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Optimization of reverse osmosis networks for seawater desalination
Computers chem. Engng Vol. 20, Suppl., pp. $345-$350, 1996
Pergamon
S0098-1354(96)00068-3
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354/96 $15.130+ 0.00
O P T I M I Z A T I O N OF REVERSE OSMOSIS NETWORKS FOR SEAWATER DESALINATION
N. VOROS, Z.B. MAROULIS, D. MARINOS-KOURIS National Technical University of Athens, Chemical Engineering Department Zografou Campus, 157 80 Athens, GREECE
Abstract - A design method for reverse osmosis seawater desalination systems has been developed. It incorporates rigorous mathematical models for the prediction of the performance of the various process units (reverse osmosis modules, pumps, energy recovery turbines) employed in the flowsheet and takes into account the network structure using an appropriate superstructure. Therefore, it offers a flexible representation of the reverse osmosis network. Cost equations that relate the capital and operating costs to the design variables as well as the structural variables of the designed network have been introduced. The total cost of the plant is minimized in order to determine the optimal operating and structural variables. The model is accurate enough to describe the process and yet simple enough to be used for design purposes. During the simulation and optimization studies several structures for multistage reverse osmosis systems have been investigated. Results concerning the economics of the process are presented. Optimal results have also been used for the derivation of design curves concerning the effect of quality and quantity of produced water to the total annualized cost of the plant. INTRODUCTION Membrane processes have found widespread application, due to their low consumption of energy when compared to conventional separation methods (thermal). They are mostly used for the recovery of water as in desalination of seawater and brackish water (Johnson et al., 1966) and waste water minimization (Channabasappa, 1970; Halwagi, 1992), as well as for the recovery of solutes in concentration (Murakami and Igarashi, 1981). Reverse osmosis membranes with various separation characteristics have been developed and are available on the market. New composite reverse osmosis membranes that operate at low to moderate pressures have also been developed. Reverse osmosis with these membranes becomes an energy saving process (Osada and Nakagawa, 1992). Much of the research in this area deals with the mathematical modeling of transport phenomena across the membranes (Lonsdale et al., 1965; Sourirajan, 1970; Pusch 1977, Evangelista, 1985, Voros et al., 1995). While the behavior of individual reverse osmosis modules has been extensively studied during the last two decades, less work has been conducted in analyzing systems of multiple reverse osmosis modules (reverse osmosis network design). Considerable efforts for the developmnet of new design methods of industrial plants (Tweedle et al., 1980; Sirkar et al., 1982) and for the optimization of reverse osmosis plants using mathematical programming techniques (Hatfield and Graves, 1970; van Dijk et al., 1984; Halwagi, 1992) have also been made. The scope of the current work is to provide an integrated methodology for the design and optimization of reverse osmosis systems used in the production of desalinated water. Both simple explicit equations and detailed models for the prediction of mass transport across high salt rejecting membranes may be implemented. Furthermore, the novel notion of synthesizing reverse osmosis networks as originally presented by Halwagi (1992), has been adopted and properly modified for seawater desalination applications. Modifications have been made in terms of proposing an alternative superstructure, which via proper handling of the design variables consists only of real variables for the representation of operational schemes and structures of the RON. As a result, the optimization problem may be tackled now as a non-linear programming problem (generally such problems are more preferable than mixed-integer NLP problems; the optimization procedure is easier), the solution of which provides both the optimum operational and structural characteristics of the plant. In this way a systematic procedure for tackling RON synthesis problems has been developed. In addition, a more general framework is presented in terms of tackling the RON synthesis problem by adding one more degree of freedom. The raw feed (seawater or brackish water) flowrate is considered as a design variable, and as a consequence the plant overall conversion ratio is not fixed. The design schemes that have been developed (both structural and operational) correspond to the optimum value of the aforementioned parameter that may be achieved for the specific set of product quality and demand constraints. Various alternative structures may be investigated and evaluated from the economical point of view. Parametric studies of the systems' performance has also been carried out. Initially we studied the performance and characteristics of a reverse osmosis network for seawater desalination as far as the quality and quantity of produced desalinated water is concerned. The influence of these paramaters on the operational and structural schemes of the designed RON is $345
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European Symposium on Computer Aided Process Engineering----6.Part A
presented. Furtermore, a systematic study has been carried out in order to investigate the influence of feed stream concentration on the unit cost of produced water. Small to medium size capacity plants have been investigated. The methodology may be applied for any type of RON to be designed while it may also be extended to membrane selection purposes. MATHEMATICAL FORMULATION A proper design methodology should include: a) a detailed mathematical model relating the design variables to the behavior and performance of the physical system, b) and effective structure representation of the reverse osmosis network to be designed c) a realistic economic model which relates the various operational and capital cost elements to the design variable values and d) an optimization procedure that should take into account all operational, technical and environmental constraints of the problem (Fan et al., 1969). The problem of synthesising the optimal network of RO-modules, pumps and energy recovery devices may be fomulated by means of a superstructure. The proposed superstructure offers adequate representation of the physical process in terms of all potential structural schemes, i.e. it is able to embed all possible network configurations. Problem Statement Given a raw seawater stream and a specific type of membrane modules to be employed during the design process, it is desired to synthesise a cost-effective network of RO-modules, pumps and energy recovery devices in order to produce potable water under certain demand constraints. The latter are the minimum flowrate of the lean (product) stream as well as its maximum desirable concentration. The design variables to the network-synthesis problem are: the raw seawater mass flowrate F S, the number of pressurization/depressurization and reverse osmosis stages, the intermediate stream splits (they determine the links between RO-mudules, pumps, turbines as well as stream mixing nodes), and the operating pressure of each reverse osmosis stage. Considering that the reverse osmosis network consists of N O pressurization/depressurization stages and NRO reverse osmosis stages the number of stream junctions employed are N0+NRo+2 (2 accounts for the brine and product streams leaving the network). The reverse osmosis network configuration may be efficiently represented by means of two sets of stream nodes: the pumping/expansion (PE) set of junctions and the reverse osmosis (RO) set of junctions. Each PE-junction represents a stream connection to a pump or a turbine, while each RO-junction represents a stream connection to a reverse osmosis stage (multiple parallel reverse osmosis modules operating at the same pressure and with the same feed stream characteristics). The superstructure and the variables employed for the representation of the RON are presented in Figure 1. Each stream of the network (the input seawater stream plus the brine and permeate streams leaving all reverse osmosis stages) may be linked to all the PE-junctions and therefore the stream properties (flowrate, concentration and pressure) leaving the first set of junctions are calculated as follows: NRo NRO F0, i = FSXS,i + ~FB,jXB,ij + y Fp,jxp,ij (1) j=l j=l NRO NRO C0,i = (FsCsxs, i + ~FB,jCB,jXB,ij + ~ F p , j C p , j x p , i j ) / F 0 , i j=l j=l
i=1,2 .... N O + 2
(2)
AS far as the pressure of the stream leaving the jth PE-junction is concerned, the minimum pressure of all substreams connected to this junction is accounted for: P0,i = min{Ps,ij,PB,ij,PP,ijl i = 1,2 ..... NO+ 2, j = 1,2 ..... NRO+ 1}
(3)
The streams leaving the N 0 pumping/expansion stages are distributed to NR0 RO-junctions. For the calculation of the properties of the streams leaving the jth RO-junction the following equations may be used: No FRO, i = ~F0,jxR,ij j=l No CRO, i = ( ~-'F0,jC0,jXR,ij) / FRO, i j=l
(4)
i=1,2 .... NR0, j=l,2 .... N O
(5)
The pressure values of the stream leaving each RO-junction PRO,i are design variables as previously stated. In addition, the operating pressure of each PE-stage Pi, is calculated as the maximum value of all substreams linked to that junction. Moreover, it is essential to have a set of equations relating the flowrates and composition of the brine and permeate streams leaving a RO-stage to the flowrate, pressure and concentration of the stream entering the stage: Fp,i = fl (FRO,i,CRo,i,PRO,i,Ni) (6) FB, i = f2(FRo,i,CRo,i,PRo,i,Ni) (7) Cp, i = f3(FRO,i,CRo,i,PRo,i,Ni) (8)
European Symposiumon Computer Aided Process Engineering--6. Part A
PE-Junctions :
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~,ij
P0,i
Fpji. " ,~-------
Fp
F~
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I / ~ / . / ~ / ] -- FROd
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Fp.i,cPJ Figure 1. Representation of the reverse osmosis network via the superstructure. CB, i = f4(FRo,i,CRo,i,PRo,i,Ni) (9) The previous formulation accounts for a rigorous mathematical description of any potential configuration of RONs, which in fact is a superstructure that may embed all possible structural schemes a RON may take. Given a set of design varibles, i.e. values for the stream splits (Xs, i, XB,ij, xp, ij and XR,ij), operating pressures for all RO-stages PRO,i, and feed stream flowrate, the set of Eqs(1-9) define the mathematical model of the specific configuration, and may be solved in order that the whole flowsheet converges. A successive substitution method for updating the iteration variables FRO,i, CRO,i has been proved adequate for the purpose of converging any type of flowsheet created via this superstructure. The problem of obtaing the optimum cost-effective RON may be formulated now as a non-linear programming task of minimizing the total annualized cost of the plant. The total annualized cost of the network is the sum of the annualized fixed cost (all RO units, pumps and turbines) and the operating cost (cost of power necessary for pumping less the cost of power generated by turbines, etc.): NRO No No CT = Y.cmN i + y~a(Fo,iAPi)b +CEtY~(Fo,iAPi)/pn TM i=1 i=1 i=l
(10)
where m equals to 1 if AP i < 0 (pump) or -1 if AP i > 0 (turbine) and C T is the objective function of the NLP optimization problem. The constraints of the problem account for the various environmental and technical aspects, such as product demand constraints and product quality constraints: F0,N0+I -> Fmin , C0,N0+I < Cma x (11) In addition technical constraints are introduced, as far as the operation of each reverse osmosis stage is concerned. Specifically we define a maximum allowable operating pressure for each RO-module assembly, which is usually dictated by membrane manufacturers: PRO,i < Pmax (12) At the optimum all values for the intermediate substream flowrates, concentrations, pressures as well as the optimum operating pressures for each RO stage will be available. The structural optimization may take place in terms of eliminating all unnecessary RO or pumping stages. This procedure is carried out by accounting an excessive number of RO and pumping stages as an initial guess for the solution, while at the optimum certain design variables are either set to zero (stream splits: mass flowrates) or to a value that indicates the absence of the specific stage. For instance, if the optimum operating pressure of a reverse osmosis stage is found to be less than the osmotic pressure of the feed stream entering that stage, the RO module assembly initially chosen eliminates to a single stream. In the same manner if the pressure difference along a pressurization/depressurization stage is zero, the specific stage eliminates to a single stream. The objective function value is sensitive to such alterations, since both some of the capital cost terms (number of parrallel reverse osmosis modules, capital cost of pumps and turbines as a function of pressure and pressure difference), and some of the operational cost terms (energy as a function of the flowrate and pressure drop at each PD stage) eliminate to zero. As a result there is no need for incorporating integer or binary variables representing pumps, turbines and reverse osmosis units, and the problem may be easily handled as any NLP problem. The E04UCF routine of the NAG Fortran library (a quite common NLP optimizer) has been used for this purpose. CASE STUDY The proposed methodology has been applied for the investigation of the operational and structural characteristics of reverse osmosis desalination plants that utilize DuPont's B-IO hollow fiber modules. For purposes of comparison to
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European Symposium on Computer Aided Process Engineering---6. Part A
previous research studies (Evangelism, 1985; Halwagi 1992), the specific type of membranes has been selected in our work. A simple two-parameter model describing the mass transport across the specific hollow fiber membranes has been introduced, which along with the mass balances for each reverse osmosis module comprises the mathematical model of each RO stage. It is described by the following equations: Mass transport across the membrane: Fp, i = NiASm(APi - zti)¥ (13) KCm,i CP, i = A(Ap i _ rci) Y (14) Material balances: FRO,i = FB, i + Fp, i
(15)
FRo,iCRo,i = FB,iCB, i + Fp,iCp, i (16) Eqs(13)-(16) are employed as the functions defined in Eqs (6)-(9) in order that prediction of the performance of each reverse osmosis stage is provided. The input data and the membrane characteristics accounted for in the design investigation of the current work are briefly outlined in Table 1. The set of cost data used is also supplied in the same table. In Table 2 comparative results between alternative design methodologies are briefly presented. The results of the current work for the specfications outlined in this case do not differ mucg from those presented in the work of El-Halwagi (1992). A slight improvement to the total annualized cost of about 3,5% has been observed while the plant conversion ration is almost 10% higher and the RO-module distribution between the stages is considerably different. Initially the optimization of the superstructure has been carried out for differerent values of the product demand and quality constraints. For desalinated product concentration varying between 250-600 ppm (TDS as NaCI) and for product flowrates 2-20 kg/s at a constant raw feed stream concentration of 34800 ppm (seawater feed) the results (product costs) are graphically presented in Figure 3. The following conclusions may be drawn as far as the effect of the aforementioned variables are concerned: a) The total production cost is analogous to the product flowrate, b) The total annualized cost increases as the composition of produced desalinated water decreases, c) A significant influence of the product demand on the optimum structure of the designed network has not been observed. Nevertheless, economy of scale has been identified as Figure 3 clearly indicates, d) The optimal structure of the RON is strongly dependent on the required product concentration. As indicated in Figure 3, there are certain regions of product concentrations at which specific structural schemes of the reverse osmosis network dominate. Six different optimal structures have been identified for the region of interest. The optimal RON structures identified for the specific type of modules may vary from two-stage permeate configuration with recycle to single stage schemes and two brine staging structures with or without bypasses. The different structures identified are presented in flowsheets in Figure 2. For each optimal scheme the operational variables are varying within its optimality range. As far as the optimum operating pressure range is concerned, for all the optimal solutions the maximum allowable (according to membrane module suppliers' specifications) pressure has been accounted for. This may be attributed to the fact that the selected type of membrane modules and the economic environment assumed favors the maximization of the plant conversion ratio (Y = Fp/FS) and the minimization of total membrane area for all the value range of the environmental constraints assumed. The variation of the overall plant conversion ratio with permeate concentration is presented in Figure 4. Product demand constraints (minimum allowable permeate flowrate) has almost no effect to the optimal operating scheme for each structure of the RON. For constant product specifications (500 ppm TDS, 20 kg/s) the total annualized plant cost as a function of the Table 1. Design andcost data Feed Composition, CS (ppm) 2000-35000 Product demand, Fmin (kg/s) 2-20 Product composition,Cmax(ppm) 250-550 Maximum feedpresstre per module emax(bar) 68.8 Pressure drop per module (bar) 0.22 Pt~e water permeability, A(kg/sN) 1.2x10-10 Solute transport parameter, K(kg/m2s) 4x 106 Effective membrane area per module, Sm (m2) 152 Annualized cost of membrane modules ($/yr) 1,450 Electric power cost (S/kWh) 0.1 Cost ofpumps, Eq.(10) a = 0.0157, b =0.79 Cost of turbines, Eq.(10) a = 0.4182, b = 0.47 Mechanical efficiency of pt~nps, n 65%
. ...............
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Table 2. Comparative results between alternative methodologies Evangelista E1-Halwagi This work .................................. .(1.9.8..5~...............Q..9.9.2~.................................... Fs (kg/s) 19.29 19.29 17.62 Cs (ppm) 34800 34800 34800 Fp (kg/s) 5.787 5.790 5.790 Cp (ppm) 570 470 470 N1 72 73 66 N2 59 33 38 XB1,2 XB1,3 CT($/y )
1.0 0.0 310,226
0.59 0.41 261,935
0.75 0.25 253,133
European Symposium on Computer Aided Process Engineering---6. Part A Scheme (a)
F!~°'lm,[" / / :'1 Fg't- [ ~
_
S c h e m e (b)
$349
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.,~
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Fs., Figure 2. Variation of RON with permeate concentration. Scheme (a):Two-Stage Permeate Processing with intermediate recycles,Scheme (b):Two-Stage Reject Processing with permeate recycle, Scheme (c):Two stage Tapered Reject Processing with bypass of brine, (d) Two-Stage straightthrough rejectprocessing, (e)-(f)Two stage rejectprocessing will feed pybass (and brine recycle). input feed concentration (20130-50000 ppm TDS) is presented in Figure 5. The design scheme may again vary from (f) to (a) as food concentration increases. The design curves for constant conversion ratio,varying from 22-40% are also plotted on the same diagram. As indicated the optimal design curve is indeed the envelope of all curves Hat correspond to a constant value of the plant conversion ratio.This was Hough expected since the feed stream flowarte has been introduced as a design variable. The areas at which each design scheme applies are also indicated on the same Figure, while the variation of the unit cost as a function of the apparent salt rejection (R = (Cs-Cp)/Cs) is graphically presented in Figure 6. It is obvious that the unit cost increases with increasing the apparent plant salt rejection. 2.4
: : Structure:
(a) (b):
:: (c)
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(e)
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300
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',
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Figure 3. Product cost versus permeate concentration.
(f)
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l
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,
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,
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Figure 4. Plant overall conversion ratio versus permeate concentration.
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,
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,
,
0'.8 0.85 0'.9 0.95 Apparent Salt Rejection
Figure 6. Unit cost as a function of the apparent salt rejection.
$350
European Symposiumon Computer Aided Process Engineering--6. Part A
CONCLUSIONS The design method has been successfully applied to the optimization of various multistage reverse-osmosis systems. Optimal operational conditions along with correct structural investigation have been provided. The representation of the physical problem via the superstructure proposed for such types of systems offers extensive flexibility towards synthesizing and optimizing various types of reverse osmosis networks (straight-through reject processing, permeate processing, tapered reject-processing, etc.) and thus may be used for the selection of flexible structural and operating schemes. The solution of the problem or of each set of subproblems derived by the superstructure includes optimal arrangement of the reverse osmosis modules, pumps, energy recovery turbines, and optimum operating conditions. The method proposed, provides a useful design tool that will help the process engineer determine better structures and operation schemes of desalination plants using reverse osmosis technology. Notation a,b = cE = cm
=
constants of Eq.(10) unit cost of power (electricity), S/kWh annualized cost of membrane module, S/module
C0,i, CRO,i =
concentration of the substream entering the ith PE-junction and ith RO-stage respectively, ppm
CB,j, Cp,j =
concentration of the brine and prermeate stream leaving thejth RO-stage, ppm
Cm, i CS
average salt concentration in the high pressure side of any module, ppm concentration of sewater feed stream, ppm
= =
CT = F0,i, FRO,i = FB,j, Fp,j =
total annualized cost of RO plant, $/y flowrate of the substream entering the ith PE-junction and ith RO stage respectively, kg/s flowrate of the brine and permeate stream leaving thejth RO-stage, kg/s
FS
=
flowrate of sewater feed stream, kg/s
K n Ni N0, NRO
= = = =
P0,i
=
solute transport parameter (kg/m2s) mechanical efficiency ofpums or turbines number of parallel modules of the ith RO-stage number of pumping/expansion stages and number of reverse osmosis stages respectively pressure of the substream entering the ith PE-junction (bar)
Pi ,PRo.i
=
operating pressure of the ith PE-stage and the ith RO-stage respectively, bar
t = Xs,i = XB,ij , xp,ij =
operating time (hr/y) stream fraction of the seawater substream entering the ith PE-junction, stream fractions of the brine and permeate substreams respectively, leaving thejth RO-stage and
XR,ij,
=
being linked to the ith PE junction stream fraction of the substream leaving the ith PE-junction and being linked to thejth RO-junction
= = = =
constant of Eq.(17) (dependent on the geometrical characteristics of each module) average pressure diferrence across each module of the ith RO-stage, bar average osmotic pressure on the high pressure side of the membrane module, bar density of saline stream (kg/l)
Greek letters ¥ AP i ~i Q
REFERENCES Chanabasappa, K.C., 1970. Chem. Eng. Prog. Symp. Ser., 67, 107. Evangelista, F., 1985. Ind. Eng. Chem. Process Des. Dev., 24, 211. Fan, L.T., Cheng, C.Y., Ho, L.Y.S., Hwang, C.L., and L.E. Erickson, 1969. Desalination, 6, 131. EI-Halwagi, M.M., 1992. AIChE J., 38(8), 1185. Johnson, J.S., Dresner, L., Kraus, K.A., 1966. Principles of Desalination, Academic Press Inc. Hatfield, G.B., Graves, G.W., 1970. Desalination, 7, 147. Lonsdale, H.K., Merten, U., Riley, R.L., 1965. J. Appl. Polym. Sci., 9, 1341. Murakami, H., Igarashi N., 1981 Ind. Eng. Chem. Prod. Res. Dev, 20,501. Osada, Y., Nakagawa, T., 1992. Membrane Science and Technology, Marcel Dekker Inc. Pusch, W., 1977. Bunsenges. Phys. Chem., 81,854. Sirkar, K.K., Dang, P.T., Rao, G.H., 1982. Ind. Eng. Chem. Process Des. Dev., 21, 517. Sourirajan, S., 1970. Reverse Osmosis, Logos Press. Tweddle, T.A., Thayer, W.C., Matsuura, T., Hsieh, F., Sourirajan, S., 1980. Desalination, 32, 181. van Dijk, J.C., de Moel, P.J., van den Berkmortel, H.A., 1984. Desalination, 52, 57. Voros, N., Maroulis, Z.B., Marinos-Kouris, D., 1995. Desalination, in press.
Pergamon
S0098-1354(96)00068-3
Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354/96 $15.130+ 0.00
O P T I M I Z A T I O N OF REVERSE OSMOSIS NETWORKS FOR SEAWATER DESALINATION
N. VOROS, Z.B. MAROULIS, D. MARINOS-KOURIS National Technical University of Athens, Chemical Engineering Department Zografou Campus, 157 80 Athens, GREECE
Abstract - A design method for reverse osmosis seawater desalination systems has been developed. It incorporates rigorous mathematical models for the prediction of the performance of the various process units (reverse osmosis modules, pumps, energy recovery turbines) employed in the flowsheet and takes into account the network structure using an appropriate superstructure. Therefore, it offers a flexible representation of the reverse osmosis network. Cost equations that relate the capital and operating costs to the design variables as well as the structural variables of the designed network have been introduced. The total cost of the plant is minimized in order to determine the optimal operating and structural variables. The model is accurate enough to describe the process and yet simple enough to be used for design purposes. During the simulation and optimization studies several structures for multistage reverse osmosis systems have been investigated. Results concerning the economics of the process are presented. Optimal results have also been used for the derivation of design curves concerning the effect of quality and quantity of produced water to the total annualized cost of the plant. INTRODUCTION Membrane processes have found widespread application, due to their low consumption of energy when compared to conventional separation methods (thermal). They are mostly used for the recovery of water as in desalination of seawater and brackish water (Johnson et al., 1966) and waste water minimization (Channabasappa, 1970; Halwagi, 1992), as well as for the recovery of solutes in concentration (Murakami and Igarashi, 1981). Reverse osmosis membranes with various separation characteristics have been developed and are available on the market. New composite reverse osmosis membranes that operate at low to moderate pressures have also been developed. Reverse osmosis with these membranes becomes an energy saving process (Osada and Nakagawa, 1992). Much of the research in this area deals with the mathematical modeling of transport phenomena across the membranes (Lonsdale et al., 1965; Sourirajan, 1970; Pusch 1977, Evangelista, 1985, Voros et al., 1995). While the behavior of individual reverse osmosis modules has been extensively studied during the last two decades, less work has been conducted in analyzing systems of multiple reverse osmosis modules (reverse osmosis network design). Considerable efforts for the developmnet of new design methods of industrial plants (Tweedle et al., 1980; Sirkar et al., 1982) and for the optimization of reverse osmosis plants using mathematical programming techniques (Hatfield and Graves, 1970; van Dijk et al., 1984; Halwagi, 1992) have also been made. The scope of the current work is to provide an integrated methodology for the design and optimization of reverse osmosis systems used in the production of desalinated water. Both simple explicit equations and detailed models for the prediction of mass transport across high salt rejecting membranes may be implemented. Furthermore, the novel notion of synthesizing reverse osmosis networks as originally presented by Halwagi (1992), has been adopted and properly modified for seawater desalination applications. Modifications have been made in terms of proposing an alternative superstructure, which via proper handling of the design variables consists only of real variables for the representation of operational schemes and structures of the RON. As a result, the optimization problem may be tackled now as a non-linear programming problem (generally such problems are more preferable than mixed-integer NLP problems; the optimization procedure is easier), the solution of which provides both the optimum operational and structural characteristics of the plant. In this way a systematic procedure for tackling RON synthesis problems has been developed. In addition, a more general framework is presented in terms of tackling the RON synthesis problem by adding one more degree of freedom. The raw feed (seawater or brackish water) flowrate is considered as a design variable, and as a consequence the plant overall conversion ratio is not fixed. The design schemes that have been developed (both structural and operational) correspond to the optimum value of the aforementioned parameter that may be achieved for the specific set of product quality and demand constraints. Various alternative structures may be investigated and evaluated from the economical point of view. Parametric studies of the systems' performance has also been carried out. Initially we studied the performance and characteristics of a reverse osmosis network for seawater desalination as far as the quality and quantity of produced desalinated water is concerned. The influence of these paramaters on the operational and structural schemes of the designed RON is $345
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European Symposium on Computer Aided Process Engineering----6.Part A
presented. Furtermore, a systematic study has been carried out in order to investigate the influence of feed stream concentration on the unit cost of produced water. Small to medium size capacity plants have been investigated. The methodology may be applied for any type of RON to be designed while it may also be extended to membrane selection purposes. MATHEMATICAL FORMULATION A proper design methodology should include: a) a detailed mathematical model relating the design variables to the behavior and performance of the physical system, b) and effective structure representation of the reverse osmosis network to be designed c) a realistic economic model which relates the various operational and capital cost elements to the design variable values and d) an optimization procedure that should take into account all operational, technical and environmental constraints of the problem (Fan et al., 1969). The problem of synthesising the optimal network of RO-modules, pumps and energy recovery devices may be fomulated by means of a superstructure. The proposed superstructure offers adequate representation of the physical process in terms of all potential structural schemes, i.e. it is able to embed all possible network configurations. Problem Statement Given a raw seawater stream and a specific type of membrane modules to be employed during the design process, it is desired to synthesise a cost-effective network of RO-modules, pumps and energy recovery devices in order to produce potable water under certain demand constraints. The latter are the minimum flowrate of the lean (product) stream as well as its maximum desirable concentration. The design variables to the network-synthesis problem are: the raw seawater mass flowrate F S, the number of pressurization/depressurization and reverse osmosis stages, the intermediate stream splits (they determine the links between RO-mudules, pumps, turbines as well as stream mixing nodes), and the operating pressure of each reverse osmosis stage. Considering that the reverse osmosis network consists of N O pressurization/depressurization stages and NRO reverse osmosis stages the number of stream junctions employed are N0+NRo+2 (2 accounts for the brine and product streams leaving the network). The reverse osmosis network configuration may be efficiently represented by means of two sets of stream nodes: the pumping/expansion (PE) set of junctions and the reverse osmosis (RO) set of junctions. Each PE-junction represents a stream connection to a pump or a turbine, while each RO-junction represents a stream connection to a reverse osmosis stage (multiple parallel reverse osmosis modules operating at the same pressure and with the same feed stream characteristics). The superstructure and the variables employed for the representation of the RON are presented in Figure 1. Each stream of the network (the input seawater stream plus the brine and permeate streams leaving all reverse osmosis stages) may be linked to all the PE-junctions and therefore the stream properties (flowrate, concentration and pressure) leaving the first set of junctions are calculated as follows: NRo NRO F0, i = FSXS,i + ~FB,jXB,ij + y Fp,jxp,ij (1) j=l j=l NRO NRO C0,i = (FsCsxs, i + ~FB,jCB,jXB,ij + ~ F p , j C p , j x p , i j ) / F 0 , i j=l j=l
i=1,2 .... N O + 2
(2)
AS far as the pressure of the stream leaving the jth PE-junction is concerned, the minimum pressure of all substreams connected to this junction is accounted for: P0,i = min{Ps,ij,PB,ij,PP,ijl i = 1,2 ..... NO+ 2, j = 1,2 ..... NRO+ 1}
(3)
The streams leaving the N 0 pumping/expansion stages are distributed to NR0 RO-junctions. For the calculation of the properties of the streams leaving the jth RO-junction the following equations may be used: No FRO, i = ~F0,jxR,ij j=l No CRO, i = ( ~-'F0,jC0,jXR,ij) / FRO, i j=l
(4)
i=1,2 .... NR0, j=l,2 .... N O
(5)
The pressure values of the stream leaving each RO-junction PRO,i are design variables as previously stated. In addition, the operating pressure of each PE-stage Pi, is calculated as the maximum value of all substreams linked to that junction. Moreover, it is essential to have a set of equations relating the flowrates and composition of the brine and permeate streams leaving a RO-stage to the flowrate, pressure and concentration of the stream entering the stage: Fp,i = fl (FRO,i,CRo,i,PRO,i,Ni) (6) FB, i = f2(FRo,i,CRo,i,PRo,i,Ni) (7) Cp, i = f3(FRO,i,CRo,i,PRo,i,Ni) (8)
European Symposiumon Computer Aided Process Engineering--6. Part A
PE-Junctions :
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Fp.i,cPJ Figure 1. Representation of the reverse osmosis network via the superstructure. CB, i = f4(FRo,i,CRo,i,PRo,i,Ni) (9) The previous formulation accounts for a rigorous mathematical description of any potential configuration of RONs, which in fact is a superstructure that may embed all possible structural schemes a RON may take. Given a set of design varibles, i.e. values for the stream splits (Xs, i, XB,ij, xp, ij and XR,ij), operating pressures for all RO-stages PRO,i, and feed stream flowrate, the set of Eqs(1-9) define the mathematical model of the specific configuration, and may be solved in order that the whole flowsheet converges. A successive substitution method for updating the iteration variables FRO,i, CRO,i has been proved adequate for the purpose of converging any type of flowsheet created via this superstructure. The problem of obtaing the optimum cost-effective RON may be formulated now as a non-linear programming task of minimizing the total annualized cost of the plant. The total annualized cost of the network is the sum of the annualized fixed cost (all RO units, pumps and turbines) and the operating cost (cost of power necessary for pumping less the cost of power generated by turbines, etc.): NRO No No CT = Y.cmN i + y~a(Fo,iAPi)b +CEtY~(Fo,iAPi)/pn TM i=1 i=1 i=l
(10)
where m equals to 1 if AP i < 0 (pump) or -1 if AP i > 0 (turbine) and C T is the objective function of the NLP optimization problem. The constraints of the problem account for the various environmental and technical aspects, such as product demand constraints and product quality constraints: F0,N0+I -> Fmin , C0,N0+I < Cma x (11) In addition technical constraints are introduced, as far as the operation of each reverse osmosis stage is concerned. Specifically we define a maximum allowable operating pressure for each RO-module assembly, which is usually dictated by membrane manufacturers: PRO,i < Pmax (12) At the optimum all values for the intermediate substream flowrates, concentrations, pressures as well as the optimum operating pressures for each RO stage will be available. The structural optimization may take place in terms of eliminating all unnecessary RO or pumping stages. This procedure is carried out by accounting an excessive number of RO and pumping stages as an initial guess for the solution, while at the optimum certain design variables are either set to zero (stream splits: mass flowrates) or to a value that indicates the absence of the specific stage. For instance, if the optimum operating pressure of a reverse osmosis stage is found to be less than the osmotic pressure of the feed stream entering that stage, the RO module assembly initially chosen eliminates to a single stream. In the same manner if the pressure difference along a pressurization/depressurization stage is zero, the specific stage eliminates to a single stream. The objective function value is sensitive to such alterations, since both some of the capital cost terms (number of parrallel reverse osmosis modules, capital cost of pumps and turbines as a function of pressure and pressure difference), and some of the operational cost terms (energy as a function of the flowrate and pressure drop at each PD stage) eliminate to zero. As a result there is no need for incorporating integer or binary variables representing pumps, turbines and reverse osmosis units, and the problem may be easily handled as any NLP problem. The E04UCF routine of the NAG Fortran library (a quite common NLP optimizer) has been used for this purpose. CASE STUDY The proposed methodology has been applied for the investigation of the operational and structural characteristics of reverse osmosis desalination plants that utilize DuPont's B-IO hollow fiber modules. For purposes of comparison to
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European Symposium on Computer Aided Process Engineering---6. Part A
previous research studies (Evangelism, 1985; Halwagi 1992), the specific type of membranes has been selected in our work. A simple two-parameter model describing the mass transport across the specific hollow fiber membranes has been introduced, which along with the mass balances for each reverse osmosis module comprises the mathematical model of each RO stage. It is described by the following equations: Mass transport across the membrane: Fp, i = NiASm(APi - zti)¥ (13) KCm,i CP, i = A(Ap i _ rci) Y (14) Material balances: FRO,i = FB, i + Fp, i
(15)
FRo,iCRo,i = FB,iCB, i + Fp,iCp, i (16) Eqs(13)-(16) are employed as the functions defined in Eqs (6)-(9) in order that prediction of the performance of each reverse osmosis stage is provided. The input data and the membrane characteristics accounted for in the design investigation of the current work are briefly outlined in Table 1. The set of cost data used is also supplied in the same table. In Table 2 comparative results between alternative design methodologies are briefly presented. The results of the current work for the specfications outlined in this case do not differ mucg from those presented in the work of El-Halwagi (1992). A slight improvement to the total annualized cost of about 3,5% has been observed while the plant conversion ration is almost 10% higher and the RO-module distribution between the stages is considerably different. Initially the optimization of the superstructure has been carried out for differerent values of the product demand and quality constraints. For desalinated product concentration varying between 250-600 ppm (TDS as NaCI) and for product flowrates 2-20 kg/s at a constant raw feed stream concentration of 34800 ppm (seawater feed) the results (product costs) are graphically presented in Figure 3. The following conclusions may be drawn as far as the effect of the aforementioned variables are concerned: a) The total production cost is analogous to the product flowrate, b) The total annualized cost increases as the composition of produced desalinated water decreases, c) A significant influence of the product demand on the optimum structure of the designed network has not been observed. Nevertheless, economy of scale has been identified as Figure 3 clearly indicates, d) The optimal structure of the RON is strongly dependent on the required product concentration. As indicated in Figure 3, there are certain regions of product concentrations at which specific structural schemes of the reverse osmosis network dominate. Six different optimal structures have been identified for the region of interest. The optimal RON structures identified for the specific type of modules may vary from two-stage permeate configuration with recycle to single stage schemes and two brine staging structures with or without bypasses. The different structures identified are presented in flowsheets in Figure 2. For each optimal scheme the operational variables are varying within its optimality range. As far as the optimum operating pressure range is concerned, for all the optimal solutions the maximum allowable (according to membrane module suppliers' specifications) pressure has been accounted for. This may be attributed to the fact that the selected type of membrane modules and the economic environment assumed favors the maximization of the plant conversion ratio (Y = Fp/FS) and the minimization of total membrane area for all the value range of the environmental constraints assumed. The variation of the overall plant conversion ratio with permeate concentration is presented in Figure 4. Product demand constraints (minimum allowable permeate flowrate) has almost no effect to the optimal operating scheme for each structure of the RON. For constant product specifications (500 ppm TDS, 20 kg/s) the total annualized plant cost as a function of the Table 1. Design andcost data Feed Composition, CS (ppm) 2000-35000 Product demand, Fmin (kg/s) 2-20 Product composition,Cmax(ppm) 250-550 Maximum feedpresstre per module emax(bar) 68.8 Pressure drop per module (bar) 0.22 Pt~e water permeability, A(kg/sN) 1.2x10-10 Solute transport parameter, K(kg/m2s) 4x 106 Effective membrane area per module, Sm (m2) 152 Annualized cost of membrane modules ($/yr) 1,450 Electric power cost (S/kWh) 0.1 Cost ofpumps, Eq.(10) a = 0.0157, b =0.79 Cost of turbines, Eq.(10) a = 0.4182, b = 0.47 Mechanical efficiency of pt~nps, n 65%
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Table 2. Comparative results between alternative methodologies Evangelista E1-Halwagi This work .................................. .(1.9.8..5~...............Q..9.9.2~.................................... Fs (kg/s) 19.29 19.29 17.62 Cs (ppm) 34800 34800 34800 Fp (kg/s) 5.787 5.790 5.790 Cp (ppm) 570 470 470 N1 72 73 66 N2 59 33 38 XB1,2 XB1,3 CT($/y )
1.0 0.0 310,226
0.59 0.41 261,935
0.75 0.25 253,133
European Symposium on Computer Aided Process Engineering---6. Part A Scheme (a)
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European Symposiumon Computer Aided Process Engineering--6. Part A
CONCLUSIONS The design method has been successfully applied to the optimization of various multistage reverse-osmosis systems. Optimal operational conditions along with correct structural investigation have been provided. The representation of the physical problem via the superstructure proposed for such types of systems offers extensive flexibility towards synthesizing and optimizing various types of reverse osmosis networks (straight-through reject processing, permeate processing, tapered reject-processing, etc.) and thus may be used for the selection of flexible structural and operating schemes. The solution of the problem or of each set of subproblems derived by the superstructure includes optimal arrangement of the reverse osmosis modules, pumps, energy recovery turbines, and optimum operating conditions. The method proposed, provides a useful design tool that will help the process engineer determine better structures and operation schemes of desalination plants using reverse osmosis technology. Notation a,b = cE = cm
=
constants of Eq.(10) unit cost of power (electricity), S/kWh annualized cost of membrane module, S/module
C0,i, CRO,i =
concentration of the substream entering the ith PE-junction and ith RO-stage respectively, ppm
CB,j, Cp,j =
concentration of the brine and prermeate stream leaving thejth RO-stage, ppm
Cm, i CS
average salt concentration in the high pressure side of any module, ppm concentration of sewater feed stream, ppm
= =
CT = F0,i, FRO,i = FB,j, Fp,j =
total annualized cost of RO plant, $/y flowrate of the substream entering the ith PE-junction and ith RO stage respectively, kg/s flowrate of the brine and permeate stream leaving thejth RO-stage, kg/s
FS
=
flowrate of sewater feed stream, kg/s
K n Ni N0, NRO
= = = =
P0,i
=
solute transport parameter (kg/m2s) mechanical efficiency ofpums or turbines number of parallel modules of the ith RO-stage number of pumping/expansion stages and number of reverse osmosis stages respectively pressure of the substream entering the ith PE-junction (bar)
Pi ,PRo.i
=
operating pressure of the ith PE-stage and the ith RO-stage respectively, bar
t = Xs,i = XB,ij , xp,ij =
operating time (hr/y) stream fraction of the seawater substream entering the ith PE-junction, stream fractions of the brine and permeate substreams respectively, leaving thejth RO-stage and
XR,ij,
=
being linked to the ith PE junction stream fraction of the substream leaving the ith PE-junction and being linked to thejth RO-junction
= = = =
constant of Eq.(17) (dependent on the geometrical characteristics of each module) average pressure diferrence across each module of the ith RO-stage, bar average osmotic pressure on the high pressure side of the membrane module, bar density of saline stream (kg/l)
Greek letters ¥ AP i ~i Q
REFERENCES Chanabasappa, K.C., 1970. Chem. Eng. Prog. Symp. Ser., 67, 107. Evangelista, F., 1985. Ind. Eng. Chem. Process Des. Dev., 24, 211. Fan, L.T., Cheng, C.Y., Ho, L.Y.S., Hwang, C.L., and L.E. Erickson, 1969. Desalination, 6, 131. EI-Halwagi, M.M., 1992. AIChE J., 38(8), 1185. Johnson, J.S., Dresner, L., Kraus, K.A., 1966. Principles of Desalination, Academic Press Inc. Hatfield, G.B., Graves, G.W., 1970. Desalination, 7, 147. Lonsdale, H.K., Merten, U., Riley, R.L., 1965. J. Appl. Polym. Sci., 9, 1341. Murakami, H., Igarashi N., 1981 Ind. Eng. Chem. Prod. Res. Dev, 20,501. Osada, Y., Nakagawa, T., 1992. Membrane Science and Technology, Marcel Dekker Inc. Pusch, W., 1977. Bunsenges. Phys. Chem., 81,854. Sirkar, K.K., Dang, P.T., Rao, G.H., 1982. Ind. Eng. Chem. Process Des. Dev., 21, 517. Sourirajan, S., 1970. Reverse Osmosis, Logos Press. Tweddle, T.A., Thayer, W.C., Matsuura, T., Hsieh, F., Sourirajan, S., 1980. Desalination, 32, 181. van Dijk, J.C., de Moel, P.J., van den Berkmortel, H.A., 1984. Desalination, 52, 57. Voros, N., Maroulis, Z.B., Marinos-Kouris, D., 1995. Desalination, in press.