Optimal design and scheduling of flexible reverse osmosis networks

journal of MEMBRANE

S C I E N C E

ELSEVIER Journal of Membrane Science 129 (1997) 161-174

Optimal design and scheduling of flexible reverse osmosis networks °a:~

Ming j i e Z h u a, Mahmoud M. E1-Halwagl ' , Malik A1-Ahmad b

a Department of Chemical Engineering, Auburn University, Auburn, AL 36849, USA b Department of Chemical Engineering, King Saud University, Riyadh, Saudi Arabia

Received 8 March 1996; received in revised form 15 October 1996; accepted 17 October 1996

Abstract

In this paper, we present a systematic technique for the optimal design and scheduling of flexible reverse osmosis networks that can accommodate a given range of potential variations in the characteristics of the feed and system performance. Flexibility provisions are made to allow the system to meet operational tasks as performance declines due to membrane fouling and scaling. In addition to designing the system, decisions are made regarding the scheduling of maintenance. The design technique is based on formulating the problem as a mixed-integer nonlinear programming (MINLP) problem whose objective is to minimize the total annualized cost, while incorporating thermodynamic, technical and flexibility constraints. An iterative solution procedure is proposed to implement the mathematical programming problem. The merits of the proposed strategy procedure is illustrated by solving an industrial case study.

Keywords: Reverse osmosis; Economics; Fouling; Water treatment; Design

1. Introduction

Reverse osmosis (RO) membrane processes are finding growing industrial interest. They can be effec- tively used for seawater desalination, waste-water treatment, and material recovery. For instance, in the area of desalination, the RO membrane processes have grown steadily and overtaken much of the share of market of multistage flash (MSF) distillation by the end of the 1980s [1]. In addition, industries that use high-value chemicals are recognizing the efficiency of RO systems in reusing/recovering these chemicals for an environmental and economic benefit.

*Corresponding author. Tel.: +1 334 844 2064; fax: +1 334 844 2063; e-mail: [email protected]

0376-7388/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. P I I S 0 3 7 6 - 7 3 8 8 ( 9 6 ) 0 0 3 1 0 - 9

Several research efforts have been directed towards membrane modeling [2], and multi-stage RO system analysis and design [3-8]. Several process configurations have been industrially implemented. For instance, the system shown in Fig. la includes three membrane stages with sequential permeate staging. This configuration recovers high purity solvent at the permeate. Another structure, called the tapered flow arrangement (see Fig. lb) consists of several stages in series with decreasing number of modules in parallel. Based on a number of heuristic rules, several short-cut methods were proposed for the design of straight-through and tapered RO systems [6-10].

Despite the usefulness of the foregoing membrane- hybrid systems, each of these methods suffers from at least one of the following limitations:

162 M. Zhu et al./Journal of Membrane Science 129 (1997) 161-174

Feed ReJect)

(a)

m

Reject)

Permeate

(b)

Fig. 1. A state-space representation of RO membrane system. (a) Permeate-staging RO system; (b) tapered reject-staging RO scheme.

(1) Inadequate structural representation. All the above-mentioned works have been based on some empirical analysis without presenting a systematic system framework that can embed all structural possibilities.

(2) Energy-recovery device. For high pressure and high-flow treatment, use of energy recovery devices may result in very cost-effective design. Most of the above-mentioned works have not exploited this economic potential.

(3) Single feed/single component separations. All of previous works addressed single feed/single component separations. None of them tackled more than one component simultaneously.

In order to tackle these limitations, a systematic procedure was proposed [11] for the design of RO systems for waste reduction. This procedure features minimization of total annualized cost and embodiment of all structural possibilities.

So far, all aforementioned works have been based on the nominal operating conditions without considering variations in characteristics of feed and system performance. An important issue in designing RO

membrane systems is flexibility. Regardless of the nature and magnitude of feed variations membrane fouling, etc. the system must realize the assigned separation tasks. Hence, there is a great need for a systematic procedure that can address the problem of designing flexible RO membrane systems.

The area of designing flexible chemical processes has attracted several research efforts over the past decade. Grossmann and Morari [12] outlined some useful concepts involved in integrating flexibility with design. Grossmann and co-workers [13,14] formulated the flexibility problem as a mathematical programming problem. Significant efforts have been directed toward the synthesis of flexible heat- exchange networks [15-18]. Design of mass exchange networks with a specified uncertainty probability distribution has been recently addressed by Zhu and E1- Halwagi [19].

The present work addresses the problem of synthesizing a flexible RO system using a combination of RO membranes, pumps, and energy-recovery turbines for the separation of solvent and solute from aqueous streams. The proposed method simultaneously inte- grates the aspects of flexibility with cost effectiveness. The problem is formulated as a mixed-integer nonlinear programming (MINLP) model for minimizing the total annualized cost (TAC) subject to thermodynamic, technical, fouling, and flexibility constraints. Owing to the complexity of the devised optimization program, we propose a solution algorithm to solve the mathematical formulation. This algorithm is based on iteratively solving a sequence of optimization or algebraic problems so as to yield the optimal solution. Finally, a case study is presented to illustrate the effectiveness of the methodology.

2. Problem description and formulation

The problem of designing a flexible RO system for solvent-solute separation, as shown in Fig. 2, can be described as follows:

Given: 1. R = { i l i= 1,NR}: a set of feed (solute-rich)

streams. The solute-laden streams have flow rates G = {Gili ~ R} and should be brought from a set of supply compositions y~ = {~[i C R} to a set of specified target compositions yt = {yll i E R}

M. Zhu et al./Journal of Membrane Science 129 (1997) 161-174 163

Feed Streams

R= {ill= I,NR} G={GilieR } yS={y~lia R}

Permeate Uncertainties yt={y~ti~ R]

O= { Oklk= l,Nu}

- - Re:te::e r

Concentrated from Feed Streams

Fig. 2. Examples of RO system configuration.

which correspond to the desired level of separation.

2. A set of membrane modules, booster pumps, and energy-recovery turbines are available for the con- struction of the system for the specified separation. Each unit has a known cost estimation function which relates the cost to its type and capacity.

3. O = { O k l k = l , N , } : the process uncertainties expressed as a set of parameters with given probability distribution functions, and lower and upper bounds such that ~ _< Ok < 0 v, k = 1,N, , where Ok can be the changes of flow rates, supply compositions, or performance degradation due to the membrane fouling and scaling. For the decline in membrane performance, the fouling models and the membrane regeneration/maintenance costs are also available.

The objective is to design a flexible RO system that can separate certain solute from the feed streams via RO membrane separation at a minimum total annualized cost while maintaining a feasible operation over the entire uncertainty region. This task involves the identification of the optimum usage of all membrane modules, as well as the types and sizes of all units, and the determination of optimum membrane maintenance schedule, and the network configuration.

2.1. Problem representation via state-space model

In order to systematically embed all system configurations, we employ the generalized state-space approach, which has been recently introduced and used in synthesizing heat-exchange, mass-exchange,

distillation and membrane-separation networks [ 11,20-23].

Based on the state-space approach, E1-Halwagi [11] developed a structural representation which embed RO systems including membrane modules, pumps and turbines. According to this representation, the RO system can be fully described via two sets of boxes: a set of distribution boxes (DBs) and a set of matching boxes (MBs). The distribution boxes are responsible for stream mixing, bypass and recycle. On the other hand, the matching boxes are used to determine the optimal assignment of streams to units. Therefore, the RO system can be conceptualized as being composed of two DBs and two MBs as shown in Fig. 3: a pressurization/depressurization stream-distribution box (PDSDB), a pressurization/depressurization matching box (PDMB), a membrane unit stream-distribution box (MUSDB), and a membrane unit matching box (MUMB). Using the same termi- nology of the state-space concept in control systems, variables associated with substreams leaving MBs are called state variables, and the corresponding variables entering MBs are called state inputs. All variables entering the DBs (supply variables) are called inputs, while all variables leaving the DBs for discharge (target variables) are called outputs. The units in the MBs are called state operators. The combination of the DB and MB embeds all potential structural possibilities including series and parallel unit arrange- ments, and stream distribution such as splitting, mix-

G ~ G t yt

Y [J c ~

~ | MUSDB

Pjb Pjp jbESb

~j~ j~S ~ MUMB ~'~ ~

Y;-P ~ o;m j,~S m

el° y; j~S m

Fig. 3. A schematic representation of the flexible RO system synthesis problem.


ing, recycle and bypass. Several examples of embed- ding various RO system configurations using the state-space approach are given by E1-Halwagi [11].

2.2. @stem description and modeling

Before exploring the system design and optimization, it is first necessary to invoke the appropriate modeling equations that can satisfactorily predict the membrane performance with reasonable computational complexity. Several models exist for modeling RO membrane modules (see Ref. [24] for an overview of these models). Lonsdale et al. [2] developed a mass transfer solution-diffusion model for the case of per- fect mixing on both sides of the membrane, and for a cross flow configuration. A local behavior model was proposed [6] for highly rejecting membranes described by the preferential water sorption model [25]. These models are mainly based on two parameters: the pure water permeability, Q, and the solute transport parameter, ( D 2 u / K r ) (some models use the rejection coefficient instead). According to the above models, the solvent and solute local fluxes are pre- dicted as follows:

N a = Q[p - pP - (Tr w - 7rP)] (1)

N b = (D2M/K6)(C w - C e) (2)

where N a and N b denote mass fluxes of solvent and solute, respectively, while Q and D2M/K6 are solvent permeability and solute transport parameter, respectively. 7r w is osmotic pressure at the wall concentration C w, and 7r p and C p are corresponding variables for permeation. Fig. 4 indicates a schematic representation of a membrane unit and the variables.

These equations apply to tubular, hollow fiber as well as spiral membranes. For hollow fiber mem-

pr

Pf - v A I L / / / / / / / / / / / / ~ Reject Feed C f G ~- C r G r

I I PPl Cp Gp

Permeate

Fig. 4. A schematic representation of a membrane unit.

branes, these local fluxes can be integrated [8,26] to predict the permeate flow rate through the membrane with area A as follows:

G p = AQ.y(ZIP - 7r) (3)

where 7/

"Y = 1 + 16A#roIls~l/r 4 (4)

tanh[( 1 6 a # r o / r 2) 1/2 (I/ri)] (5)

z l = [(16A#ro/r2)l/2(1/ri)]

and Ap = (pf + p r ) / 2 - PP. The osmotic pressure 7r is estimated at the average bulk concentration (C j + C~)/2.

The solute flux can be approximated by:

N b ( D 2 M ~ ( C f + c r ) (6) = \ - ~ ) 2

and the permeate concentration is approximated by:

N b C P = - - ( 7 )

N a

In addition, overall and component material balances are given by:

G f = G p + G r (8)

G f c f = GPC p q- G T c T (9)

2.3. Membrane foul ing and maintenance scheduling

Fouling and scaling are major factors affecting the RO system performance. It is significantly important from both the operational and design point of view to have a reliable model to predict and evaluate the fouling extent in the RO membrane system.

Foulants exist in different forms, such as suspended particles (e.g. silica and iron oxides), inorganic com- pounds (e.g. calcium sulphate, calcium carbonate, magnesium hydroxide, and magnesium sulphates). Other foulants are just organic or biological com- pounds which can drastically deteriorate the membrane itself and change its performance over time. Several approaches have appeared in literature to model membrane fouling. These approaches include modeling the cake formation (build up of fouling layers) [27,28], and modeling pore blockage and constriction [29-31].


A major effect of fouling is the deterioration of membrane permeability. Reverse osmosis membranes can also be chemically or mechanically regenerated so that their performance can be recovered. One of the objectives of this paper is to determine the optimal maintenance schedule at the design stage so that the resulting plant is not only optimum in terms of fixed cost, but also optimum in terms of operating cost which includes maintenance cost. In order to address the scheduling of maintenance, the following terms are defined:

Inter-maintenance period: Time between two con- secutive membrane regeneration operations. Within a membrane regeneration operation, some (not necessary all) modules are regenerated.

Maintenance cycle: Periodical time during which all modules are regenerated once. One maintenance cycle may contain several inter-maintenance periods.

Therefore, several questions need to be answered: What is the frequency of maintenance cycle? How many periods are there in a cycle? Which membranes should be regenerated at the end of each period? And, what is the best sequencing of these periods in the cycle?

2.4. Mathematical formulation

Having invoked the system representation and models, we are now in a position to develop an optimization design formulation which aims at minimizing the total annualized cost (TAC) of the network while satisfying the operational, physical, thermodynamic and environmental considerations for the network. The TAC of the network consists of two terms: annual operating cost and annualized investment cost. The annual operating cost is a function of the operating condition that includes the cost of power necessary for pumping less the value of power generated by turbines, the cost of membrane regeneration, replacement, etc., while the annualized investment cost is for all membrane module units, pumps and energy-recovery devices, and is fixed once the configuration of the network is determined. The following are the set of necessary constraints for the formulation.

It should be noted that these constraints should be repeated for each stream or component for multi- stream or multi-component problem.

2.4.1. Constraints for the PDSDB Following are the material balances for the inlet

junctions of the PDSDB:

G" = Z GB + GD (10) jbES b

£es b

- , = V " c, ar . + GT; ' G). ~ --Jm ~]b

jm E S m

jm E S m

(11)

(12) ~Es b

The superscripts A, B, C and D denote the stream distribution directions as follows: A - from MB outlets (or state variables) to their inlets (or state inputs); B - from fresh-feed supplies (or network inputs) to MB inlets; C - from MB outlets to targets for discharging (or network outputs); and D - from fresh-feed supplies (or network inputs) to targets. The lower case superscripts are defined as follows: b for booster pumps, m for membrane modules, p for membrane permeates, and r for membrane retentates, etc. Therefore, bm indicates the distribution from b to m, and rb from r to b, etc.

For the material balances of the outlet junctions of the PDSDB, the following equations can be obtained:

c ' = (13) j,,ES ~

Gtyt = Z (GC: ypjm -}- GC~yrjm) + GDy" (14) jmEs"

6;j~,---- Z ( ~ f ~ b + G A j ~ ; ~ ) + a ~ b , j b E S b (15) j~cs m

Gjb:Vj, = Z (G:j:b,~. + G:2bby;,~) + G~Y ", jbES b j,,, E S"

(16)

The component material balances such as Eq. (14) should be employed in the MUSDB whenever stream- mixing occurs to transfer composition variables to the relevant substreams.

Whenever stream-mixing occurs, the following isobaric-mixing constraints are necessary since only streams with equal pressures can be mixed together:

(P jb -P ' )G~=O, jb E S b (17)

(Pj, -- P~jm)G~j~b = O, jb E S b, in E S m (18)


(ejb r Arb (19) - P}~)G~db = O, jb C Sb, jm E S m

Similar constraints should be applied to the MUSDB wherever the stream-mixing occurs to transfer pressure variables to all substreams.

2.4.2. Constraints for the PDMB Since it is illogical to pressurize a stream and

immediately depressurize it, each stream entering the PDMB will pass through either a booster pump or a turbine. The identification of the existence or absence of a pump or turbine can be done by defining the binary integers (1,0) bj~ and tj~. Therefore, the following constraints identifying the values of the binary integers are necessary:

Ubje > PJb - PJ~ >- Lbjb (20)

Utjb >-- PJb -- PJ~ >- Ltje (21)

bib + tjb < 1 (22)

bj~,tjb = 1,0; jo C S o

where U and L are arbitrary large and small positive numbers, respectively. The foregoing constraints will force bjb to be one and tjb to be zero if PJb is larger than Pjb. Similarly, if Pjb is larger than PJb, then tjb will be one and bj~ zero. The constraint (22) ensures that there exists at most one booster pump or one turbine for each stream in the PDMB.

The following equations will predict the energy consumptions for the booster pumps or recovered energy of turbines:

{ GJ~ " P£) if bib 1 W b. = eo---p (Pjb - =

gb etGjb .. (23) (PJb-PJ~) i f t j b = l

where eo and et are the efficiencies of booster pumps and turbines, respectively.

2.4.3. Constraints for the MUSDB Material balances for the inlet junctions of the

MUSDB:

~ P = Z ( T A b m S O ~jbd,.' jb E (24) j~cS m

Material balances for the outlet junctions of the MUSDB:

GJm = ~ GAbOn jm C S m (25) Jbdm ~

Jb ~ S b

Gj.YJm = E GAbj myjb' jm E S m (26) jb cS b

The following isobaric-mixing constraints will ensure that only the streams of the same pressure are allowed to mix. They are also necessary for transferring the pressures variables from the PDSDB to the inlets of the MUMB:

(Pj~ - e. "~G Abm = 0, jm E S m, jb E S b (27) Jb I jb,]m

2.4.4. Constraints for the MUMB The following equations are simplified from

Eqs. (1) and (2). The modification is based on the consideration of simplification of computational complexity, while possessing adequate accuracy for design and optimization purpose. The high pressure side concentration is assumed to have the value of the arithmetic average of the feed and retentate concentrations. This assumption is mild due to the relatively small pressure drop per module compared to the feed pressure. For instance, in a typical hollow fiber RO module, the pressure drop is normally less than one atmosphere compared to a feed pressure that usually ranges from 30 to 60 atm. Hence, the maximum error in pressure driving force will be less than -t-1.67 per cent. The following equations will relate the flow rates and compositions on the reject and the permeate leaving a stage to the flow rate, pressure and composition of the stream entering that stage as well as the pressures of the permeate and the reject leaving that stage.

Q[P +er 2

7rj= _ (Yj= + 52) /2

7r0 Yo

G~P = Mj AfNj a

_u O

-/ffJm - r% 1 (28)

+Y)r2 ~ ' ) (29)

(30)

(31)

(32)


Gjm =~jjm + ~ r jm (33)

Gj.~j. + G'j,y), (34) = aJP~JJm - r r

j m E S m

H e r e , Mj,, is the number of membrane modules parallel in the stage jm. N a and N b are the mass fluxes for the permeated solvent and the solute, respectively, while ~ and yP are the corresponding flow rate and concentration through the membrane. The parameter 7r ° is the osmotic pressure at a given concentration Y0 and 7 can be calculated from Eq. (4). For a given entering flow rate and concentration, there are eight un- known variables for these seven equations. If the feed pressure or membrane area is fixed, then all leaving flow rates and concentrations will be determined. The design procedure is aimed at determining the optimal system configuration and operating conditions.

2.5. Feasibility test

The feasibility test is designed to check whether the derived network can be feasibly operated over the entire uncertainty region. This test can be mathema- tically formulated as a max-min-max problem [13] as follows:

x(d) = max na~n OEF

where d, z and 0

max fAd, z,O)

are vectors representing the design variables, control variables and uncertain parameters, respectively. J is the set of indices of inequality constraints formalized as fj(d,z,q) <_ O,j E J. F is the uncertainty region, or uncertainty hyper-rectangle, and is defined as follows:

/" = {0k[~ <_ Ok _< O~,k = 1, Nu}

The feasibility of operation for all j E J can be ensured by the constraint:

x(,/) _ o (35)

2.6. Design objective function

The design task is aimed at minimizing the TAC of the network which is the summation of the annualized investment cost and the annual operating cost. The investment cost of the network includes all relevant equipments, such as membrane modules, booster

pumps, turbines, and necessary pipe lines. The operating cost includes energy consumptions for pumps, and membrane regeneration/maintenance. The operating cost for the entire flexibility region can be expressed as an expectation over the entire uncertainty region as follows:

COSTVr d ~ n J ~ (o,) " "

[o~y(o,,o2,...,oN.-1) cUr ( o)jb(O)dON. "'" dOzdO1 J o~"(O1,02,...,ON. --1)

fV cVT ( o)Jb ( O)dO (36)

where 01,02,.. . are uncertainty variables, while 02 (01), o~ax(01), "'" correspond to lower 0~un 0]nax, min

and upper bounds which describe the boundary of feasible region F which is defined as:

F = {0IV0 e F 3z : operating constraints are satisfied}

The function jb(O) is the joint probability distribution function of the uncertainty variables and CUr(0) is the operating (utility) cost under uncertainty 0. Eq. (36) can be discretized by employing a Gaussian quadrature formula as follows:

N, C O S T w = y p(O")cUT(O ") (37)

m=l

where Np denotes the number of quadrature points within F A / ' , andp(0m) denotes the occurrence probability for a small uncertainty region around the mth quadrature point 0 m such that p(0 ~) = W,db(Om), where Wm is the Gaussian quadrature weight. This discretization transforms the optimization problem from the integral form into the more manageable algebraic form. Another motivation for employing this discretization is that it allows the handling of the continuous operation via a multi-period representation. Within each period, the average performance of the system is employed instead of the transient behavior. Therefore, the system performance can be described by algebraic equations as opposed to dynamic partial differential equations. This approach renders the optimization program in a form which is much easier to solve compared to a formulation which involves algebraic and differential equations. The accuracy of the formulation can be preserved to any


desired extent by simply increasing the number of quadrature points. Attention should be given not to excessively increase the number of quadrature points to the extent leading to a very large size of the optimization program.

2.7. Design formulation

The optimal design of flexible RO systems can be formulated as a mixed-integer non-linear programming (MINLP) problem which has the TAC as the objective function and Eqs. (10)-(35) as its constraints. The solution for this formulation features a minimized TAC and is feasible to tackle the entire

uncertainty region. At the same time, thermodynamic, technical, and fouling constraints are satisfied. The MINLP can be solved using the software LINGO [32].

3. Design approach

The objective of the design procedure is to develop an RO system that features a minimum TAC and is flexible to tackle the specified uncertainties. The major uncertainties are assumed to be performance declines due to the fouling. Therefore, the optimal membrane maintenance schedule should be determined at the design stage. The proposed design pro-

Preselect Schedules i= l,Ns

1st. Schedule: i = 1 I.

-Fi~xil~D~;i~~t;~oiim .............. ~ .......................................

P= { OklkeU~}

d = arglDESIGN(P)} "~ TAC' = UT'(d) + IV(d) | '(d) = ,~upW(8 k) UT(8 ~, d)~

1 Adjust Points and Weights:~ (Flexibility Test ~ ( k=k+l ~'

P={ P'OC} j

N_ . . . . . . . . . . . . . . . . . . . . . . . .

Change Schedule L i=i+l

I ~(d) = EP~rlUT(0, d)} 1

Screen the Best 1 Network & Schedule

TAC = rain

1

Fig. 5. Flow diagram of the solution strategy for flexible RO system scheduling and design.


cedure, as shown in Fig. 5, consists of two main problems: the scheduling problem that predetermines a number of maintenance schedules for membrane regeneration (the uncertainty ranges are also fixed for each schedule), and the flexible design problem that produces the network configuration based on the input information and uncertainty information from the scheduling problem.

3.1. Flexible design problem

This procedure is based on a given flexibility framework. Although the synthesis problem is formulated as an MINLP, the problem can not be directly solved due to two reasons. First, the feasibility test itself is an optimization problem which cannot be solved before the design has been derived. Secondly, the operating cost term of the TAC is an integral over time which cannot be represented analytically. Therefore, a solution procedure (see the part inscribed inside the dashed-line box in Fig. 5) which consists of the following major steps is proposed:

(1) Select a finite number of parameter points as periods of operation. The nominal operating condition, and some extreme operating conditions may be considered as an initial set of candidates.

(2) Carry out the optimal design procedure, DESIGN, to determine the optimal network configuration for the selected points. The derived network features the minimal TAC for selected periods. In evaluating the operating cost, we employ weight coefficients which can be determined from the uncertainty probability distribution function.

(3) Perform the feasibility test procedure, F-TEST, to determine whether the derived network is feasible to operate over the entire uncertainty region. This can be done by inspecting Eq. (35). Since the design variables have been determined in the previous step, the network can be configured and the index x(d) can be evaluated. If the test is satisfied (which corresponds to x(d) being non-positive), go to the next step. Other- wise, add the critical point to the set of operating points, then return to step 2.

(4) Perform the Gaussian quadrature for the operating cost within the uncertainty region for the design, which will give the accurate expectation of the operating cost term of the TAC. If the difference is within the tolerance, stop. The optimal solution has been

obtained. Otherwise, modify the weights of the selected points and return to step 2.

Although the designed network is based upon a finite number of operating points, the feasible operation for the infinite number of variations contained in the uncertainty region is ensured by the flexibility step of the synthesis strategy.

3. 2. Scheduling problem

It is now appropriate to include the scheduling of membrane maintenance/regeneration into the design procedure. The fouling is the major uncertainty affecting the system performance. Due to the deterioration of membrane properties over time, maintenance should be regularly scheduled to regenerate the membranes. The optimization of maintenance scheduling is not a straightforward task. More frequent the regeneration, better is the performance but higher the operating cost. The scheduling problem is composed of the following steps:

(1) Preselect a number of membrane maintenance schedules. The corresponding uncertainty distribu- tions and ranges are consequently fixed for each selected schedule. The regeneration of a module unit entails down time for that unit along with the con- sequent reduced capacity, incurred regeneration cost and improved membrane performance.

(2) For each maintenance schedule, call the flexible design sub-procedure (inside the dashed-line box of Fig. 5) to get an RO system which is optimal under the specified uncertainties representing the selected maintenance schedule.

(3) Among all resulting RO systems, each representing an uncertainty framework, select a system that has the lowest TAC. The corresponding maintenance schedule is optimal.

4. Case study

This example deals with the desalination of seawater using DuPont B-10 hollow-fiber RO modules. It is desired to produce fresh water from seawater. The minimum acceptable flowrate of freshwater (permeate) is 21.6 m 3 h -1 . The maximum allowable composition (mass fraction) of salt in the permeate is 0.00057. The data for nominal case (no feed changes,

170 M. Zhu et aL /Journal of Membrane Science 129 (1997) 161-174

Table 1 Geometrical Properties of the DuPont B-10 RO modules

Property Value

Fiber length, l, m 0.750 Fiber seal length, ls, m 0.075 Outer radius of fiber, ro, m 50x 10 -6 Inner radius of fiber, ri, m 21 x 10 6 Membrane area, A, m 2 152.0

Table 2 Input data for the seawater desalination example

Feed composition, yS 0.03480 Maximum permeate compostion, yP 0.00057 Minimum water capacity, m 3 h -l >21.6 Minimum flow rate per module, m 3 s -~ 2.1 x 10 -4 Maximum flow rate per module, m 3 s -1 2.7x 10 4 Pressure drop per module, N m -2 0.22x 105 Solute transport parameter, D2M/K6, kg/s m 2 4.0× 10 -6

Initial water permeability, Q0, kg/s N 3.0× l0 lO

no fouling) are taken from Refs. [7,11]. Geometrical properties of the modules are listed in Table 1. The input data for the design are summarized in Table 2. Due to the fouling and scaling, the major dynamic variation is the decline of water permeability. Without regenerating membranes, it is estimated that after one year of operation, water permeabil i ty changes from the initial value 3.0 × 10 -1° down to 1.0 x 10 -1° kg/ s N. An exponentially decaying model is used to describe it, as shown below:

Q = Qoe-t/T

where Qo = 3.0 × 10 - l ° kg/s N and ~- = 328 d. In addition, the feed flowrate and pressure are

allowed to vary so that the water capacity can be maintained at the desired level.

The membrane modules have an upper pressure bound of 70 atm, a maximum unit flowrate per module of 2.7 × 10 -4 m 3 s -1, and a minimum unit flowrate per module of 2.1 x 10 -4 m 3 s 1. The high pressure side bulk concentration is assumed to have the value of the arithmetic average of the feed and retentate concentrations. Turbines may be used to recover energy from high pressure retentate. The decisions of selecting the units and determining the system configuration are to be made at the design stage. The opt imum RO maintenance schedule will also be decided.

The following cost data were employed in calculat- ing the TAC. Membrane costs were assumed to be $2700/module. Each time a maintenance is scheduled for regenerating the modules, the cost involved fixed charges and variable charges. The variable charges are typical ly a linear function of the number of regenerated modules. The following expression is used:

Membrane regeneration cost = $10000 ÷ 450

x (number of modules)

The equipment costs of pumps and turbines are obtained from the following equations:

Pump cost = $2590(power in kW) 0"79

Turbine cost = $830(power in kW) 0'47

The corresponding electricity cost was chosen as $0.07/kW h. The equipment costs were adjusted to account for installation and contingencies by a Lang factor of 2.8. A three-year linear depreciation scheme was used with no salvage value.

By employing the proposed design procedure, the resulting design, as shown in Fig. 6, was found to include 94 RO membrane modules, a feed pump and an energy-recovery turbine. This is a two-stage tapered-flow scheme. There are 54 and 40 modules in the first and second stage, respectively. The design solution also identified the optimal pol icy for the feed pressure to vary between 62 and 70 atm depending on the status of fouling, while the feed seawater flowrate is controlled to fluctuate from 45 to 51 m 3 h -1 (see Fig. 7). The water capacity is always maintained above 21.6 m 3 h -1. The total annualized cost of the network is $518 000/yr.

The opt imum maintenance schedule of the network is found to have two maintenance cycles per year. This solution indicates that all membrane modules are to be regenerated once every six months. The opt imum inter-maintenance period of the network is found to be three months. At the end of the first three-month period, half of the modules (27 in the first stage and 20 in the second stage) are regenerated and the other half (marked as gray modules to indicate different fouling status from the first half) are regenerated at the end of the second period, thereby completing a regeneration cycle. Fig. 7 presents the optimal operating profiles of the designed system during one operating year. At the beginning, all modules are assumed to be new (or just


Fresh Water

2 21.6 m3/l~ -

~ 570 ppm

45 ? ; 1 : 3 ~ ~ % K ppm

201

T

W271 IT

Fig. 6. Op t imum RO system for seawater desalination.

Optimal Profile of Feed Pressure

r0 I I ) / I r~ I / /1 I I I t I I p

. . . . . . . . . . . . . i . . . . . t . . . . . :; . . . . . . . . . . . ! . . . . . 6 6 - - I I I ~ _ ~ I _ _ _ I _1 . . . . .

,, ,, ,, ,,, ,, / 62 i i i i

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0

Optimal Profile of Feed Flowrate 52 [ [ ~ I I I I

I I I t ~ r i ~50 . . . . . - I . . . . . . L . . . . . J . . . . J- . . . . . . I . . . . .L . . . . . . L . . . . . I J ~ i J I i I i i t i t t t i t t I i i I t I t t ~ t

4 8 . . . . . I . . . . . . r . . . . . " 1 - - - 1 - . . . . . ~ . . . . " r . . . . . . . . . . . . t l t J t i

I J t t t i

t t t !

4 ~ . ~ t t i l t l 'i

0 50 100 150 200 250 300 350 400 D a y s

Fig. 7, Opt imal profiles o f opera t ing condit ions.

regenerated). Hence, the feed pressure can be maintained at lower value (62 atm). Because of fouling the feed pressure has to be increased, while the feed flowrate can be slightly decreased. Once the feed

TAC

$1,000 630

620

610

600

590

58d

570

560

55o

540

530

520

510

j ' v

Inter-Maintenance Periods Per Cycle

Fig. 8. Opt imal main tenance schedule.

pressure reaches its upper bound (70 atm), the feed flowrate is increased so that the water capacity is ensured. After regenerating the membranes at the end of each period, the feed pressure is maintained at a lower level. This maintenance schedule results in a minimum total annualized cost for the system.

In order to display the effect of selecting non- optimal scheduling on the cost, Fig. 8 compares the total annualized costs for different designs. The plots indicate the TACs over maintenance periods for a number of preselected schedules. As has been seen, the schedule with two maintenance cycles per year and two maintenance periods per cycle features the lowest TAC among all designs.

5. Conclusions

This work has presented a systematic methodology for the optimal design of flexible RO systems under variable feed conditions and system performance (varying permeability caused by fouling). The design task has been formulated as an MINLP which seeks to minimize the TAC of the network while considering the thermodynamic, modeling, economic, environmental, and feasibility constraints. The optimum RO maintenance schedule is also determined in the


design stage. An iterative solution procedure has been proposed to implement the mathematical programming approach. The effectiveness of this design methodology has been demonstrated by solving a seawater desalination case study.

6. Nomenclature

A b

cUT

[D2M / K~5] Gi

J

L l

mjo

N

Nb N~ Up NR N~ P

Q

R

s b

S m

area of membrane module, m 2 set of binary integers (1,0) to identify the existence/absence of booster pumps. b = {bjblbj~ = 0 , 1,jb E S b} utility or operating cost of network which is a function of operating conditions. cuT(o) means operating cost under uncertainty 0 solute transport parameter, kg/s m 2. mass flow rate of the ith entering solute rich stream, kg/s. G = {Gil i E R} mass flow rates of substreams going through the MBs, kg/s set of indices of inequality constraints describing the network arbitrary small positive number fiber length, m. ls for fiber seal length number of membrane modules parallel in the stage jm mass flux through membrane. N a and N b denote mass fluxes of solvent and solute, respectively number of booster pumps and turbines number of membrane stages number of uncertainty points number of entering solute rich streams number of uncertainty parameters inlet pressure for the MBs, N m -2. P is the corresponding outlet pressure solvent permeability through membrane, kg/s N. Q0 denotes the initial value. set of indices of entering rich streams. R = {ili = 1, Ng} radius of fiber, m. ri and ro for inner and outer radius, respectively set of indices of booster pumps or turbines. S b ~- { jb~b = 1, Nb }

set of indices of membrane stages. S m ----- {jm~,n = 1, Nm}

U

W

yS

yt

set of binary integers (1/0) to identify the e x i s t e n c e / a b s e n c e of tu rb ines , t = {tjbltje ---- 0/1, jb E S b} arbitrary large positive number, or set of indices of uncer ta in ty pa ramete r s . u -- {klk = 1, Nu} work, kW inlet mass composition of rich substream for the MBs. y is the corresponding outlet composition set of supply mass compositions, of entering rich streams, yS = {~[i E R} set of mass compositions, or target compositions, of exiting lean streams.

6.1. Greek letters

F

7 # 7f

P T

0

o L

0 U

defined in Eq. (5) energy efficiency, eb for booster pumps and et for energy-recovery turbines region of uncertainty. F = {0kl~ _< Ok <_O~,k~ u} defined in Eq. (4) viscosity, kg/m s osmotic pressure, N m -2 density, kg m -3 membrane performance decline time con- stant, d set of uncer ta in ty parameters . 0 =

{o~1,~ ~ u} set of lower bounds of uncertainty parameters. 0 L = {~ lk E U} set of upper bounds of uncertainty parameters. 0 U = {O~lk C U}

6.2. Subscripts

i

J jb jm

index of entering rich streams, i E R index of inequality constraints, j C J index of booster pumps or turbines, jb C S b index of membrane stages. Jm C S m

6.3. Superscripts

A distribution direction of flowrate inside the DB: from the inlet junctions connecting to the outlets of the MB to the outlets leading to the inlets of the MB


B

b C

D

m

P F

s

t

d i s t r ibu t ion d i rec t ion o f f lowra te ins ide the

DB: f r o m the in le t j u n c t i o n s c o n n e c t i n g to

the en t e r i ng s t r eams to the out le ts l ead ing

to the in le ts o f the M B . For co ld s t reams,

en t e r ing s t r eams c o m e f r o m the coo led

r ich s t r eams

boos t e r p u m p s or tu rb ines



the out le ts o f the M B to the out le t s for

ex i t ing



the en t e r i ng s t r eams to the out le ts for

ex i t ing

m e m b r a n e m o d u l e s

p e r m e a t e s ides o f m e m b r a n e

re jec t ( re tenta te) s ides of m e m b r a n e

supp ly

ta rge t

Acknowledgements

T h e f inanc ia l Suppo r t o f the Na t iona l Sc i ence

F o u n d a t i o n (grant # N S F - C T S - N Y I - 9 4 5 7 0 1 3 ) is

g ra te fu l ly a c k n o w l e d g e d .

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Optimal design and scheduling of flexible reverse osmosis networks

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