Environ. Sci. Technol. 2002, 36, 1107-1112
Optimal Design of Zero-Water Discharge Rinsing Systems JORG THO ¨ MING* Chemical Engineering Department, Federal University of Rio Grande do Sul, Rua Rua Luiz Englert s/no, CEP 90040-040, Porto Alegre RS, Brazil
This paper is about zero liquid discharge in processes that use water for rinsing. Emphasis was given to those systems that contaminate process water with valuable process liquor and compounds. The approach involved the synthesis of optimal rinsing and recycling networks (RRN) that had a priori excluded water discharge. The total annualized costs of the RRN were minimized by the use of a mixed-integer nonlinear program (MINLP). This MINLP was based on a hyperstructure of the RRN and contained eight counterflow rinsing stages and three regenerator units: electrodialysis, reverse osmosis, and ion exchange columns. A “large-scale nickel plating process” case study showed that by means of zero-water discharge and optimized rinsing the total waste could be reduced by 90.4% at a revenue of $448 000/yr. Furthermore, with the optimized RRN, the rinsing performance can be improved significantly at a low-cost increase. In all the cases, the amount of valuable compounds reclaimed was above 99%.
Introduction The aims of water treatment systems have been discussed controversially with reference to comprehensive environmental assessment of production processes. The “zero discharge” of pollutants has been defined as the ultimate goal of environmental protection by the United States Federal Water Pollution Control Act Amendments since 1972. Discussions have begun in the past decade about the pros and cons of zero-water discharge and “green water utilization” in production plants. However, a solution to zero liquid discharge has not been attempted as yet (1). This is because of a dilemma that is linked to any application of desalination equipment. The advantage of a reduction of wastewater discharge requires the operation of high-technology water treatment devices that produce “hidden wastes” in the form of the consumption of energy and chemicals. A solution to the dilemma is to balance the two aspects by finding the optimum between them. One approach for such an optimization is to perform an impact assessment of conceptual design alternatives. However, only few applications of this methodology have been applied to chemical processes and their designs (2). With respect to water treatment systems, the method used was net-waste-reduction analysis (3). This analysis allowed the assessment of alternative water treatment and recycle systems. It was based on an expansion of the system boundaries to include hidden wastes, e.g., such as those produced by generation of the required energy. Alternative conceptual designs of water treatment * Present address: University of Bremen, Institute of Environmental Process Engineering, FB4, FG 22, P.O. Box 330440, D28334 Bremen, Germany; e-mail:
[email protected]. 10.1021/es010754t CCC: $22.00 Published on Web 01/26/2002
2002 American Chemical Society
and recovery systems were assessed in the aforementioned case study with respect to an application in a chromic acid electroplating line. The findings were that the closed-loop systems, using (a) evaporation and (b) ion exchange, showed an increase of total waste of 31% and 115%, respectively, when compared to a conventional triple-stage counterflow rinse. As a consequence, the authors have suggested not to concentrate any further on zero-water discharge but instead to concentrate on conventional pollution-reduction techniques. However, the investigated closed-loop systems may have been far away from an optimum design and thus environmentally much more expensive than need be. The hypothesis presented in this work is based upon an already optimized process design and shows that a comparative net-waste-reduction analysis gives reasonable results. The second group of optimization approaches involves the process integration methodologies (4). Generally they are embedded into the design procedure and may produce design alternatives for a final comparative assessment. Reviews of design procedures for water networks have been recently published (1, 5). A hybrid of both approaches, which is the application of life cycle assessment to process optimization, has been presented recently (6). It integrates impact assessments such as environmental considerations and socioeconomic factors with process synthesis by means of multiobjective optimization. The main disadvantage of this hybrid methodology is that it requires quantitative weighing factors. The present work follows the process integration approach mentioned above and also includes zero-water discharge as a constraint for the synthesis of a closed-loop recycling system. Emphasis was put on rinsing and recycling systems because they are widespread in industry (7). Rinsing has to be applied if a film of the process solution clings to the workpieces (so-called drag-out) and has to be diluted. An important reason for rinsing is to avoid contamination of the next process step with the previous process solution. The task of a recovery system is to reverse the dilution performed by the rinse. To avoid large amounts of hazardous waste from the rinse, several separation techniques have been proposed for regeneration of rinsing liquids (8). This was also done for heuristically developed structural solutions for improving the recycling of regenerated process solutions (9). However an extensive application of such proposals has been limited by the high costs of these measures. Therefore, a further reduction of costs is needed. Little attention has been given to systematic cost optimization of process structures in galvanic techniques and in hydrometallurgy as a whole (10).
Methods Structural Representation of a Zero-Water Discharge Rinsing System. This investigation describes a schematic hydrometallurgical process that consists of a reactor, a rinsing and regeneration system, and the purification of recycled concentrate. To model the rinsing and recycling network (RRN), it was necessary to separate it from the process for convenience as illustrated in Figure 1. The process streams entering the partition are the drag-out (D0) from the production process (e.g., a galvanic plating bath) and the water makeup (W). The streams that leave the system are the drag-out from the final rinse (D1) and the concentrate (C), which is recycled back to the production process. As a priori claim, the circuit system works without liquid discharge. To achieve this aim, the concentrate (Cc) needs to equal the loss of the plating bath that results from the VOL. 36, NO. 5, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Plating process that combines a plating bath with a rinsing and recycle network. The marked area indicates the investigated rinsing and recycling network (RRN).
of tradeoff parameters such as flow rates or concentrations. The optimization is restricted to a feasible region that is defined by a set of constraints such as equipment characteristics and process fundamentals (mass and heat balances, thermodynamic equilibria, product properties). Operational Equations. For all rinsing units (Rr), operational equations of the type shown in eq 1 were used where Kr is the rinsing equilibrium constant, Xr is the composition of the lean stream leaving rinsing unit r, and Yr is the composition of the rich stream leaving the same unit r:
Xout ) KrYr ∀ r ∈ R r
(1)
The separation target was implemented in form of the rinsing criterion RC (eq 2):
RC ) Y0/Y1
(2)
The performances of the concentrators, which are explained in the case study section latter, were considered by eq 3:
Xout c ) KcXc ∀ c ∈ CU
(3)
The zero discharge condition is represented by eq 4:
∑C
Dr )
c
∀ c ∈ CU, r ∈ R
(4)
c
A set of rinsing stage conditions was added to avoid overflow of the tanks (eqs 5-6): FIGURE 2. Hyperstructure of the partition defined in Figure 1. Straight lines illustrate potential rinsing streams. Dashed lines illustrate the drag-out and an utility stream. Resin flow rate, L. Rinsing stages Ri ) 3-6 are represented by the stage R...; CU, concentrator unit; c, reverse osmosis and/or electrodialysis; IX, ion exchange; D, dragout; C, concentrate; W, water makeup; S, splitter; M, mixer. initial drag-out (D0). Furthermore a water makeup is needed to replace the drag-out (D1) of the final rinsing stage. The composition (Y1) represents the purity of the workpiece after the rinse and is determined by the rinsing criterion (RC). This parameter is the ratio of the initial and final concentrations of target compounds present in the drag-out associated with the workpiece. The most frequent value of RC is 1000, and this value has been reported for many applications (7). To assess the effect of this criterion on optimal design and costs, the RC used in this study was varied between 500 and 50 000. The RRN consists of a set of eight countercurrent rinsing stages that are combined with a regeneration system. This system consists of a reverse osmosis unit, an electrodialysis unit, and/or an ion exchange unit for stream splitting and purification. This regeneration system can easily be extended by the addition of separators, for example, electrolysis or evaporation equipment. The RRN was represented by a hyperstructure (Figure 2). It contains the stream interconnections within the system that define all the possible configurations including regeneration, stream splitting, recycle of dilute and concentrate solution, stream mixing, and bypass streams. There is one heuristic rule implemented into the hyperstructure that shows that water makeup is restricted to the final rinsing stage. By this means, a crossflow behavior of the rinsing is excluded in favor of countercurrent characteristics. Mathematical Model for RRN Optimization. The procedure of constrained optimization is considered as a method by which the best solution for a problem can be found within a feasible region (11, 12). To quantify the quality of a solution, an objective function is used. It describes the objective, e.g., minimum costs or minimum waste discharge, as a function 1108
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Qr e Qmaxyr ∀ r ∈ R, yr ∈ {0,1}n Qc e
∑y Q r
max
∀ c ∈ RO, yr ∈ {0,1}n
(5) (6)
r
Material balances were formulated for the carrier liquids for each concentrator unit (CUc), i.e., electrodialysis (ED) and reverse osmosis (RO) (eq 7), for all rinsing stages Rr (eq 8), for each mixer (Mi,i) (eqs 9-11), and for all splitters (Si,i) (eqs 12 and 13). The same was done for the compound material balances (eqs 14-17) except for splitters because splitters show identical composition for inlet and outlet streams. Finally, nonnegativity constraints were applied for all unit operations (UO) on the flow rates and concentrations (inequalities 18-21). Material Balances For all concentrator units (CUc):
Qc ) Qout c + Cc ∀ c ∈ CU, c * IX
(7)
Qc + L ) Qout c + Cc ∀ c ) IX
(8)
For all rinsing stages (Rr): out Qin ∀r∈R r ) Qr
(9)
For all mixers (Mr,c):
Qc )
∑Q
r,c
+
r
∑Q
c′,c
∀ c, c′ ∈ CU, c′ * c
(10)
c′
For all mixers (Mc,r):
Qr )
∑Q
c,r
+ Qr+1,r ∀ r ∈ R, r * 8
(11)
+ W ∀ r ∈ R, r ) 8
(12)
c
Qr )
∑Q c
c,r
For all splitters (Sc,r):
Qout c
)
∑Q
+
c,r
r
∑Q
CCc ) CPc,0 c,c′
∀ c, c′ ∈ CU, c′ * c
(13)
c′
For all splitters (Sr,c):
Qr )
∑Q
r,c
+ Qr,r+1 ∀ r ∈ R, r * 8
(14)
c
Compound Material Balances For all concentrator units (CU): out QcXc ) CcYc + Qout c Xc ∀ c ∈ CU
(15)
For all rinsing stages (Rr):
Dr+1Yr+1 + QrXr ) DrYr + QrXout ∀r∈R r
(16)
For all mixers (Mr,c):
QcXc )
∑Q
r,cXr
+
r
∑Q
out c′,cXc′
∀ c, c′ ∈ CU, c′ * c
(17)
c′
For all mixers (Mc,r):
QrXr )
∑Q
c,rXc
out + Qr+1,rXr+1 ∀r∈R
(18)
c
Nonnegativity
Xi g 0 ∀ i ∈ UO
(19)
Yi g 0 ∀ i ∈ UO
(20)
Qi g 0 ∀ i ∈ UO
(21)
Cc g 0 ∀ c ∈ CU
(22)
The feasibility region of the optimization problem is described by both equations 1-18 and inequalities 19-22. The objective function (eq 23) describes how the total annualized cost (TAC) depends on the operational cost (OCi) of unit operation (UOi), which is annualized by the conversion factor (CF ) 8000 h a-1) (eq 24). The variable CPi is the cost parameter, and Qi is the flow rate of that stream i to which CPi refers:
∑
min TAC )
i
(
OCi +
)
CCi AD
∀ i ∈ UO
OCi ) CPiQiCF ∀ i ∈ UO
(23) (24)
The TAC also depends on the capital costs (CCi) that are annualized by the annual depreciation factor (AD ) 5 a) (eqs 25-27). The variables CCc are calculated using the exponent Rc and two reference values, CPi,0 and Qi,0, for the costs and the flow rate, respectively. All these data are determined empirically (13, 14):
CCr ) CPr,0
∑y
r
∀ r ∈ R, y ∈ {0,1}n
(25)
r
(
CCc ) nCPc,0
QcXc
ts (n - 1)Qc,0 CAP
)
Rc
∀ c ∈ IX
(26)
( ) Qc Qc,0
Rc
∀ c ∈ {RO,ED}
(27)
The minimization of the objective function that is subject to the constraints which form the feasibility region is referred to as a mixed-integer nonlinear program (MINLP). This is due to (a) the nonlinearity of eqs 15-18, 26, and 27 and (b) the binaries in the eqs 5, 6, and 25. The MINLP is recognized as the most sophisticated type of optimization program (11). It is usually difficult to solve especially if the variables are not properly initialized (15). The solution of the MINLP, modeled in GAMS version 19.6 (16), gives the unit interconnections, the flow rates, and the concentration of each stream in the hyperstructure and the number of rinsing stages. Case Study: Large-Scale Nickel Plating Plant. To demonstrate the synthesis approach, the hyperstructure of Figure 2 was taken as a model for the following case study: A largescale nickel plating process produces a turnover of 100 m2 of plated workpiece surface per hour. The rinsing system applies overflow from one tank into another. Additionally, there is controlled pumping needed to run the recycling network. For the plating bath (Watts Nickel) and the contoured surfaces of the workpieces, a dragout rate of D0 ) 16 kg h-1 was assumed (17). Because of the high initial concentration of Ni in the initial drag-out (Y0 ) 300 g kg-1), the rinsing criterion (RC) used should not be less than RC ) 500. The following four RC values were used in this study: RC ) 500, 1000, 10 000, and 50 000. Spray rinsing, which was applied at all the stages, assumes that Kr ) 1.0 for the equality condition. For the operational costs, a value of CPr ) $5 × 10-6/kg of rinse water was used. For the capital costs, a value of Cr ) $3 × 103 was used for each rinsing stage. Electrodialysis (ED) has been used successfully in rinsing water regeneration (18). The ED is in the form of a stack of N cells with a volume of Vcell ) 0.25 L, a hydraulic load of H ) 0.005 m s-1, and a residence time for the dilute chamber of tED ) 10 s. Since electrodialysis is considered to work economically with 90% separation (KED ) 0.1) for inlet concentrations in the range of XED = 0.05 mol L-1 (19), an ED-specific constraint was set for the MINLP (XED g 3 g kg-1). Operational costs (CPED) ) $1.3 × 10-4/kg of dilute were taken into account plus capital costs (CED,0) ) $1.81 × 106 for Q0 ) 4700 m3/day. For the power-law expression (eq 27), an exponent of RED ) 0.47 was used. All these cost parameters were estimated on the basis of literature data (19). For reverse osmosis (RO), hollow fiber modules are used. In the present study, design calculations were performed using the approach described in ref 20. They show that the RO modules work economically with a 99% separation (KRO ) 0.01). For operational costs, the factor CPRO ) $1.3 × 10-4/ kg of dilute is used while capital cost data were taken from ref 15: CRO,0 ) $720 × 105 when QRO,0 ) 400 m3/day and RRO ) 0.47 when VRO ) 3-400 m3/day. The electricity required was approximated with 5 kWh m-3 for pumping (19). Three ion exchanger columns (IX) operate continuously as fixed beds, two of them work in parallel while the other one undergoes maintenance. The service time was set to ts ) 1 day and allowed 300 bedvolumes before breakthrough with an outlet concentration higher than leakage of 5% (i.e., KIX ) 0.1, and thus a bed volume of VIX ) tIX × Q × 300-1). All data used for the design are referred to S940 (Purolite), a resin with chelating aminophosphate functional groups that are especially suited to the remove of nickel from process solutions (21). For nickel, the capacity of the resin is 30 g L-1. This value was selected to quantify the maximum load of the resin. An operational costs factor of CPIX ) $0.05/kg of resin and a capital cost factor of CIX,0 ) $423 × 105 (13) was used when VIX ) 1 m3 and an exponent of RIX ) 0.67 for bed volumes in the range of VIX ) 0.3-30 m3 was used. The electricity required was estimated with 0.05 kWh m-3 for pumping. VOL. 36, NO. 5, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Comparison of local optima for three different rinsing criteria (RC). The symbols indicate the solutions as they resulted from the MINLP. The lines serve as a guide for the eyes. Those solutions, which contain two or three rinsing stages only, are supplied just by one RO unit, whereas the solutions with four to six rinsing stages are supplied additionally by an IX unit. In this example, any addition of further chemicals to the recycle stream had to be avoided, including those for the regeneration of the IX resin. As a consequence, eq 8 was not included, i.e., CIX was not considered as a recycle stream (that conserved the key component Ni) but as a purge stream from the plating process. All capital cost factors given are for labor and material cost (14), i.e., all cost to be covered including the purchase of equipment as well as the supply of all materials. Furthermore, the costs of labor to receive, uncrate, and install the equipment are included. To compare energy demand and waste production of the optimized RRN with numbers of a base case, a 3-fold counterflow rinsing system without regeneration was designed using eqs 1, 2, 9, 16, and 28. The latter considers the absence of any regeneration unit and simply states that the outlet of unit r is directly connected with the inlet of unit r + 1:
Xr )
out Xr+1
∀r∈R
Results and Discussion One of the most difficult problems with solving a nonconvex MINLP is that there is no guarantee to have found the “global” optimum. Because the described model is such a nonconvex problem, the choice of initial values is important. The initial levels of the key parameters yr, CED, CIX, Yr, and Qr,c were set to arbitrary estimates. The initial flow rate levels of the concentrates, for instance, were set to CED ) CIX ) 0.5D0. The levels of Qr,c and yr were varied between 0 and Qmax and 0 and 1, respectively, to find the different local optima. The other variables were expressed as functions of the ones first initialized. This procedure enables to create an initial point in which 71% of the equations were already satisfied. By this means, infeasibility as well as suboptimal solutions (with objective values that were up to 1 order of magnitude higher than the final solutions) could be avoided. In Figure 3, a circumstance is illustrated which shows that several local optima do exist for each rinsing criterion. The solutions of highest TAC values contain fewer rinsing stages and only one RO unit. With an increased number of rinsing stages, the TAC drops down until a minimum is reached. 9
TABLE 1. Flow Rate Data for Solutions a-c As Illustrated in Figure 4 flow rate data (kg h-1) solution
QRO
a b c
43 52
QED
Q1,IX
Q2,IX
108
41 1731 1746
4.9 57
Q1,2
Q2,3
Q3,4
16
43 16 16
43 52 16
(28)
This base case was chosen because of common industrial practice as well as a net-waste-reduction analysis (3) that favored this system. The electricity required for its wastewater treatment was approximated to 0.75 Wh L-1, while the waste production in the precipitation step was assumed to be 0.04 kg L-1 (3).
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FIGURE 4. Structural representations of global optima for (a) RC ) 1000 and (b) 50 000. A local optimum that favors ED above RO is shown for RC ) 50 000 (c). Flow rate data are given in Table 1.
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 5, 2002
Starting with the optimum number of rinsing stages (RC ) 500: 6 stages, RC ) 1000, 10 000 and 50 000: 7 stages) a reduction in the optimum number of stages causes an increase in the TAC. The greater the RC value, the more significant the influence. For a greater than the optimum number of stages the opposite is true for the influence of the RC value. An interesting finding for zero liquid discharge systems is that the TAC only increases for 14% if the RC increases by 2 orders of magnitude. An improvement from RC ) 10 000 to RC ) 50 000 is proportional to an increase in the TAC of only 0.1%. As a result, it can be stated that the higher the RC value needed to obtain the desired purity of the workpiece after the rinse, the better the cost efficiency of the RRN. The optimum design for a RC of up to 10 000 requires seven rinsing stages, one RO unit that provides the recycle for the galvanic bath, and one IX for producing high-quality rinse recycling. The structural representations of the global optimums for RC ) 1000 and 50 000 are given in Figure 4,panels a,b, and the corresponding flow rate data are given in Table 1. In each case, there was a need to use an IX unit for the regeneration of rinsing water that was received from stage R1 for an RC of up to 10 000. Only for RC ) 50 000, the two stages R1 and R2 were served by the IX. As a second regenerator, the RO always receives the rich stream from the initial rinse and serves the stage before the polishing rinse. This would be significantly different if an ED was used, which served a stage that is linked to the polishing rinsing by an intermediate stage. This is because of the separation factor
KED, which is relatively low as compared to KRO. A local optimum that favors an ED above RO is shown in Figure 4c for RC ) 50 000. What is the reason for the preference of the RO in this case study? A sensitivity analysis of the exponent RED of the cost functions was performed for RC ) 10 000. It showed a significant impact of RED on the final design. An increase in the original value of RED ) 0.47 to the arbitrary value R′ED ) 0.7 resulted in a cost reduction of the new optimum design by 40%. This solution contained a replacement of the RO module by an ED. The former RO supplied the second stage of 7-fold rinsing, and the new ED unit supplied the third stage of 5-fold rinsing after the replacement. The data used for capital cost calculation of the ED are based on the reference flow rate of QED,0 ) 196 000 kg h-1, which is 3 orders of magnitude higher than the flow rate range applied here. For this reason, the ED cost data are uncertain, particularly the exponent. An improvement could be made to the rinsing model by considering the nonideal operation by adding a “backwash factor”, because counterflow rinsing may show a backwashing behavior (7). The optimal process structures that were identified clearly show that the recycling systems are actually not fully operating with a closed loop. There are two “purge” streams in the RRN, the final drag-out (D1) and the stream (CIX). The first is highly diluted (depending on the RC) and the latter has a small flow rate. For instance, in the optimal system with an RC ) 1000, the purges for the key component Ni are 0.1% and 0.3%, respectively. The waste production of this RRN is equal to these purges that are equal to 0.019 kg h-1 Ni. Conversely the base case produces a wastewater stream of 155 L h-1 Ni (RC ) 1000) that yields 6.6 kg h-1 of precipitation chemicals plus 4.8 kg h-1 of nickel. The advantage is that the base case requires 116 W for a filter press as compared to 217 W by the RRN. The net-waste-balance (3), which includes the hidden waste of the energy generation, favors the RRN with a total production rate of 1.15 kg h-1 (9.58%), as opposed to the base case that has a rate of 12 kg h-1 (100%). If the wastewater is also considered as waste, this difference increases further with 167 kg h-1 of total waste production in the base case. A further advantage of an optimized RRN is its significant revenue as compared to the base case. The recovery of 4.795 kg h-1 nickel (D ) 16 kg h-1, Y0 ) 300 g kg-1) equals 38 t a-1 nickel. For a price of $12/kg of electrolytic nickel (Brazil), the revenue is $448 000 a-1 (RC ) 1000). Those closed-loop systems that were investigated and assessed already (3) are located on the left side of Figure 3. Consequently, these systems were not close to their optimum design and hence are environmentally much more expensive than they should be. The conclusion made by the authors must be revised here: closed-loop systems may be environmentally benign and favor the above-mentioned conventional counterflow systems. In principle, the hyperstructure of Figure 2 can be extended to any number of rinsing stages and a number of regeneration processes such as evaporation, electrolysis, or liquid-liquid extraction. Furthermore, the hyperstructure is valid for a large number of different processes that consist of a reactor followed by a subsequent rinsing-recycle network as shown in Figure 1. The optimum design is highly dependent on the index used to measure the quality of the solution, i.e., the functional expression of the objective variable. If a cost function is used as in this work, then statistically determined cost parameters are used. Thus the actual cost might vary by a certain degree from the calculated value. However, the optimal design found for each RC value is only a first step in the final design. In any case there will always be the need for a finalized design
based on the identified process structure. For this purpose, the commercial process simulator packages are the most convenient. Even if the rinsing systems are comparatively simple (often they can be characterized by only one key component), their streams vary over a large scale in process liquor composition. The more diversified the water treatment tasks are, the higher the economical impact of process integration becomes. This is particularly true for zero discharge systems.
Acknowledgments The author thanks Jorge Trierweiler of Universidade Federal do Rio Grande do Sul, who made this work possible; Arne Pietsch for reading the manuscript; and three anonymous reviewers for helpful comments. The financial support from the Fundac¸ a˜o Coordenac¸ a˜o de Aperfeic¸ oamento de Pessoal de Nı´vel Superior (CAPES) and the German Academic Exchange Service (DAAD) is greatly acknowledged.
List of Symbols AD
annual depreciation (a)
c
concentrator index
c′
concentrator index
Cc
concentrated recycle stream of c (g h-1)
CAP
capacity of IX resin (g m-3)
CCi
capital costs of i ($ a-1)
CF
conversion factor (h a-1)
CPi
cost parameter of i ($ h-1)
CPi,0
reference cost parameter of i ($ a-1)
CUc
concentrator unit c
D0
drag-out entering the initial rinse (kg h-1)
Dr
drag-out off the rinse r (kg h-1)
ED
electrodialysis
f(x,y)
objective function ($ a-1)
g
inequality constraints
h
equality constraints
i
operation index
IX
ion exchange
Ki
equilibrium constant of i
L
flow rate of IX regeneration liquid (kg h-1)
m
mixer index
Mi,j
mixer after i ahead of j
n
number of parallel IX columns
OCi
operation costs of i ($ a-1)
p
number of inequality constraints
q
number of equality constraints
Qi
flow rate into i (kg h-1)
Qi,0
reference flow rate (kg h-1)
Qi,j
flow rate from i into j (kg h-1)
Qout c
flow rate of retentate leaving c (kg h-1)
r
rinsing stage index
Rr
rinsing stage r
RC
rinsing criterion (kg h-1)
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RO
reverse osmosis
s
splitter index
Si,j
splitter after i ahead of j
t
time (s)
ts
service time of IX column (h)
TAC
total annualized costs ($ a-1)
UO
unit operations
V
volume (L)
W
water makeup (kg h-1)
Xi
key component composition in the lean stream entering i (g kg-1)
Xout i
key component composition in the lean stream leaving i (g kg-1)
x
solution vector of continuous variables
Yi
key component composition in the rich stream leaving i (g kg-1)
y
solution vector of integer variables
R
exponent of cost function
Literature Cited (1) Bagajewicz, M. Comput. Chem. Eng. 2000, 24, 2093-2113. (2) Shonnard, D. R.; Hiew, D. S. Environ. Sci. Technol. 2000, 34, 5222-5228. (3) Cohen Hubal, E. A.; Overcash, M. R. J. Cleaner Prod. 1995, 3, 161-167. (4) Rossiter, A. P.; Kumana, J. D. In Waste Minimization through Process Design; Rossiter, A. P., Ed.; McGraw-Hill: New York, 1995; pp 43-49. (5) Alva-Arga´ez, A.; Kokossis, A. C.; Smith, R. Comput. Chem. Eng. 1998, 22, S741-S744.
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Received for review March 16, 2001. Revised manuscript received October 26, 2001. Accepted November 7, 2001. ES010754T
