Optimization of Reverse Osmosis Networks with Spiral-Wound

Article pubs.acs.org/IECR

Optimization of Reverse Osmosis Networks with Spiral-Wound Modules Yawei Du,†,‡ Lixin Xie,*,†,‡ Yuxin Wang,†,‡ Yingjun Xu,§ and Shichang Wang†,‡ †

Chemical Engineering Research Center, School of Chemical Engineering and Technology, and ‡Tianjin Key Laboratory of Membrane Science and Desalination Technology, Tianjin University, Tianjin 300072, PR China § State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, PR China S Supporting Information *

ABSTRACT: An optimization of reverse osmosis (RO) networks for seawater desalination with spiral-wound modules (SWM) was presented in this work. The membrane transport model, which was based on the mass and momentum transport equations, took into consideration the longitudinal variation of the velocity, the pressure, and the salt concentration in the membrane modules. The pressure exchanger (PX) was included in the RO superstructure, and salinity increase caused by volumetric mixing in the PX was considered. The results obtained from the presented model were compared with the actual plant operational data from literature and found to be in good agreement with relative errors of 0.81%∼2.15% and 0.01%∼0.09%, in terms of water recovery and salt rejection, respectively. The optimum design problem was formulated as a mixed integer nonlinear programming (MINLP) problem. The variation of feed salinity was studied using the RO networks model. For the feed concentration higher than 32 kg/m3, one-stage RO system is favored. When the feed concentration is below 28 kg/m3, two-stage RO system is the better choice. The unit product cost increases with the decreases of permeate concentration requirement. For the looser permeate concentration requirement (0.30 kg/m3), one-pass configuration can meet the required quality of desalted water. When the lower permeate quality requirement of concentration is from 0.050−0.20 kg/m3, a two-pass system is more suitable. The influence of system recovery rate on the plant performance was discussed. Finally, sensitivity analysis showed that the total annualized cost is highly sensitive to the feed flow rate, the operating pressure, and electricity cost, while the energy consumption is highly sensitive to the operating pressure, the feed salinity, and the feed temperature.

1. INTRODUCTION Reverse osmosis (RO) has been widely used for brackish and seawater desalination. RO has several advantages over other desalination processes such as distillation and electro-dialysis.1 The main advantages of RO are its no phase change, lower energy consumption, modularity, flexibility, and the ability to construct small size plants.2−4 Developments in membrane materials and module design, process design, feed pretreatment and energy recovery device lead to significant cost reductions.5 Most membranes to be installed recently in RO membrane desalination are flat sheet membranes in a spiral-wound module (SWM) configuration, because SWMs could offer a good balance in terms of permeability, packing density, fouling control, and ease of operation.6 RO seawater desalinations are usually multiple stage processes, and each RO stage contains many parallel arrays of pressure vessels (PV), which containing several RO membrane modules connected in series. The optimization of a RO system includes the optimal design of both the individual module structure and the network configuration, which needs designing the process layout and defining operating conditions to meet given project specifications. This requires making choices over a vast number of options that rely on both continuous (e.g., flow rates) or discontinuous (choice of equipments, interconnections, etc.) variables. All these have made the design of RO process more flexible. Many researches of the optimum RO system have been made.7−10 Geraldes et al.11 developed a numerical model to optimize the module configuration and the operating conditions © 2012 American Chemical Society

of a medium-sized SWRO with SWMs. The longitudinal variation of the velocity, the pressure, and the salt concentration in the membrane modules were taken into consideration. Marcovecchio et al.12 developed a novel global optimization algorithm to find the global optimal design of RO seawater desalination with hollow fiber modules. On the basis of the state-space approach, El-Halwagi13 optimized the hollow fiber RO network problem as a mixed integer nonlinear programming (MINLP). Maskan et al.14 used directed graph and connectivity matrix to represent the RO network superstructure. A variable reduction technique was introduced to accelerate the computational process. Voros and Maroulis15,16 simplified the El-Halwagi’s representation by reducing the distribution boxes to junctions. Consequently, the model was formulated as a nonlinear programming (NLP) problem. Saif et al.17,18 formulated the RO design network based on a superstructure which embeds all possible alternatives of a potential treatment network for water and wastewater streams. Lu et al.19 optimized the RO system with SWMs under different feed concentration and product specification. Stream split ratio and isobaric-mixing constraints were introduced in the model of superstructure. Sassi and Mujtaba20 optimized the RO process with membrane fouling. A variable fouling profile along the membrane stages was introduced to investigate the effect of Received: Revised: Accepted: Published: 11764

March 11, 2012 August 3, 2012 August 10, 2012 August 10, 2012 dx.doi.org/10.1021/ie300650b | Ind. Eng. Chem. Res. 2012, 51, 11764−11777

Industrial & Engineering Chemistry Research

Article

Figure 1. Scheme of the rectangular channel model of spiral-wound module feed channel.11.

fouling on different process parameters such as water flux and water recovery. Vince et al.21 developed a multiobjective optimization (MOO) method using a flexible superstructure. The RO process configurations were evaluated by economical (investment and operating costs), technical (energy requirement, water recovery rate), and environmental performance indicators (Life Cycle Assessment). Guria et al.22 developed a MOO method using genetic algorithm (GA) for the desalination of brackish and seawater with spiral-wound or tubular modules. The binary coded elitist nondominated sorting genetic algorithm (NSGA-II) is used to obtain the solutions. In this paper, a numerical model describing the membrane transport of reverse osmosis in SWMs is introduced. The transport phenomena of solute and water through the membrane are described by a set of algebraic and differential equations. A RO superstructure with pressure exchanger (PX) is presented, and salinity increase caused by mixing between the brine and seawater in PX is considered. Discrete constraints were introduced to make sure the numbers of membrane elements in each PV and the number of PVs employed in the RO stage were integers. The model also includes system flow constraints such as the maximum concentration polarization factor (CPF), the average permeates flux and the maximum permeate flux in PV. The formulated MINLP is solved to determine the optimal system structure, the proper type membrane module used, and the operating conditions.

(Lm). The seawater enters in the channel with an average velocity Vin, a pressure Pin, and a feed concentration Cin. The material balances to the infinitesimal control volume are given by V dV = −2 w dz h

dVCch,b dz

= −2

(1)

VwCch,p (2)

h

where z is the distance to the channel inlet [m]; V is the superficial velocity in the feed channel [m/s]; Vw is the permeate velocity [m/s]; h is the feed channel thickness [m]; and Cch,b, Cch,p are the concentration in the feed and permeate channels, respectively [kg/m3]. A momentum balance to the infinitesimal control volume gives25 ρ V2 dP = −λ dz de 2

(3)

λ = Kλ6.23Re−0.3

(4)

where P is the feed-side pressure [MPa]; ρ is the fluid density [kg/m3], λ is the friction factor; and de the equivalent diameter of the feed channel [m]. The factor Kλ is constant relation to pressure losses in the feed tubes and module fittings.11 The above equations are solved using the method of finite differences and detailed in Table 1. The boundary conditions are as follows:

2. PROCESS MODELING 2.1. The Model for Spiral-Wound Modules. A model for a spiral-wound RO system using a solution diffusion model is based on the one described by Geraldes et al. and Avlonitis.11,23 The basic assumptions of this model are as follows: (i) neglect the curvature of the channels; the membrane modules are made up of flat channels with constant geometrical shape, because the channel thickness are much lower than the module radius; (ii) assume plug flow in feed channels; (iii) the pressure decrease in the permeate channels is negligible; (iv) the flow in the porous substructure of the asymmetric membrane is unhindered; (v) the concentration polarization is quantified by the film theory;24 (vi) negligible component of the concentrate and permeate flow velocities along the x and y axis, respectively; (vii) there are no sparingly soluble salts that can precipitate on the membrane surface. The schematic diagram of a flat feed channel with a thickness h and dimensions of LPV (length) × W (width) is depicted in Figure 1. The length LPV is the number of modules in series in a pressure vessel (nm) multiplied by the length of each module

at

z = 0,

V = Vin ,

C = C in ,

P = Pin

(5)

The membrane water permeability A is approximated as a function of feedwater temperature T [°C], trans-membrane osmotic pressure Δπ [MPa], and fouling factor FF by the following relation:21 A = A ref (Δπ ) ·FF·TCF

(20)

where Aref (Δπ) donates the reference permeability at T0 =25 °C without fouling, TCF is the temperature correction factor at T, and FF is the fouling factor. The temperature correction factor TCF which denotes the influence of temperature on membrane permeability is expressed as follow:26 ⎡e⎛ 1 ⎞⎤ 1 ⎟⎥ TCF = exp⎢ ⎜ − ⎝ ⎣ R 298 273 + T ⎠⎦ 11765

(21)

dx.doi.org/10.1021/ie300650b | Ind. Eng. Chem. Res. 2012, 51, 11764−11777


Article

Table 1. Model Equations for Water and Solute Transport through a PVa equation

Jw, l = A(Pl − (πch,mw, l − πch,p, l))

J s, l =B(Cch,mw, l − Cch,p, l)

Vw, l =

description local permeate flux through subelement l of a PV

(6)

local solute flux through subelement l of a PV

(7)

Jw, l + Js, l

Cch,p, l =

ρp

permeate velocity

(8)

Js, l

× 1000

Vw, l

Cch,mw, l − Cch,p, l Cch,b, l − Cch,p, l

⎛ Vw, l ⎞ = exp⎜ ⎟ ⎝ Kl ⎠

Kl = 0.04Rel 0.75Scl 0.33

local salt concentration in the permeate

(9)

Ds de

concentration polarization24

(10)

mass transfer coefficient19

(11)

Q ch,b, l + 1 = Q ch,b, l − 3600Vw, lSl

flow rate at subelement l + 1

(12)

Q ch,b, l + 1Cch,b, l + 1 − Q ch,b, lCch,b, l = − 3600Vw, lSlCch,p, l

Q p,n =

∑ 3600Vw,lSl l

Q p,nCp,n =

∑ 3600Vw,lSlCch,p,l

total quantity of permeate of a PV salt concentration of permeate of a PV

(15)

ρ V2 ρ V 2⎞ Δz ⎛ ⎜− λl b l − λl + 1 b l + 1 ⎟ 2 ⎝ de 2 de 2 ⎠

λl = Kλ6.23Rel−0.3

Q f,n = Q b,n + Q p,n

salt concentration at subelement l + 1

(14)

l

Pl + 1 = Pl +

(13)

feed side pressure drop in feed channels

(16)

friction factor11

(17) (18)

Q f,nCf,n = Q b,nC b,n + Q p,nCp,n

total material balance for the PV

(19)

total mass balance for the PV

Symbols: A, water permeability coefficient [kg/m ·s·Pa]; B, solute transport coefficient [kg/(m2·s)]; Cmw, brine concentration at the membrane feed side surface [kg/m3]; π, local osmotic pressures of the solutions [Mpa]; Jw, local permeate flux [kg/(m2·s)]; Js, local solute flux [kg/(m2·s)]; Vw, permeate velocity [m/s]; ρp, density of the permeate [kg/m3]; K, local mass transfer coefficient [m/s]; de is the feed channel equivalent diameter [m]. Sl, membrane area of subelement l for a PV [m2], (Sl = Sm·nm/L), where Sm is the membrane area per element [m2], nm is the number of membrane elements in each PV, L is the number of discretization grid points; Re, Reynolds number (Re = ρVde/μ), where μ is liquid viscosity (Pa·s); Sc, Schmidt number (Sc = μ/ρDs), where Ds is the solute diffusivity [m2/s]; Q, flow rate [m3/h]; V, superficial velocity in the feed channel [m/s], (V = Q/(3600Sfcsεsp), Sfcs is the feed cross-section open area [m2], εsp is the void fraction of the spacer; Qp,n, total quantity of permeate [m3/h]; Cp,n, average concentration of permeate [kg/m3]; Δz, the integration step [m]; l, index number along z-axis on the PV; (subscripts: ch, feed, or permeate channel in PV; b, brine stream; f, feed stream; p, permeate stream; mw, membrane wall). a

2

where T is the water temperature [°C]; R is the universal gases constant equal to 8.314 J/(mol·K); e is the membrane activation energy [J/mol], which is estimated for all RO membranes at 25 000 J/mol when T ≤ 25 °C and at 22 000 J/mol when T > 25 °C.21,27 For each membrane, the influence of the osmotic pressure Δπ on the reference pure water permeability Aref (Δπ) has been measured experimentally at T0 = 298 K and FF = 1 by membrane producers. However in most cases, this relation is not provided. In a first approach, Aref (Δπ) is therefore considered to be constant and labeled Aref.21 For a RO desalination plant design, the decrease in permeate flux caused by fouling and salt passage increase in years should be taking into consideration:

FF = (1 − FFd)Nmlp

It is assumed that the model solution for the brackish and seawater is an equivalent solution of sodium chloride with the same osmotic pressure. The osmotic pressure was calculated from a nonlinear correlation derived from seawater osmotic pressure data:14 π = 4.54047(103C / Msρ)0.987

(0.0238 kg/m 3 < C < 119 kg/m 3) (24)

where Ms is the molecular weight of solute. The viscosity of the concentrated seawater was determined by the following relation:28 μ = (1.4757 × 10−3 + 2.4817 × 10−6C + 9.3287 × 10−9C 2) exp( −0.02008T ) × (0 kg/m 3 < C < 100 kg/m 3, 20°C < T < 45°C)

(22)

(25)

B = Bref (1 + Bin )

Nmlp

There is negligible variation in the value of diffusion coefficient Ds for the range of NaCl concentrations from 0 to 95.5 kg/m3, in this work Ds was estimated as 1.61 × 10−9 m2/s.29 2.2. RO Network Model. The superstructure approach to process design provides a systematic framework for simultaneous optimization of process configuration and operating

(23)

where FFd donates the decrease in permeate flux per year, Bin donates the salt passage increase per year, Bref donates the reference permeability at T0 = 25 °C without fouling, and Nmlp donates the design period [year]. 11766



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conditions.30 For RO systems, the basic components of the superstructure are RO modules, pumps, energy recovery device (ERD), stream mixers, and stream splitters. The RO network as presented by Liu et al19 is introduced in this paper. Pressurization stages Nps and RO stages NRO are the key structures of reverse osmosis networks (RON) shown in the Figure 2. The number of

NRO

Q ps, iCps, i =

j=1

(32)

stream junctions employed is Nps + 2, where 2 indicated the brine and product streams leaving the network. Each stream node among the Nps nodes represents a stream connected to a pump. The streams pressurized by high pressure pump (HPP) or not are connected to a corresponding RO stages. The RO stages consist of multiple parallel reverse osmosis PVs operating at the same conditions. Each stream of the network (the brine and permeate streams leaving all reverse osmosis stages) may be linked to all the Nps + 2 nodes. The complete mathematical model that describes the superstructure is presented as follow by means of the appropriate relationships between the variables (mass balance equations, technical and operational constraints): NRO

NRO

Q ps,1Cps,1 = Q f Cf +

(26) NRO

(34)

Q RO,1 = Q hpp + Q pxhout

(35)

Q RO,1C RO,1 = Q hppC hpp + Q pxhoutCpxhout

(36)

Q pxhout = Q pxlin

(37)

Q pxhin = Q pxlout

(38)

Lpx Q pxhin /100 = Q pxhin − Q pxhout

(39)

(27)

Mix = 6.0057 − 0.3559OF + 0.0084OF2

(28) NRO

Q pxhin =

∑ Q b,px,j

OF[%] = 100 × (29)

j=1

∑ Q b,px,jC b,px,j (30)

j=1 NRO

Q ps, i =

NRO

∑ Q b,i ,j + ∑ Q p,i ,j j=1

j=1

(41)

(for ERI PX‐220) (42)

Q pxhin − Q pxhout Q pxhin

(43)

where Mix and OF donate the volumetric mixing and overflush of the PX, respectively. The range of overflush is −10% to 15%. The concentration of the low-pressure brine leaving the PX is calculated as follow:

NRO

Q pxhinCpxhin =

(for ERI PX‐220)

Cpxhout = Mix (Cpxhin − Cpxlin) + Cpxlin

j=1

Pps,1 = Pf

Q ps,1Cps,1 = Q hppC hpp + Q pxlinCpxlin

Direct contact between the brine and seawater streams occurs inside the PX. As a result, these streams mix slightly. Because of the volumetric mixing, there is a salinity increase at the outlet of the PX. Volumetric mixing in a PX device is obtained by fitting the relationship between overflush and mixing.32 According to the solute material balance, the concentration of the high-pressure water leaving the PX is calculated with the following equations:

∑ Q b,1,jC b,1,j + ∑ Q p,1,jCp,1,j j=1

(33)

(40)

NRO j=1

Q ps,1 = Q hpp + Q pxlin

Lpx [%] = 0.3924 + 0.01238Ppxhin

∑ Q b,1,j + ∑ Q p,1,j j=1

i = 2, ..., Nps

j=1

where Qps,1 [m3/h], Cps,1 [kg/m3] denote the flow rate and concentration of the first pressurization stage, Qf [m3/h], Cf [kg/ m3] denote the feed flow rate and feed concentration of the RO network, Qb,px [m3/h], Cb,px [kg/m3] are the flow rate and concentration of high pressure brine entering the pressure exchanger, respectively. Qps,i [m3/h], Cps,i [kg/m3] denote the flow rate and concentration of the ith pressurization stage, respectively. Qb,i,j [m3/h], Cb,i,j [kg/m3] denote the brine flow rate and concentration of the jth RO stage being linked to the ith pressurization stage and its concentration, respectively. Qp,i,j [m3/h], Cp,i,j [kg/m3] denote the permeate flow rate and concentration of the jth RO stage being linked to the ith pressurization stage and its concentration, respectively. Pf, Pps,1 donate the feed pressure of the RO network and the inlet pressure of the first pressurization stage, respectively [MPa]. Qpxhin [m3/h], Cpxhin [kg/m3] denote the flow rate and concentration of the high-pressure brine water entering the PX, respectively. The material balance relationships for high pressure pump and PX are presented as eqs 33− 40, lubrication Lpx in a PX device is obtained by fitting the relationship with Ppxhin.31

Figure 2. Superstructure representation of the RO network.19

Q ps,1 = Q f +

NRO

∑ Q b,i ,jC b,i ,j + ∑ Q p,i ,jCp,i ,j

i = 2, ..., Nps

Cpxlout =

(31) 11767

Q pxlinCpxlin + Q pxhinCpxhin − Q pxhoutCpxhout Q pxlout

(44)



Article Kt

Furthermore, it is assumed that mixing is not allowed between streams that have different pressure values. (Pps, i − Pb, j)Q b, i , j = 0 (Pps, i − Pp, j)Q p, i , j = 0

i = 1, 2, ..., Nps , i = 1, 2, ..., Nps ,

(Ppxhin − Pb, j)Q b,px, j = 0

j = 1, 2, ..., NRO j = 1, 2, ..., NRO

j = 1, 2, ..., NRO

hj =

j = i,

j = 1, 2, ..., NRO

(45)

(47)

Pj − Pk ,max ≤ U (1 − yj , k ),

(49)

PRO, j = P′ps , i ,

j = i,

j = 1, 2, ..., NRO

(50)

j = 1, 2, ..., NRO

(51)

∑

Q p, i , j

j = 1, 2, ..., NRO

i

∑ yj ,k Ak ,

j = 1, 2, ..., NRO

k=1

∑ yj ,k Bk ,

j = 1, 2, ..., NRO

k=1

(61)

∑ yj ,k Sk ,

j = 1, 2, ..., NRO

∑ Q b, i , j + Q pxlout (62)

NRO

Q bC b =

∑ Q b,i ,jC b,i ,j + Q pxloutCpxlout (63)

j=1 NRO

Qp =

∑ Q p,i ,j (64)

j=1 NRO

Q pCp =

∑ Q p,i ,jCp,i ,j (65)

j=1

where Qb [m /h], Cb [kg/m ] are the flow rate and concentration of the brine leaving the RO network, respectively. Qp [m3/h], Cp [kg/m3] are the flow rate and concentration of the product water, respectively. Other constraints are imposed on the exit streams from the network. The flow rate of the final permeate streams cannot be less than a minimum desirable product flow rate, and the product concentration cannot exceed maximum allowable product concentration. 3

(53)

(54)

3

Q p ≥ Q plo

(66)

Cp ≤ Cpup

(67)

where refers to the minimum desirable product flow rate [m3/h], Cpup refers to the maximum allowable product concentration [kg/m3]. Qlop

Kt k=1

Q f Cf = Q bC b + Q pCp

j=1

Kt

Sj =

(60)

(52)

Kt

Bj =

(59)

Qf = Qb + Qp

Qb =

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 2−8 membrane elements per PV consist of a RO stage. Three different types of spiral-wound FilmTec reverse osmosis membrane elements have been considered. In the present work, it was assumed that the membrane elements employed in the PV in the same RO stage are the same. The membrane intrinsic properties (A, B) are assumed to be constant. According to its performance characteristics and the design requirements of a specific desalination application, the optimum selection of types of the membrane element employed in each PV can be determined by the following equations: Aj =

j = 1, 2, ..., NRO

NRO

Nps + 2

Q p, j =

(58)

where Ak is the pure water permeability of the kth membrane element [kg/(m2·s·Pa)]. Bk is the pure salt permeability of the kth membrane element [kg/(m2·s)]. Sk is the active membrane area of the kth membrane element [m2]. hk is the feed spacer thickness of the kth membrane element [m]. Cm,k is the price of the kth membrane element [$]. yj,k is a 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. Equation 58 is used to restrict the maximum pressure allowed for different type membrane elements. Pk,max denotes the maximum operating pressure for the kth type membrane elements [MPa]. U is a sufficiently large number. Kt is the total number types of RO membrane elements. The overall material balances for the RO network are presented as follows:

Nps + 2

Q b, i , j + Q b,px, j

≤1

k=1

3

i

j = 1, 2, ..., NRO

Kt

∑ yj ,k

where QRO,j [m /h], CRO,j [kg/m ], PRO,j [MPa] denote the feed flow rate, concentration, and operation pressure of the jth RO stage, respectively. Pps,i ′ is the outlet pressure of the ith pressurization stage [MPa]. The brine and permeate streams of the jth RO stage are split into several exit streams:

∑

(57)

k=1

(48)

j = 1, 2, ..., NRO

Q b, j =

∑ yj ,k Cm,k

(46)

j = i,

(56)

Kt

Cm , j =

C RO, j = Cps, i ,

3

j = 1, 2, ..., NRO

k=1

where Pps,i denotes the inlet pressure of the ith pressurization stage [MPa]. Pb,j, Pp,j denote the brine and permeate pressure of the jth RO stage, respectively [MPa]. Ppxhin denote the pressure of the high pressure flow entering in the PX [MPa]. The streams leaving the ith pressurization stages are correspondingly connected to the jth RO stages. Therefore the following equations can be obtained: Q RO, j = Q ps, i ,

∑ yj ,k hk ,

(55) 11768



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pass 2; minimum brine flux in the PV is 3.6 m3/h for seawater desalination (20 kg/m3∼ 45 kg/m3), and 2.4 m3/h for brackish desalination (3 kg/m3∼16 kg/m3); maximum CPF in the PV is 1.2 for pass 1 and 1.3 for pass 2.

2.3. Discrete Variable Constraints. Normally, a PV is capable of holding from two to eight SWM elements. In this work, the element number nm is expressed in terms of binary variables:30,33 nm, j = nmlo +

Nb

∑

2km − 1Zj ,km

3. SOLUTION METHODOLOGY The optimization design problem is formulated as a mixed integer nonlinear programming (MINLP) for minimizing the total annualized cost or energy consumption 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 cost equations are presented as follows:

(68)

km = 1

⎧ log(n up − n lo) ⎫ m m ⎬ Nb = 1 + int⎨ log(2) ⎭ ⎩ ⎪

⎪

⎪

⎪

(69)

where Zj,km is binary variable, nm,j donates the number of membrane elements employed in the PV in the jth RO stage. Nb lo is the minimum number of binary variable needed. nup m ,nm donate maximum and minimum number of elements allowed in a PV, respectively. The number of pressure vessels employed in the jth RO stage could also formulated in this method. n pv, j =

lo n pv

Nn

∑

+

2 kpv − 1Zj ,kpv + svj − svbj

kpv = 1

up lo ⎫ ⎧ ⎪ log(n pv − n pv ) ⎪ ⎬ Nn = 1 + int⎨ ⎪ ⎪ log(2) ⎩ ⎭

(70)

CC hpp = 52(ΔPhppQ hpp)

(74)

CC bp = 52(ΔPbpQ pxhin)

(75)

CCpx = 3134.7Q pxhin 0.58

(76)

NRO

NRO

∑ Cm,jnm,jnpv,j+ ∑ Cpvnpv,j j=1

(77)

j=1

(71)

TCC = 1.411(CCswip + CC hpp + CCpx + CC bp + Cm) (78)

OCm = 0.2Cm

(79)

⎛ P Q 24 PhppQ hpp24 swip f OCe = Cefc ⎜⎜ + ηhppηmoter ⎝ ηswipηmoter +

Cch,mw, l Cch,b, l

(73)

Cm =

where Zj,kpv is binary variable, npv,j donates the number of lo pressure vessels employed in the jth RO stage. nup pv , npv donate the maximum and minimum number of elements allowed in jth RO stage, respectively. Nn is the minimum number of binary variable needed. Note that eq 69 and 71 are used only to calculate the value of Nb and Nn; they are not used as constraint equations. In the lo following case studies, nlom = 2 and nup m = 8, which yield Nb = 3; npv up = 0 and npv = 200, which yield Nn = 8. To avoid mathematical infeasibility if the RO stage is nonexisting, the design specifications are relaxed using limited slack variables sv and svb, which are introduced as an additional set of terms in the objective function to be minimized. The weight chosen for the slacks are generally small. 2.4. System Flow Constraints. The concentration polarization factor (CPF) in the module feed channels can be defined as a ratio of salt concentration at the membrane surface Cch,mw,l to bulk concentration Cch,b,l, where CPF =

CCswip = 996(Q f 24)0.8

(72)

Membranes can only operate efficiently if the CPF in the module feed channels is lower than a given safety value. The feed pressure, the average permeates flux, and the maximum permeate flux are also imposed by the module manufacturer. Another constraint is that the brine concentration in the feed channels must be lower than 90 kg/m3.34 On the basis of the TORAY design guideline,35 the following constraints were added into the RO network model: maximum system average permeate flux is 20 L/(m2·h) for seawater desalination (20 kg/m3∼45 kg/m3), and 25 L/(m2·h) for brackish desalination (3 kg/m3 ∼16 kg/ m3); maximum allowable pressure drop of the pressure vessel is 0.35 MPa; maximum allowable permeate flux of the pressure vessel is 16 m3/h for SW30XLE −400 and SW30HR-380, 17 m3/ h for BW30−400; maximum permeate flux in the first membrane module in the PV is 35 L/(m2·h) for pass 1 and 48 L/(m2·h) for

(Phpp − Ppxhout)Q pxhout 24 ⎞ ⎟ ⎟ ηbpηmoter ⎠

(80)

OCinsrce = 0.005TCC

(81)

OC labor = Q p· 24· 365·fc ·0.01

(82)

OCch = Q f · 24· 365·fc ·0.0225

(83)

OCmaint = Q p· 24· 365·fc ·0.01

(84)

OCO&M = OCinsrce + OC labor + OCch + OCmaint

(85)

AOC = OCm + OCe + OCO&M

(86)

ηpx =

PpxhoutQ pxhout + PpxloutQ pxlout PpxhinQ pxhin + PpxlinQ pxlin

TAC = TCC/crf + AOC +

× 100% (87)

∑ (svj + svbj) (88)

j

upc =

EW =

TCC/crf + AOC Q p·24·365

PswipQ f ηswipηmoter

+

PhppQ hpp ηhppηmoter

(89)

+

(Phpp − Ppxhout)Q pxhout ηbpηmoter (90)

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4. CASE STUDY The proposed methodology for RO system optimization design was applied to the investigation of the operational and structural characteristics of two-stage RO system. Three types of FilmTec reverse osmosis membrane elements from DOW were included in the design studies of the current work, named as SW30XLE400, SW30HR-380, and BW30-400. The characteristics of the elements are given in Table 2. The parameters for calculation are listed in Table 3.

Equations 73− 78 refer to capital cost. CCswip, CChpp, CCbp, and CCpx were the capital cost of the seawater intake and pretreatment (SWIP), the high pressure pumps, the booster pumps, and the pressure exchangers, respectively [$]. The cost functions referred to the paper8 were introduced to calculate the capital cost for the CCswip, CChpp and CCbp. The PX cost function was assessed by Lu et al.19 ΔPhpp and ΔPbp was the pressure difference across the high pressure pump and booster pump, respectively [MPa]. Cm was the total membrane module cost [$]. Ck was the price of the kth membrane element and Cpv was the price of the pressure vessel [$]. Then, the plant equipment cost (CCequip) was the sum of CCswip, CChpp, CCbp, CCpx and Cm. Equation 78 gives the total capital cost (TCC). The factor 1.411 represents site development cost and indirect cost connected to the capital cost.8 OCm was the membrane replacement cost [$]. OCe was the energy cost [$]. Equations 79−85 refer to annual operation and maintenance cost (OCO&M). The OCO&M was considered to be the sum of the annual labor cost OClabor, the annual maintenance cost OCmaint, the annual chemical cost OCch, and the annual insurance cost OCinsrce [$]. These costs were calculated using the value reported in the papers.36−38 Phpp was the feed pressure of the high pressure pump [MPa]. ηswip, ηhpp, ηbp, ηmotor, and ηpx were the efficiency of the intake pump, high pressure pump, booster pump, electric motor and the PX, respectively. fc was the RO plant load factor. Then, the annual operating cost AOC was given by eq 86. The total annualized cost TAC was given by eq 88. sv and svb were introduced slack variables. Finally, the fresh water cost in $/m3 was given by eq 89. The capital recovery factor (crf) was

Table 2. Characteristics of FilmTec Spiral-Wound Reverse Osmosis Membrane Elements.19,20 SW30XLE-400 active area [ft2 (m2)] length of the element [in. (m)] diameter of the element [in. (m)] effect length of the element [m] feed cross-section open area, Sfcs [m2] feed space, h [mil]a void fraction of the spacer, εsp feed channel equivalent diameter, de [m] feed flow rate range [m3/h] maximum operating pressure [psig] ([MPa]) pure water permeability constant, Aref [kg/ (m2·s·Pa)] salt permeability constant, B [kg/ (m2·s)] membrane element cost [$] (estimation)

nLT

crf =

(Ir + 1) − 1 Ir(Ir + 1)nLT

(91)

where Ir is the interest rate set at 5%, nLT the plant lifetime set at 25 years. Equation 90 referred to energy consumption (the rate of pump work). The terms of capital cost items take into account the number and type of RO modules utilized for each RO stage, and the number and capacity of each pump and PX. The existence of the specific device may be determined indirectly by operational variables, such as the input flow rate to the unit, the pressure of the pressurization stage or RO stage. The structural optimization may take place in terms of eliminating all unnecessary pressurization stages or RO stages. This procedure is carried out by introducing an excessive number of units as an initial guess, while at the optimum certain design variables, such as the number of pressure vessels employed in the RO stage, are either set to zero or to a value, which indicates the absence or presence of the specific stage. All the examples were formulated using GAMS39 and solved on an Intel 1.83 GHz Linux machine with 2 GB memory. The outer approximation algorithm was employed, which consisted of iterating between reduced nonlinear programming (NLP) subproblems and a mixed-integer linear programming (MILP) master problem. The solvers GAMS/CPLEX and GAMS/MINOS were used to solve the MILP and NLP problems, respectively.39,40 Since the MINLP optimization problem was nonconvex, the global optimal solution could not be guaranteed. Several starting points were used to obtain the best possible solution.

a

SW30HR-380

BW30-400

400 (37.2) 40 (1.016)

380 (35.3) 40 (1.016)

400 (37.2) 40 (1.016)

7.9 (0.201)

7.9 (0.201)

7.9 (0.201)

0.88

0.88

0.88

0.0150

0.0147

0.0150

28 0.9

28 0.9

28 0.9

8.126 × 10−4

8.126 × 10−4

8.126 × 10−4

0.8−16

0.8−16

0.8−17

1,200 (8.3)

1000(6.9)

600 (4.1)

3.5 × 10−9

2.7 × 10−9

9.39 × 10−9

3.2 × 10−5

2.3 × 10−5

5.65 × 10−5

1200

1000

900

1 mil = 0.0254 mm.

Table 3. Parameters for the Reverse Osmosis Desalination Model feed water temperature, T [°C] average brine density, ρ [kg/m3] universal gases constant, R [J/(mol·K)] molecular weight of solute, Ms SWIP outlet pressure, Pswip [MPa] intake pump efficiency, ηswip high pressure pump efficiency, ηhpp pressure exchanger efficiency, ηpx electric motor efficiency, ηmotor RO plant load factor, fc cost of electricity, Ce [$ /(kWh)−1] pressure vessel cost (estimation) [$] friction factor correction parameter, Kλ

20 102019 8.31421 58.521 0.512 75%19 75%19 95% 98%19 0.9 0.0819 100019 2.411

The integration step was set to Δz = 1/30 of the total pressure vessel length LPV. The optimization of the superstructure was carried out using the following specifications: • potable water - daily production capacity: 120 m3/h • water resource - expected decrease in flux per year: 7% 11770



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- expected salt passage increase per year: 10% - design average membrane lifetime: 5 year - design period: 3 year - temperature T: 20 °C In general, the MINLP model contents 28 binary variables and 1636 continuous variables. Solving the mathematic programming problems takes between 27607 and 73667 iterations, 18.608 and 100.557 s CPU time to finish, depending on the initial values used for the optimized variables. To verify the accuracy of the model, the operational data from the paper11 were simulated by the RON without PX. The results were compared with the plant operational data and the output obtained from the powerful Dow/FilmTec ROSA 7.2 software package. As can be seen from Table 4, the overall results obtained

at same levels. The total production capacity was 120 m3/h, and the maximum allowed salt concentration was 0.50 kg/m3. For the relatively higher feed concentrations varying from 32 to 45 kg/m3, the results of the RO system optimization design are presented in Table 5. (The entire set of optimization results is fully described in Supporting Information Table S1.) One stage structure is favored (shown in Figure 3) for this range of

Table 4. Comparison Operational Data with Desalination Unit from Porto Santo Island11 and This Work Results

Figure 3. Optimum RO system for feed concentration varying from 32 to 45 kg/m3.

temperature [°C]

feed pressure [MPa]

feed flow [L/s]

system recovery rate [%]

salt rejection [%]

31-Dec-00a ROSA7.2 model

19

5.90

22.50

30.93 31.22 31.18

99.24 99.56 99.29

31-Jan-01a ROSA7.2 model

19

5.90

22.90

32.14 30.90 32.83

99.28 99.56 99.29

28-Feb-01a ROSA7.2 model

19

5.90

20.20

32.67 33.16 33.30

99.31 99.53 99.24

31-Mar-01a ROSA7.2 model

18

5.71

19.50

30.82 31.73 31.45

99.27 99.52 99.18

concentrations. With an increase of feed concentration, the osmotic pressures increase, and the RO operation pressure increases correspondingly. Meanwhile, the system recovery decreases, and the numbers of elements in each PV decrease slightly. The selected type of membrane element is the SW30XLE-400 that offers very high productivity and rejection, enabling the lowest total cost of water for desalination. For the medium feed concentrations varying from 20 to 28 kg/ m3, the design results are presented in Table 6. (The entire set of optimization results is fully described in Supporting Information Table S2.) The two-stage structure in which the brine coming from stage 1 enters fully to stage 2 has been identified in this range of concentrations. An interstage booster pump is not favored (shown in Figure 4). The selected membrane element is SW30XLE-400 for both stages. For the lower feed concentrations (3−16 kg/m3), the design results are presented in Table 7. (The entire set of optimization results is fully described in Supporting Information see Table S3.) The two-stage RO system with brine reprocessing is favored (shown in Figure 5). The first stage RO rejection entering into the second stage could be pressurized up to higher pressure, so the system recovery can be enhanced. BW30-400 is selected by both RO stages. In general, for different feed concentration (varying from 3 to 45 kg/m3), the design results (shown as Tables 5−7) indicate that the unit product cost is proportional to the feed concentration. With increasing feed concentration the system recovery rate decreases and the product water quality deteriorates. For processing higher feed concentration, the high operating pressure is necessary, and the simple one-stage structure is more favored.

a

Desalination unit from Porto Santo Island, Portugal with a single RO stage (12 pressure vessels with 4 membrane modules (FilmTec SW30HR-380, Dow Chemical Company, Midland, MI)). The feed concentration was 38 kg/m3.

from the presented RON model are in good agreement with the actual plant. The model yielded an overall water recovery and salt rejection having relative errors of 0.81%∼2.15% and 0.01%∼0.09%, respectively, compared to those produced by the actual plant. 4.1. Optimization of the System Total Annualized Cost. 4.1.1. The Study of Varying Feedwater Concentration. Several cases were solved, in which the feedwater concentration varied while the product demand and quality constraints maintained

Table 5. Design and Optimization Results for Feed Concentration Varying from 32 to 45 kg/m3 process flow system feed flow [m3/h] system overall recovery [%] product concentration [kg/m3] membrane type in stage 1 number of elements per PV number of PV in stage 1 operation pressure [MPa] unit product cost [$/m3]

45 kg/m3

42 kg/m3

38 kg/m3

35 kg/m3

32 kg/m3

Figure 3 293.4 40.9 500 SW30XLE-400 5 32 7.62 0.597

Figure 3 265.9 45.1 500 SW30XLE-400 6 27 7.53 0.570

Figure 3 229.4 52.3 500 SW30XLE-400 6 27 7.49 0.538

Figure 3 217.9 55.1 500 SW30XLE-400 7 24 7.22 0.519

Figure 3 210.3 57.1 490 SW30XLE-400 7 25 6.84 0.502

11771



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The longitudinal variations of the pressure in the PV are depicted in Figure 7a and Figure 7b. Pressure is the required driving force for the separation in the reverse osmosis process. Because of pressure drop due to friction, feed side pressure decreases along the feed side channel. The maximum pressure difference is in the feed entrance position and its minimum is in the end position in the PV. As the feedwater flows through the channel, permeate water is produced. With the increasing of brine concentration (shown in Figure 7c,d), the osmotic pressure increased, so the net driving force (ΔP − Δπ) of RO process decreased. If the membrane channel is long enough, salt concentration reaches its equilibrium value, where the osmotic pressure equals the driving pressure and the transport process stops. At this point the permeate flux diminishes to zero. The longitudinal variations of the CPF in pass 1 and pass 2 (Figure 7e,f) are quite different. The maximum CPF occurs at the entrance of the PV and decreases along the PV in pass 1. The concentration polarization is directly proportional to the rate of

Table 6. Design and Optimization Results for Feed Concentration Varying from 20 to 28 kg/m3 28 kg/m3 process flow system feed flow [m3/h] system overall recovery [%] product concentration [kg/m3] membrane type in stage 1 membrane type in stage 2 number of elements per PV in stage 1 number of elements per PV in stage 2 number of PV in stage 1 number of PV in stage 2 operating pressure in stage 1 [MPa] operating pressure in stage 2 [MPa] unit product cost [$/m3]

24 kg/m3

20 kg/m3

Figure 4 205.5

Figure 4 199.3

Figure4 197.6

58.4

60.2

62.7

0.45

0.41

0.35

SW30XLE-400

SW30XLE-400

SW30XLE-400

SW30XLE-400

SW30XLE-400

SW30XLE-400

6

6

5

5

5

7

25 6 6.28

25 8 5.67

27 8 5.06

6.09

5.52

4.98

0.478

0.455

0.433

Table 7. Design and Optimization Results for Feed Concentration Varying from 3 to 16 kg/m3 process flow system feed flow [m3/h] system overall recovery [%] product concentration [kg/m3] membrane type in stage 1 membrane type in stage 2 number of elements per PV in stage 1 number of elements per PV in stage 2 number of PV in stage 1 number of PV in stage 2 operating pressure in stage 1[MPa] operating pressure in stage 2[MPa] unit product cost [$/m3]

4.1.2. The Study of Varying Product Concentration. For different values of the product quality constraints varying between 0.050 and 0.30 kg/m3, the optimization of the superstructure was carried out at a constant raw feed concentration of 35 kg/m3 and the product capacity of 120 m3/h. The design results are presented in Table 8. (The entire set of optimization results is fully described in Supporting Information Table S4.) The optimal structure employed in the design is strongly dependent on the required product concentration. For the looser permeate concentration requirement (0.30 kg/m3), one stage configuration (shown in Figure 3) could meet the required quality of product water. When the permeate quality requirement of concentration is in the range from 0.050−0.2 kg/m3, the twopass system with permeate reprocessing and brine recycle (shown in Figure 6) is selected, the permeate coming from the pass 1 is reprocessed in pass 2. The selected membrane element is SW30XLE-400 for pass 1 and BW30-400 for pass 2. For the seawater concentration of 35 kg/m3, the temperature of 20 °C and the permeate concentration of 0.050 kg/m3, the longitudinal variation along a pressure vessel of several relevant variables are shown in Figure 7. (The longitudinal variation of the feed velocity and permeate flux in the PV are shown in the Supporting Information, see Figure S1.)

16 kg/m3

12 kg/m3

6 kg/m3

3 kg/m3

Figure5 183.9 65.3 0.44

Figure 5 171.3 70.1 0.37

Figure5 150.5 79.7 0.20

Figure5 137.1 87.5 0.12

BW30-400 BW30-400 4

BW30-400 BW30-400 6

BW30-400 BW30-400 6

BW30-400 BW30-400 8

7

6

7

6

20 10 3.23

17 10 2.86

16 7 2.17

13 7 1.61

3.81

3.41

2.69

2.31

0.353

0.322

0.270

0.240

Figure 5. Optimum RO system for feed concentration varying from 3 to 16 kg/m3.

Figure 4. Optimum RO system for feed concentrations varying from 20 to 28 kg/m3. 11772



Article

an additional constraint in the optimization procedure. Then the minimum unit product cost and the energy consumption under certain system recovery rates were obtained. The unit product cost and the corresponding energy consumption are represented as functions of the system recovery rate, and the results are displayed in Figure 8. It appears that there are two groups of solutions. For instance, when the feed seawater concentration is 32 kg/m3, one-stage system (shown in Figure 3) is selected when system recovery rate is below 60%. When the recovery rate is 60% or higher, a twostage system with interstage pump (shown in Figure 5) is needed to enhance the recovery rate. In this case, the maximum admissible brine concentration of 90 kg/m3 is achieved. Further increase of the water recovery rate is restricted by the maximum brine concentration. As shown in Figure 8, for the feed seawater concentration is 32 kg/m3, the energy consumption of product water drops to a minimum value of 2.418 kWh/m3. The water cost decreases as the water recovery rate increases, up to water recovery rates of 0.57. Beyond this value, the water cost increases again. As a consequence, there is an optimal value of the water recovery rate, which is 0.57 for this particular case. 4.3. Sensitivity Analysis. In this section, the effects of increasing or decreasing some model parameters were studied. The outer approximation algorithm was implemented for the resolution. When an optimization problem was solved, the objective function was minimized or maximized. Let y be the objective function. The marginal value of a variable x, also known as dual variable or Lagrangian multiplier, was defined as: ∂y/∂x evaluated at the optimal value. This partial derivative considered all the explicit and implicit influence of the variable x over the variable y, given by the proper objective function definition as well as the model equations. The marginal values for a representative example are presented. The solved case corresponds to 35 kg/m3 of seawater concentration, 20 °C of feed temperature, and 0.050 kg/m3 of requirement permeate concentration. Table 9 shows the relative marginal values for the total annual cost (TAC) and the rate of pump work (Ew) for the RO system, they are: (∂(TAC)/∂p)(p/ TAC) and (∂(Ew)/∂p)(p/Ew) for the main model parameters. The relative marginal values are more appropriated to evaluate the influence of the change at each variable over the whole process. Note that operating conditions (Pf, T, C, Qf), membrane intrinsic properties (A, B), and the cost of electricity (Ce) were investigated. All of these parameters were defined as “variables” in the GAMS models, and their values were fixed before solving the models. In this way, the solver computed the numerical derivative for the objective function and constraint with respect to these fixed variables and reports their values at optimality. The relative marginal values were arranged according their absolute values. The positive sign of relative marginal values indicates that the objective function increases its value when the corresponding parameters are increased. On the contrary, a negative relative marginal value indicates that increasing the parameter decreases the objective value. The marginal values mean how much we can benefit from modifying a particular parameter. Then, if a parameter p is increased from its current value p* by a small amount Δp, the net change for the objective function y will be (Δp∂y/∂p) approximately. As Table 9 shows, the feed flow rate Qf is more influential over the TAC. In the second place, the operating pressure Pf in the first pass is proportional to the TAC. Electricity cost Ce

Table 8. Design and Optimization Results for Different Product Concentrations at a Constant Raw Feed Concentration of 35 [kg/m3] 0.30 [kg/m3] 0.20 [kg/m3] 0.10 [kg/m3] 0.050 [kg/m3] process flow system feed flow [m3/h] system overall recovery [%] membrane type in pass 1 membrane type in pass 2 number of elements per PV in pass 1 number of elements per PV in pass 2 number of PV in pass 1 number of PV in pass 2 operating pressure in pass 1 [MPa] operating pressure in pass 2 [MPa] unit product cost [$/m3]

Figure 3 294.8

Figure 6 237.4

Figure 6 233.7

Figure6 233.6

40.7

50.5

51.4

51.4

SW30HR380

SW30XLE400 BW30-400

SW30XLE400 BW30-400

SW30XLE400 BW30-400

5

6

6

7

8

8

8

31

32

32

8

10

12

6.88

6.92

6.64

1.65

1.96

1.86

0.626

0.668

0.689

37

6.85

0.567

Figure 6. Optimum RO system for product concentration varying from 0.050 to 0.20 kg/m3.

accumulation of solute on the membrane surface. Although the feed flow rate decreases along the PV, the first module is more prone to membrane scaling than the last one in the PV, despite the reduction of the feed velocity from the first to the last membrane module. In pass 2 the decrease in the volumetric permeates flux along the length of the feed channel was very small. The lower feed concentration and operating pressures resulted in the CPF, which was actually increasing along the PV and reaches its maximum at the outlet of the PV. The definition of apparent salt rejection is (1 − Cch,p,l/Cch,b,l). As shown in Figure 7 panels g and h, the intrinsic rejection coefficient deteriorates along the pressure vessel, on one hand, because of the increase of the feed concentration and, on another hand, because of the reduction of the permeate flux, which reduces the intrinsic rejection factor from 99.52% in the channel inlet to 97.64% in the channel outlet in pass 1. The variation of apparent salt rejection is similar to the intrinsic salt rejection. 4.2. The Study of Varying System Recovery Rates. Because the system recovery rate influences both the energy and capital costs, it is important to study the influence of this parameter on the plant performance. The effects of system recovery rate on the RO system performance were studied for the seawater concentration range from 32 to 38 kg/m3. In this case, the total annualized cost was minimized with the system recovery rate as 11773



Article

Figure 7. Longitudinal variation of the relevant variables in the PV for the feed concentration of 35 kg/m3, the temperature of 20 °C, and the permeate concentration of 0.050 kg/m3: (a) pressure in pass 1; (b) pressure in pass 2; (c) concentration in pass 1; (d) concentration in pass 2; (e) CPF in pass 1; (f) CPF in pass 2; (g) rejection coefficients in pass 1; (h) rejection coefficients in pass 2;.

decrease the operating pressure. Although the feed salinity Cf is not always a parameter feasible to manipulate, its rise would both increases the TAC and Ew.

is another important factor proportional to the TAC. The operating pressure Pf is the parameter more influential over the Ew. To obtain lower energy consumption, it would be possible to 11774



Article

Figure 8. Effects of system recovery rate on the RO system for the feed concentration range from 32 to 38 kg/m3: (a) energy consumption; (b) unit product cost.

factor, and maximum system average permeate flux), the results of optimization could be different. Finally, a sensitivity analysis was performed. This analysis shows that the total annual cost of the process is highly sensitive to the feed flow rate Qf, the operating pressure Pf and electricity cost Ce; the energy consumption is highly sensitive to the operating pressure Pf, the feed salinity Cf and the feed temperature T. Therefore, a careful effort should be made to accurately calculate all the technological coefficients involved in the process.

Table 9. Relative Marginal Values for the Total Annualized Cost and the Rate of Pump Work for the RO System (∂(TAC) /∂p)(p/TAC) Qf Pf (pass1) Ce Cf A (pass 1) Pf (pass 2) T B (pass 2) A (pass 2) B (pass 1)

0.5128 0.4382 0.4303 0.3447 −0.1243 0.1147 −0.09886 0.004938 −0.002875 0.002647

(∂(Ew)/∂p)(p/Ew) Pf (pass 1) Cf Pf (pass 2) T A (pass 2) Qf A (pass 1) B (pass 1) B (pass 2)

0.6499 0.6490 0.2073 −0.1971 −0.1703 0.1360 −0.09798 0.02689 0.008470

■

ASSOCIATED CONTENT

S Supporting Information *

The entire set of design and optimization results obtained by the proposed model. The longitudinal variation of the feed velocity and permeate flux in the PV. This material is available free of charge via the Internet at http://pubs.acs.org.

In the proposed model, feed temperature is affecting both the physical properties of seawater and the membrane water permeability A. With the feed temperature increases the membrane water permeability increases. At the same time, water viscosity decreases and water passes easily through the membrane. As a result, the effects of a raise in the feed temperature are both negative to decrease the TAC and Ew. 4.4. Conclusion. An optimization model of RO networks was established with SWMs. The objective was to determine a few modules/pumps/ERD optimal system configuration and operating conditions under system flow constraints that could be used in the actual design of industrial seawater RO plants. The variation of salinity of feed seawater was studied using the RON model. For the feed concentration higher than 32 kg/m3, onestage RO system is favored. When the feed concentration is below 28 kg/m3, two-stage RO system is the better choice. The unit product cost increases with the decreases of permeate concentration requirement. For the looser permeate concentration requirement (0.30 kg/m3), one-pass configuration can meet the required quality of desalted water. When the lower permeate quality requirement of concentration from 0.050−0.20 kg/m3, two-pass system is more suitable. The influence of system water recovery on the plant performance was discussed. For the seawater concentration of 32 kg/m3, the energy consumption of product water drops to a minimum value of 2.418 kWh/m3. The water cost decreases as the water recovery increases, up to water recovery rates of 0.57. Further increase of the water recovery rate is restricted by the maximum brine concentration. It is possible that if the capital and operating costs, or system flow constraints are different for the local context (e.g., seawater temperature, membrane fouling

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 (022)27404347. Fax: +86 (022)87891367. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We were able to complete this work with the kind assistance of the State Key Laboratory of Earth Surface Processes and Resource Ecology (Beijing Normal University).

■

11775

NOMENCLATURE A = water permeability [kg/(m2·s·Pa)] B = solute transport parameter [kg/(m2·s)] C = concentration of solute [kg/m3] CCbp = capital cost of the booster pumps [$] CCequip = plant equipment cost [$] CChp = capital cost of the high pressure pumps [$] CCswip = capital cost of the seawater intake and pretreatment [$] CCpx = capital cost of the pressure exchangers [$] Ce = electricity cost [$/(kWh)] Cm,k = price of the kth membrane element [$] Cmw = concentration at the membrane wall [kg/m3] Cpv = price of the pressure vessel [$] DCC = direct capital cost [$] de = feed channel equivalent diameter [m] dx.doi.org/10.1021/ie300650b | Ind. Eng. Chem. Res. 2012, 51, 11764−11777


■

Ds = solute diffusivity [m2/s] h = feed spacer thickness [m] ICC = indirect capital cost [$] Ir = interest rate Js = salt flux [kg/(m2·s)] Jw = water flux [kg/(m2·s)] K = mass transfer coefficient [m/s] Kλ = friction factor correction parameter l = index number along z-axis on the PV L = number of discretization grid points Lpv = length of the pressure vessel [m] Lm = length of a membrane element [m] Mix = volumetric mixing in PX nLT = plant lifetime [year] nm = number of membrane elements in each PV npv,j = number of pressure vessels employed in the jth RO stage OCbp = energy cost of the booster pump [$] OCswip = energy cost of the intake pump [$] OChpp = energy cost of the high pressure pump [$] OCs = spares cost [$] OCch = chemical treatment cost [$] OCom = operation and maintenance costs [$] OF = overflush of the PX P = operating pressure [MPa] Q = flow rate [m3/h] Re = Reynolds number Sc = Schmidt number Sm = membrane area per element [m2] T = temperature [°C] TAC = total annualized cost [$] TCC = total capital cost [$] V = superficial velocity in the feed channel [m/s] Vw = permeate velocity [m/s] W = membrane width [m] Zj,km = binary variable used to express discrete membrane module number Zj,kpv = binary variable used to express discrete PV number z = distance to the pressure vessel inlet [m]

Article

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Greek Symbols

λ = friction factor μ = seawater viscosity [Pa·s] π = seawater osmotic pressure [MPa] ρ = seawater density [kg/m3] η = efficiencies of a pump or a PX Subscripts

b = brine stream b, j = the brine stream of the jth RO stage bp = booster pump f = feed stream hpp = high pressure pump swip = seawater intake and pretreatment k = kth element type motor = electric motor p = permeate stream p,j = permeate stream of the jth RO stage ps,i = ith pressurization stage PX = pressure exchanger pxhin = high-pressure water entering the PX pxlin = low-pressure water entering the PX pxhout = high-pressure water leaving the PX pxlout = low-pressure water leaving the PX RO, j = jth RO stage 11776



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Optimization of Reverse Osmosis Networks with Spiral-Wound

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