Superstructure of Alternative Configurations of the Multistage Flash

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Ind. Eng. Chem. Res. 2006, 45, 7190-7203

Superstructure of Alternative Configurations of the Multistage Flash Desalination Process Sergio F. Mussati, Pio A. Aguirre, and Nicola´ s J. Scenna* INGAR-Instituto de Desarrollo y Disen˜ o, AVellaneda 3657, (3000) Santa Fe, Argentina

This paper addresses the optimal synthesis and design of multistage flash (MSF) evaporator systems. A detailed nonlinear programming (NLP) model based on a superstructure developed previously by Mussati et al. [Ind. Eng. Chem. Res. 2003, 42, 4828-4829] has been appropriately reformulated to include more alternative configurations for the process. The new superstructure includes the number of stages and the stream flow patterns (distillate, feed and discharge brine, extraction points, and recycle flow patterns). Therefore, the superstructure simultaneously embeds an enormous number of alternatives, including, of course, the three commonly operating modes for the evaporator: MSF-BR, MSF-M, and MSF-OT systems and their combinations, which, to date, have not been analyzed systematically. In addition, a rigorous mathematical model by stage and component, which also includes the geometric design of each stage (length, height, and width), number of tubes in the preheaters, fluid-dynamic equations for the streams among others, is applied. An attractive configuration for the MSF system resulted from the proposed superstructure. This structure differs from the conventional structure, because it considers distillate extractions, two recycle streams, and new allocations for the cooling seawater, blow-down brine, and recycle. The mathematical model and the solution procedure have been implemented in GAMS [Brooke et al., GAMS: A User’s Guide; Scientific Press: 1992]. Two study cases are presented, to illustrate the model and solution procedure capabilities. A complete description of the novel configuration, detailed comparison between different “sub-optimal” structures, and a sensitivity analysis on the main process variables are summarized. I. Introduction The objective of the industrial process design is to generate the best alternative configurations and to obtain one or more solutions that satisfy specified criteria and restrictions. Recently, substantial changes have taken place in this problematic situation that have forced the generation of new methodologies. The complexity of desalination processes, such as multistage flash (MSF) and multieffect desalination (MED) systems, and dualpurpose plants (DPP), and the necessity of a continuous improvement of the competitiveness, has raised new challenges. Many reviews have been published on this topic. Much effort has been exerted to reduce the cost of water produced from MSF plants. This have been achieved through the increase of the plant capacity, improving the thermal performance of plants, better materials, and good practices of reliable operation and scale control. These developments are a consequence of better understanding and good modeling and simulation of the MSF process.1-3 Nevertheless, there is no research evidence on the simultaneous optimization of structural configuration and operating conditions to improve the efficiency and/or the economy of the process. No significant improvements through the analysis of the streamflow patterns have been achieved to increase the performance of the process. Different rigorous models for MSF desaltors have been presented in previous works.2-5 These models consider the conventional configuration for the streams (well-known stream flow patterns) and the number of stages as parameters of the problem. The optimal design of the system consisted of determining the flashing chamber dimensions (length, width, and height) and the distributions of heat-transfer areas and operating conditions * To whom correspondence should be addressed. Tel.: 00 54 342 4534451. Fax: 00 54 342 4553439. E-mail: [email protected].

(profiles of pressure, temperature, composition, among others) to minimize the total annual cost. Mussati et al.6 recently developed a mathematical model based on a superstructure of alternative configurations. In this model, in contrast with the aforementioned models, the optimal number of stages is obtained as a result of the problem that does not require a parametric procedure. Although novel stream flow patterns resulted from the proposed superstructure, only one operating mode of the recycle has been taken into account: the MSF-mixer configuration). The most common configuration used in desalination process is the multistage flash unit that uses brine recycle (MSF-BR). The recycle is used to avoid pretreatment cost (chemical additives). In a recent interesting work presented by El-Dessouky et al.,3 all these structures (MSF-mixer and MSF-BR) have been analyzed comparatively; however, a systematic approach to determine the optimal arrangement of the process was not presented. The inclusion of other possibilities (the combination of all possible flow patterns for recycles, extraction points of distillate, discharge, and cooling streams) demands a new and more complex superstructure that considers all back recycles, because, in our previous model, only one recycle position (the MSF-mixer) was considered.6 In this paper, a new superstructure, the mathematical model, and a solution procedure that is capable of handling all alternative configurations for the process are presented. The resulting mathematical model takes into account the most important variables: the number of stages, stream flow patterns, number of tubes in the preheaters, geometry of stages, heat consumption, heat-transfer area, distillate alternative extraction points, and different recycles options (optional operating modes). The paper is outlined as follows. Section II briefly describes the process. Section III introduces the problem formulation. Section IV summarizes the solution procedure and the math-

10.1021/ie051053y CCC: $33.50 © 2006 American Chemical Society Published on Web 09/08/2006

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7191

on the cold end of the plant. Operating a plant at the higher temperature limits of 120 °C has a tendency to increase the efficiency, but it also increases the potential for detrimental scale formation and accelerated corrosion of metal surfaces. III. Problem Definition

ematical model. Section V presents case studies and discusses the obtained results. Section VI presents the conclusions and future work.

The problem can be stated as follows. The seawater conditions, freshwater production, and cost data are given parameters. The optimization problem is to determine the optimal structure and operating conditions needed to reduce the water production cost. Precisely, the operating mode of the recycle, optimal number of stages, stream flow patterns (recycle, distillate, feed, and discharged brine), and the optimal-stage geometric design are determined by minimizing the total annual cost (TAC), which is composed of investment capital and operating costs.

II. Description of the Multistage Flash Conventional (MSF-BR) Process

IV. Superstructure Proposed for the Process. Rigorous Model

Multistage flash desalination systems have been commercially available since the early 1960s. The world’s largest desalination systems use the MSF process. Figure 1 shows a diagram of an MSF unit. The incoming seawater (feed) passes through the heating stage(s) and is heated further in the heat recovery sections (RS) of each subsequent stage. After passing through the last heat recovery section, and before entering the first stage, where flash-boiling (or flashing) occurs, the feed water is further heated in the brine heater (BH), using externally supplied steam. This raises the feed water to its highest temperature (Tmax), after which it is passed through the various stages where flashing occurs. The vapor pressure in each of these stages is controlled so that the heated brine enters each chamber at the proper temperature and pressure (each lower than the preceding stage) to cause instantaneous and sudden boiling/evaporation. The fresh water (distillate) is formed by condensation of the water vapor, which is collected on the distillate tray at each stage and passed from stage to stage in parallel with the flashing brine. At each stage, the product water is also flash-boiled, so that it can be cooled and the surplus heat recovered for preheating the feed water. Because of the large amount of flashing brine required in an MSF plant, a portion (50%-75%) of the brine from the last stage is often mixed with the incoming feed water, recirculated through the heat recovery sections of the brine heater, and flashed again through all of the subsequent stages. A facility of this type is often referred to as a “brine recycle” plant (MSFBR). This mode of operation reduces the amount of waterconditioning chemicals that must be added and can significantly affect operating costs. On the other hand, it increases the salinity of the brine at the product end of the plant, raises the boiling point, and increases the danger of corrosion and scaling in the plant. To maintain a proper brine density in the system, a portion of the concentrated brine from the last stage is discharged to the sea. The discharge flow rate is controlled by the brine concentration at the last stage. There are limits on the number of stages, because one of the major factors that affect the thermal efficiency of MSF plants is the temperature difference between the brine heater and the final condenser at the cold end of the plant. Typically, an MSF plant can contain from 4 to ∼40 stages. The MSF plants usually operate at the maximum operating temperatures (after the brine heater) of 90-120 °C. One of the factors that affect the thermal efficiency of the plant is the difference in temperature from the brine heater to the condenser

A detailed representation of a general stage j used to model the superstructure is illustrated in Figure 2. As is shown, the input stream to the preheater of stage j may be composed by the following streams (point C): (1) fresh seawater (feed); (2) recycle stream from flashing chamber j; (3) recycles coming from flashing chambers placed downstreams (j + 1, j + 2, ..., N); and, eventually, (4) the mixture between streams 1 and 2 or streams 1 and 3, or streams 2 and 3. On the other hand, the input stream at the flashing chamber on stage j may be composed by the following streams (point B): (1) incoming seawater from stages j, j - 1, j - 2, j - 3, ..., and (2) its mixtures with the other streams. No constraints on the number of streams and its placements are introduced. Splitters and mixers on preheaters and flashing chambers are placed appropriately to allow, in the solution strategy, the elimination of stages, instead of bypassing them. Finally, partial or total extraction of distillate may happen in each stage-splitter (D). According to this, all the potential combinations of liquid flow patterns are considered in the optimization process. IV.1. Model Assumptions. The resulting mathematical model takes into account the more relevant functionalities between the main process variables to obtain a realistic optimization model. In fact, irreversibilities (flashing efficiency and boiling point elevation) that are dependent on the geometry of stages, the salinity, and the temperature, are modeled in a rigorous way using specific correlations. Also, hydraulic constraints to describe the flashing flow rate, taking into account the geometry of the flashing chamber, are considered. Briefly, the rigorous mathematical model for the superstructure (Figure 2) was derived using the following assumptions. (1) The total annual cost (TAC) is the objective function to be minimized. (2) Investment capital of preheaters (heat transfer area), stage area (flashing chambers) and pumps are considered. A linear relation between the total recirculation pumping cost (investment and operating cost) and the recycle flow rate is considered. On the other hand, the unit costs of heat-transfer area (preheaters) and stage area (flashing chambers) are related by the “weight” factor (F). This factor takes into account resistance to corrosion, tendencies to form scale, and mechanical strength, which is dependent on the material used for the chamber construction and its geometry. Indeed, it also includes chamber instrumentations, roller-expanding, and drilling the tubes and welding. Typical values of F estimated in this way usually are within

Figure 1. Multistage flash (MSF) conventional system (brine recirculation unit, MSF-BR).

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Figure 2. Representation of stage j.

the range of 10-40.7 This model offers the possibility to select different materials for each part of the evaporator, depending on the salinity and temperature and to calculate a detailed cost for each stage. (3) Operating cost of pretreatment, hot utility, and recirculation pumping are taken into account. (4) Detailed model by stage and component. (5) Alternative configurations for the flow-pattern streams (feed, distillate, blowdown, and recycle streams). (Figure 2 shows a stage representation.) Distillate extraction (total or partial) in all stages and different recycle configurations (including the MSF-R, MSF-OT, and MSF-Mixer) are included in the model. (6) A maximum number of stages of NS ) 40 is assumed. The optimal number of stages is obtained as a result of the model. (7) The dependence of heat capacity, boiling point elevation, and latent heat of evaporation on temperature and concentration is considered by rigorous correlations. (8) Dependence of the overall heat-transfer coefficient on brine velocity, fouling factor, temperature, and tube diameter is considered. (9) Nonequilibrium allowance is taken into account according to the correlation developed by El-Dessouky et al.3 (10) Hydraulic correlations given by ref 3 are adopted. Equations to describe the interstage flow rate of the flashing brine are considered. (11) Chen’s approximation is used to compute the driving force for the heat-transfer area calculation. This equation avoids convergence problems when some stages (uneconomic stages) must be deleted from the superstructure. Nevertheless, the error introduced by this approximation is not important on the value range of our interest. (12) Preheater tube configuration is arranged perpendicular to the brine flow. (13) Stage geometric design (length, width, and height) is considered. (14) Demisters and noncondensable gases are not contemplated. (15) The main brine heat-transfer area is not considered.

Table 1. Problem Data parameter

value

water production maximum operating temperature, Tmax seawater temperature, T Finl tube diameter, di pitch heat-transfer area cost utility cost, CQDes capital recovery factor, CRF

1000 ton/h 390 K 298 K 35.48 mm 1.2 $50/m2 $0.252/106 BTU 0.16/yr

The most critical model parameters that influence on the final configuration of the system are the maximum admissible temperature, the factor F, the investment capital, and the operating costs. Investment and operating costs are computed using classical and detailed cost equations. The investment capital includes the preheaters, flashing chambers, and pumps, whereas operating costs include the hot utility, pretreatment, and pumping costs. A specific feature of the model is the use of the factor F, which allows one to compute the investments of preheaters and flashing chambers in a detailed and disaggregated way. A parametric analysis on the main variables of the model is presented in Section V.2. Table 1 lists the data of main parameters adopted for the optimization problem. In the next section, the mathematical model is presented. IV.2. Mathematical Model. To obtain a mathematical representation for the proposed superstructure, the same approach as that followed by ref 6 is used. Several new constraints and variables are added in the mathematical formulation, to consider all potential structural combinations. As shown in Figure 2, the stage model involves four generic compartments, namely, the brine pool (primary flashing chamber), the product tray and vapor space (secondary flashing chamber), and the tube bundle (preheater). The complete mathematical model that describes the steadystate model for the MSF system can be found in Appendix A. Some variables are defined over two indices, k and j, which are defined over the possible number of stages to be selected (1-40). As will be seen in the model formulation, this definition is necessary to indicate the interconnection among stages. For example, variable WREC(k,j) refers the flow rate of the brine


Figure 3. Optimal structure for study case 1.

recycle from stage k to stage j. The problem constraints are formulated over the index k. Correlations that are used are listed in Appendix B. The model was implemented in the General Algebraic Modeling System (GAMS).8 The generalized reduced gradient algorithm CONOPT 2.041 was used as a nonlinear programming (NLP) solver.9 All studied cases were solved using a computer with a Pentium III processor (733 MHz and 256 MB RAM). The mathematical model involves more than 3124 variables and 3017 constraints. An efficient solution procedure is needed to solve the resulting equation system efficiently. The model solution is difficult, because contains many variables and nonlinear constraints. The optimal number of stages is determined “indirectly” from the optimal location of the feed stream. The proposed superstructure and constraints, as well as the solution strategy, force the stages to be deleted from stage 40 to stage 1 in a sequential way. For example, if the feed is located in stage 28, it means that 12 stagessstage 29 to stage 40sare deleted from the superstructure and the optimal number of stages is 28. In addition, numerical values of representative variables for the stages to be deleted (vapor and liquid flow-rates) are forced to be zero for the converged solution. The model convergence is facilitated by a “preprocessing phase” that allows one to obtain a feasible solution within a few iterations. A complete description of this procedure can be found in ref 6. Briefly, the preprocessing phase uses an optimal solution obtained from a simplified model. This solution then is used as the initialization to solve the rigorous model, increasing the robustness of the optimization algorithm. Basically, the procedure involves two phases: a preprocessing phase and a “rigorous optimization” phase. Precisely, a preprocessing phase provides the initial values and some critical bounds to solve the rigorous model (superstructure). Recall that the superstructure initialization must contemplate 40 stages, because this is the starting number of stages. Good prediction of the candidate stages in which the feed and recycle flow rates should be placed (among others) are obtained by this phase. V. Study Cases In this section, numerical results and the influence of some critical parameters on the optimal solution are presented. Study case 1 describes a typical MSF design problem. It is described the novel configuration of the evaporator resulting from the proposed superstructure after solving the synthesis problem. The main characteristics of this configuration are discussed. Also, the novel design is compared to other alternatives. Study case 2 presents a sensitive analysis on the main variables of the model. V.1. Study Case 1. The operating parameters (seawater composition, maximum temperature, seawater temperature, and

water production rate) of Distiller No. 5 of the Umm Al Nar East Plant, located in the Emirate of Abu Dhabi (UAE) have been adopted (see Table 1). According to the definition problem, the goal is to find the best configuration and optimal operating conditions to satisfy a given water production and minimize the TAC. Figure 3 shows the main characteristics of the achieved optimal configuration. This structure (hereafter named structure A) is different from the conventional configuration reported in the literature (hereafter named structure B) and from the “modified mixer” configurationspreviously presented in Mussati et al.6s(hereafter named structure C). The differences are present in (a) product extraction, (b) allocation of rejected stream, (c) cooling water, and (d) recycle stream flow patterns. Each characteristic is discussed in the following sections. V.1.1. Product Extraction. In the new solution (design), the desaltor operates at stages j ) 1 to j ) 21, according to the MSF conventional theory (no stream splitting). The total distillate accumulated from stage j ) 1 to j ) 21 then is completely extracted there and the entire vapor produced for j ) 22 to j ) 29 is extracted in each chamber, as indicated in Figure 3. Distillate extractions improve the plant performance ratio (PR), which is classically defined as the ratio between the water production and heat consumption. Typical PR values usually range between 5 kg h-1/kcal h-1 and 10 kg h-1/kcal h-1, depending on the operating costs and investments. For equal water production and heat consumption (PR ) 5), the total area corresponding to structure A (with distillate extraction) is 7.24% lower than that of structure B (without extraction). Therefore, structure A is more efficient than structure B, because, for the same water production and heat consumption, it requires less total area. To analyze more deeply the effect of the distillate extraction on the evaporator’s performance, a new simplified problem by relaxing the hypothesis set listed in Section IV is here studied. Briefly, the hypothesis assumed by this simple model is as follows: (a) the boiling point elevation (BPE) effect is not considered, (b) chamber geometry and recycle are not taken into account, (c) the overall heat-transfer coefficient and evaporation latent heat are assumed to be constant values, and (d) eight stages are adopted. It is clear that the resulting model is not realistic, because it involves only water; however, it is useful to determine if the resulting flow pattern is still preserved as an optimal solution under these conditions. Thus, the following two optimization problems (P1 and P2) were solved for two alternatives: with and without distillate extraction. (i) In regard to optimization problem P1, the heat-transfer area given to the feed flow rate (8290 ton/h) and heat consumption (200 Gcal/h) was minimized. Fresh water production and

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Table 2. Comparison of Vapor Production, Extracting Distillate, and Heat Recovery between Structures A and Ba Heat Recovery (Gcal/h) Vapor Production (ton/h) stage 1 2 3 4 5 6 7 8

Extracting Distillate (ton/h)

structure A

structure B

structure A

132.05 130.15 127.90 125.81 123.59 126.34 120.25 113.91

128.38 126.45 123.70 120.98 120.10 117.85 116.11 113.99

0 0 0 0 640 126.34 120.25

structure B

total 1000 967.56 886.59 a QDes ) 200 Gcal/h. Feed rate is 8290 ton/h (eight stages).

Structure A

Structure B

Qfresh

Qreflash

Qtotal

Qfresh

Qreflash

Qtotal

76.588 75.488 74.182 72.968 71.685 73.277 69.745 66.066

1.215 2.409 3.582 4.729 0 0 0

76.588 76.703 76.591 76.550 76.414 73.277 69.745 66.066

74.461 73.338 71.747 70.169 69.657 68.353 67.344 66.114

1.147 2.262 3.338 4.440 5.488 6.536 7.542

74.461 74.485 74.010 73.507 74.096 73.841 73.880 73.655

579.999

11.936

591.935

561.182

30.753

591.935

Table 3. Comparison of Heat-Transfer Area and ∆Tml between Structures A and Ba Heat Transfer Area (m2) Structure A ATotal

AFresh

25.411 50.319 74.813 98.773 0 0 0

1604.421 1604.199 1599.830 1598.718 1596.012 1575.382 1552.740 1532.681

total 12414.667 249.316 12663.983 QDes ) 200 Gcal/h. Feed rate is 8290 ton/h.

1 2 3 4 5 6 7 8

1604.421 1578.788 1549.511 1523.905 1497.239 1575.382 1552.740 1532.681

∆Tml (K)

Structure B

ARe-flash

stage

AFresh

ARe-flash

ATotal

structure A

structure B

1536.865 1511.803 1487.379 1463.597 1441.454 1417.940 1396.043 1374.746

23.513 46.645 69.397 91.775 113.789 135.452 156.781

1536.865 1535.316 1534.024 1532.994 1532.229 1531.729 1531.495 1531.527

19.094 19.125 19.150 19.153 19.150 18.606 17.967 17.243

19.243 19.280 19.331 19.369 19.317 19.315 19.289 19.280

11629.827

637.352

12266.179

a

Table 4. Comparison of Vapor Production, Extracting Distillate, and Heat Recovery between Structures A and B. Heat Recovery (Gcal/h) Vapor Production (ton/h) stage 1 2 3 4 5 6 7 8

Extracting Distillate (ton/h)

structure A

structure B

structure A

132.05 130.15 127.90 125.81 123.59 126.34 120.25 113.91

132.01 130.08 128.08 126.04 123.99 121.94 119.93 117.93

0 0 0 0 640.20 126.34 120.25

structure B

total 1000 1000 886.79 a QDes ) 200 Gcal/h. Production rate is 1000 ton/h (eight stages).

discharge temperature are the problem variables. Tables 2 and 3 compare both solutions. (ii) In optimization problem P2, the same objective function for a given fresh water production (1000 ton/h) and heat consumption (200 Gcal/h) was minimized. Here, the feed flow rate is an optimization variable. Tables 4 and 5 compare the solutions achieved for both structures. Solving optimization problem P1 for the structure with distillate extractions, the solution is characterized by the extraction of the total distillate formed in stages j ) 1 to j ) 5 on stage j ) 5 and the distillate produced from stage j ) 6 to j ) 8 is completely extracted in each corresponding stage. According to Tables 2 and 3, this structure produces 1000 ton/h, requiring 12663.983 m2 of heat-transfer area, whereas the other structure (without distillate extraction) produces 967.56 ton/h, requiring 12266.179 m2 of heat-transfer area. The total heat recovery (Qrec) and heat consumption (QDes) in both structures are 591.935 and 200 Gcal/h, respectively. The temperatures of the blow-down stream are 325.83 and 328.89 K for the structure with and without distillate extractions, respectively.

Structure A

Structure B

QFresh

QRe-flash

QTotal

QFresh

QRe-flash

QTotal

76.588 75.488 74.182 72.968 71.685 73.277 69.745 66.066

1.215 2.409 3.582 4.729 0 0 0

76.588 76.703 76.591 76.550 76.414 73.277 69.745 66.066

76.564 75.447 74.289 73.106 71.913 70.724 69.559 68.400

1.183 2.350 3.497 4.622 5.726 6.809 7.870

76.564 76.630 76.639 76.602 76.535 76.450 76.368 76.270

579.999

11.936

591.935

580.002

32.057

612.059

The total heat-transfer area of the second structure is smaller than the first, because of a driving force increment; however, its production is lower. Despite the fact that the total heat recovery is the same in both structures, the ratio between the 8 total heat recovered from “fresh” vapor ∑j)1 Qfresh (which conj tributes to the water production) and total vapor produced by 8 the distillate re-flashing ∑j)1 Q re-flash (which does not contribj ute to the production) is different. This ratio is higher when distillate extraction is permitted, increasing the performance ratio. In this process, unlike classical problems of heat-exchange networks, the water production should be considered. There is a tradeoff between water production, heat recovered from distillate re-flashed, and the heat-transfer area. Therefore, the difference between both alternatives is the “re-flashing” operation of the distillate. Table 3 also shows, for the structure with distillate extractions, a significant decrease in both the driving force for heat transfer on the preheaters and heat-transfer area from stage j ) 6, where the distillate extraction begins, whereas a flat profile is obtained for the other structure.

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7195 Table 5. Comparison of Heat-Transfer Area and ∆Tml between Structures A and Ba Heat-Transfer Area (m2) Structure A stage 1 2 3 4 5 6 7 8 total a

QDes

AFresh

∆Tml (K)

Structure B

ARe-flash

ATotal

AFresh

ARe-flash

ATotal

structure A

1604.421 1578.788 1549.511 1523.905 1497.239 1575.382 1552.740 1532.681

25.411 50.319 74.813 98.773 0 0 0

1604.421 1604.199 1599.830 1598.718 1596.012 1575.382 1552.740 1532.681

structure B

1644.503 1617.549 1591.223 1565.545 1540.529 1516.181 1492.711 1469.276

25.372 50.331 74.877 99.013 122.748 146.119 169.061

1644.503 1642.921 1641.554 1640.422 1639.542 1638.929 1638.830 1638.337

19.094 19.125 19.150 19.153 19.150 18.606 17.967 17.243

18.623 18.657 18.675 18.679 18.672 18.658 18.639 18.622

12414.667

249.316

12663.983

12437.517

687.435

13124.952

) 200 Gcal/h. Production rate is 1000 ton/h.

Table 4 compares both structures for equal water production (1000 ton/h) and same heat consumption (200 Gcal/h). As indicated in Table 4, to produce 1000 ton/h, according to the structure without distillate extraction, an increment of 1.1% (13124.95 m2) in the heat-transfer area and an increment of 3.06% (8548 ton/h) in the feed flow rate are required. According to this, for the same production and heat external consumption, both the heat-transfer area and feed flow rate for the structure with distillate extraction are lower than those for the other structure. Finally, according to the results presented in Tables 2-5, it is possible to conclude that the evaporator operating with distillate extraction is more attractive than the classic configuration, by decreasing not only the heat-transfer area (investments) but also the seawater flow rate (operating costs, and, more precisely, chemical pretreatment cost). It must be mentioned that, in this simple problem, only the distillate stream is allowed to be split. Splittings of the feed and blow-down streams are avoided. The achieved solution of this study case presents a similar behavior, in regard to the decrease in heat-transfer area from stage j ) 21, where distillate extractions are performed. The decrease in the heat-transfer area is not only due to the distillate extractions but also due to different seawater flow rates on the preheaters, which, in turn, are functions of the brine recycle streams. V.1.2. Discharged Streams Allocation. Another distinguishing characteristic, compared to the “conventional” and “mixer” structures, is that the rejected stream (W blow-down) is not located at the last stage. The rejected stream is located at stage j ) 27 (see Figure 3). This allocation produces a slight reduction of the size of the stage j ) 28 (the desaltor investment). V.1.2.1. Recycle Streams. The purpose of the brine recycle is to reduce the chemical additive consumption and minimize the pretreatment cost. In our design, the recycle flow pattern is composed of two recycle streams (different from the conventional one). Precisely, the optimal allocations of the recycle streams are in stages j ) 26 and j ) 29. Part of the flashing brine is mixed in stages j ) 26 and j ) 29 with seawater incoming from stages j ) 23 and j ) 26, respectively, and the resulting streams then are recirculated from stage j ) 26 to stage j ) 22 and from stage j ) 29 to stage j ) 25 (see Figure 3). Figure 4 illustrates the driving force for heat transfer on the preheaters and the heat-transfer area (HTA) distributions through the flashing chambers for the achieved solution. It can be shown that the ∆Tml profile increases in the stages where the recycle and distillate extraction occur (stages j ) 22 to j ) 29), while the HTA profile decreases considerably in these stages and, consequently, the desaltor investment is reduced.

Figure 4 also shows a flat profile for both ∆Tml and Atub from stage j ) 1 to stage j ) 21. As was mentioned previously, the desaltor operates from stage j ) 1 to stage j ) 21, according to the classic desalination theory (that is, with a constant flow rate on the preheaters and without distillate extraction). From stage j ) 1 to stage j ) 22, the flow rate in the preheaters is 8636 ton/h. For stage j ) 23, the flow rate on the preheater decreases (5 608 ton/h) and then it increases for stages j ) 24 and j ) 25 (7544 ton/h). Finally, the flow rate decreases to a constant flow rate in the last four stages (3664 ton/h). From stage j ) 22 to stage j ) 29, the evaporator does not operate traditionally and distillate extractions are performed. In fact, the distillate produced and accumulated from stage j ) 1 to stage j ) 21 is totally extracted in stage j ) 21 and, from this stage, all the distillate produced in each remaining stage is completely extracted. The optimal allocations of the recycle reduce the irreversibility of the mixing process and the driving force for the heattransfer operation is increased whereby reducing the heat-transfer area (see Figure 4). V.1.2.2. Cooling Water. Finally, the allocation of the cooling seawater and its composition are also different from the conventional structure (commonly used). Seawater intake is used as a cooling utility in the conventional configuration. In this novel structure, the composition of the cooling seawater is higher than the conventional structure, because it is composed, in part, of the seawater intake and flashing brine (see Figure 3). Up to this point, the structural characteristics of the achieved configuration have been analyzed and interpreted. On the other hand, Figure 5 shows the length, the gate height, and the width stage distribution through the flashing chambers. The tubes number, the overall heat transfer, and the lineal velocity distributions through the preheaters are illustrated in Figure 6. It is clear that the achieved solution is useful for the detailed design of the process.

Figure 4. Distribution of heat-transfer area and ∆Tml through the stages.

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Figure 5. Distribution of length, width, and gate height through the stages.

Figure 6. Distribution of brine velocity, number of tubes, and heat-transfer coefficient through the stages.

In the following section, the new design will be compared with the optimal MSF-mixer alternative. V.1.3. Comparison between the New Design and the Optimal MSF-Mixer Configuration. The MSF-mixer optimization was described in ref 6. Both solutions are obtained by applying the problem data given in Table 1. The MSF-mixer configuration is obtained here by fixing only one recycle (at the last stage). Figures 7a and 8a illustrate the resulting flowsheet for each alternative. For clarity, Figures 7b and 8 b only include the distribution of ∆Tml, product production, overall heat-transfer coefficient, heat recovery, and heat-transfer area through the stages to distinguish specific characteristics of both designs and to visualize the differences between both configurations. The complete distribution of these variables through the stages for both structures is shown in Figures 9-12. As is shown in Figures 7b and 8b, each configuration involves different flow patterns for the recycle stream, as well as different distributions of ∆Tml, produced vapor, total heat recovery, and heat-transfer areas through all the stages. Distillate extraction is also performed in both configurations, producing the same effect on the performance of the desaltor. This design clearly presents the optimal distribution and allocation for the recycle streams, which, in turn, reduces the heat-transfer area by increasing the driving force for the heattransfer process. On the other hand, the “mixer” configuration is characterized by an inefficient ∆Tml, recirculation flow rates, and vapor production profiles. Contrary to the new MSF configuration, the seawater is preheated not only by the latent heat of the condensing vapors (as observed in the new MSF design), but also by the mixing process between the seawater stream and part of the flashing brine, which increases both its temperature and composition. This recycle strategy considerably reduces the driving force for the heat-transfer process and increases the

Figure 7. (a) Optimal structure for study case 1. (b) Detailed description of the last stages for the optimal structure for the study case 1.


Figure 8. (a) Structure derived previously by Mussati et al.6 (b) Detailed description of the last stages for the structure derived from the model presented in Mussati et al.6

Figure 9. Distribution of recovered heat through the stages.

Figure 10. Distribution of vapor production through the stages.

heat-transfer area, with respect to the novel MSF configuration. Also, a different vapor profile in the last stages can be observed, as a consequence of the mixing process (see Figures 7b and 8b). Finally, Table 6 shows the optimal values for each design. Despite the fact that both structures involve different arrangments, its TAC values are slightly similar. In the next section, a sensitivity analysis is presented to show the influence of the critical variables on the achieved optimal structure. V.2. Study Case 2. In this section, as a second study case, several optimization problems are solved for the same water production (1000 ton/h) to analyze the influence of the most important parameters such as the maximum admissible temperature, investment capital and operating costs, and the factor F on the optimal structure. V.2.1. Influence of Maximum Admissible Temperature on the Performance of the Desaltor. The optimization problem is solved for different values for Tmax and the same data given in Table 1. Table 7 shows the main values achieved by adopting Tmax ) 388, 377, and 360 K.

According to the results presented in Table 7, QDes, the total heat-transfer area, and the number of stages are strongly influenced by Tmax, as was expected. For low values of Tmax, the TAC, the total area (heat-transfer area and stage area), the feed flow rate, and the hot utility increase while the optimal number of stages decreases. On the other hand, regarding to the optimal structure, it should be mentioned that solutions presented in Table 7 preserve characteristics similar to those analyzed in study case 1 but the locations of the distillate extractions, recycles, and cooling seawater streams are modified depending on Tmax. V.2.2. Influence of Investment and Operating Costs on the Performance of the Desaltor. Investment and operating costs strongly influence the final configuration of the system. To know the impact on the optimal solution, many study cases were solved and analyzed for different cost ratios by perturbing the cost data listed in Table 1 (study case 1). In fact, the optimization problem has been solved for different F values, which are used to compute the investment of flashing

7198


Figure 11. Distribution of ∆Tml through the stages.

Figure 12. Distribution of recovered heat through the stages.

Table 6. Optimal Values for the Novel and MSF-Mixer Configurations

VI. Conclusions and Future Works

variable

novel structure

MSF-mixer configuration

total annual cost, TAC hot utility total heat recovery stage chamber area heat-transfer area feed total recirculation flow rate number of tubes per stagea

$1.275 × 106/yr 82.088 Gcal/h 573.546 Gcal/h 27935.144 m2 4.0098 × 106 m2 3664 ton/h 16180 ton/h 1129.093

$1.302 × 106/yr 83.863 Gcal/h 573.913 Gcal/h 28432.117 m2 4.2299 × 106 m2 3508 ton/h 5096 ton/h 1193.647

a

Average value.

chambers. The results indicate that the optimal number of stages decreases considerably in the range of 10 e F e 20, whereas for values >20, the optimal NS decreases asymptotically to NS ) 25; in all cases, the optimal configurations (recycles and distillate flow patterns) are still preserved. On the other hand, the influence of the recirculation pumping cost on the optimal solution was studied. From the obtained results, we can conclude that the TAC and the system configuration are not strongly influenced by the recirculation pumping cost. Summarizing, the final process is dependent on the ratio between total investments and operating costs, but optimal solutions preserve similar characteristics. In fact, the optimal number of stages and stream flow patterns are different, depending on costs; however, their existence is still preserved.

A detailed and rigorous nonlinear programming (NLP) model based on a superstructure of alternatives for the process has been developed. The mathematical formulation and superstructure recently proposed by ref 6 has been appropriately reformulated to include more alternative configurations for the process. The mathematical model is formulated by stage and component, including the detailed geometric design of each stage (length, height, and width), the number of tubes in the preheaters, and the fluid-dynamic equations for the streams, among other factors. It also considers the number of stages and the stream flow patterns (distillate, feed, recycle, and discharge brine) as structural variables. A novel configuration for the multistage flash (MSF) system resulted, solving the problem based on the proposed superstructure. This new structure differs from the conventional structure, in regard to the flow patterns of the distillate recycle, cooling seawater, and blow-down brine streams. From the examples presented in this work, it is possible to conclude that the novel arrangement is still preserved by adopting different values, perturbing the most critical parameters of the process. That is, the distribution of distillate extractions along the evaporator, the blow-down stream location (not placed at the last stage), and back recycles are the more relevant improvements presented in this paper on the configuration of the process. However, for a deep analysis of the new structure and its feasibility, other aspects such as controllability, operability, and

Table 7. Optimal Solutions for Different Tmax Valuesa Value

a

variable

Tmax ) 388 K

Tmax ) 377 K

Tmax ) 360 K

total annual cost, TAC feed hot utility number of stages total recovered heat total stage chamber area total heat-transfer area stage chamber areab heat-transfer areab stage lengthb stage widthc stage heightb ∆Tml per stageb brine velocity in preheaterb number of tubes per stageb iteration (rigorous model) CPU time (rigorous model)

$1.275 × 106/yr 3664.32 ton/h 82.088 Gcal/h 29 573.546 Gcal/h 27935.144 m2 4.0098 × 106 m2 963.27 m2 137931.03 m2 0.820 m 12.21 m 0.507 m 6.590 K 2.74 m/s 1129.09 2689 70.490



Water production rate is 1000 ton/h. b Average value. c Optimization variable assuming the same length for all existing stages.


reliability must be taken into account, and they may be important in the selection of the optimal configuration. Therefore, the final decision must be analyzed deeply, including all of the aforementioned aspects. Indeed, a new pumping cost model that explicitly includes investment cost should be used to verify the influence on the recycle flow pattern in optimal solution. A mixed-integer nonlinear programming (MINLP) model will be presented in future work for this analysis. Summarizing, through incorporation of the analysis of different stream flow patterns (recirculations, distillate and cooling seawater, and discharges), a new research direction is opened from the results here presented. This is also important for hybrid desalination processes. For example, the possible integration of streams that result from MSF evaporators (distillate, recycles) with the RO process on hybrid desalination systems is an interesting and attractive point to be investigated. On the other hand, power plants can also be coupled to MSF and multieffect desalination (MED) desalination systems to produce electricity and fresh water more efficiently.

Primary Flashing Chamber (Pool Brine). For the primary flashing chamber (pool brine), the mass and energy balances are given as follows: 1

WCONVj,k + W B,inl - W B,out - V pj ∑ j j j

Superstructure of Alternative Configurations of the Multistage Flash

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