increase by 9.2 ¢/m3, 8.4 ¢/m3, 8.2 ¢/m3, and 8.2 ¢/m3 respectively for a unit increase in labor cost multiplier for the said sequence of processes. For a variation in spares cost multiplier, the DE based optimal cost is 1.66% lower than that reported in the literature. 5.5.8. Effect of top brine temperature Fig. 5(h) presents the variation of optimal water production cost with variation in top brine temperature for various cases. As shown, for a variation in TBT from 90 to 110 °C, the costs varied linearly from 1.1051 to 1.2251 and from 1.0845 to 1.1980 $/m3 for MSF-OT and MSF-M processes. However, for the MSF-BR processes (reported in this work and in literature), up to a temperature of 100 °C, the optimal water production cost was not at all affected with variation in TBT. Above 100 °C, the optimal water production cost increased with increasing TBT. Based on these increasing trends, the slopes of the various plots have been evaluated (5.9 × 10−3, 5.6 × 10− 3, 2.5 × 10−3, and 2.8 × 10−3 respectively for the said sequence of processes). Thus, for a unit increase in top brine temperature, the water cost will increase by 0.59 ¢/m3, 0.56 ¢/m3, 0.25 ¢/m3, and 0.28 ¢/m3 for the said sequence of processes. Overall, the lowest data trends have been obtained for the MSF-BR and DE algorithm case. For a variation in TBT from 95 to 110 °C, the DE based optimal cost is 1.66% lower than that reported in the literature. In summary, the DE based sensitivity analysis of MSF-BR enables one to infer that the slope based ranking of various process and operating parameters is as per the following order: steam cost multiplier (43 ¢/m3) N labor cost multiplier (10 ¢/m3) N power cost multiplier (8.2 ¢/m3) N spares cost multiplier (8.2 ¢/m3) N chemical cost multiplier (7.1 ¢/m3) N TBT (0.25 ¢/m3) N feed concentration (fairly constant). In other words, the optimal freshwater production cost for the MSF processes is highly sensitive to steam cost multiplier, marginally sensitive with all other process parameters but not feed concentration and TBT. Since the MSF process is highly energy intensive, the highest sensitivity of the MSF-BR process optimal cost with steam multiplier is expected.

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6. Conclusions Based on the global optimization approach involving differential evolution algorithm, this work provided significant insights and inferences with respect to the comparative assessment of alternate MSF processes, non-deterministic/deterministic optimization methods and pertinent sensitivity analysis. The modeling approach adopted in this work might refer to design solutions under stringent uncertainty, given the inability to generate feasible solutions with methods other than DE. The following conclusions are applicable from the insights deduced in this work. Firstly, DE has been proven to be effective to generate feasible optimal design variable values for MSF-BR, MSF-M but not MSF-OT processes. Compared to the literature optimal value, DE provided a reduction of about 2.31% in the optimal freshwater production cost. This is due to the identification of better optimal decision variable value set of [2794.4 m3/h, 1.0499, 7.62 m, 3.359 kW/m2 ∙ K, 3.297 kW/m2 ∙ K, 3.042 kW/m2 ∙K and 22] for variable set [WM, RH, LT, UB, UR, Uj and NR] respectively where the optimal freshwater production cost corresponds to 1.0785 $/m3. Secondly, the deterministic optimization algorithms such as SQP, MS-SQP and DE-SQP could not provide better solution than the DE. This is primarily due to the dependence of the optimal variable value set and objective function on the initial guess values. Thus, compared to other optimization methods, DE would provide better initialization strategies and is expected to serve better for problems with greater complexity in terms of decision variables. Another important insight that has been deduced in this work is that the MATLAB based optimization toolbox uses default optimization algorithm parameters and they cannot be as such used for MSF optimization problems studied in this work. Thirdly, the sensitivity analysis affirmed that DE based analysis provided 1.41–3.99% better profiles than those available in the literature. While such improvement could be regarded to be optimal, it is important to note that the freshwater production cost related improvement is significant, given the fact that optimization studies that allow even 1% reduction in water production cost could turn out in terms of a huge amount of savings. Fourthly, inequality constraint resolution appears to be better tackled by DE than any other optimization method. A further resolution of the generated solutions has also been demonstrated in this work i.e., to alter certain design parameter value for chamber width. Fifthly, the chosen literature might be relatively old in the existing state-of-the-art, but the trends obtained in this work appear to be generic to affirm upon the efficacy of DE as the most versatile optimization method to yield feasible solutions under strong conditions of uncertainty. In summary, it is inferred that DE based optimization is highly effective to obtain feasible global optimization solutions in conjunction with SQP, MS-SQP and DE-SQP. It is anticipated that DE would be able to provide confidence in the solutions generated with complex and hybrid process configurations involving MSF process configurations. This will be addressed in subsequent research articles. Nomenclature Abbreviations CR cross over ratio DE differential evolution ED electrodialysis F mutation factor DICOPT++ DIscrete and Continuous OPTimizer GA Genetic algorithm GAMS general algebraic modeling system GOR gained output ratio GRG generalized reduced gradient ID inside diameter of condenser tubes, m IDA International Desalination Association

MEE MS-SQP MSF MSF-BR MSF-M MSF-OT NG NLP NP OF OPR ppm RO sA sF sWCW SQP TA TBT TDM TDS TPR USA VC

multi-effect evaporator multistart-sequential quadratic programming multi-stage flash brine recycle (BR) multistage flash system (MSF) brine-mixing (M) multistage flash system (MSF) once through (OT) multistage flash system (MSF) maximum number of generations non-linear programming population size objective function overall plant recovery parts per million reverse osmosis specific heat transfer area specific feed flow rate specific cooling feed flow rate sequential quadratic programming Thomas algorithm top brine temperature tridiagonal matrix total dissolved solids thermal performance United States of America vapor-compression

Symbols

a AB Aj AR AT CDCC CDCLIC CIC CICC Clast CMBD CMF CMR CMSMF COMC Cp CPR Cpj ID LT M1 Nj NR OD RH S1–2 sWCW T1 T2 T3 T4

coefficient to account for using average latent heat of vaporization heat transfer area of brine heater, m2 heat transfer area of the rejection section, m2 heat transfer area of recovery section, m2 total heat transfer area of the MSF process, m2 direct capital investment, $ direct capital less intake investment, $ annual plant intake-outfall cost, $ indirect capital investment, $/y concentration of brine stream from last stage of the heat rejection section, ppm concentration of reject stream leaving the heat rejection section, ppm feed (seawater) concentration, ppm concentration of recycle stream (from splitter to heat recovery section), ppm concentration of feed stream to the MSF-M process, ppm operating and maintenance cost, $/y heat capacity, kJ/kg∙K avg. specific heat capacity, recovery section, kJ/kg∙K avg. specific heat capacity, rejection section, kJ/kg∙K inside diameter of condenser tubes, m tube length, m mixer 1 number of rejection stages number of stages in heat recovery stages outside diameter of condenser tubes, m specific heat ratio (WMRCpR/WMFCpj) splitter 1–2 respectively specific cooling water flow rate temperature of reject coolant stream in the MSF-BR process, °C temperature of brine stream entering the brine heater in the MSF process, °C top brine temperature, oC temperature of brine stream leaving last stage in the heat recovery section of the MSF-BR process, °C

C.S. Bandi et al. / Desalination 394 (2016) 30–43

T5

temperature of brine stream leaving last stage in the heat rejection section of the MSF-BR process, °C temperature of recycle stream in the MSF-BR process, °C T6 temperature of reject stream in the MSF-BR process, °C TBD temperature of distillate product stream in the MSF process, TD °C steam temperature, °C TS seawater temperature/feed temperature, °C TSea overall heat transfer coefficient in brine heater, kW/m2 ∙K UB overall heat transfer coefficient in the rejection section, Uj kW/m2 ∙ K overall heat transfer coefficient in recovery section, kW/m2 ∙K UR calculated overall heat transfer coefficient in brine heater, UBcal kW/m2 ∙K calculated overall heat transfer coefficient in the rejection Ujcal section, kW/m2 ∙K calculated overall heat transfer coefficient in recovery section, URcal kW/m2 ∙K brine velocity in brine heater, m/s VB brine velocity in the rejection section, m/s Vj brine velocity in recovery section, m/s VR flow rate of flashing brine stream leaving the last stage of the Wlast heat rejection section in the MSF-M and MSF processes, m3/h flow rate of makeup stream in the MSF process, m3/h WM flow rate of rejected stream in the MSF process, m3/h WMBD flow rate of reject coolant stream in the MSF process, m3/h WMCW flow rate of total potable water product stream in the MSF WMD process, m3/h feed flow rate in the MSF process, m3/h WMF feed flow rate of seawater (before mixing) of MSF-M, m3/h WMSC WMSMF feed flow rate of MSF-M, m3/h flow rate of recycle stream in MSF-M and MSF-BR processes, WMR m3/h WMRWBD flow rate of brine stream leaving mixer M1 and entering splitter S2 in the MSF-BR process, m3/h flow rate of steam fed to brine heater, m3/h WS yearly capacity, m3/y WY Greek symbols

¢ αR αj λ ρB ρW ψ

1/100th $ (cent) average boiling point rise, heat recovery section, °C average boiling point rise, heat rejection section, °C average latent heat of vaporization, kJ/kg average brine density, kg/m3 average pure water density, kg/m3 objective function (cost), $/m3

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