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The design of reverse osmosis systems with multiple-feed and multiple-product
Desalination 307 (2012) 42–50
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The design of reverse osmosis systems with multiple-feed and multiple-product Yanyue Lu a,⁎, Anping Liao a, Yangdong Hu b a School of Chemistry and Chemical engineering, Key Laboratory of Chemical and Biological Transformation Process of Guangxi Higher Education Institutes, Guangxi University for Nationalities, Nanning, 530006, China b College of Chemistry and Chemical Engineering, Ocean University of China, Qingdao, 266003, China
H I G H L I G H T S ► ► ► ► ►
A process synthesis-based optimization technique has been developed. Membrane units were approximated by the pressure vessel model. The module model takes into account the pressure drop and concentration changes. The stream split ratios and logical expressions of stream mixing were employed. The design results present optimal structure and the optimal streams distribution.
a r t i c l e
i n f o
Article history: Received 1 June 2012 Received in revised form 21 August 2012 Accepted 22 August 2012 Available online 25 September 2012 Keywords: Reverse osmosis Desalination Optimum design Mathematic model Process synthesis
a b s t r a c t A reverse osmosis (RO) desalination process with multiple-feed and multiple-product is the main focus of this work. A process synthesis-based optimization technique has been developed for the design of the RO system. The adoption of this approach provides an economically attractive desalination scheme. Membrane separation units employing the spiral wound reverse osmosis elements were approximated by the pressure vessel model presented in this paper, which takes into account the pressure drop and concentration changes in the membrane channel. A simplified superstructure that contains all the feasible design for this desalination problem has also been presented. In this structure representation, the stream split ratios and logical expressions of stream mixing were employed, which can make the mathematical model to be easily handled. The optimum design problem is formulated as a mixed-integer non-linear programming (MINLP) problem, which minimizes the total annualized cost of the RO system. The cost equation relating the capital and operating cost to the design variables, as well as the structural variables has been introduced in the objective function. The solution of the problem includes the optimal system structure and operating conditions, and the optimal streams distribution. The design method could also be used for the optimal selection of the type of membrane elements in each stage and the optimal number of membrane elements in each pressure vessel. The effectiveness of this design methodology has been demonstrated by solving a desalination case. The comparisons of several alternate schemes indicate that the feed position of streams and outlets of the system are the critical variables that should be optimized for the RO system design. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Desalination of sea and brackish waters is the main source for supplying fresh water in the regions suffering from the scarcity of natural fresh water supply. Since 1960s, due to the development of new reverse osmosis (RO) modules and membranes, RO is experiencing growing applications in the desalination field. Now it has become a major technology for large-scale desalination plants for both seawater and brackish water sources [1–5]. The interest in RO is due to its low energy consumption (as compared to multistage flash distillation process), high product recovery ⁎ Corresponding author. Tel.: +86 0771 3260558. E-mail address: [email protected] (Y. Lu). 0011-9164/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2012.08.025
and quality. The other attractive feature of RO process is their modular plant design and ease of operation. Membrane plants are often more compact, can be scaled up easily and installed more quickly than thermal separations plants. Also, it makes the maintenance of RO systems easier. Another advantage of the RO process is that it is able to meet varying feed water concentration and varying production water quantity and quality requirement through the change of system configuration and operation condition. RO membrane manufacturers have developed various membrane types to precisely meeting the varying needs of a wide range of industrial, municipal, commercial and drinking water application. This includes high flux, high rejection, fouling resistant, low pressure, and high rejection membrane, etc [6–10]. These features allow the design of RO processes to be more flexible.
Y. Lu et al. / Desalination 307 (2012) 42–50
Considerable efforts have been made for the research of the optimum design of RO system [6,11–14]. El-Halwagi [15] investigated the synthesis of RO networks which involve multiple feed streams for waste reduction. Based on the state-space approach, a structural representation of RO networks was introduced. The RO networks were described using four boxes, i.e. a pressurization/depressurization stream-distribution box, a pressurization/depressurization matching box, a RO stream-distribution box, a RO matching box. The function of the distribution boxes is to represent all possible combinations of stream splitting, mixing, bypass and recycle. The matching boxes locate all possible stream assignments to units. With this formulation, all possible structure arrangements are represented. The mathematical model was formulated as a mixed integer nonlinear programming (MINLP). In their later work, Zhu et al. [16] included a factor for flux decline over time based on their earlier work [15]. Voros et al. [17] simplified El-Halwagi's representation by reducing the distribution boxes to junctions. Consequently, the model was formulated as a nonlinear programming (NLP) model by using a variable split ratio. Maskan and Wiley [19] used a directed graph and connectivity matrix to represent the RO networks superstructure. In the mathematical model of the superstructure, a variable reduction technique is performed to accelerate the computational process. Nemeth [20] studied the behavior of the ultra-low pressure RO membranes in the full-scale system and presented recommendations to improve system design. Van der Meer [21,22] and Wessels [23] developed a simplified mathematical model to optimize the performance of NF and RO membrane filtration plants. The study showed that the productivity of nanofiltration plants can be significantly improved by installing a reduced number of membrane elements serially in pressure vessels (PV) and by changing system configuration. Malek et al. [24] provided a realistic economic model that relates the various operational and capital cost elements to the design variable values. In this work, an RO-based desalination process is considered for the production of fresh water from three raw feeds (seawater, brackish and regenerated water). A systematic methodology is presented for the optimal design of RO desalination system that processes multiple feed streams simultaneously, and at the same time, supplies various product streams (water) of different quality. The adoption of this approach can provide an economically attractive desalination scheme. This leads to significant energy and raw-materials saving and generates income from the sales of multiple grades of water products. First, a simplified superstructure representation that contains all feasible designs for this desalination problem is presented. Then a pressure vessel model is also developed. The model could be used for the optimum selection of types and number of membrane element, according to its performance characteristics, the prices, and the design requirements of a specific desalination application. Therefore the optimal design of RO systems was formulated as a mixed integer nonlinear programming problem (MINLP). The objective is to determine the optimal system structure, stream distribution and operating conditions; subject to the constraints of the multiple-feed and multiple-product system. The solution to the problem also includes the most appropriate choice of the type of membrane elements in each stage and the optimal number of membrane elements in each PV. 2. RO unit model 2.1. The mass transfer model of RO process Numerous models to predict membrane performance have been introduced [7,25,26]. They are derived from different theories and all of them may be simplified to the solution diffusion model, as shown in Eqs. (1) and (2). For the RO system design and optimization, it is necessary to adopt the appropriate modeling equations that can satisfactorily predict the membrane performance with reasonable computational complexity. Therefore the solution diffusion model is
43
among the most commonly used model in RO system design. The model is mainly based on two parameters, i.e. water permeability (A) and solute transport parameter (B). Values for these parameters are usually specified by membrane manufacturers. According to the model, the pure water flux, Jw (kg/m 2.s), and the salt flux, Js (kg/m 2.s), are given as follow: Jw ¼ A
ΔP 6 P f −P p − f − πw −πp 10 2
J s ¼ B C w −C p π¼
ð2Þ
0:2641 C ðT þ 273Þ 1:0 106 −C
ð3Þ
Jw þ Js ρp
ð4Þ
Js 1000 Vw
ð5Þ
Vw ¼
Cp ¼
ð1Þ
where Pf and Pp (Mpa) denote feed and permeate pressure, respectively; ΔPf is the pressure drop in the membrane channel; ∏w (Mpa) is the osmotic pressure of the brine at the membrane wall concentration Cw (ppm), and ∏p and Cp are corresponding variables for the permeate; ρp denotes the density of the permeate. Vw (m/s) is the permeate velocity. In mass transfer process, the variation of concentration on membrane surface should be considered. The change can be represented by Eqs. (6) to (8). Cw ¼ Cp þ
V w Cf þ Cb −C p e K 2
ð6Þ
K (m/s) is the mass transfer coefficient, which can be calculated from empirical relations such as: 0:75
K ¼ 0:04 Re Re ¼
0:33
Sc
Ds d
V ρd μ
ð7Þ ð8Þ
where Re and Sc are the Reynold's and the Schmidt numbers and Ds is the solute diffusivity. d is the feed spacer thickness, ρ is the feed side solution density and μ is the water viscosity. V denotes the flow velocity that was calculated using the averaged values of the inlet and outlet flow rates in the membrane channel. 2.2. The model for RO module In the practical process, the multiple stages RO configuration would be used, one RO stage consists of multiple parallel RO pressure vessels operating at the same conditions. Each pressure vessel contains several membrane elements that are connected in series. The concentrate of the first element becomes the feed to the second, and so on. The products tubes of all elements are coupled and connected to the module permeate port [21–23]. For different application a suitable hydraulic design can be made (2, 3, 4, 5, 6, 7, 8 serial elements), based on the actual situation. Fig. 1 shows the schematic representation of a RO module. Al-Bastaki and Abbas [25,26] presented the models of the spiral-wound and hollow-fiber membrane elements, which took into account the pressure drop and concentration changes in the membrane channel. The PV performance can be approximately simplified to the performance of the membrane elements connected in series. Therefore, based on the membrane element models, a pressure vessel model is presented as follows which
44
Y. Lu et al. / Desalination 307 (2012) 42–50
Feed
Pf
Membrane module Brine
Q b Cb Pb
Qf Cf
Permeate
Qp Cp Pp Fig 1. The schematic diagram of a RO unit.
would be used for the optimal selection of the membrane element type and number. For a spiral-wound membrane module, each of the permeate and feed side flows can be considered as a flow between two parallel plates with a length L, a width W and a spacing d. Hence the pressure drop on the feed side can be calculated as follows: ΔP f ¼
0:0033 Q a Lpv μ
ð9Þ
W d3
Lpv ¼ m Lm
ð10Þ
ΔP f ≤0:35
ð11Þ
Q f Cf ¼ Q b Cb þ Q p Cp L
ð15Þ
U
Q rated ≤Q f ≤Q rated
ð16Þ
where Q Lrated and Q Urated are respectively the lower bound and upper bound of rated flow rate of pressure vessel. When a RO stage consists of multiple parallel pressure vessels, and operates at the same conditions, the flow rate of stream entering and leaving this RO stage can be calculated as following: Q ROf ¼ n Q f
ð17Þ
Q ROp ¼ n Q p
ð18Þ
Q ROf ¼ Q ROp þ Q ROb
ð19Þ
3
where Qa (m /h) is the flow rate that was calculated using the averaged values of the inlet and outlet flow rates in the membrane channel; Lpv and Lm denote the length of the PV and the length of a membrane element, respectively; m is the number of membrane elements in each PV. In order to reduce the computation complexity, m is considered to be a continuous variable, and the approximate result would be obtained by rounding the variable. The maximum allowable pressure drop of the pressure vessel is 0.35 Mpa. The technical constraint is usually specified by membrane manufacturers. For the spiral-wound membrane element, the membrane width (W), can be calculated by the membrane area(Sm) and the number of leaves(Nl) : S m ¼ W Lm N l
where QROf, QROp and QROb are respectively the feed flow rate, permeate flow rate and brine flow rate entering and leaving a RO stage. n is the number of pressure vessel in a RO stage which is considered to be a continuous variable, and the approximate result would be obtained by rounding the variable. 3. RO system model 3.1. Problem description
ð12Þ
For a pressure vessel, the feed flow rate, Qf (m 3/h), the permeate flow rate, Qp (m 3/h), the brine flow rate, Qb (m 3/h), and the corresponding concentration, Cf, Cp, Cb (ppm), can be calculated from the mass and salt balance equations: Q p ¼ 3600 V w Sm m
ð13Þ
Qf ¼ Qb þ Qp
ð14Þ
For the RO desalination system with multiple-feed and multipleproduct requirement, the design objective is to identify the most cost effective RO network configuration, the optimal streams distribution and operating conditions, and the optimal arrangement of the membrane elements. Fig. 2 represents a potential multiple stages RO configuration for this desalination problem. It is necessary to develop a structure representation that contains all feasible designs of RO network synthesis. The network representation based on the state-space approach was presented earlier by El-Halwagi
membrane module
Q p,1 C
high pressure pump
Q f,1 Cf,1 turbine
Q b,1 C b,1
high pressure pump
Q p,2
Q f,2 Cf,2 Q b,2 C
b,2
Fig. 2. RO process configuration with multiple-feed and multiple-product.
p,1
Y. Lu et al. / Desalination 307 (2012) 42–50
[15,16] and Voros et al. [17]. Here we adopt and properly modify the approach, and in turn present a simplified superstructure that incorporates all the feasible process flow for the RO desalination system with multiple-feed and multiple-product. As shown in Fig. 3, an RO network consists of Nps pressurization stages and NRO reverse osmosis stages. In this configuration, there are four sets of stream nodes employed: a set WT={q| q=1, Nw} of inlet junctions for the feed streams; a set PS={j| j=1, Nps} of stream mixing junctions; a set R={i| i=1, NRO} of reverse osmosis junctions; a set PD={r| r=1, Np} of outlet junctions for the product streams. The junction of Nps +1 indicated the brine stream leaving the network. Each mixing junction in the PS set indicates that the feed streams, the brine and the permeate streams leaving all reverse osmosis stages are mixed at the node. The mixing streams pressurized by high pressure (HP) pump or not are connected to the corresponding reverse osmosis stages. The RO stages consist of multiple parallel pressure vessels operating at the same conditions.
The complete mathematical model that describes the superstructure is presented as follows by means of the appropriate relationships between the variables (material and energy balance equations, technical and operational constraints). Nw NRO X X Q f ;q xf ;i;q þ Q ROb;j xb;i;j q¼1
j¼1
ð20Þ
N RO X þ Q ROP;j xp;i;j
i ¼ 1; 2; …; Nps
j¼1
Q ps;i C ps;i ¼
Nw X
Q f ;q xf ;i;q C f ;q þ
q¼1
þ
NRO X
Q ROb;j xb;Nps þ1;j
ð22Þ
j¼1
Q ps;Nps þ1 C ps;Nps þ1 ¼
NRO X
Q ROb;j xb;Nps þ1;j C ROb;j
ð23Þ
j¼1 N ps X
xf;i;q ¼ 1
q ¼ 1; 2; …; Nw
ð24Þ
j ¼ 1; 2; …NRO
ð25Þ
j ¼ 1; 2; …NRO
ð26Þ
i¼1 Nps þ1
X
xb;i;j ¼ 1
i¼1 Nps þN p
X
xp;i;j ¼ 1
i¼1
3.2. Mathematical formulation
Q ps;i ¼
Q ps;Nps þ1 ¼
45
N RO X Q ROb;j xb;i;j C ROb;j
i ¼ 1; 2; …; N ps
j¼1
ð21Þ
NRO X Q ROp;j xp;i;j C ROp;j j¼1
where Qps,i and Cps,i denote the flow rate and concentration of the ith pressurization stage, respectively. Qf, q Cf, q denote the qth feed flow rate and concentration of the RO network. QROb, j, CROb, j denote the brine flow rate and concentration of the jth RO stage, QROp, j and CROp, j denote the permeate flow rate and concentration of the jth RO stage, respectively. xf, i, q, xb, i, j and xp, i, j indicate the stream split ratios of the feed, the brine and permeate, respectively. The values determine the flow rates of brine and permeate leaving the jth RO stage and being linked to the ith pressurization stage. All streams connected to ith pressurization stage firstly mix in the mixer. The outlet pressure from the mixer is the average feed pressure. When the raw streams enter the RO system from interstage, it must be pressurized to the outlet pressure of the pressurization stage by high pressure pump. The stream split ratios and the logical expression of stream mixing are employed in this paper, these techniques reduced the number of binary variable and the solving space, therefore the mathematical model may be easily handled. Following are the stream-mixing constraints. L1 1−yb;i;j ≤xb;i;j −xc ≤U 1
PS stage
Q f,1 Cf,1 feed
yb;i;j −ε
ð27Þ i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO
1 1
L1 1−yp;i;j ≤xp;i;j −xc ≤U 1
mixer
q
2 i
Nw
Nps
yp;i;j −ε
Qb Cb
N ps +1 1 2
p
p
Q1 C 1 p p Q2 C 2
yb;i;j þ yp;i;j ≤1
ð28Þ i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð29Þ
L2 1−yb;i;j ≤ P b;j −P b;i;j ≤U 2 1−yb;i;j
i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO
P b;i;j ≤U 2 yb;i;j
i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð31Þ
L2 1−yp;i;j ≤ P p;j −P p;i;j ≤U 2 1−yp;i;j
i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO ð32Þ
P p;i;j ≤U 2 yp;i;j
i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð33Þ
ð30Þ
Np RO stage NRO j
stage 2
permeate
stage 1
2
1
Q p,1 Cp,1 Q b,1 Cb,1 brine Fig. 3. Representation of the RO network via the superstructure.
NRO X P b;i;j þ P p;i;j
P ps;i ¼
j N RO X yb;i;j þ yp;i;j j
i ¼ 1; 2; …; N ps ð34Þ
46
Y. Lu et al. / Desalination 307 (2012) 42–50
where L1 and L2 are the arbitrary small numbers, U1 and U2 are large enough numbers. y is the binary variable. xc and ε denote the small positive number. It is assumed that the streams link to the ith pressurization stage when the stream split ratios, x, is larger than xc, hence y = 1, whereas if x is smaller than xc, then y = 0. Pb,j and Pp.j are the brine and permeate pressure of the j th RO stage, respectively. Pb,i, j, Pp. i, j denote the streams pressure which link to the ith pressurization stage. Pps,i denotes the inlet pressure of the ith pressurization stage. The streams leaving the ith pressurization stages are correspondingly connected to the jth RO stages. Therefore the following equations can be obtained:
Qp, j = 0. By this way the module type and number of stage are chosen simultaneously when the optimization design is performed. The overall material balances for the RO network and a set of product quantity and quality constraints concerning the minimum desirable product flow rate, and the maximum allowable product concentration are presented as follow: Nw X
b
Q f ;q ¼ Q þ
q¼1
Nw X
b
ð35Þ
C ROf ;j ¼ C ps;i
j ¼ i; j ¼ 1; 2; …; N RO
ð36Þ
Q ¼ Q ps;Nps þ1
j ¼ i; j ¼ 1; 2; …; NRO
ð37Þ
C ¼ C ps;Nps þ1
where QRO f, j, CROf, j and PRO, j denote the feed flow rate, concentration, and operation pressure of the jth RO stage, respectively. P 'ps,i is the outlet pressure of the ith pressurization stage. The mathematical models that predict the performance of each RO stage have been presented in detail in the previous section. These model equations relate the flow rate and concentration of the brine and permeate leaving an RO stage to the flow rate, concentration, and pressure of the stream entering the stage. The arrays of pressure vessels (PV) with 2 up to 8 membrane elements per PV consist of a RO stage. In this paper, the optimal PV structure has been researched. Four different types of spiral wound FilmTec reverse osmosis membrane elements have been considered. According to its performance characteristics and the requirements of a specific desalination application, the optimal selection of types of the membrane element employed in each PV can be determined by the following equations: 4 X
Z j;k ≤1
j ¼ 1; 2; …; NRO
ð38Þ
k¼1
! 4 X 1− Z j;k ≤Q ROf ;j −xc ≤U 3
L3
ð39Þ
k¼1
4 X Z j;k −ε
j ¼ 1; 2; …; NRO
k¼1
Q p;j ¼
4 X
Q p;j;k
j ¼ 1; 2; …; N RO
ð40Þ
Q p;j;k ≤U 4 Z j;k
j ¼ 1; 2; …; NRO
ð41Þ
mj;k ≤U 5 Z j;k
j ¼ 1; 2; …; N RO
ð42Þ
k¼1
where Zj,k is the binary variable. It takes the value of 1 when the kth element type is utilized in the PV in the jth RO stage, otherwise, it takes the value of 0. Here, it is assumed that the same membrane element is employed in the same RO stage. L3 and U3, are the arbitrary small and large number, respectively. For the Eq. (39), when QROf, j takes some value which is larger than xc, it means the jth RO stage is present and one element type should be chosen, otherwise if QROf, j = 0 it means the jth RO stage is absent. Qp, j is the permeate flow rate of pressure vessel. While Qp, j, k denote the permeate flow rate of pressure vessel when the kth element type is utilized in this PV, it can be calculated by Eq. (13). mj,k is the number of the kth membrane element. U4 and U5 are large enough positive number so that Q p, j, k and mj,k are not restricted if Zj,k = 1. In terms of Eqs. (38), (40) and (41), Qp, j is equal to one of Qp, j, k when Zj,k j = 1, otherwise
ð43Þ
b
Q f ;q C f ;q ¼ Q C þ
j ¼ i; j ¼ 1; 2; …; NRO
0
p
Qr
r¼1
Q ROf ;j ¼ Q ps;i
P RO;j ¼ P ps;i
Np X
Np X
q¼1
p
p
Q r Cr
ð44Þ
r¼1
b
ð45Þ
b
ð46Þ
p
Qr ¼
NRO X
Q ROp;j xp;r;j
r ¼ 1; 2; …; Np
ð47Þ
r ¼ 1; 2; …; Np
ð48Þ
r ¼ 1; 2; …; Np
ð49Þ
r ¼ 1; 2; …; Np
ð50Þ
j¼1
p
p
Q r Cr ¼
N RO X
Q ROp;j xp;r;j C ROp;j
j¼1 p
p
Q r ≥Q r;min p
p
C r ≤C r; max
where Q b and C b are the flow rate and concentration of the brine leaving the RO network, respectively. Qrp and Crp are the flow rate and concentration of the rth product water required, respectively. xp, r, j are the outlet stream split ratios. Q pr,min refers to the minimum desirable flow rate of the r th product required, C pr,max refers to the maximum allowable concentration of the rth product required. The optimization design problem is formulated as an MINLP model for minimizing the total annualized cost subject to thermodynamic, technical, and flexibility constraints. The total annualized cost (TAC) of the system consists of two terms: annual operating cost (OC) and annualized capital cost (CC). The annual operating cost includes the energy cost required for the pump, membrane module maintenance cost (OCm). The annualized capital cost is for initial membrane module, pumps and energy recovery devices. The objective function is presented as follows: TAC ¼ CCin þ CChpp þ CCTb þ Cm 1:411 0:08 þ OCin þ OChpp OCTb þ OCm 0:96 CChpp ¼ 52 ΔP Q hpp
Cm ¼
NRO X j¼1
NRO X 4 X
C k mj;k nj
ð52Þ
ð53Þ
j¼1 k¼1
P Q Ce f c 3:6 ηhpp ηmotor
ð54Þ
P Q ηTb C e f c 3:6
ð55Þ
OChpp ¼
OCTb ¼
C pv nj þ
ð51Þ
where CCin, CChpp and CCTb are the capital cost of the seawater intake pump, the high pressure pump, and the turbine, respectively. OCin, OChpp and OCTb are the energy cost required for these pumps and the saving cost generated by turbine. Their functions refer to the
Y. Lu et al. / Desalination 307 (2012) 42–50 Table 1 Characteristics of FilmTec spiral wound reverse osmosis membrane elements. Element type
SW30XLE-400 SW30HR-380 SW30HR-320 BW30-400
Active area, ft2(m2) Length of the element, inch (mm) Diameter of the element, inch (mm) Feed space, mil⁎
400 (37.2) 40 (1016)
Feed flow rate range, m3/h Permeate flow, gpd (m3/d) Stabilized salt rejection, % Maximum operating pressure, psig (Mpa) a Pure water permeability constant A, (kg/m2.s.Pa) b Salt permeability constant B, (kg/m2.s) Membrane element cost, $ (estimation) a, b
380 (35.3) 40 (1016)
320 (29.7) 40 (1016)
400 (37) 40 (1016)
47
The solver DICOPT under the GAMS environment [27] is used to solve the MINLP problem by decomposing it into a series of nonlinear (NLP) sub-problems and mixed integer (MIP) master problems. Several starting points are used to obtain the best possible solution. 4. Case study
7.9 (201)
7.9 (201)
7.9 (201)
7.9 (201)
28 0.8–16
28 0.8–16
34 0.8–14
34 0.8–19
9000(34.1)
6000(22.7)
6000(22.7)
10,500(40)
99.70
99.70
99.75
99.5
1200 (8.3)
1200 (8.3)
1200 (8.3)
600 (4.5)
3.5 × 10−9
2.7 × 10−9
3.1 × 10−9
7.5 × 10−9
3.2 × 10−5
2.3 × 10−5
2.2 × 10−5
6.2 × 10−5
1200
1000
1400
800
Refer to the Refs. [10] and [18].
papers [2,6,16,24]. Cm denotes the total membrane module cost. Ck is the price of the kth membrane element and Cpv is the price of the pressure vessel. The nj is the number of pressure vessel employed in the jth RO stage. 1.411 is the coefficient that is used to calculate the practical investment. 0.08 is the capital charge rate. Ce is the cost of electricity and fc is the load factor. ηhpp, ηmotor and ηTb are the efficiency of HP pump, electric motor, and turbine, respectively. In this MINLP model, the concentrations of feed water and the product specification are defined, the variables include flow rates, concentrations and pressures of the intermediate sub-stream, number of pressure vessels, type and number of the membrane elements, operating pressures for each RO stage, stream split ratios (x), and binary variables (y, Z) used decision. This procedure is carried out by introducing an excessive number of stages as an initial guess, while at the optimum certain design variables, such as stream split ratios, are either set to zero or to a value that indicate the absence or presence of the specific stage. Therefore the structural optimization may take place in terms of eliminating all unnecessary pressurization stages or RO stages. The existence of the specific device may be determined indirectly by operational variables, such as the input flow rates to the unit, the pressure of the pressurization stage or RO stage.
Table 2 The parameters for calculation. Feed concentration of seawater, ppm Feed concentration of brackish, ppm Maximum brackish flow rate, m3/h Feed concentration of regenerated water, ppm Maximum regenerated water flow rate, m3/h Feed water average temperature, °C Average brine density,ρ, kg/m3 The average brine viscosity,μ, kg/m s Average diffusion coefficient, Ds, m2/s High pressure pump efficiency, ηhpp Turbine efficiency, ηTb Electric motor efficiency, ηmotor The cost of electricity, Ce, $/kw.h The pressure vessel cost, (estimation) $
42000 12000 100 5000 50 25 1020 1.09 × 10−3 1.35 × 10−9 75% 80% 98% 0.08 1000
The proposed methodology for optimum RO system design is applied to a desalination problem of three water sources: seawater, brackish and regenerated water. In this case, there are three water product outlets that satisfy different permeate quantity and quality requirement. The minimum desirable product flow rate for these outlets are 200 m 3/h, 100 m 3/h, 50 m 3/h, while the corresponding maximum allowed salt concentration are 100 ppm, 300 ppm, 500 ppm, respectively. The best water quality requirement is the major product constraint. Four different types of FilmTec reverse osmosis membrane elements from DOW have been included in the design studies of the current work. They are the low energy, high productivity element SW30XLE-400, the high rejection, high productivity element SW30HR-380, the high rejection, fouling resistant element SW30HR-320, and the high productivity, high rejection brackish RO element BW30-400. The geometrical properties and membrane characteristics (membrane pure water permeability, solute transport parameter) of these elements are given in Table 1 [10]. The necessary input data for this study case are summarized in Table 2 [6,16,24]. In general, the mathematic programming problem in this case includes 246 continuous variables and 21 discrete variables. The problem was solved by taking between 0.92 and 6.02 CPU time (sec), 1996 and 8489 iterations on the latest PC. It is assumed that seawater is the major feed water source. The feed flow rate of seawater is a variable that should be optimized. Whereas, the feed flow rates of brackish and regenerated water must subject to the maximum flow rate constraints. The results of the RO system optimization design are presented in Table 3. The three-stage RO configuration was employed in design (shown as Fig. 4). The data shown on each stream represent flow rate, concentration and the stream split ratio, respectively. The seawater with 495 m 3/h flow rate enters the system from stage 1, while the mixing Table 3 Design and optimization results for the study case. Process flow Seawater feed flow, Qf,1 (m3/h) Brackish feed flow, Qf,2 (m3/h) Regenerated water feed flow, Qf,3 (m3/h) Flow rate of the first product water, Qp1 (m3/h) Salt concentration of the first product water, Cp1 (ppm) Flow rate of the second product water, Qp2 (m3/h) Salt concentration of the second product water, Cp2 (ppm) Flow rate of the third product water, Qp3 (m3/h) Salt concentration of the third product water, Cp3 (ppm) Membrane type in stage 1 Number of elements per PV in stage 1 Number of PV in stage 1 Operating pressure in stage 1, P1 (Mpa) Membrane type in stage 2 Number of elements per PV in stage 2 Number of PV in stage 2 Operating pressure in stage 2, P2 (Mpa) Membrane type in stage 3 Number of elements per PV in stage 3 Number of PV in stage 3 Operating pressure in stage 3, P3 (Mpa) The total annualized cost, ($)
Three-stage RO system, show as Fig. 4 495 100 50 200 100 100 300 50 300 SW30XLE-400 5 82 7.3 SW30HR-380 3 33 4.4 SW30XLE-400 4 25 4.5 1,412,000
48
Y. Lu et al. / Desalination 307 (2012) 42–50
245.5 m 3 /h 300 ppm
540 m 3 /h 41000 ppm
xp,6,1 =0.611
xp,2,1=0.264
seawater
brackish
74.4 m 3/h 120 ppm
164 m 3 /h 17840 ppm
regenerated water
permeate 3 150 m /h 300 ppm
x p,5,1=0.125
97.1 m 3 /h 30 ppm
3
214.8 m /h 7410 ppm
permeate 200 m 3 /h 100 ppm
xb,3,3=0.488 89.6 m 3 /h 32620 ppm
x b,1,3=0.512 3
294.6 m /h 75000 ppm
brine
Fig. 4. The optimal configuration of RO system for the study case.
solution, the common approach of mixing three feed streams prior to treatment is employed. As shown in Fig. 6, the stage 2 consists of two units which process the brine coming from stage 1 with the same operating pressure. The selected type of membrane elements are the SW30HR-320 in stage 1, the SW30HR-380 in stage 2, respectively. Three kind of permeate are produced in this system, which salt concentration are 100 ppm, 300 ppm and 500 ppm, respectively. The corresponding total annualized cost is $1,743,900 per year which is 24% more expensive than the cost of the optimal RO system. The comparisons of these alternate schemes suggest that the feed position of streams and outlet of the system are the critical variable that should be optimized for the RO system design. The optimal structural scheme employed during design is different when the feed position of streams is variable. In this study case, the three-staged with brine recycle and the two-staged structure with brine recycle are employed in the optimal solution and the first suboptimal solution (shown as Fig. 5), respectively. The recycle streams can increase the utilized ratio of the brine and decrease the feed concentration of the RO stage. Since the overall recovery ratio of the three-staged scheme is larger than that of the two-staged scheme, therefore, the unit production cost of optimal solution is lower. In the second suboptimal solution (shown as Fig. 6), the high salinity brine coming from stage 1 is reprocessed with high pressure in stage 2, which leads to the higher total annualized cost. The comparison results indicated that the feed position of streams also affect the concentration of inner streams, therefore, the choice of the membrane element is different in the three schemes. For the output of the RO system, the high quality water product is prior to output, then the other grade water product are provided in
brackish and regenerated water enter the system from stage 2. The optimal product outlets are two kinds of permeate of different concentration (100 ppm and 300 ppm). In this RO system, the partial permeate, which leaves stage 1 and comes into the stage 2 (xb,2,1 = 0.234), decreases the feed concentration of stage 2 and increases the output of water product. The low concentration brine leaving stage 3 are split into two recycle streams. One stream is fed to stage 1 and the other recycle around stage 3. These recycle streams decrease the feed concentration of stage 1 and increase the overall recovery of the system. The total annualized cost of this scheme is $1,412,000 per year. In this case study, several initialization points were used in order to achieve the best possible solution. It is interesting to compare some local optimal solutions obtained. Following are design results of two suboptimal solutions with different configurations. One of the solutions employs the two-stage construct (shown as Fig. 5). The data shown on each stream represent flow rate, concentration and the stream split ratio, respectively. The mixing seawater and brackish enter the RO system from stage 1. The feed streams of stage 2 include regenerated water, the partial permeate coming from stage 1 and brine recycle stream around stage 2. The selected type of membrane elements are the SW30XLE-400 in stage 1, the BW30-400 in stage 2, respectively. The optimal product outlets are two kinds of permeate that salt concentration are 100 ppm and 300 ppm, respectively. The corresponding total annualized cost is $1,546,500 per year, which is 9.5% more expensive than the cost of the optimal RO system. This result indicates that, when the outlets of the system are the same, the effect of feed position on annualized cost is remarkable for multiple feed streams. While for the other
3
seawater brackish
727.8 m /h 36590 ppm
330.8 m 3 /h 300 ppm xp,2,1=0.445
xp,6,1 =0.453 x p,5,1=0.102
355 m 3/h 4920 ppm
regenerated water
166.3 m 3 /h 60 ppm
xb,2,2=0.826
permeate 150 m 3 /h 300 ppm permeate 3
200 m /h 100 ppm
3
188.7 m /h 9200 ppm
x b,1,2=0.174
397 m 3 /h 66800 ppm Fig. 5. The configuration of the local optimal solution with two kinds of permeate output.
brine
Y. Lu et al. / Desalination 307 (2012) 42–50
seawater
49
261.4 m 3/h 100 ppm
611.6 m 3 /h 34070 ppm
xp,5,1 =0.765
brackish
xp,6,1 =0.235 xb,2,1=0.176
regenerated water
3
350.2 m /h 59430 ppm
3
23.2 m 3/h 500 ppm
38.4 m /h 95080 ppm
xp,6,3 =0.24
permeate 200 m 3 /h 100 ppm permeate 100 m 3 /h 300 ppm
3
65.2 m /h 500 ppm
xb,3,1=0.494
permeate 50 m 3/h 500 ppm
107.8 m 3/h 95080 ppm xb,4,1=0.33
brine
Fig. 6. The configuration of the local optimal solution with three kinds of permeate output.
turn. In this case, under the conditions of meeting all of the product quantity and quality requirement, the optimal solution provided two kinds of product with less salinity. These different grades of product can satisfy the requirement of water quality on different occasions. The high quality water product can be widely used in many industrial processes, therefore it is the main product in this RO system with multiple-product output, and it is more profitable product. When producing this main product, the RO system can simultaneously provide the other grade by-product. 5. Conclusions The optimization method based on process synthesis concept has been applied to design the RO system with multiple-feed and multiple-product required. A structural representation that incorporates all feasible arrangements for this desalination problem of multiple water sources was presented. This superstructure was used to develop the general mathematical model by means of the appropriate relationships between the variables. The design task is formulated as an MINLP model which minimizes the total annualized cost of the RO system. The solution of the problem includes the optimal system structure and operating conditions, the optimal streams distribution, and the optimal feed position of streams and outlet of the system. The design method could also be used for the optimal selection of type and number of membrane elements in each stage and in each pressure vessel. A desalination case is solved to demonstrate the effectiveness of this design method. Symbols A
water permeability, kg/m 2 s Pa
B Ce Cm C Ck Cpv Cw CCin CChpp CCTb Ds d Jw Js K Lpv
solute transport parameter, kg/m 2 s electricity cost, $/(kw .h) membrane module cost, $ concentration of solute, ppm the price of the kth membrane element, $ the price of the pressure vessel, $ concentration at the membrane wall, ppm capital cost of the seawater intake pump, $ capital cost of the high pressure pump, $ capital cost of the turbine, $ the solute diffusivity, m 2/s the feed spacer thickness, m water flux, kg/m 2 s salt flux, kg/m 2 s the mass transfer coefficient, (m/s) length of the pressure vessel, m
Lm m n Nl Nw Np Nps NRO OCin OChpp OCTb OCm P ΔPf PD PS Q Qp Qrp Crp Re R Sc Sm T TAC U L Vw W WT xf,i,q
xc y ,Z
the length of a element, m the number of membrane elements in each PV the number of pressure vessel the number of leaves in a membrane element inlet junctions of feed streams outlet junctions of product streams mixing junctions, reverse osmosis junctions energy cost of the intake pump, $ energy cost of the high pressure pump, $ the saving cost generated by turbine, $ the cost of membrane module maintenance, $ operating pressure, Mpa the pressure drop in the membrane channel, Mpa set of outlet junctions for the product streams set of stream mixing junctions flow rate, m 3/h the permeate flow rate of pressure vessel, m 3/h the flow rate of the r th product water required, m 3/h the concentration of the r th product water required, ppm Reynold's number set of reverse osmosis junctions Schmidt number the membrane area per element, m 2 temperature, °C total annualized cost, $ arbitrary large numbers arbitrary small numbers the permeate velocity, m/s. membrane width, m set of inlet junctions for the feed streams the stream split ratio of the qth feed stream linked to the ith pressurization stage the stream split ratio of the brine leaving the jth RO stage and being linked to the ith pressurization stage the stream split ratio of the permeate leaving the jth RO stage and being linked to the ith pressurization stage the small positive number binary integer
Greek ∏ ∏w μ ρ
osmosis pressure, Mpa osmosis pressure of the brine at the membrane wall, Mpa the water viscosity, kg/m s density, kg/m 3
xb,i,j xp,i,j
50
η ε
Y. Lu et al. / Desalination 307 (2012) 42–50
pump efficiency the small positive number
Subscripts f feed stream p permeate stream b brine stream in intake seawater ps,i the i th pressurization stage RO,j the jth RO stage ROf,j the feed stream of the j th RO stage ROb,j the brine stream of the jth RO stage ROp,j the permeate stream of the jth RO stage k the kth element type hpp high pressure pump Tb turbine i index of reverse osmosis junctions j index of stream mixing junctions r index of outlet junctions for the product streams q index of inlet junctions for the feed streams
Acknowledgements Financial support from the Natural Science Foundation of Guangxi (No. 2012GXNSFAA053025), Guangxi Higher Education Institutes Talent Highland Innovation Team Scheme (GJR201147-12) are gratefully acknowledged. Many thanks to Jinxian Liu and Dominic Foo Chwan Yee for help with the English. References [1] E. El-Zanati, S. Eissa, Development of a locally designed and manufactured small-scale reverse osmosis desalination system, Desalination 165 (2004) 133–139. [2] A. Villafafila, I.M. Mujtaba, Fresh water by reverse osmosis based desalination: simulation and optimization, Desalination 155 (2003) 1–13. [3] N.M. Wade, Distillation plant development and cost update, Desalination 136 (2001) 3–12. [4] M. Wilf, C. Bartels, Optimization of seawater RO systems design, Desalination 173 (2005) 1–12.
[5] M. Wilf, Design consequences of recent improvements in membrane performance, Desalination 113 (1997) 157–163. [6] M.G. Marcovecchio, P.A. Aguirre, N.J. Scenna, Global optimal design of reverse osmosis networks for seawater desalination: modeling and algorithm, Desalination 184 (2005) 259–271. [7] A. Abbas, N.M. Al-Bastaki, Modeling of an RO water desalination unit using neural networks, Chem. Eng. J. 114 (2005) 139–143. [8] J.A. Redondo, A. Casanas, Designing seawater RO for clean and fouling RO feed. Desalination experiences with the FilmTec SW30HR-380 and SW30HR-320 elements — Technical — economic review, Desalination 134 (2001) 83–92. [9] M. Busch, W.E. Mickols, Reducing energy consumption in seawater desalination, Desalination 165 (2004) 299–312. [10] Membrane Technical Information, http://www.dow.com/liquidseps/service/ lm_techinfo.htm. [11] Y.Y. Lu, Y.D. Hu, Optimum design of reverse osmosis seawater desalination system considering membrane cleaning and replacing, J. Membr. Sci. 282 (2006) 7–13. [12] Y.Y. Lu, Y.D. Hu, Optimum design of reverse osmosis system under different feed concentration and product specification, J. Membr. Sci. 287 (2007) 219–229. [13] I.M. El-Azizi, Design criteria of 10,000 m3/d SWRO desalination plant of Tajura, Libya, Desalination 153 (2002) 273–279. [14] G. Al-Enezi, N. Fawzi, Design consideration of RO units: case studies, Desalination 153 (2002) 281–286. [15] M.M. El-Halwagi, Synthesis of reverse osmosis networks for waste reduction, AICHE J. 38 (1992) 1185–1198. [16] M. Zhu, M.M. El-Halwagi, Optimal design and scheduling of flexible reverse osmosis networks, J. Membr. Sci. 129 (1997) 161–174. [17] N.G. Voros, Z.B. Maroulis, Short-cut structural design of reverse osmosis desalination plants, J. Membr. Sci. 127 (1997) 47–68. [18] H.J. See, V.S. Vassiliadis, D.I. Wilson, Optimization of membrane regeneration scheduling in reverse osmosis network for seawater desalination, Desalination 125 (1999) 37–54. [19] F. Maskan, D.E. Wiley, Optimal design of reverse osmosis module networks, AICHE J. 46 (2000) 946–954. [20] J.E. Nemeth, Innovative system designs to optimize performance of ultra-low pressure reverse osmosis membranes, Desalination 118 (1998) 63–71. [21] W.G.J. Van der Meer, C.W.A. Averink, Mathematical model of nanofiltration systems, Desalination 105 (1996) 25–31. [22] W.G.J. Van der Meer, M. Riemersma, Only two membrane modules per pressure vessel hydraulic optimization of spiral-wound membrane filtration plants, Desalination 119 (1998) 57–64. [23] L.P. Wessels, W.G.J. Van der Meer, Innovative design of nano- and ultrafiltration plants, Desalination 119 (1998) 341–345. [24] A. Malek, M.N.A. Hawlader, Design and economics of RO seawater desalination, Desalination 105 (1996) 245–261. [25] N.M. Al-Bastaki, A. Abbas, Predicting the performance of RO membranes, Desalination 132 (2000) 181–187. [26] N.M. Al-Bastaki, A. Abbas, Modeling an industrial reverse osmosis unit, Desalination 126 (1999) 33–39. [27] A. Brooke, D. Kendrick, A. Meeraus, GAMS: A User's Guide, Scientific Press, Palo Alto, CA, 1992.
Contents lists available at SciVerse ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
The design of reverse osmosis systems with multiple-feed and multiple-product Yanyue Lu a,⁎, Anping Liao a, Yangdong Hu b a School of Chemistry and Chemical engineering, Key Laboratory of Chemical and Biological Transformation Process of Guangxi Higher Education Institutes, Guangxi University for Nationalities, Nanning, 530006, China b College of Chemistry and Chemical Engineering, Ocean University of China, Qingdao, 266003, China
H I G H L I G H T S ► ► ► ► ►
A process synthesis-based optimization technique has been developed. Membrane units were approximated by the pressure vessel model. The module model takes into account the pressure drop and concentration changes. The stream split ratios and logical expressions of stream mixing were employed. The design results present optimal structure and the optimal streams distribution.
a r t i c l e
i n f o
Article history: Received 1 June 2012 Received in revised form 21 August 2012 Accepted 22 August 2012 Available online 25 September 2012 Keywords: Reverse osmosis Desalination Optimum design Mathematic model Process synthesis
a b s t r a c t A reverse osmosis (RO) desalination process with multiple-feed and multiple-product is the main focus of this work. A process synthesis-based optimization technique has been developed for the design of the RO system. The adoption of this approach provides an economically attractive desalination scheme. Membrane separation units employing the spiral wound reverse osmosis elements were approximated by the pressure vessel model presented in this paper, which takes into account the pressure drop and concentration changes in the membrane channel. A simplified superstructure that contains all the feasible design for this desalination problem has also been presented. In this structure representation, the stream split ratios and logical expressions of stream mixing were employed, which can make the mathematical model to be easily handled. The optimum design problem is formulated as a mixed-integer non-linear programming (MINLP) problem, which minimizes the total annualized cost of the RO system. The cost equation relating the capital and operating cost to the design variables, as well as the structural variables has been introduced in the objective function. The solution of the problem includes the optimal system structure and operating conditions, and the optimal streams distribution. The design method could also be used for the optimal selection of the type of membrane elements in each stage and the optimal number of membrane elements in each pressure vessel. The effectiveness of this design methodology has been demonstrated by solving a desalination case. The comparisons of several alternate schemes indicate that the feed position of streams and outlets of the system are the critical variables that should be optimized for the RO system design. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Desalination of sea and brackish waters is the main source for supplying fresh water in the regions suffering from the scarcity of natural fresh water supply. Since 1960s, due to the development of new reverse osmosis (RO) modules and membranes, RO is experiencing growing applications in the desalination field. Now it has become a major technology for large-scale desalination plants for both seawater and brackish water sources [1–5]. The interest in RO is due to its low energy consumption (as compared to multistage flash distillation process), high product recovery ⁎ Corresponding author. Tel.: +86 0771 3260558. E-mail address: [email protected] (Y. Lu). 0011-9164/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.desal.2012.08.025
and quality. The other attractive feature of RO process is their modular plant design and ease of operation. Membrane plants are often more compact, can be scaled up easily and installed more quickly than thermal separations plants. Also, it makes the maintenance of RO systems easier. Another advantage of the RO process is that it is able to meet varying feed water concentration and varying production water quantity and quality requirement through the change of system configuration and operation condition. RO membrane manufacturers have developed various membrane types to precisely meeting the varying needs of a wide range of industrial, municipal, commercial and drinking water application. This includes high flux, high rejection, fouling resistant, low pressure, and high rejection membrane, etc [6–10]. These features allow the design of RO processes to be more flexible.
Y. Lu et al. / Desalination 307 (2012) 42–50
Considerable efforts have been made for the research of the optimum design of RO system [6,11–14]. El-Halwagi [15] investigated the synthesis of RO networks which involve multiple feed streams for waste reduction. Based on the state-space approach, a structural representation of RO networks was introduced. The RO networks were described using four boxes, i.e. a pressurization/depressurization stream-distribution box, a pressurization/depressurization matching box, a RO stream-distribution box, a RO matching box. The function of the distribution boxes is to represent all possible combinations of stream splitting, mixing, bypass and recycle. The matching boxes locate all possible stream assignments to units. With this formulation, all possible structure arrangements are represented. The mathematical model was formulated as a mixed integer nonlinear programming (MINLP). In their later work, Zhu et al. [16] included a factor for flux decline over time based on their earlier work [15]. Voros et al. [17] simplified El-Halwagi's representation by reducing the distribution boxes to junctions. Consequently, the model was formulated as a nonlinear programming (NLP) model by using a variable split ratio. Maskan and Wiley [19] used a directed graph and connectivity matrix to represent the RO networks superstructure. In the mathematical model of the superstructure, a variable reduction technique is performed to accelerate the computational process. Nemeth [20] studied the behavior of the ultra-low pressure RO membranes in the full-scale system and presented recommendations to improve system design. Van der Meer [21,22] and Wessels [23] developed a simplified mathematical model to optimize the performance of NF and RO membrane filtration plants. The study showed that the productivity of nanofiltration plants can be significantly improved by installing a reduced number of membrane elements serially in pressure vessels (PV) and by changing system configuration. Malek et al. [24] provided a realistic economic model that relates the various operational and capital cost elements to the design variable values. In this work, an RO-based desalination process is considered for the production of fresh water from three raw feeds (seawater, brackish and regenerated water). A systematic methodology is presented for the optimal design of RO desalination system that processes multiple feed streams simultaneously, and at the same time, supplies various product streams (water) of different quality. The adoption of this approach can provide an economically attractive desalination scheme. This leads to significant energy and raw-materials saving and generates income from the sales of multiple grades of water products. First, a simplified superstructure representation that contains all feasible designs for this desalination problem is presented. Then a pressure vessel model is also developed. The model could be used for the optimum selection of types and number of membrane element, according to its performance characteristics, the prices, and the design requirements of a specific desalination application. Therefore the optimal design of RO systems was formulated as a mixed integer nonlinear programming problem (MINLP). The objective is to determine the optimal system structure, stream distribution and operating conditions; subject to the constraints of the multiple-feed and multiple-product system. The solution to the problem also includes the most appropriate choice of the type of membrane elements in each stage and the optimal number of membrane elements in each PV. 2. RO unit model 2.1. The mass transfer model of RO process Numerous models to predict membrane performance have been introduced [7,25,26]. They are derived from different theories and all of them may be simplified to the solution diffusion model, as shown in Eqs. (1) and (2). For the RO system design and optimization, it is necessary to adopt the appropriate modeling equations that can satisfactorily predict the membrane performance with reasonable computational complexity. Therefore the solution diffusion model is
43
among the most commonly used model in RO system design. The model is mainly based on two parameters, i.e. water permeability (A) and solute transport parameter (B). Values for these parameters are usually specified by membrane manufacturers. According to the model, the pure water flux, Jw (kg/m 2.s), and the salt flux, Js (kg/m 2.s), are given as follow: Jw ¼ A
ΔP 6 P f −P p − f − πw −πp 10 2
J s ¼ B C w −C p π¼
ð2Þ
0:2641 C ðT þ 273Þ 1:0 106 −C
ð3Þ
Jw þ Js ρp
ð4Þ
Js 1000 Vw
ð5Þ
Vw ¼
Cp ¼
ð1Þ
where Pf and Pp (Mpa) denote feed and permeate pressure, respectively; ΔPf is the pressure drop in the membrane channel; ∏w (Mpa) is the osmotic pressure of the brine at the membrane wall concentration Cw (ppm), and ∏p and Cp are corresponding variables for the permeate; ρp denotes the density of the permeate. Vw (m/s) is the permeate velocity. In mass transfer process, the variation of concentration on membrane surface should be considered. The change can be represented by Eqs. (6) to (8). Cw ¼ Cp þ
V w Cf þ Cb −C p e K 2
ð6Þ
K (m/s) is the mass transfer coefficient, which can be calculated from empirical relations such as: 0:75
K ¼ 0:04 Re Re ¼
0:33
Sc
Ds d
V ρd μ
ð7Þ ð8Þ
where Re and Sc are the Reynold's and the Schmidt numbers and Ds is the solute diffusivity. d is the feed spacer thickness, ρ is the feed side solution density and μ is the water viscosity. V denotes the flow velocity that was calculated using the averaged values of the inlet and outlet flow rates in the membrane channel. 2.2. The model for RO module In the practical process, the multiple stages RO configuration would be used, one RO stage consists of multiple parallel RO pressure vessels operating at the same conditions. Each pressure vessel contains several membrane elements that are connected in series. The concentrate of the first element becomes the feed to the second, and so on. The products tubes of all elements are coupled and connected to the module permeate port [21–23]. For different application a suitable hydraulic design can be made (2, 3, 4, 5, 6, 7, 8 serial elements), based on the actual situation. Fig. 1 shows the schematic representation of a RO module. Al-Bastaki and Abbas [25,26] presented the models of the spiral-wound and hollow-fiber membrane elements, which took into account the pressure drop and concentration changes in the membrane channel. The PV performance can be approximately simplified to the performance of the membrane elements connected in series. Therefore, based on the membrane element models, a pressure vessel model is presented as follows which
44
Y. Lu et al. / Desalination 307 (2012) 42–50
Feed
Pf
Membrane module Brine
Q b Cb Pb
Qf Cf
Permeate
Qp Cp Pp Fig 1. The schematic diagram of a RO unit.
would be used for the optimal selection of the membrane element type and number. For a spiral-wound membrane module, each of the permeate and feed side flows can be considered as a flow between two parallel plates with a length L, a width W and a spacing d. Hence the pressure drop on the feed side can be calculated as follows: ΔP f ¼
0:0033 Q a Lpv μ
ð9Þ
W d3
Lpv ¼ m Lm
ð10Þ
ΔP f ≤0:35
ð11Þ
Q f Cf ¼ Q b Cb þ Q p Cp L
ð15Þ
U
Q rated ≤Q f ≤Q rated
ð16Þ
where Q Lrated and Q Urated are respectively the lower bound and upper bound of rated flow rate of pressure vessel. When a RO stage consists of multiple parallel pressure vessels, and operates at the same conditions, the flow rate of stream entering and leaving this RO stage can be calculated as following: Q ROf ¼ n Q f
ð17Þ
Q ROp ¼ n Q p
ð18Þ
Q ROf ¼ Q ROp þ Q ROb
ð19Þ
3
where Qa (m /h) is the flow rate that was calculated using the averaged values of the inlet and outlet flow rates in the membrane channel; Lpv and Lm denote the length of the PV and the length of a membrane element, respectively; m is the number of membrane elements in each PV. In order to reduce the computation complexity, m is considered to be a continuous variable, and the approximate result would be obtained by rounding the variable. The maximum allowable pressure drop of the pressure vessel is 0.35 Mpa. The technical constraint is usually specified by membrane manufacturers. For the spiral-wound membrane element, the membrane width (W), can be calculated by the membrane area(Sm) and the number of leaves(Nl) : S m ¼ W Lm N l
where QROf, QROp and QROb are respectively the feed flow rate, permeate flow rate and brine flow rate entering and leaving a RO stage. n is the number of pressure vessel in a RO stage which is considered to be a continuous variable, and the approximate result would be obtained by rounding the variable. 3. RO system model 3.1. Problem description
ð12Þ
For a pressure vessel, the feed flow rate, Qf (m 3/h), the permeate flow rate, Qp (m 3/h), the brine flow rate, Qb (m 3/h), and the corresponding concentration, Cf, Cp, Cb (ppm), can be calculated from the mass and salt balance equations: Q p ¼ 3600 V w Sm m
ð13Þ
Qf ¼ Qb þ Qp
ð14Þ
For the RO desalination system with multiple-feed and multipleproduct requirement, the design objective is to identify the most cost effective RO network configuration, the optimal streams distribution and operating conditions, and the optimal arrangement of the membrane elements. Fig. 2 represents a potential multiple stages RO configuration for this desalination problem. It is necessary to develop a structure representation that contains all feasible designs of RO network synthesis. The network representation based on the state-space approach was presented earlier by El-Halwagi
membrane module
Q p,1 C
high pressure pump
Q f,1 Cf,1 turbine
Q b,1 C b,1
high pressure pump
Q p,2
Q f,2 Cf,2 Q b,2 C
b,2
Fig. 2. RO process configuration with multiple-feed and multiple-product.
p,1
Y. Lu et al. / Desalination 307 (2012) 42–50
[15,16] and Voros et al. [17]. Here we adopt and properly modify the approach, and in turn present a simplified superstructure that incorporates all the feasible process flow for the RO desalination system with multiple-feed and multiple-product. As shown in Fig. 3, an RO network consists of Nps pressurization stages and NRO reverse osmosis stages. In this configuration, there are four sets of stream nodes employed: a set WT={q| q=1, Nw} of inlet junctions for the feed streams; a set PS={j| j=1, Nps} of stream mixing junctions; a set R={i| i=1, NRO} of reverse osmosis junctions; a set PD={r| r=1, Np} of outlet junctions for the product streams. The junction of Nps +1 indicated the brine stream leaving the network. Each mixing junction in the PS set indicates that the feed streams, the brine and the permeate streams leaving all reverse osmosis stages are mixed at the node. The mixing streams pressurized by high pressure (HP) pump or not are connected to the corresponding reverse osmosis stages. The RO stages consist of multiple parallel pressure vessels operating at the same conditions.
The complete mathematical model that describes the superstructure is presented as follows by means of the appropriate relationships between the variables (material and energy balance equations, technical and operational constraints). Nw NRO X X Q f ;q xf ;i;q þ Q ROb;j xb;i;j q¼1
j¼1
ð20Þ
N RO X þ Q ROP;j xp;i;j
i ¼ 1; 2; …; Nps
j¼1
Q ps;i C ps;i ¼
Nw X
Q f ;q xf ;i;q C f ;q þ
q¼1
þ
NRO X
Q ROb;j xb;Nps þ1;j
ð22Þ
j¼1
Q ps;Nps þ1 C ps;Nps þ1 ¼
NRO X
Q ROb;j xb;Nps þ1;j C ROb;j
ð23Þ
j¼1 N ps X
xf;i;q ¼ 1
q ¼ 1; 2; …; Nw
ð24Þ
j ¼ 1; 2; …NRO
ð25Þ
j ¼ 1; 2; …NRO
ð26Þ
i¼1 Nps þ1
X
xb;i;j ¼ 1
i¼1 Nps þN p
X
xp;i;j ¼ 1
i¼1
3.2. Mathematical formulation
Q ps;i ¼
Q ps;Nps þ1 ¼
45
N RO X Q ROb;j xb;i;j C ROb;j
i ¼ 1; 2; …; N ps
j¼1
ð21Þ
NRO X Q ROp;j xp;i;j C ROp;j j¼1
where Qps,i and Cps,i denote the flow rate and concentration of the ith pressurization stage, respectively. Qf, q Cf, q denote the qth feed flow rate and concentration of the RO network. QROb, j, CROb, j denote the brine flow rate and concentration of the jth RO stage, QROp, j and CROp, j denote the permeate flow rate and concentration of the jth RO stage, respectively. xf, i, q, xb, i, j and xp, i, j indicate the stream split ratios of the feed, the brine and permeate, respectively. The values determine the flow rates of brine and permeate leaving the jth RO stage and being linked to the ith pressurization stage. All streams connected to ith pressurization stage firstly mix in the mixer. The outlet pressure from the mixer is the average feed pressure. When the raw streams enter the RO system from interstage, it must be pressurized to the outlet pressure of the pressurization stage by high pressure pump. The stream split ratios and the logical expression of stream mixing are employed in this paper, these techniques reduced the number of binary variable and the solving space, therefore the mathematical model may be easily handled. Following are the stream-mixing constraints. L1 1−yb;i;j ≤xb;i;j −xc ≤U 1
PS stage
Q f,1 Cf,1 feed
yb;i;j −ε
ð27Þ i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO
1 1
L1 1−yp;i;j ≤xp;i;j −xc ≤U 1
mixer
q
2 i
Nw
Nps
yp;i;j −ε
Qb Cb
N ps +1 1 2
p
p
Q1 C 1 p p Q2 C 2
yb;i;j þ yp;i;j ≤1
ð28Þ i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð29Þ
L2 1−yb;i;j ≤ P b;j −P b;i;j ≤U 2 1−yb;i;j
i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO
P b;i;j ≤U 2 yb;i;j
i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð31Þ
L2 1−yp;i;j ≤ P p;j −P p;i;j ≤U 2 1−yp;i;j
i ¼ 1; 2; …; Nps ; j ¼ 1; 2; …NRO ð32Þ
P p;i;j ≤U 2 yp;i;j
i ¼ 1; 2; …; N ps ; j ¼ 1; 2; …NRO ð33Þ
ð30Þ
Np RO stage NRO j
stage 2
permeate
stage 1
2
1
Q p,1 Cp,1 Q b,1 Cb,1 brine Fig. 3. Representation of the RO network via the superstructure.
NRO X P b;i;j þ P p;i;j
P ps;i ¼
j N RO X yb;i;j þ yp;i;j j
i ¼ 1; 2; …; N ps ð34Þ
46
Y. Lu et al. / Desalination 307 (2012) 42–50
where L1 and L2 are the arbitrary small numbers, U1 and U2 are large enough numbers. y is the binary variable. xc and ε denote the small positive number. It is assumed that the streams link to the ith pressurization stage when the stream split ratios, x, is larger than xc, hence y = 1, whereas if x is smaller than xc, then y = 0. Pb,j and Pp.j are the brine and permeate pressure of the j th RO stage, respectively. Pb,i, j, Pp. i, j denote the streams pressure which link to the ith pressurization stage. Pps,i denotes the inlet pressure of the ith pressurization stage. The streams leaving the ith pressurization stages are correspondingly connected to the jth RO stages. Therefore the following equations can be obtained:
Qp, j = 0. By this way the module type and number of stage are chosen simultaneously when the optimization design is performed. The overall material balances for the RO network and a set of product quantity and quality constraints concerning the minimum desirable product flow rate, and the maximum allowable product concentration are presented as follow: Nw X
b
Q f ;q ¼ Q þ
q¼1
Nw X
b
ð35Þ
C ROf ;j ¼ C ps;i
j ¼ i; j ¼ 1; 2; …; N RO
ð36Þ
Q ¼ Q ps;Nps þ1
j ¼ i; j ¼ 1; 2; …; NRO
ð37Þ
C ¼ C ps;Nps þ1
where QRO f, j, CROf, j and PRO, j denote the feed flow rate, concentration, and operation pressure of the jth RO stage, respectively. P 'ps,i is the outlet pressure of the ith pressurization stage. The mathematical models that predict the performance of each RO stage have been presented in detail in the previous section. These model equations relate the flow rate and concentration of the brine and permeate leaving an RO stage to the flow rate, concentration, and pressure of the stream entering the stage. The arrays of pressure vessels (PV) with 2 up to 8 membrane elements per PV consist of a RO stage. In this paper, the optimal PV structure has been researched. Four different types of spiral wound FilmTec reverse osmosis membrane elements have been considered. According to its performance characteristics and the requirements of a specific desalination application, the optimal selection of types of the membrane element employed in each PV can be determined by the following equations: 4 X
Z j;k ≤1
j ¼ 1; 2; …; NRO
ð38Þ
k¼1
! 4 X 1− Z j;k ≤Q ROf ;j −xc ≤U 3
L3
ð39Þ
k¼1
4 X Z j;k −ε
j ¼ 1; 2; …; NRO
k¼1
Q p;j ¼
4 X
Q p;j;k
j ¼ 1; 2; …; N RO
ð40Þ
Q p;j;k ≤U 4 Z j;k
j ¼ 1; 2; …; NRO
ð41Þ
mj;k ≤U 5 Z j;k
j ¼ 1; 2; …; N RO
ð42Þ
k¼1
where Zj,k is the binary variable. It takes the value of 1 when the kth element type is utilized in the PV in the jth RO stage, otherwise, it takes the value of 0. Here, it is assumed that the same membrane element is employed in the same RO stage. L3 and U3, are the arbitrary small and large number, respectively. For the Eq. (39), when QROf, j takes some value which is larger than xc, it means the jth RO stage is present and one element type should be chosen, otherwise if QROf, j = 0 it means the jth RO stage is absent. Qp, j is the permeate flow rate of pressure vessel. While Qp, j, k denote the permeate flow rate of pressure vessel when the kth element type is utilized in this PV, it can be calculated by Eq. (13). mj,k is the number of the kth membrane element. U4 and U5 are large enough positive number so that Q p, j, k and mj,k are not restricted if Zj,k = 1. In terms of Eqs. (38), (40) and (41), Qp, j is equal to one of Qp, j, k when Zj,k j = 1, otherwise
ð43Þ
b
Q f ;q C f ;q ¼ Q C þ
j ¼ i; j ¼ 1; 2; …; NRO
0
p
Qr
r¼1
Q ROf ;j ¼ Q ps;i
P RO;j ¼ P ps;i
Np X
Np X
q¼1
p
p
Q r Cr
ð44Þ
r¼1
b
ð45Þ
b
ð46Þ
p
Qr ¼
NRO X
Q ROp;j xp;r;j
r ¼ 1; 2; …; Np
ð47Þ
r ¼ 1; 2; …; Np
ð48Þ
r ¼ 1; 2; …; Np
ð49Þ
r ¼ 1; 2; …; Np
ð50Þ
j¼1
p
p
Q r Cr ¼
N RO X
Q ROp;j xp;r;j C ROp;j
j¼1 p
p
Q r ≥Q r;min p
p
C r ≤C r; max
where Q b and C b are the flow rate and concentration of the brine leaving the RO network, respectively. Qrp and Crp are the flow rate and concentration of the rth product water required, respectively. xp, r, j are the outlet stream split ratios. Q pr,min refers to the minimum desirable flow rate of the r th product required, C pr,max refers to the maximum allowable concentration of the rth product required. The optimization design problem is formulated as an MINLP model for minimizing the total annualized cost subject to thermodynamic, technical, and flexibility constraints. The total annualized cost (TAC) of the system consists of two terms: annual operating cost (OC) and annualized capital cost (CC). The annual operating cost includes the energy cost required for the pump, membrane module maintenance cost (OCm). The annualized capital cost is for initial membrane module, pumps and energy recovery devices. The objective function is presented as follows: TAC ¼ CCin þ CChpp þ CCTb þ Cm 1:411 0:08 þ OCin þ OChpp OCTb þ OCm 0:96 CChpp ¼ 52 ΔP Q hpp
Cm ¼
NRO X j¼1
NRO X 4 X
C k mj;k nj
ð52Þ
ð53Þ
j¼1 k¼1
P Q Ce f c 3:6 ηhpp ηmotor
ð54Þ
P Q ηTb C e f c 3:6
ð55Þ
OChpp ¼
OCTb ¼
C pv nj þ
ð51Þ
where CCin, CChpp and CCTb are the capital cost of the seawater intake pump, the high pressure pump, and the turbine, respectively. OCin, OChpp and OCTb are the energy cost required for these pumps and the saving cost generated by turbine. Their functions refer to the
Y. Lu et al. / Desalination 307 (2012) 42–50 Table 1 Characteristics of FilmTec spiral wound reverse osmosis membrane elements. Element type
SW30XLE-400 SW30HR-380 SW30HR-320 BW30-400
Active area, ft2(m2) Length of the element, inch (mm) Diameter of the element, inch (mm) Feed space, mil⁎
400 (37.2) 40 (1016)
Feed flow rate range, m3/h Permeate flow, gpd (m3/d) Stabilized salt rejection, % Maximum operating pressure, psig (Mpa) a Pure water permeability constant A, (kg/m2.s.Pa) b Salt permeability constant B, (kg/m2.s) Membrane element cost, $ (estimation) a, b
380 (35.3) 40 (1016)
320 (29.7) 40 (1016)
400 (37) 40 (1016)
47
The solver DICOPT under the GAMS environment [27] is used to solve the MINLP problem by decomposing it into a series of nonlinear (NLP) sub-problems and mixed integer (MIP) master problems. Several starting points are used to obtain the best possible solution. 4. Case study
7.9 (201)
7.9 (201)
7.9 (201)
7.9 (201)
28 0.8–16
28 0.8–16
34 0.8–14
34 0.8–19
9000(34.1)
6000(22.7)
6000(22.7)
10,500(40)
99.70
99.70
99.75
99.5
1200 (8.3)
1200 (8.3)
1200 (8.3)
600 (4.5)
3.5 × 10−9
2.7 × 10−9
3.1 × 10−9
7.5 × 10−9
3.2 × 10−5
2.3 × 10−5
2.2 × 10−5
6.2 × 10−5
1200
1000
1400
800
Refer to the Refs. [10] and [18].
papers [2,6,16,24]. Cm denotes the total membrane module cost. Ck is the price of the kth membrane element and Cpv is the price of the pressure vessel. The nj is the number of pressure vessel employed in the jth RO stage. 1.411 is the coefficient that is used to calculate the practical investment. 0.08 is the capital charge rate. Ce is the cost of electricity and fc is the load factor. ηhpp, ηmotor and ηTb are the efficiency of HP pump, electric motor, and turbine, respectively. In this MINLP model, the concentrations of feed water and the product specification are defined, the variables include flow rates, concentrations and pressures of the intermediate sub-stream, number of pressure vessels, type and number of the membrane elements, operating pressures for each RO stage, stream split ratios (x), and binary variables (y, Z) used decision. This procedure is carried out by introducing an excessive number of stages as an initial guess, while at the optimum certain design variables, such as stream split ratios, are either set to zero or to a value that indicate the absence or presence of the specific stage. Therefore the structural optimization may take place in terms of eliminating all unnecessary pressurization stages or RO stages. The existence of the specific device may be determined indirectly by operational variables, such as the input flow rates to the unit, the pressure of the pressurization stage or RO stage.
Table 2 The parameters for calculation. Feed concentration of seawater, ppm Feed concentration of brackish, ppm Maximum brackish flow rate, m3/h Feed concentration of regenerated water, ppm Maximum regenerated water flow rate, m3/h Feed water average temperature, °C Average brine density,ρ, kg/m3 The average brine viscosity,μ, kg/m s Average diffusion coefficient, Ds, m2/s High pressure pump efficiency, ηhpp Turbine efficiency, ηTb Electric motor efficiency, ηmotor The cost of electricity, Ce, $/kw.h The pressure vessel cost, (estimation) $
42000 12000 100 5000 50 25 1020 1.09 × 10−3 1.35 × 10−9 75% 80% 98% 0.08 1000
The proposed methodology for optimum RO system design is applied to a desalination problem of three water sources: seawater, brackish and regenerated water. In this case, there are three water product outlets that satisfy different permeate quantity and quality requirement. The minimum desirable product flow rate for these outlets are 200 m 3/h, 100 m 3/h, 50 m 3/h, while the corresponding maximum allowed salt concentration are 100 ppm, 300 ppm, 500 ppm, respectively. The best water quality requirement is the major product constraint. Four different types of FilmTec reverse osmosis membrane elements from DOW have been included in the design studies of the current work. They are the low energy, high productivity element SW30XLE-400, the high rejection, high productivity element SW30HR-380, the high rejection, fouling resistant element SW30HR-320, and the high productivity, high rejection brackish RO element BW30-400. The geometrical properties and membrane characteristics (membrane pure water permeability, solute transport parameter) of these elements are given in Table 1 [10]. The necessary input data for this study case are summarized in Table 2 [6,16,24]. In general, the mathematic programming problem in this case includes 246 continuous variables and 21 discrete variables. The problem was solved by taking between 0.92 and 6.02 CPU time (sec), 1996 and 8489 iterations on the latest PC. It is assumed that seawater is the major feed water source. The feed flow rate of seawater is a variable that should be optimized. Whereas, the feed flow rates of brackish and regenerated water must subject to the maximum flow rate constraints. The results of the RO system optimization design are presented in Table 3. The three-stage RO configuration was employed in design (shown as Fig. 4). The data shown on each stream represent flow rate, concentration and the stream split ratio, respectively. The seawater with 495 m 3/h flow rate enters the system from stage 1, while the mixing Table 3 Design and optimization results for the study case. Process flow Seawater feed flow, Qf,1 (m3/h) Brackish feed flow, Qf,2 (m3/h) Regenerated water feed flow, Qf,3 (m3/h) Flow rate of the first product water, Qp1 (m3/h) Salt concentration of the first product water, Cp1 (ppm) Flow rate of the second product water, Qp2 (m3/h) Salt concentration of the second product water, Cp2 (ppm) Flow rate of the third product water, Qp3 (m3/h) Salt concentration of the third product water, Cp3 (ppm) Membrane type in stage 1 Number of elements per PV in stage 1 Number of PV in stage 1 Operating pressure in stage 1, P1 (Mpa) Membrane type in stage 2 Number of elements per PV in stage 2 Number of PV in stage 2 Operating pressure in stage 2, P2 (Mpa) Membrane type in stage 3 Number of elements per PV in stage 3 Number of PV in stage 3 Operating pressure in stage 3, P3 (Mpa) The total annualized cost, ($)
Three-stage RO system, show as Fig. 4 495 100 50 200 100 100 300 50 300 SW30XLE-400 5 82 7.3 SW30HR-380 3 33 4.4 SW30XLE-400 4 25 4.5 1,412,000
48
Y. Lu et al. / Desalination 307 (2012) 42–50
245.5 m 3 /h 300 ppm
540 m 3 /h 41000 ppm
xp,6,1 =0.611
xp,2,1=0.264
seawater
brackish
74.4 m 3/h 120 ppm
164 m 3 /h 17840 ppm
regenerated water
permeate 3 150 m /h 300 ppm
x p,5,1=0.125
97.1 m 3 /h 30 ppm
3
214.8 m /h 7410 ppm
permeate 200 m 3 /h 100 ppm
xb,3,3=0.488 89.6 m 3 /h 32620 ppm
x b,1,3=0.512 3
294.6 m /h 75000 ppm
brine
Fig. 4. The optimal configuration of RO system for the study case.
solution, the common approach of mixing three feed streams prior to treatment is employed. As shown in Fig. 6, the stage 2 consists of two units which process the brine coming from stage 1 with the same operating pressure. The selected type of membrane elements are the SW30HR-320 in stage 1, the SW30HR-380 in stage 2, respectively. Three kind of permeate are produced in this system, which salt concentration are 100 ppm, 300 ppm and 500 ppm, respectively. The corresponding total annualized cost is $1,743,900 per year which is 24% more expensive than the cost of the optimal RO system. The comparisons of these alternate schemes suggest that the feed position of streams and outlet of the system are the critical variable that should be optimized for the RO system design. The optimal structural scheme employed during design is different when the feed position of streams is variable. In this study case, the three-staged with brine recycle and the two-staged structure with brine recycle are employed in the optimal solution and the first suboptimal solution (shown as Fig. 5), respectively. The recycle streams can increase the utilized ratio of the brine and decrease the feed concentration of the RO stage. Since the overall recovery ratio of the three-staged scheme is larger than that of the two-staged scheme, therefore, the unit production cost of optimal solution is lower. In the second suboptimal solution (shown as Fig. 6), the high salinity brine coming from stage 1 is reprocessed with high pressure in stage 2, which leads to the higher total annualized cost. The comparison results indicated that the feed position of streams also affect the concentration of inner streams, therefore, the choice of the membrane element is different in the three schemes. For the output of the RO system, the high quality water product is prior to output, then the other grade water product are provided in
brackish and regenerated water enter the system from stage 2. The optimal product outlets are two kinds of permeate of different concentration (100 ppm and 300 ppm). In this RO system, the partial permeate, which leaves stage 1 and comes into the stage 2 (xb,2,1 = 0.234), decreases the feed concentration of stage 2 and increases the output of water product. The low concentration brine leaving stage 3 are split into two recycle streams. One stream is fed to stage 1 and the other recycle around stage 3. These recycle streams decrease the feed concentration of stage 1 and increase the overall recovery of the system. The total annualized cost of this scheme is $1,412,000 per year. In this case study, several initialization points were used in order to achieve the best possible solution. It is interesting to compare some local optimal solutions obtained. Following are design results of two suboptimal solutions with different configurations. One of the solutions employs the two-stage construct (shown as Fig. 5). The data shown on each stream represent flow rate, concentration and the stream split ratio, respectively. The mixing seawater and brackish enter the RO system from stage 1. The feed streams of stage 2 include regenerated water, the partial permeate coming from stage 1 and brine recycle stream around stage 2. The selected type of membrane elements are the SW30XLE-400 in stage 1, the BW30-400 in stage 2, respectively. The optimal product outlets are two kinds of permeate that salt concentration are 100 ppm and 300 ppm, respectively. The corresponding total annualized cost is $1,546,500 per year, which is 9.5% more expensive than the cost of the optimal RO system. This result indicates that, when the outlets of the system are the same, the effect of feed position on annualized cost is remarkable for multiple feed streams. While for the other
3
seawater brackish
727.8 m /h 36590 ppm
330.8 m 3 /h 300 ppm xp,2,1=0.445
xp,6,1 =0.453 x p,5,1=0.102
355 m 3/h 4920 ppm
regenerated water
166.3 m 3 /h 60 ppm
xb,2,2=0.826
permeate 150 m 3 /h 300 ppm permeate 3
200 m /h 100 ppm
3
188.7 m /h 9200 ppm
x b,1,2=0.174
397 m 3 /h 66800 ppm Fig. 5. The configuration of the local optimal solution with two kinds of permeate output.
brine
Y. Lu et al. / Desalination 307 (2012) 42–50
seawater
49
261.4 m 3/h 100 ppm
611.6 m 3 /h 34070 ppm
xp,5,1 =0.765
brackish
xp,6,1 =0.235 xb,2,1=0.176
regenerated water
3
350.2 m /h 59430 ppm
3
23.2 m 3/h 500 ppm
38.4 m /h 95080 ppm
xp,6,3 =0.24
permeate 200 m 3 /h 100 ppm permeate 100 m 3 /h 300 ppm
3
65.2 m /h 500 ppm
xb,3,1=0.494
permeate 50 m 3/h 500 ppm
107.8 m 3/h 95080 ppm xb,4,1=0.33
brine
Fig. 6. The configuration of the local optimal solution with three kinds of permeate output.
turn. In this case, under the conditions of meeting all of the product quantity and quality requirement, the optimal solution provided two kinds of product with less salinity. These different grades of product can satisfy the requirement of water quality on different occasions. The high quality water product can be widely used in many industrial processes, therefore it is the main product in this RO system with multiple-product output, and it is more profitable product. When producing this main product, the RO system can simultaneously provide the other grade by-product. 5. Conclusions The optimization method based on process synthesis concept has been applied to design the RO system with multiple-feed and multiple-product required. A structural representation that incorporates all feasible arrangements for this desalination problem of multiple water sources was presented. This superstructure was used to develop the general mathematical model by means of the appropriate relationships between the variables. The design task is formulated as an MINLP model which minimizes the total annualized cost of the RO system. The solution of the problem includes the optimal system structure and operating conditions, the optimal streams distribution, and the optimal feed position of streams and outlet of the system. The design method could also be used for the optimal selection of type and number of membrane elements in each stage and in each pressure vessel. A desalination case is solved to demonstrate the effectiveness of this design method. Symbols A
water permeability, kg/m 2 s Pa
B Ce Cm C Ck Cpv Cw CCin CChpp CCTb Ds d Jw Js K Lpv
solute transport parameter, kg/m 2 s electricity cost, $/(kw .h) membrane module cost, $ concentration of solute, ppm the price of the kth membrane element, $ the price of the pressure vessel, $ concentration at the membrane wall, ppm capital cost of the seawater intake pump, $ capital cost of the high pressure pump, $ capital cost of the turbine, $ the solute diffusivity, m 2/s the feed spacer thickness, m water flux, kg/m 2 s salt flux, kg/m 2 s the mass transfer coefficient, (m/s) length of the pressure vessel, m
Lm m n Nl Nw Np Nps NRO OCin OChpp OCTb OCm P ΔPf PD PS Q Qp Qrp Crp Re R Sc Sm T TAC U L Vw W WT xf,i,q
xc y ,Z
the length of a element, m the number of membrane elements in each PV the number of pressure vessel the number of leaves in a membrane element inlet junctions of feed streams outlet junctions of product streams mixing junctions, reverse osmosis junctions energy cost of the intake pump, $ energy cost of the high pressure pump, $ the saving cost generated by turbine, $ the cost of membrane module maintenance, $ operating pressure, Mpa the pressure drop in the membrane channel, Mpa set of outlet junctions for the product streams set of stream mixing junctions flow rate, m 3/h the permeate flow rate of pressure vessel, m 3/h the flow rate of the r th product water required, m 3/h the concentration of the r th product water required, ppm Reynold's number set of reverse osmosis junctions Schmidt number the membrane area per element, m 2 temperature, °C total annualized cost, $ arbitrary large numbers arbitrary small numbers the permeate velocity, m/s. membrane width, m set of inlet junctions for the feed streams the stream split ratio of the qth feed stream linked to the ith pressurization stage the stream split ratio of the brine leaving the jth RO stage and being linked to the ith pressurization stage the stream split ratio of the permeate leaving the jth RO stage and being linked to the ith pressurization stage the small positive number binary integer
Greek ∏ ∏w μ ρ
osmosis pressure, Mpa osmosis pressure of the brine at the membrane wall, Mpa the water viscosity, kg/m s density, kg/m 3
xb,i,j xp,i,j
50
η ε
Y. Lu et al. / Desalination 307 (2012) 42–50
pump efficiency the small positive number
Subscripts f feed stream p permeate stream b brine stream in intake seawater ps,i the i th pressurization stage RO,j the jth RO stage ROf,j the feed stream of the j th RO stage ROb,j the brine stream of the jth RO stage ROp,j the permeate stream of the jth RO stage k the kth element type hpp high pressure pump Tb turbine i index of reverse osmosis junctions j index of stream mixing junctions r index of outlet junctions for the product streams q index of inlet junctions for the feed streams
Acknowledgements Financial support from the Natural Science Foundation of Guangxi (No. 2012GXNSFAA053025), Guangxi Higher Education Institutes Talent Highland Innovation Team Scheme (GJR201147-12) are gratefully acknowledged. Many thanks to Jinxian Liu and Dominic Foo Chwan Yee for help with the English. References [1] E. El-Zanati, S. Eissa, Development of a locally designed and manufactured small-scale reverse osmosis desalination system, Desalination 165 (2004) 133–139. [2] A. Villafafila, I.M. Mujtaba, Fresh water by reverse osmosis based desalination: simulation and optimization, Desalination 155 (2003) 1–13. [3] N.M. Wade, Distillation plant development and cost update, Desalination 136 (2001) 3–12. [4] M. Wilf, C. Bartels, Optimization of seawater RO systems design, Desalination 173 (2005) 1–12.
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