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Ind. Eng. Chem. Res. 2007, 46, 7497-7512

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Wastewater Treatment Plant Synthesis and Design Noelia Alasino, Miguel C. Mussati, and Nicola´ s Scenna* INGAR Instituto de Desarrollo y Disen˜ o (CONICET-UTN), AVellaneda 3657, (S3002GJC) Santa Fe, Argentina

The most used process for biological nitrogen removal from municipal and industrial wastewaters is the activated sludge process. Because of the importance of this process, as well as the large number of existing facilities, a lot of research effort has been focused on optimizing the operation strategies or improving the individual plant design. However, the systematic optimization of the process structure (process synthesis) and operation conditions based on rigorous process models has not been presented in the literature. The objective of this work is to address the simultaneous optimization of the process configuration and equipment dimensionssi.e., process synthesis and designsand the operation conditions of activated sludge wastewater treatment plants for nitrogen removal based on a superstructure model. The model embeds up to five reactors and a secondary settler, and allows flow distribution of the main process streams, i.e., nitrate and sludge recycle streams and fresh feed, along the reaction zone. The objective function is to minimize the net present value formed by investment and operating costs, while verifying compliance with the effluent permitted limits. The investment cost computes the reaction tanks, aeration systems, secondary settler, influent pumping station, and sludge pump costs. The operation cost computes the cost for pumping, aeration, dosage of an external carbon source, excess sludge treatment for disposal, and fines according to pollution units discharged. Influent wastewater flowrate and composition are assumed to be known. The activated sludge model no. 3 and the Taka´cs model are selected to describe the biochemical processes and the secondary settler, respectively. This results in a highly nonlinear system with nonsmooth functions. Because of the problem complexity, in this first approach, a nonlinear programming (NLP) problem (specifically a nonlinear programming with discontinuous derivatives (DNLP) problem) is proposed and solved to obtain some insights for future models. It was implemented and solved using general algebraic modeling system (GAMS). Results for case studies are presented and discussed. 1. Introduction The most used process for biological nitrogen (N) removal from municipal and industrial wastewaters is the activated sludge process (ASP). Because of the importance of the ASPs and the large number of existing facilities, a lot of research effort has been focused on optimizing the operation strategies or improving the individual plant design but not on the systematic optimization of the process structure based on the process model. According to Rigopoulos and Linke,1 this lack of effort can be attributed mainly to the fact that general-purpose and comprehensive mechanistic models of these processes involve a large number of biochemical reactions with highly nonlinear kinetics, thus posing a computational barrier to the most widely used deterministic optimization algorithms. In addition, biochemical systems are quite difficult to control, resulting in research focused on how to control existing systems rather than on optimizing basic flowsheet structures. Recent advances in modeling biochemical reactions in activated sludge systems, as well as the fact that process systems engineering is now a mature area, have opened up the possibility of investigating ASP design and operation based on models. However, in the wastewater treatment field, few papers addressed the process synthesis. Rigopoulos and Linke1 and Linke and Kokossis2 presented a systematic design of optimal activated sludge wastewater treatment plants (WWTPs), based on the activated sludge model no. 1 (ASM1).3 They proposed a superstructure and used stochastic search techniques in the form of simulated annealing for optimization. Mussati et al.4 presented a mixed-integer nonlinear programming (MINLP) model for the simultaneous optimization of the * Corresponding author. E-mail: [email protected] Tel.: +54342-4534451. Fax: +54-342-4553439.

process configuration and the operation conditions of WWTPs for nitrogen removal, introducing a superstructure that embeds the three most used ASP configurations, i.e., predenitrification (PreDN), postdenitrification (PostDN), and pre-postdenitrification (PrePostDN) systems, but without allowing streamflow distribution patterns. In the studies by Alasino et al.,5,6 that model was extended to account for flow distribution of the main process streams (nitrate and sludge recycle streams and fresh feed stream) along the reaction zone. However, reactor volumes were fixed and investment costs were not included in the model. Other published works on activated sludge systems using ASM-type models and the double-exponential model for settling tanks have focused on the achievement of the best combination of operation variables by means of simulating two or three alternative designs and choosing the one with the lowest cost.7-11 Gillot et al.7 used the net present value (NPV), integrating investments and operating costs of a wastewater treatment plant to standardize a cost-calculation procedure, and used these cost models for comparison of different treatment scenarios by simulation. A software tool for economic evaluation of a WWTP over its life cycle was also developed.8 Espı´rito Santo et al.9 optimized the design and operation variables of four WWTPs consisting of one aeration tank (described by the ASM1 model) and one secondary settler (described by the model proposed by the Abwassertechnische Vereinigung e.V. -ATV model-) using four different optimizers, but they did not deal with process synthesis. In summary, the rigorous modeling for optimal process synthesis, design, and operation of WWTPs by mathematical programming contemplating all structural possibilities; embedding reaction compartments, secondary settler, and stream interconnections; and aiming at minimizing NPV is a difficult task and has not been addressed in the literature. The biochemi-

10.1021/ie0704905 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/16/2007

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Figure 1. Conventional activated sludge wastewater treatment plants.

cal process rates and settler model include highly nonlinear functions. Moreover, the settler model includes nonsmooth functions such as min/max that lead to special models called nonlinear programming with discontinuous derivatives problems (DNLPs), which, in general, may cause numerical problems. To avoid this, a possibility is the introduction of binary variables to model the nonsmooth functions. As mentioned, to contemplate all structural possibilities, a superstructure must be introduced. The natural approach to handle superstructure models is the MINLP methodology. Moreover, rigorous investment cost functions usually consider fixed costs, which are related to the existence or not of a process unit. Those functions are discontinuous, having different parameter values for different ranges of the equipment characteristic dimension. Finally, in a design problem, the secondary settler dimensions (depth and transversal area) and feed allocation point might be optimization variables. To consider all these aspects, additional binary variables must be introduced into the model. The objective of this work is to address the simultaneous optimization of the process configuration and equipment dimensionssi.e., process synthesis and designsand the operation conditions of activated sludge wastewater treatment plants for nitrogen removal based on a superstructure model. In this first approach, a DNLP problem is proposed to overcome most of the above-mentioned difficulties and to obtain some insights for the construction of more rigorous models in the future. The model embeds up to five reaction compartments and a secondary settler and allows for flow distribution of the main process streams, i.e., nitrate and sludge recycle and fresh feed streams and external carbon source dosage along the reaction zone. The objective function is to minimize the NPV considering investment and operating costs. In all case studies, a chain of reaction compartments in series followed by a decanter are the available pieces of process equipment, whose dimensions (continuous design variables) are to be optimized. The selection of a compartment type, i.e., an aerated or anoxic unit, is to be chosen with a continuous variable: the aeration flowrate. When continuous variables (reaction compartment volume, decanter area, and flowrate of aeration, fresh wastewater, recycles, and external carbon source dosage to each reaction compartment) take a zero value at a solution point, the corresponding unit and/or stream is removed from the superstructure. In future works, new mathematical models based on MINLP or general disjunctive programming (GDP) methodologies will be presented. 2. Process Description In general, activated sludge systems employ aerated, tubular, well-mixed reactors or a series of well-mixed reactors to

accommodate the biological reactions. Feeding strategies including stream distribution to any point of the reaction zone and water recycles from one zone to another are a common feature of combined stabilization-denitrification systems. The stream leaving the reaction zone is generally fed into a sedimentation basin to separate the stream into the cleaned effluent and the sludge, a fraction of which is recycled back to the reaction zone. A fraction of sludge, called waste sludge, is purged from the recycle line to compensate for the increase in biomass concentration due to biomass growth during the biodegradation processes. According to Van Haandel et al.,1,12 the most used activated sludge wastewater treatment plant (ASWWTP) configurations providing the different environmental conditions for biological nitrogen removal and organic matter oxidation are presented in Figure 1. The reduction of carbonaceous matter and the nitrification process (ammonium is converted to nitrate by autotrophic bacteria) are favored by aerobic conditions, while the denitrification process (nitrate is converted to nitrogen gas by heterotrophic bacteria) is favored by anoxic conditions if readily biodegradable organic matter is available. Anoxic zones can be placed either at the beginning (predenitrification configuration) or at the end of the reaction zone (postdenitrification). In a predenitrifying system, an internal recirculation flow is usually introduced to transport the nitrate-rich liquid back to the anoxic zone. The anoxic zone may require an external carbon dosage to facilitate denitrification if the influent readily biodegradable matter is consumed by aerobic microorganisms in the nitrification zone. This is even worse when the fresh wastewater stream has a low carbon/nitrogen ratio. Two basic process design classes can be identified, depending on the type of energy source that is utilized by the heterotrophic organisms to accomplish denitrification: internal or selfgenerated. In processes that use an internal energy source, the organisms make use of the influent biodegradable material to extract the energy they need. This requires that the anoxic reaction compartment has to be placed first in the network. Since nitrate is not present in the influent, it has to be brought into the anoxic reactor by recycling part of the effluent of an aerobic reactor. A process known as Ludzack Ettinger (Figure 1a) utilizes two reactors with partial communication, with the sludge being recycled to the aerobic part. Recirculation of nitrified liquor takes place in a rather indeterminate fashion by the mixing action in the two reactors. As a consequence, the performance of this process with respect to the reduction in total nitrogen content is very poor. Another process, the modified Ludzack Ettinger (MLE) (Figure 1b), uses separate anoxic and aerobic reactors and recycles both sludge and nitrified liquor to the

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7499

Figure 2. (a) WWTP superstructure; (b) representation of the WWTP superstructure. Table 1. ASM3 Model Compounds symbol

component

unit

SI SS XI XS XH XA XSTO XSS SO SNOX SN2 SNH SALK

soluble inert organics readily biodeg. substrates inert particulate organics slowly biodeg. substrates heterotrophic biomass autotrophic biomass internal storage product suspended solids dissolved oxygen nitrate and nitrite N dinitrogen ammonia and ammonium N alkalinity

(g of COD) m-3 (g of COD) m-3 (g of COD) m-3 (g of COD) m-3 (g of COD) m-3 (g of COD) m-3 (g of COD) m-3 (g of SS) m-3 (g of O2) m-3 (g of N) m-3 (g of N) m-3 (g of N) m-3 (g of COD) m-3

anoxic reactor in order to achieve a better performance. By default, the processes using an internal energy source cannot produce nitrate-free effluents. This is because both recycle and effluent flows come from the aerated reactor and contain a concentration of nitrate. Figure 1 depicts other process configurations attempting to obtain low nitrate concentrations in the effluent by using different recycle strategies and by alternating aerobic and anoxic zones. Here, a model for optimal synthesis and design that considers a superstructure that embeds a vast number of activated sludge process configurations, given specified design criteria and cost data, is presented. 3. Problem Definition The problem addressed in this paper is the simultaneous optimization of the system structure (process configuration), the design (equipment dimensions, i.e., reaction compartments volume and secondary settler transversal area), and the operating conditions (flowrates of aeration, recycles, and fresh feed wastewater to each reaction compartment and external carbon source dosage) of activated sludge WWTPs for nitrogen removal, aimed at minimizing the net present value (NPV), given the following: (1) defined influent wastewater specifications (composition and flowrate), (2) effluent permitted limits,

(3) a process superstructure model with a maximum of five reaction compartments and one secondary settler, and (4) a defined cost model computing operation and investment costs. In this first approach, a NLP (DNLP) problem is proposed and solved for different case studies. 4. Process Model The considered process superstructure is shown in Figure 2a. In order to facilitate the model description, a more detailed superstructure is given in Figure 2b. Basically, the plant superstructure model consists of a maximum of five reaction compartments, a secondary settler, pumps and stream mixers, and splitters. The design variables (the volume of each compartment (Vi) and the secondary settler cross-area (Asett)) are to be optimized together with the operation variables (flowrate of aeration and process streams). The T ), the recycle streams, and the external carbon fresh feed (Qfresh source dosage (uECSD) can be distributed into one or more of the five reaction compartments. The superstructure has two possible internal (nitrate) recycle streams and one external T (sludge) recycle stream. The external recycle stream (Qr,ext ) pumps a fraction of sludge from the secondary settler underflow back to the reaction zone. The two possible internal (nitrate) recycle flowrates are QTr,int,1 and QTr,int,2. The first recycles a fraction of the mixed liquor from the last to the rest of the reaction compartments, and QTr,int,2 recycles from the fourth to the preceding compartments, as is shown in Figure 2b. The reaction compartment volumes can range from zero to a given arbitrary maximum value. A zero reaction compartment volume indicates that it is eliminated from the superstructure. Finally, reaction compartments can operate under anoxic or aerobic conditions, depending on the optimal value computed for the oxygen transfer coefficient kLa. If the kLa value for a given compartment is zero, an anoxic reactor is selected. The preference for a given plant configuration over the others depends on the influent wastewater flowrate and composition and the used cost functions, as well as on several economical

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Table 2. ASM3 Processes Fk

process

F1

hydrolysis

F2

aerobic storage of SS

F3

aerobic storage of SS

process rate

[

]

XS/XH X KX + XS/XH H

F 1 ) kH

[

SS KS + S S

F2 ) kSTO

[

F3 ) kSTO‚ηNOX F4

aerobic growth

F5

anoxic growth (denitrification)

F6

aerobic endogenous respiration

F7

anoxic endogenous respiration

[ [

F9

anoxic respiration of XSTO

][ ][ [ ]

anoxic growth of XA (nitrification)

F11

aerobic endogenous respiration

F12

anoxic endogenous respiration

SO X KO 2 + S O H

[

[

[

F10 ) µA

(1)

∀i,x

][

]

(2)

where Fk,i is the kth process rate in reactor i and υk,c are the stoichiometric coefficients.

]

]

]

[

][

]

KO 2 SNO X KO2 + SO KNOX + SNO STO

][ ]

][

]

SO SNH SALK X KA,O2 + SO KA,NH + SNH KA,ALK + SALK A

[

SO X KA,O2 + SO A

[

KA,O2

][

]

SNO X KA,O2 + SO KA,NOX + SNO A

The ASM3 model considers 13 compounds (Cx), which are divided into soluble compounds and particulate compounds, for which concentrations are indicated by S and X, respectively (Table 1), i.e., in eq 1, C can be S or X. The ASM3 involves 12 transformation processes, which are listed in Table 2 together with their process rate equations. For dissolved oxygen, eq 1 is modified to account for gasliquid mass transfer,

Qi (S - SO,i) + kLai(SO,sat - SO,i) + rSO,i ) 0, ∀i Vi O,i,in where SO,sat is the oxygen saturation constant at 15 °C (SO,sat ) 8 (g of O2)‚m-3). Finally, the volume of reaction compartment i is defined as a positive variable. The following constraint is introduced,

Vi e Vmax, ∀i

where Qi is the volumetric flowrate that enters and leaves reaction compartment i, Vi is the volume of reaction compartment i, Cx,i and Cx,i,in are the concentrations of component x inside and at the inlet of the reactor i, respectively. The reaction term rx,i, for each compound x and reactor i is computed as follows, x

][

SO X KO2 + SO STO

F8 ) bSTO,O2

and technological aspects and trade-offs. It is clear that conventional processes described in Section 2 are embedded in the superstructure and, hence, are candidates for the optimal flowsheet resulting from the assumed hypotheses. 4.1. Reactor Model. For the aeration tanks, steady-state continuous stirred tank reactors (CSTRs) are considered, and it is assumed that no biological reactions take place in the secondary settler. The activated sludge model no. 3 (ASM3)12 is chosen as the biological process model. This model considers both the elimination of the carbonaceous matter and the removal of the nitrogen compounds. The ASM3 is presently the most widely accepted model for description of biological nitrogen removal in activated sludge systems. For each model component x and reactor i,

∑K υk,c .Fk,i,

][ ][

KO2 SNO X KO2 + SO KNOX + SNO H

F12 ) bA,NOX

rx,i )

][ ][

SO SNO SNH SALK XSTO/XH X KO2 + SO KNOX + SNO KNH + SNH KALK + SALK KSTO + XSTO/XH H

F11 ) bA,O2

Qi (C - Cx,i) + rx,i ) 0, ∀i,x * O2 Vi x,i,in

][ ]

F 5 ) µH

F9 ) bSTO,NOX F10

][

KO 2 SNO SS X KO2 + SO KNOX + SNO KS + SS H

SO SNH SALK XSTO/XH X KO2 + SO KNH + SNH KALK + SALK KSTO + XSTO/XH H

F7 ) bH,NOX aerobic respiration of XSTO

]

SO X KO 2 + S O H

F 4 ) µH

F6 ) bH,O2

F8

][

(3)

where Vmax is a sufficiently large upper bound for reactor volumes. To avoid numerical problems (e.g., division by zero), very small lower bounds for reaction compartment volumes are set (Vi,min ) 0.01 m3); however, when an optimal reaction volume Vi reaches the lower bound, it is considered as a zero volume reaction compartment and is consequently “deleted” from the superstructure. Finally, the following constraints are considered for the mass transfer coefficient kLai in each compartment i, which is defined as a positive variable,

kLai e kLai,max, ∀i

(4)

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Xx,m Xx,sett,in ) , ∀x,m Xsett,in Xm

(5)

where Xx,sett,in and Xx,m are the x particulate compound concentrations in the feed layer and layer m, respectively, and Xsett,in and Xm are the corresponding total suspended solids concentrations. The particulate and soluble compound flux due to the bulk movement of the liquid is straightforward to assess, being equal to the product of the concentration Cx,m and the bulk velocity of the liquid, which can be up or down (νup or νdn) depending on the position of the layer with respect to the feed point,

νdn ) Qbottom/Asett

(6)

νup ) Qef/Asett

(7)

where Qbottom and Qef are the volumetric flowrates in sedimentation and clarification zones, respectively. Taka´cs double-exponential settling velocity function is expressed by eq 8, whose parameters are listed in Table 3,

νs,m(X) ) max{0,min[νo′,νo(e-rh(Xm-Xmin) e-rp(Xm-Xmin))]}, ∀m (8) Xmin ) fnsXsett,in

Figure 3. Settler model scheme.

where kLai,max is a maximum operating limit (kLai,max ) 360 d-1). Analogously, to avoid numerical problems, very small lower bounds for oxygen mass transfer coefficients are chosen (kLai,min ) 0.0001 d-1); however, when an optimal kLai value reaches the lower bound, it is considered zero, and consequently, the corresponding reaction compartment is assumed to be anoxic. 4.2. Secondary Settler Model. The secondary settler is modeled as a nonreactive settling tank subdivided into 10 layers of equal thickness, using the double-exponential settling velocity model of Taka´cs et al.14 According to a comparative study of several sedimentation models,15 this settler model provides the most reliable results. Here, both a fixed settler depth of 4 m and a feed point allocation at the sixth layer from the bottom are adopted.11 However, its cross-area (Asett) results from optimization. In order to shorten the notation, the total suspended solids concentration XSS is renamed to X in the settler model equations. Figure 3 schematizes the settler model, which consists of five different groups of layers depending on their relative position to the feed point. It also shows the streams due to the bulk movement of the liquid and to gravity settling involved in the mass balance around each layer. This balance depends on whether the component is particulate or soluble. The movement of soluble compounds across the settler is only due to the bulk movement of the liquid, whereas the movement of particulate compounds is due to the bulk liquid movement and to gravity settling. The particulate component flux depends on the solid concentration but not on the solid composition. For any particulate compound concentration Xx, the following holds

(9)

where νs,m is the settling velocity in layer m, Xm is the suspended solids concentration in layer m, Xmin is the minimum attainable suspended solids concentration, Xsett,in is the mixed-liquor suspended solids concentration entering the settling tank, and fns is the nonsettleable fraction. The particulate compound flux due to gravity settling, J, depends on the position relative to the feed point. For the layers under the feed layer (m ) 2-6), the sedimentation flux, Jsed,m, is given by

Jsed,m ) min(νs,mXm,νs,m-1Xm-1), m ) 2, ..., 6

(10)

and for the layers above the feed point (m ) 7-10), the clarification flux, Jclar,m, is given by

Jclar,m ) νs,mXm if Xm-1 e Xt , m ) 7, ..., 10 (11) min (νs,mXm,νs,m-1Xm-1) otherwise

{

}

The threshold concentration Xt is adopted in such a form to limit the solids downward flux to that which can be handled by the layer below. For example, above the feed layer, the flux leaving layer m is restricted, if the concentration in layer m 1 is greater to or equal than some threshold value (Xt), in which case the flux leaving layer m is set equal to the min (νs,mXm,νs,m-1Xm-1). According to Taka´cs model,14 Xt is equal to 3000 g m-3. The resulting steady-state compound balances around each layer are the following:

Mass balances for the sludge (particulate components) For the feed layer (Qsett,inXsett,in/Asett) + Jclar,m+1 (νup + νdn)Xm - Jsed,m , m)6 0) hm

(12)

where Qsett,in and Xsett,in are the volumetric flowrate and the

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Table 3. Settler Model Parameters and Default Values description parameter

symbol

units

value

maximum settling velocity maximum Vesilind settling velocity hindered zone settling parameter flocculant zone settling parameter nonsettleable fraction

ν0′ ν0 rh rp fns

m d-1 m d-1 m3 (g of SS)-1 m3 (g of SS)-1

250 474 0.000 576 0.002 86 0.002 28

particulate compound concentration fed to the settler, respectively. Asett and hm (0.4 m) are the settler cross-area and the layer height, respectively.

0)

νdn(Xm+1 -Xm) + Jsed,m+1 - Jsed,m , m ) 2, ..., 5 hm

(13)

νdn(Xm+1 - Xm) + Jsed,m+1 , m)1 hm

(14)

For the intermediate clarification layers above the feed layer 0)

νup(Xm-1 - Xm) + Jclar,m+1 - Jclar,m , m ) 7, ..., 9 hm

Qr,int,2,i ∑ i)1

(23)

The volumetric flowrates Qr,int,1,i and Qr,int,2,i are defined as positive variables. For the external recycle,

∑I Qr,ext,i

(24)

For the stream leaving the fourth reaction compartment,

νup(Xm-1 - Xm) - Jclar,m , m ) 10 hm

(25)

where Q4 is the stream flowrate leaving the fourth reaction compartment, QTr,int,2 is the stream flowrate to be recycled to the previous four compartments, and Qr,int,2,5 is the stream flowrate directed from the fourth compartment to the last one. For the stream leaving the fifth reaction compartment,

(15)

T Q5 ) Qsett,in + Qr,int,1

(16)

where Q5 is the stream flowrate that leaves the fifth compartment, QTr,int,1 is the stream flowrate to be recycled to the previous five compartments, and Qsett,in is the stream flowrate directed to the decanter. Finally, the stream leaving the decanter sedimentation zone can be divided according to

For the top layer 0)

)

T Q4 ) Qr,int,2,5 + Qr,int,2

For the bottom layer 0)

(22)

4

T Qr,int,2

T ) Qr,ext

For the intermediate layers below the feed layer

∑I Qr,int,1,i

Mass balances for the soluble components Sx (including dissolved oxygen)

(26)

Qbottom ) Qwaste + QTr,ext

(27)

For the feed layer 0)

(Qsett,inSx,sett,in/Asett) - (νup + νdn)Sx,m , m)6 hm

(17)

For the layers m ) 1-5 νdn(Sx,m+1 - Sx,m) , m ) 1, ..., 5 0) hm

T ) uECSD

(18)

For the layers m ) 7-10 0)

νup(Sx,m-1 - Sx,m) , m ) 7, ..., 10 hm

(19)

The following constraint is introduced for the secondary settler cross-area,

Asett e Asett,max

(20)

where Asett,max is a maximum design limit, which is here set at 1500 m2. 4.3. Splitter Mass Balances. For the feed stream,

QTfresh )

∑I Qfresh,i

where Qbottom is the stream flowrate that leaves the sedimentation zone of the decanter; QTr,ext is the external recycle stream; and Qwaste is the sludge stream flowrate to be wasted. T ), For the external carbon source dosage rate (uECSD

∑I uECSD,i

(28)

where uECSD,i is the external carbon source rate (in (g of CODSs)‚d-1) dosed to reactor i. uECSD,i is defined as a positive variable. 4.4. Mixer Mass Balances. Before each reaction compartment, a nonreactive mixer is modeled,

Qi,in,g ) Qi, ∑ G

∀i

(29)

where Qi,in,g is the gth entering stream flowrate and Qi is the stream flowrate leaving mixer i. The following are the component mass balances,

Qi,in,gCx,i,in,g + ux,i,in ) QiCx,i,in, ∑ G

∀i,x

(30)

(21)

T is the total volumetric feed flowrate and Qfresh,i is where Qfresh the feed flowrate directed to reactor i. For the internal recycle flowrates QTr,int,1 and QTr,int,2, the following constraints must be verified:

where Cx,i,in,g is the concentration of component x (Xx or Sx) in the gth stream entering mixer i, ux,i,in is the x mass flowrate to mixer i, and Cx,i,in is the concentration of component x in the stream Qi leaving mixer i and entering tank i. 4.5. Specification Constraints. Other model constraints to be fulfilled:

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ICsett ) bsett,1Asettδsett,1 + bsett,2Asettδsett,2

Effluent threshold values Cy e Cy,lim, ∀y

(31)

ICips ) bips,1QTfreshδips,1 + bips,2QTfreshδips,2 + bips,3QTfreshδips,3

where Cy is the concentration of the contaminant y and Cy,lim is its threshold value.

Maximum values for operation variables (32)

where OVl are the operation variables and OVl,lim are their maximum values. 4.6. Objective Function. In previous works on WWT process optimization, the total plant cost was evaluated using the present worth method.7-10 Here, the net present value (NPV) is also adopted as the objective function to be minimized. The total cost can be calculated as the sum of the investment (ICT) and operation (OCT) costs,

NPV ) IC + OC ICT ) OC ) ΓOC T

T

)Γ

n

∑ j)1

1

)

∑P OCp

annual

(35)

1 - (1 + id)-n

(1 + id)j

id

ICT ) ICt + ICa + ICsett + ICips + ICsr

(37)

where 5

btViδ ) ∑ i)1 t

(38)

5

baOxCaiδ ) ∑ i)1

ICa ) (

a

OCTa ) Γ(REEa)

(44)

OCTpump ) Γ(REEpump)

(45)

OCTEQ ) Γ(REQEQ)

(46)

T ) Γ(RSLDGDuSLDGD (kg 1000-1 g-1)) OCSLDGD

(47)

T ) Γ(RECSDuECSD (kg 1000-1 g-1)) OCECSD

(48)

where EQ, Ea, Epump, uSLDGD, and uECSD are the effluent quality index, the aeration energy demand, the pumping energy demand, the waste sludge production rate, and the external carbon source dosage rate, respectively, which are expressed as follows4,11,17 (coefficients R are the corresponding unitary annual operation costs):

Pumping energy Epump (kWh d-1) T T + Qr,int,2 + QTr,ext + Qwaste) Epump ) γ(Qr,int,1

(36)

where id is the interest rate (discount rate) and n is the life span of the WWTP. The investment cost functions ICp have the basic structure ICp ) bpZpδp, where bp and δp are cost parameters and Zp is the equipment characteristic dimension; for instance, for the reaction system investment cost, the volume Vi of each compartment i is considered, and for the aeration system, the oxygen capacities OxCai(OxCai ) BkLaiVi) are used as characteristic dimensions. For the secondary settler, the cross-area Asett is considered, computing two different investment costs, namely, the decanter tank construction cost and the corresponding electromechanical system cost. For the influent pumping station investment cost, which computes costs related to concrete, screws, and screening, the characteristic dimension is the influent wastewater flowrate T T (Qfresh ). Finally, the influent wastewater flowrate (Qfresh ) is used as the characteristic dimension for the sludge pump cost. The investment cost of the plant is computed as follows,7

ICt ) (

where OCTp is the operating cost of unit p over the WWTP life ). span, being equal to (ΓOCannual p

(34)

where ICp represents the investment cost and OCpannual represents the operating annual cost of a unit p. OCT,annual is the total operating annual cost of the plant. P is the set of units taken into account, and Γ is the updating term used to compute costs to the present value. The updating term Γ is

Γ)

(42)

OCT ) OCTa + OCTpump + OCTEQ + OCTSLDGD + OCTECSD (43)

(33)

∑P ICp

T,annual

ICsr ) bsrQTr,extδsr

(41)

The operation cost is computed as follows,16,4

OVl e OVl,lim, ∀l

T

(40)

(39)

(49)

where γ is 0.04 (kWh m-3).11 Only the internal and external recycles streams and waste sludge stream are considered. Flows among reactors are assumed to be drawn by gravity.

Aeration energy Ea (kWh d-1) Ea ) 24

∑I

[

(2267 × 10-7)

( ) kLaiVi 24

2

+

( )]

(5.612 × 10-3)

kLaiVi 24

(50)

The effluent quality index EQ, which is related to the fines to be paid due to contaminant discharge, is computed by weighting the compounds loads having an influence on the water quality that are usually included in the legislation. It is defined as

Effluent quality index ((kg of contaminating unit) d-1) EQ )

1 (1000 )(β

SSXSS,ef

+ βCODCODef + βBODBODef + βTKNTKNef + βNOSNO,ef)Qef (51)

where XSS,ef, SNO,ef, CODef, BODef, and TKNef are the concentration of suspended solids, the concentration of nitrate and nitrite nitrogen, CODef, BODef, and the total Kjendal nitrogen in the clarified effluent; Qef is the flowrate of the clarified liquid; and βy are the weighting factors to convert the contaminant y into contaminating units. In addition,

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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

CODef ) SS,ef + SI,ef + XS,ef + XI,ef + XH,ef + XA,ef + XSTO,ef (52)

Table 4. Effluent Threshold Values contaminant, Cy SNH,ef ((g of N) NTOT,efa ((g of N) m-3) BODef ((g of COD) m-3) CODef ((g of COD) m-3) XSS,ef ((g of SS) m-3)

BODef ) 0.25(SS,ef + XS,ef + 0.8(XH,ef + XA,ef + XSTO,ef)) (53)

TKNef ) (0.01SI,ef + 0.03SS,ef + SNH,ef + 0.0426XS,ef + 0.02XI,ef + 0.7(XH,ef + XA,ef)) (54) Production rate of sludge for disposal uSLDGD((g of SS)d-1) uSLDGD ) (XSS,wasteQwaste)

(55)

4.7. Model Parameters. The numerical values for the model parameters used are listed in the following tables. The used ASM3 stoichiometric coefficients and kinetic constants are interpolated to 15 °C based on the default parameter values at 10 and 20 °C and the temperature interpolation function given by Gujer et al.13 The effluent threshold values (eq 31) used as specification constraints are listed in Table 4.11 The adopted maximum values for the operation variables (eq 32) are listed in Table 5.11 A discount rate id of 0.05 and a life span n of 20 years are used (eq 36). Table 6 lists parameters b and δ for the investment cost functions (eqs 38-42) given by Gillot et al.8 The authors also report the application ranges for these cost functions. In the present work, these functions are considered valid in the whole search space defined. Annual unitary operation costs used in eqs 44-48 (coefficients R) are those proposed by Vanrolleghem and Gillot16 and Mussati et al.4 and are listed in Table 7. Weighting factors βy for the contaminating components used in eq 51 are those proposed by Vanrolleghem et al.18 and listed in Table 8. 5. Case Studies The resulting DNLP model is used for optimal synthesis and design and optimization of the operation variables of wastewater treatment plants for given influent wastewater specifications (composition and flowrate). Two different scenarios are selected as case studies. In case study I, the reaction compartment volumes and settler cross-area are optimization variables. In case study II, the reaction compartment volumes are fixed at 1333 m3, and the secondary settler cross-area is fixed at 1500 m2. As mentioned, the settler depth is fixed at 4 m in both cases. Three different influent wastewater compositions (Table 9) were used to show the model capabilities, mainly its robustness and flexibility. One of them consists of the original COST (European Cooperation in the field of Scientific and Technical Research) wastewater specifications11 for ASM1 model modified as by Mussati et al.19 in order to make it compatible with the ASM3 model. The other two influent wastewaters are characterized by lower and higher carbon/nitrogen (C/N) ratios compared to the former one. The C/N ratio is measured as the biochemical oxygen demand (BOD) to the total Kjeldahl nitrogen (TKN) of the influent wastewater. The different C/N ratios were obtained by varying the influent XS and SS content. These wastewater specifications are indicated in Table 9 as A, B, and C, respectively. On the basis of the COST benchmark treatment plant, the influent wastewater flowrate is set at 18 446 m3 d-1 for all cases. Optimal solutions resulting from each scenario for wastewater specifications A, B, and C are compared and

threshold value, Cy,lim

m-3)

a

4 18 10 100 30

NTOT,ef is the total nitrogen: NTOT,ef ) TKNef + SNO,ef ((g of N) m-3).

Table 5. Maximum Values for Operation Variables operation variable, OVl QTr,ext, m3 d-1 QTr,int,1 and QTr,int,2, Qwaste, m3 d-1

maximum value, OVl,lim T 36 892 ()2Qfresh ) T 92 230 ()5Qfresh ) T 1 844.6 ()0.1Qfresh ) 2 000 000 360

m3 d-1

uECSD, (g of COD) d-1 kLai, d-1

Table 6. Parameters b and δ for the Investment Cost Functions bt ba bsett,1 bsett,2 bips,1 bips,2 bips,3 bsr

δt δa δsett,1 δsett,2 δips,1 δips,2 δips,3 δsr

10 304 8 590 2 630 6 338 2 334 2 123 3 090 5 038

0.477 0.433 0.678 0.325 0.637 0.540 0.349 0.304

Table 7. Unitary Annual Operation Costs unitary annual cost

value

REQ (euro day (kg of PU year)-1) RE (euro day (kWh year)-1) RSLDGD (euro day (kg of SS year)-1) RECSD (euro day (kg of COD year)-1)

50 25 75 109.5

Table 8. Weighting Factors for Contaminants ((g of contaminating unit) g-1) factor

βSS

βCOD

βTKN

βNO

βBOD

value

2

1

20

20

2

Table 9. Influent Wastewater Specifications value component m-3)

SI ((g of COD) SS ((g of COD) m-3) XI ((g of COD) m-3) XS ((g of COD) m-3) XH ((g of COD) m-3) XA ((g of COD) m-3) XSTO ((g of COD) m-3) XSS ((g of SS) m-3) SO ((g of COD) m-3) SNOX ((g of N) m-3) SN2 ((g of N) m-3) SNH ((g of N) m-3) SALK ((g of COD) m-3) BOD/TKN

A

B

C

30 69.5 51.2 202.32 28.17 0 0 215.493 0 0 0 36.425 7 3.79

30 34.75 51.2 134.88 28.17 0 0 164.913 0 0 0 36.425 7 2.69

30 139 51.2 303.48 28.17 0 0 291.96 0 0 0 36.425 7 5.33

analyzed, referred to as solutions I.A, I.B, I.C, II.A, II.B, and II.C, respectively. After a lot of simulation runs, it was found that, in the clarification zone as well as in the feed layer (m ) 6-10), the suspended solid concentration XSS is less than the threshold concentration Xt defined in the Taka´cs model. This behavior can, of course, be related to the adopted fixed secondary settler depth and feed allocation, as well as influent wastewater characteristics here considered. For simplicity and to avoid the complex treatment of eqs 11, they were implemented considering only one clause of the conditional equations Jclar,m )

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7505 Table 10. Main Variables Optimal Values for Cases (a) I.A, (b) I.B, and (c) I.C contaminant effluent values

Figure 4. (a) Solution I.A: optimal configuration and main process variable values for case I.A; (b) solution I.B: optimal configuration and main process variable values for case I.B; (c) solution I.C: optimal configuration and main process variable values for case I.C.

νs,mXSS,m, which is valid when XSS < Xt. The fulfillment of the aforementioned constraint has been verified for each achieved solution. 6. Results and Discussion 6.1. Case I. The DNLP model presented in Section 4 is used for optimal synthesis and design and optimization of the operation variables of a WWTP for an influent wastewater flowrate of 18 446 m3/d and influent wastewater specifications listed in Table 9, i.e., cases I.A, I.B, and I.C. A multiple starting point strategy was adopted, and as expected, several locally optimal solutions were found for each case. The WWTP configurations that result from the proposed superstructure model showing the minimal NPV values for cases I.A, I.B, and I.C are represented in parts a, b, and c of Figure 4, respectively, named for simplicity solutions I.A, I.B, and I.C. Parts a, b, and c of Table 10 show the contaminant effluent values, main variables optimal values, and costs. A detailed list showing costs for the optimal solution for each case can be found in Table 14. As shown in Figure 4a, the optimization of case I.A resulted in three reaction compartments with volumes of 1083, 9096, and 6099 m3, respectively. That is, the optimal configuration includes three of the five available reaction compartments. The last two compartments resulted to have zero volume (in fact, the lower bound 0.01 m3 was achieved) and are not represented

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, (g of O2) m-3 CODef, (g of O2) m-3 XSS,ef, (g of SS) m-3

(a) Solution I.A 0.69 OCT,annual 2.18 ICT 1.41 NPV 48.18 14.36

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, (g of O2) m-3 CODef, (g of O2) m-3 XSS,ef, (g of SS) m-3

(b) Solution I.B 1.94 OCT,annual 3.82 ICT 0.90 NPV 49.58 15.15

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, (g of O2) m-3 CODef, (g of O2) m-3 XSS,ef, (g of SS) m-3

(c) Solution I.C 0.38 OCT,annual 1.82 ICT 1.92 NPV 47.77 14.31

costs 386 148.53 2 662 505.98 7 473 916.66

345 130.02 2 717 580.57 7 017 900.62

501 878.11 2 864 144.88 9 117 546.13

in the figure. This WWTP consists of a first small moderated aerated compartment and two larger slightly aerated ones. As a result of the large total volume used, a good effluent quality is reached. It should be noted that a lower EQ value indicates discharge of a lower contaminant load in the receiving water bodies (OCTEQ in Table 14). The optimal solution considers feed distribution: ∼56% of the influent wastewater flowrate (10 360 m3) is fed to the first compartment, and the rest is fed to the second one. No external recycle distribution is necessary, and no external carbon source is dosed. As can be appreciated in Table 10a, none of the effluent component concentrations achieves the effluent threshold value. The C/N ratio decreases from 3.79 for influent A to 2.69 for B, decreasing Xs and Ss levels. The optimization of case I.B resulted in the solution represented in Figure 4b. Table 10b shows the main variables optimal values. This solution resulted in three reaction compartments with volumes of 946, 1 247, and 22 672 m3, respectively. As in the previous case, the last two compartments have zero volume. In this configuration, the first two compartments operate moderately aerated while the last one is practically anoxic. Around 67% of the influent wastewater flowrate (12 427 m3) is fed to the first compartment, and the rest is fed to the second one. As shown in a previous work, the fresh feed distribution improves the process efficiency.6 No external recycle stream distribution is present, and no external carbon source is dosed to the process. As shown in Table 10b, none of the effluent component concentrations is at the effluent threshold value, not even SNH,ef, despite the fact that the BOD concentration is lower than in case I.A. Here, it is convenient to choose a large plant with higher ICT, lower fines, and, consequently, lower OCT rather than a small plant and SNH,ef at the effluent threshold value (increasing OCT). The influent BOD content for this case is reduced compared to that for case I.A and NPV decreases 6% (see Table 14 for comparison). While the ICT increases (2%), mainly because of an increased total reactor volume and, hence, investment tank cost (4%), the OCT decreases significantly (11%). The cost for aeration energy T demand OCTa and for treating the sludge for disposal OCSLDGD decreases 26 and 33%, respectively, while fines paid OCTEQ increase 28%. The increment in fines, as a consequence of effluent quality deterioration compared to case I.A (higher SNH,ef and NTOT,ef), is probably due to an unfavorable C/N ratio from a treatment efficiency point of view. The other cost variations are less relevant.

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Figure 5. Costs distribution for solutions I.

Figure 4c and Table 10c show the WWTP configuration and operation conditions with the minimum NPV for case I.C computed by optimization of the proposed superstructure. Solution I.C consists of four reaction compartments with volumes of 1 315, 1 527, 5 569, and 4 956 m3, respectively. Here, the last compartment resulted to have zero volume and is not represented in the figure. In this configuration, the four compartments are aerated. Around 39% of the influent wastewater flowrate (7 232 m3) is fed to the first compartment, 35% is fed to the second, and 26% is fed to the third compartment. No external recycle stream distribution is present, and no external carbon source is dosed to the process. None of the effluent component concentrations is at the effluent threshold value. Because the influent BOD concentration is higher with respect to Solution I.A, the NPV increased 22%, as expected. The increments on ICT and OCT are 8 and 30%, respectively (see Table 14 for comparison). The main difference is on the operation costs, especially on the aeration demand (39%) and costs for treating the sludge (58%). However, the achieved effluent EQ is better than that for solution I.A (OCTEQ decreases 6%). The main increment in the investment costs ICT is produced by the reaction compartments (8%), because four compartments are needed instead of three as before. Nevertheless, the total reaction volume decreases because the treatment efficiency is better for this influent C/N ratio than for the previous case. 6.1.1. Costs Distribution for Solutions I. The average costs distribution for solutions I.A, I.B, and I.C is illustrated in Figure 5. Around 65 ( 4% of the NPV corresponds to OCT and 35 ( 4% corresponds to ICT. In order of relevance, each cost contributes to the NPV as follows: fines paid represents 20 ( 6%, aeration energy demand cost 24 ( 4%, tanks investment cost 23 ( 3%, sludge treatment cost 19( 5%, settler investment cost 6 ( 1%, influent pumping station investment cost 3 ( 0.5%, aeration systems investment cost 2 ( 0.2%, pumping energy demand cost 2 ( 0.5%, and, finally, sludge recirculation pump cost 0.5 ( 0.1%. Since solvers for DNLP problems cannot guarantee a globally optimal solution and existing software for global optimization cannot cope with problems of this size, several locally optimal solutions could be found depending on the initial point used for optimization. To verify this behavior, a multiple starting point strategy was adopted. The initial points were generated combining several initial values for critical variables, such as ranging from 1000 to 5000 m3, and using different Vinitial i aeration patterns along the reaction compartments (e.g., increasing or decreasing aeration profiles or alternating aerated zones with anoxic ones). Other initial points were generated by solving first different “secondary” objective functions such as minimizing only the operation costs or the investment costs and then the objective function as formulated. As expected, several locally optimal mathematical solutions to each case study were found with the aforementioned strategy. Here, a “mathematical solution” refers to a given (unique)

solution vector, whose components are the variable values that result from optimization. As a consequence of the superstructure representation and the mathematical model formulation, different mathematical solutions showing the same objective function value may represent the same “real or physical solution”, after deleting the zero variables. A real or physical solution is here defined as a given (unique) WWTP configuration (flowsheet) with given operation conditions, i.e., a set of variable values after deleting the zero-valued variables. As a result, these different “mathematical solutions” representing the same real or physical solution with the same objective function value are here considered equiValent among them, and constitute the same solution to the synthesis, design, and optimization problem. As an example, three equivalent mathematical solutions for case study I.A (Seq(1), Seq(2), and Seq(3)) are shown in Table 11. They represent the real or physical solution I.A, which has been shown before in this section. From Table 11, it can be seen that they represent the same final flowsheet and lead to the same plant sizing and design, but some model variable values are interchanged. The effluent contaminant concentrations and costs are also the same (Table 10a, Table 14, and Figure 4a). Specifically, the reaction compartments with zero volume correspond to the last two compartments for Seq(1), to the first and second for Seq(2), and to the first and fourth ones for Seq(3). It should be noted that the resulting nonzero volume compartments are placed in the same sequence after suppressing the zero-volume ones, having identical kLa values and feed distribution pattern. All these solutions represent a plant consisting of three reaction compartments of 1 083, 9 096, and 6 099 m3, with kLa values of 218, 36, and 27 d-1, respectively. The influent wastewater flowrate is fed to the first and second reactors (59 and 41%, respectively). For example, as in Seq(2), the first two reaction compartments resulted to have zero volumes, the sum of the influent wastewater flowrate (Qfresh,i) to the first, second, and third reactors is fed to the first nonzero reactor (i.e., the third reaction compartment), which is the first one in the “real” WWTP configuration. By the other hand, several real or physical solutions were found for each case study. These solutions are alternatiVe configurations for the problem, i.e., different solutions to the synthesis, design, and optimization problem. The final flowsheet, equipment size, and operation variable value are different among these solutions, and they can show the same, similar, or quite different NPVs. So far, all the solutions shown in this section were those with the lowest NPV for each case study. As an example, two of the alternative configurations found for case study I.A are shown in Figure 6 and Table 12 and are referred to as solutions I.Aalt(1) and I.Aalt(2). Summarizing, for case study I.A, three different alternative configurations were presented: solutions I.A (Figure 4 and Table 10), I.Aalt(1), and I.Aalt(2) (Figure 6 and Table 12). It is worth mentioned that these configurations present very similar NPVs (in a range of 1%). Table 14 also shows detailed costs for Solutions I.Aalt(1) and I.Aalt(2). Figure 6a shows solution I.Aalt(1), which resembles very much solution I.A (Figure 4a), having similar characteristics. Both have a first moderately aerated reactor with a volume of 1 085 m3, followed by two less aerated reactors summing around 16 000 m3. The NPV is almost the same due to a trade-off between the OCTswhich increases 2%sand the ICTswhich decreases 3.5%. The OCT increases due to fines to be paid (OCTEQ increases 6.3%) and the ICT decreases due to tanks investment cost (ICt decreases 5.5%). Other alternative solutions having the same characteristics, i.e., a first moderately aerated

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7507

Figure 6. Two alternative configurations for case I.A: solutions I.Aalt(1) and I.Aalt(2). Table 11. Three Equivalent Solutions for the “Real or Physical Solution”, Solution I.A Seq(1)

Seq(2)

Seq(3)

Ri

Vi, m3

kLai, d-1

Qfresh,i, m3 d-1

Vi, m3

kLai, d-1

Qfresh,i, m3 d-1

Vi, m3

kLai, d-1

Qfresh,i, m3 d-1

1 2 3 4 5

1 083 9 096 6 099 0 0

218 36 27 0 0

10 360 8 086 0 0 0

0 0 1 083 9 096 6 099

0 0 218 36 27

5 690 4 009 661 8 086 0

0 1 083 9 096 0 6 099

0 218 36 0 27

10 360 0 8 086 0 0

Table 12. Main Variables Optimal Values for Solutions I.Aalt(1) and I.Aalt(2) I.Aalt(1)

I.Aalt(2)

contaminant effluent values SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, (g of O2) m-3 CODef, (g of O2) m-3 XSS,ef, (g of SS) m-3

0.87 2.53 1.45 48.44 14.56

costs OCT,annual ICT NPV

contaminant effluent values 394016.37 2570061.6 7479505.62

compartment of 1 085 m3 followed by two less aerated ones summing around 16 000 m3 were also found for this case. Solution I.Aalt(2) is a plant with two aerated reactors and feed distribution. The NPV increases only 0.86%, OCT increases 6%, and ICT decreases 8%, with respect to solution I.A. The OCT increases mainly due to an increment on the aeration energy demand cost (OCTAE increases 11%), and the decrease of the ICT is mainly attributed to the reaction tanks (ICt decreases 10%). It is interesting to note that, from the 41 initial points considered in case study I.A, 34 trials rendered feasible (7 initial points that were tried resulted infeasible). From the 34 successful trials, 29 different “mathematical solutions” were computed. After solution analysis, 6 different real or physical solutions (WWTP configurations) were found. Twelve from the 29 different mathematical solutions represent the real or physical solution I.A, which shows the lowest NPV. The solution of the problem using different initial points allows one to ensure a good locally optimal solution and to find alternative configurations and their characteristics. It is convenient for the designer to gain insight on locally optimal properties and to know when different structures or operating conditions have similar costs. It also allows the further selection among these alternatives according to several aspects that are difficult to introduce simultaneously into this DNLP model, such as flexibility (considering trends and predictions of future requirements), reliability (minimum risk of failure due to a complex treatment systems), and controllability, among others. 6.2. Case Study II. Here, the DNLP model is solved for operation variables optimization assuming given reaction compartment volumes and secondary settler cross-area for the three wastewater specifications listed in Table 9. Compartment

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, (g of O2) m-3 CODef, (g of O2) m-3 XSS,ef, (g of SS) m-3

1.11 2.69 1.33 48.11 14.24

costs OCT,annual ICT NPV

407977.51 2455421.69 7538821.52

volumes are fixed at 1 333 m3, and the influent wastewater flowrate is kept at 18 446 m3/d as in case I. The secondary settler cross-area is also fixed at 1 500 m2. Thus, three cases are optimized, named cases II.A, II.B, and II.C. Despite the fact that the reaction compartment volumes and the secondary settler cross-area are fixed, the rest of the design and operation variables result from optimization. As in previous cases, a multiple starting point strategy for optimizing each case was used, leading to different solutions. The WWTP configurations that result from the proposed model showing the minimal NPV values for cases II.A, II.B, and II.C are represented in parts a, b, and c of Figure 7, respectively, named for simplicity solutions II.A, II.B, and II.C. Parts a, b, and c of Table 13 show the contaminant effluent values, main variables optimal values, and costs. A detailed list showing costs for the optimal solution for each case can be found in Table 14. As expected, as the reaction compartment volumes and the secondary settler cross-area are fixed, the feasible solution regions are reduced with respect to cases I.A, I.B, and I.C, and consequently, the NPV values are comparatively higher than in previous cases. Figure 7a shows that, for case II.A, a feed distribution with decreasing flowrate along the first three reactors (around 44% of the flowrate is fed to the first compartment, 36% is fed to the second, and 20% is fed to the third one) is optimal. The first, second, and fourth compartments resulted aerobic, while the third and fifth ones are anoxic. That is, this solution presents aerated zones followed by anoxic zones. No external recycle distribution is present, and no external carbon source is dosed to the process. As can be appreciated in Table 13a, none of the effluent component concentrations achieve the effluent threshold value. As mentioned, the NPV increases 15% compared to that

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Figure 7. (a) Solution II.A: optimal configuration and main process variable values for case II.A; (b) solution II.B: optimal configuration and main process variable values for case II.B; and (c) solution II.C: optimal configuration and main process variable values for case II.C. Table 13. Main Variables Optimal Values for Cases II.A, II.B, and II.C contaminant effluent values SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, g m-3 CODef, (g of COD) m-3 XSS,ef, (g of SS) m-3

(a) Solution II.A 3.59 OCT,annual 6.22 ICT 2.18 NPV 50.55 16.37

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, g m-3 CODef, (g of COD) m-3 XSS,ef, (g of SS) m-3

(b) Solution II.B 4 OCT,annual 10.85 ICT 1.71 NPV 51.51 16.71

SNH,ef, (g of N) m-3 NTOT,ef, (g of N) m-3 BODef, g m-3 CODef, (g of COD) m-3 XSS,ef, (g of SS) m-3

(c) Solution II.C 2.61 OCT,annual 4.87 ICT 2.68 NPV 50.29 16.27

costs 485 265.65 2 512 861.77 8 559 271.77

513 409.46 2 469 361.72 8 866 443.59

588 803.57 2 572 254.88 9 908 747.36

in case I.A, while OCT increases 26% and ICT decreases 6% (see Table 14). The main increment is on cost for fines paid OCTEQ (71%), since the achieved effluent quality is worse than that in case I.A. The cost variation for the remaining items in

decreasing order of relevance is as follows: 22% increment in the treatment of sludge for disposal, 12% decrease for aeration energy, 9% decrease for investment in reaction tanks, and 93% increment for pumping energy. The cost variations for the other items are irrelevant. The optimization of case II.B leads to solution II.B represented in Figure 7b. The optimal solution considers feed distribution: around 55% of the influent wastewater is fed to the first compartment, and the rest is fed to the third one. The first and fourth reaction compartments operate as aerated reactors and the other three operate as anoxic; showing, as in the previous solution, aerated zones followed by anoxic zones. No external recycle stream distribution is present, and no external carbon source is dosed to the process. As shown in Table 13b, the effluent SNH,ef concentration is at the effluent threshold value. As the influent C/N ratio decreases, with respect to wastewater specification A, by lower XS and SS levels, the denitrification process is limited. Comparing solution II.B with solution I.B, the NPV increases 26%, OCT increases 49%, and ICT decreases 9%. The main increment is on effluent fines OCTEQ (93%), while the cost increment for treating the sludge for disposal is 27%, and the decrement on tank investment is 12%; finally, the increment on pumping energy is 76% and that on aeration energy is 2% (in order of relevance). In spite of the reduction in the influent XS and SS content for case II.B compared to case II.A, the NPV increases 4%, OCT increases 6%, and ICT decreases 2% with respect to solution II.A. This fact can be explained by the unfavorable C/N ratio from a treatment efficiency point of view. The main increment is on effluent fines OCTEQ (44%), but the costs for treating the sludge for disposal, for aeration energy demand, and for tank investments decrease 30, 15, and 27%, respectively. From other optimization cases (not shown), it was observed that, by varying XS and SS content in the influent wastewater (and consequently the C/N ratio) and keeping constant the rest of the compound levels, for case II in which the reaction volumes and sedimentation cross-area are fixed, the NPV exhibits a minimum between the C/N ratio of wastewater A and B. As shown in Figure 7c and Table 13c, solution II.C has feed distribution into the first four aerated compartments: around 41, 26, 30, and 3% of the influent stream flowrate is fed to the first, second, third, and fourth compartments, respectively. No external recycle distribution is presented, and no external carbon source is dosed to the process. None of the effluent component concentrations achieves the effluent threshold value. As expected, the NPV increases 9% compared to solution I.C, operation costs increase 17%, and investment cost decreases 10%. The main increment is on fines paid due to effluent quality deterioration (58%). The sludge disposal cost also increments (20%), but aeration energy cost decreases (11%) when compared to solution I.C. The tank investment cost decreases 15%, and the other cost variations are less relevant over NPV. With respect to solution II.A, NPV, OCT, and ICT increase 16, 21, and 2%, respectively. The main increments in OCT are on the cost for treating the sludge for disposal (56%) and for aeration energy demand (40%). However, fines decrease (13%) despite the fact that the influent BOD level is higher than that in case I.A, as a consequence of a more convenient C/N ratio from a treatment efficiency point of view. The main increment on ICT is due to the increment on the aeration system investments (36%). The remaining cost variations have poor impact on NPV.

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7509 Table 14. Detailed Solution Costs

I.A I.B I.C II.A II.B II.C I.Aalt(1) I.Aalt(2)

OCTEQ

OCTpump

OCTa

T OCSLDGD

OCTECSD

OCT

ICt

ICa

ICset

ICips

ICsr

ICT

NPV

1 407 398 1 806 487 1 324 757 2 412 190 3 478 411 2 098 800 1 495 701 1 519 135

149 247 188 809 147 767 287 928 331 551 278 685 158 308 144 431

1 843 952 1 359 139 2 556 543 1 631 151 1 386 819 2 285 708 1 838 564 2 055 377

1 410 813 945 885 2 224 335 1 715 142 1 200 302 2 673 300 1 416 870 1 364 457

0 0 0 0 0 0 0 0

4 811 410 4 300 320 6 253 401 6 046 410 6 397 081 7 336 492 4 909 444 5 083 399

1 746 404 1 813 635 1 885 135 1 594 116 1 594 116 1 594 116 1 650 699 1 573 881

171 258 156 587 234 335 166 524 121 207 226 379 173 913 137 025

442 671 442 671 442 671 442 671 442 671 442 671 442 671 442 671

268 985 268 985 268 985 268 985 268 985 268 985 268 985 268 985

33 188 35 703 33 019 40 566 42 383 40 104 33 793 32 860

2 662 506 2 717 581 2 864 145 2 512 862 2 469 362 2 572 255 2 570 062 2 455 422

7 473 917 7 017 901 9 117 546 8 559 272 8 866 444 9 908 747 7 479 506 7 538 821

6.2.1. Costs Distribution for Solutions II. The average costs distributions for solutions II.A, II.B, and II.C are illustrated in Figure 8. Around 72 ( 2% of NPV corresponds to OCT and 28 ( 2% corresponds to ICT. In order of relevance, each cost contributes to NPV as follows (average from solutions II.A, II.B, and II.C): fines 30 ( 9%, sludge treatment 20 ( 7%, aeration energy demand 19 ( 4%, tanks investment 18 ( 1%, settler investment 5 ( 0.5%, influent pumping station investment 3 ( 0.5%, pumping energy demand 3 ( 0.5%, aeration system investment 2 ( 0.5%, and, finally, sludge recirculation pump investment 0.5 ( 0.1%. Summarizing, for the assumptions made and the parameter values used in the model, it was shown that: • For case I, the lowest NPV value achieved corresponds to solution I.B, followed by solutions I.A and I.C. • For case II, the lowest NPV value corresponds to solution II.A (despite wastewater A not being the less contaminated one), followed by solutions II.B and II.C. • It is convenient to choose a large plant with lower fines and, consequently, lower OCT than a small plant with SNH,ef at the effluent threshold value (increasing OCT through fines). • As in previous works,5,6 it was observed that treatment performance benefits from operational features such as the use of different aeration and stream distributions patterns, mainly in cases where the reaction volumes are constrained. • The average costs distributions for solutions I.A, I.B, and I.C show that around 65 ( 4% of NPV corresponds to OCT and 35 ( 4% corresponds to ICT. On the basis of average values from solutions I.A, I.B, and I.C, the main contributors to NPV in order of relevance are as follows: fines (20 ( 6%), aeration energy demand (24 ( 4%), tanks investment (23 ( 3%), sludge treatment (19 ( 5%), settler investment (6 ( 1%), influent pumping station investment (3 ( 0.5%), aeration systems investment (2 ( 0.2%), pumping energy demand (2 ( 0.5%), and, finally, sludge recirculation pump (0.5 ( 0.1%). • The average costs distributions for solutions II.A, II.B, and II.C show that around 72 ( 2% of NPV corresponds to OCT and 28 ( 2% corresponds to ICT. Each cost item contributes to NPV as follows: fines 30 ( 9%, sludge treatment 20 ( 7%, aeration energy demand 19 ( 4%, tanks investment 18 ( 1%, settler investment 5 ( 0.5%, influent pumping station investment 3 ( 0.5%, pumping energy demand 3 ( 0.5%, aeration system investment 2 ( 0.5%, and sludge recirculation pump investment 0.5 ( 0.1%. • The NPV for solutions II with respect to solutions I increases 17 ( 9%. This fact is a consequence of a reduction of the feasible solution regions, because the reaction compartment volumes and the secondary settler cross-area are fixed. Investment cost decreases 8 ( 2% while operation cost increases 31 ( 16%. The main increment in OCT is in fines to be paid (74 ( 17%), and the main decrease in ICT is on the reaction compartments investment cost (12 ( 3%). The remaining investment costs have low impact on NPV variation. The other two operation costs that vary significantly from solution I to

solution II are the sludge treatment cost, which increases 23 ( 4%, and the aeration energy demand cost, which increases or decreases depending on the wastewater characteristics. The two last cost variations are particularly significant in the case of wastewater C. Finally, the sludge recirculation pump investment cost increases 86 ( 9% but has a low impact on the NPV variation. The model resulted in being flexible and robust and able to be used for process synthesis, for process optimization (fixing the structural variables), and also for simulation of a given plant and operating conditions. For example, the optimal configuration, design, and operating conditions obtained for influent A in case I [see Figure 4a, main optimal values: Vi ) (1083, 9096, 6099, 0, 0); Asett ) 1500; KLai ) (218, 36, 27, 0, 0); Qfresh,i ) (10360, 8086, 0, 0, 0); QTr,int,1 ) 0; QTr,int,2 ) 0; Qr,ext,i ) (11841, T ) 0] were simulated by feeding influents B 0, 0, 0, 0); uECSD and C, without imposing effluent threshold values, since simulation runs were performed instead of optimization. Effluent quality conditions were not met (effluent contaminant concentrations are higher than the effluent permitted limits), and the NPV increased 71 and 83%, with respect to the optimal configurations for those case studies (solutions I.B and I.C, respectively). On the other hand, if reaction compartments and decanter are given, the model can be used for optimization. For example, by setting volumes and decanter area to the optimal values achieved for case I.A [Vi ) (1083, 9096, 6099, 0, 0); Asett ) 1500] and optimizing the other variables, the following optimal values feeding influent B were obtained: [KLai ) (238, 19, 20, 0, 0); Qfresh,i ) (12565, 5173, 708, 0, 0); QTr,int,1 ) 0; QTr,int,2 ) T 0; Qr,ext,i ) (18042, 0, 0, 0, 0); uECSD ) 0]. Here, the effluent quality conditions were met and the NPV increased only 5% with respect to the optimal configuration for this case study (solution I.B). Finally, feeding influent C, the main optimal values were as follows: [KLai ) (256, 53, 45, 0, 0); Qfresh,i ) (9228, 9218, 0, 0); QTr,int,1 ) 0; QTr,int,2 ) 0; Qr,ext,i ) (10426, 0, T ) 0]. Again, the effluent quality conditions 0, 0, 0); uECSD were met and the NPV increased only 1% with respect to the optimal configuration for this case study (solution I.C). It is clear that convenient aeration and recycle stream patterns and a suitable influent feed distribution can significantly improve the system performance in the face of an influent composition variation. This also shows that it is important to optimize operation modes of the process under variable influent conditions. Design under uncertainty or flexible design will be addressed in a future work. The models were implemented and solved using general algebraic modeling systems GAMS.20 The model results in 536 single variables and 509 single equations. The code CONOPT21 was employed for solving the DNLP problem. The total CPU time needed to solve the models was among 8.7 and 16.3 s. The average total CPU time needed to solve the models was 12.9 s. An Intel Pentium IV 2.40GHz CPU with 248 MB of RAM was used.

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Acknowledgment The financial support from Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET) and Agencia Nacional de Promocio´n Cientı´fica y Tecnolo´gica (ANPCyT) of Argentina are gratefully acknowledged. Notation Figure 8. Costs distribution for solutions II.

7. Conclusions and Future Works In the present work, a DNLP model was developed for optimal synthesis and design as well as optimization of the operation variables of wastewater treatment plants for given influent wastewater specifications (composition and flowrate). Two different scenarios are selected as case studies. In case study I, the reaction compartment volumes and settler cross-area are optimization variables. In case study II, the reaction compartment volumes are fixed at 1 333 m3, and the secondary settler cross-areas are fixed at 1 500 m2. As mentioned, the settler depth is fixed at 4 m in both cases. Three different influent wastewater compositions characterized by different carbon/nitrogen ratios were used to show the model capabilities, mainly its robustness and flexibility. The different C/N ratios were obtained by varying the influent XS and SS content. Optimal solutions resulting from increasing C/N ratios (B < A < C) were compared and analyzed for both cases. The models were implemented in GAMS. The code CONOPT was employed for solving the DNLP problems. As a consequence of the problem nature and the solver characteristics, in order to verify the expected presence of local optimal solutions, a multiple starting point strategy was adopted. Indeed, different locally optimal solutions to the problem were found for each case study. Because of the model flexibility and robustness, it is a useful tool to help the designer to make a further selection among the solutions found considering other design aspects such as flexibility, reliability, and controllability, among others. As mentioned, a DNLP model was developed as a first step of the modeling task. Although the existence of nonsmooth functions, CONOPT solved the model quite well, providing solutions for >80% of the different initial points tried. Also, the existence and type of reaction compartments and process streams were satisfactorily handled by the use of very small lower bounds (practically zero), which introduce only a small error in the objective function calculation if the associated equipment is eliminated to achieve a practical flowsheet. However, it is interesting to note that, in some cases, solutions with small kLa values (but not the lower bound value) were computed, which may be nonsensical from an engineering point of view. This limitation will be overcome with the implementation of MINLP models. In future works, new mathematical models based on MINLP or GDP programming will be presented. This will also allow the incorporation of the settler depth and the feed allocation point as optimization variables, as well as the use of a more rigorous economic objective function. All the expertise gained and results obtained from the proposed model will be used as a starting point. Finally, design under uncertainty or flexible design will also be addressed.

A ) area (m2) b ) cost parameter B ) constant (3000-1 (kg of O2) d (h m3)-1) BOD ) biochemical oxygen demand ((g of O2) m-3) C ) component concentration (C ) S or X) COD ) chemical oxygen demand ((g of O2) m-3) E ) energy (kWh d-1) EQ ) effluent quality index ((kg of contaminating unit) d-1) h ) height (m) I ) set of reactors/set of mixers IC ) investment cost (euro) id ) interest rate (discount rate) J ) solids flux due to gravity settling K ) set of process rates kLa ) oxygen transfer coefficient (d-1) n ) life span of the WWTP (year) N ) nitrogen NPV ) net present value (euro) OC ) operation cost (euro) OV ) operation variable OxCa ) oxygen capacity ((kg of O2)/h) P ) set of equipment units Q ) volumetric flowrate (m3 d-1) r ) reaction rate (g m-3) S ) soluble component concentration (g m-3) TKN ) total Kjendal nitrogen u ) mass flowrate (g d-1) V ) volume (m3) X ) particulate component concentration (g m-3) Z ) characteristic dimension Subscripts a ) aeration system BOD ) biochemical oxygen demand bottom ) settler bottom effluent c ) component clar ) clarification zone COD ) chemical oxygen demand ECSD ) external carbon source dosage ef ) clarified effluent EQ ) effluent quality index ext ) external fresh ) influent wastewater g ) generic flowrate i ) reactor/mixer prior to reactor in ) inlet int ) internal ips ) influent pumping station k ) process l ) operation variable lim ) limit m ) settler layer max ) maximum min ) minimum concentration p ) unit pump ) pump r ) recycle

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sat ) saturation sed ) sedimentation zone sett ) settler SLDG ) sludge sr ) sludge recirculation pump SS ) suspended solids t ) tank TKN ) total Kjendal nitrogen TOT ) total tresh ) threshold concentration waste ) waste sludge x ) component y ) contaminant Superscripts annual ) annual T ) total initial ) initial Greek Letters R ) unitary annual operation cost (euro day(year)-1) β ) weighting factors for contaminant r ((g of cont. unit) g-1) δ ) cost parameter νdn ) bulk velocity of the liquid below the feed layer (m d-1) νs ) settling velocity (m day-1) νup ) bulk velocity of the liquid above the feed layer (m d-1) F ) process rate (g(d m3)-1) υ ) stoichiometric coefficient Γ ) updating term (year)

bA,O2 ) aerobic endogenous respiration rate of XA ) 0.087 (1/ d) bA,NOX ) anoxic endogenous respiration rate of XA ) 0.032 (1/d) fsI ) production of SI in hydrolysis ) 0 ((g of CODSI)/(g of CODXs)) YSTO,O2 ) aerobic yield of stored product per SS ) 0.85 ((g of CODXSTO)/(g of CODSs)) YSTO,NOX ) anoxic yield of stored product per SS ) 0.80 ((g of CODXSTO)/(g of CODSs)) YH,O2 ) aerobic yield of heterotrophic biomass ) 0.625 ((g of CODXh)/(g of CODXSTO)) YH,NOX ) anoxic yield of heterotrophic biomass ) 0.54 ((g of CODXh)/(g of CODXSTO)) YA ) yield of autotrophic biomass per NO3-N ) 0.24 ((g of CODXa)/(g of NSNOX) fXI ) production of XI in endog. respiration ) 0.20 ((g of CODXI)/(g of CODXbm)) iN,SI ) N content of SI ) 0.01 ((g of N)/(g of CODSI)) iN,Ss ) N content of SS ) 0.03 ((g of N)/(g of CODSs)) iN,XI ) N content of XI ) 0.02 ((g of N)/(g of CODXI)) iN,XS ) N content of XS ) 0.04 ((g of N)/(g of CODXs) iN,BM ) N content of biomass XH, XA ) 0.07 ((g of N)/(g of CODbm)) iSS,XI ) SS-to-COD ratio for XI ) 0.75 ((g of SS)/(g of CODXI)) iSS,XS ) SS-to-COD ratio for XS ) 0.75 ((g of SS)/(g of CODXs)) iSS,BM ) SS-to-COD ratio for biomass XH, XA ) 0.90 ((g of SS)/(g of CODXbm)) Literature Cited

Kinetic, Stoichiometric, and Composition Parameters for ASM3 (Values at 15 °C) kH ) hydrolysis rate constant ) 2.45 ((g of CODXs)/((g of CODXh) d)) KX ) hydrolysis saturation constant ) 1 ((g of CODXs)/(g of CODXh)) kSTO ) storage rate constant ) 3.54 ((g of CODSs)/((g of CODXh) d)) ηNOX ) anoxic reduction factor ) 0.6 KO2 ) saturation constant for SO ) 0.2 ((g of O2)/m3) KNOX ) saturation constant for SNOX ) 0.5 (((g of NO3) N)/m3) KS ) saturation constant for substrate SS ) 2 ((g of CODSS)/ m3) KSTO ) saturation constant for XSTO ) 1 ((g of CODXSTO)/(g of CODXh)) µH ) heterotrophic maximum growth rate of XH ) 1.4 (1/d) KNH ) saturation constant for ammonium SNH ) 0.01 ((g of N)/m3) KALK ) saturation constant for alkalinity for XH ) 0.1 ((mol of HCO3)/m3) bH,O2 ) aerobic endogenous respiration rate of XH ) 0.14 (1/d) bH,NOX ) anoxic endogenous respiration rate of XH ) 0.07 (1/d) bSTO,O2 ) aerobic respiration rate for XSTO ) 0.14 (1/d) bSTO,NOX ) anoxic respiration rate for XSTO ) 0.07 (1/d) µA ) autotrophic maximum growth rate of XA ) 0.59 (1/d) KA,NH ) ammonium substrate saturation for XA ) 1 ((g of N)/ m3) KA,O2 ) oxygen saturation for nitrifiers ) 0.5 ((g of O2)/m3) KA,ALK ) bicarbonate saturation for nitrifiers ) 0.5 ((mol of HCO3)/m3)

(1) Rigopoulos, S.; Linke, P. Systematic Development of Optimal Activated Sludge Process Designs. Comput. Chem. Eng. 2002, 26 (4-5), 585. (2) Linke, P.; Kokossis, A. Advanced Process Systems Design Technology for Pollution Prevention and Waste Treatment. AdV. EnViron. Res. 2004, 8 (2), 229. (3) Henze, M.; Grady, C., Jr; Gujer, W.; Marais, G.; Matsuo, T. ActiVated Sludge Model No. 1; IAWQ Scientific and Technical Report No. 1; IAWQ: London, 1987. (4) Mussati, M. C.; Mussati, S. F.; Alasino, N., Aguirre, P.; Scenna, N. Optimal Synthesis of Activated Sludge Wastewater Treatment Plants for Nitrogen Removal. Presented at 2nd Mercosur Congress on Chemical Engineering and 4th Mercosur Congress on Process Systems EngineeringsENPROMER 2005, Rı´o de Janeiro, Brazil, Aug 2005; pp 1418. (5) Alasino, N.; Musatti M.; Scenna N. Synthesis of Activated Sludge Wastewater Treatment Plants for Nitrogen Removal. Presented at XXII Interamerican Chemical Engineering Congress (V CAIQ), Buenos Aires, Argentina, Oct 1-4, 2006; Paper 06C(505). (6) Alasino, N.; Musatti M.; Scenna N. Optimization of the Operation Conditions for Denitrifying Wastewater Treatment Plants. Presented at International Mediterranean Modeling Multiconference (EMSS06), Barcelona, Spain, Oct 4-6, 2006; pp 427-434. (7) Gillot, S.; De Clercq, B.; Defour, D.; Simoens, F.; Gernaey, K.; Vanrolleghem, P. A. Optimization of wastewater treatment plant design and operation using simulation and cost analysis. Presented at 72nd Annual WEF Conference and Exposition, New Orleans, LA, 1999. (8) Gillot, S.; Vermeire, P.; Jacquet, P.; Grootaerd, H.; Derycke, D.; Simoens, F.; Vanrolleghem, P. A. Integration of wastewater treatment plant investment and operating costs for scenario analysis using simulation. In Proceedings 13th Forum Applied Biotechnology; Med. Fac. Landbouww. Univ. Gent: Gent, Belgium, 1999; 64/5a, 13. (9) Espı´rito Santo, I. A. C. P.; Fernandes, E. M. G. P.; Arau´jo, M. M.; Ferreira, E. C. NEOS Server Usage in Wastewater Treatment Cost Minimization. In Lecture Notes in Computer SciencesComputational Science and Its Applications; Springer-Verlag: New York, 2005; Vol. 3483, p 632. (10) Espı´rito Santo, I. A. C. P.; Fernandes, E. M. G. P.; Arau´jo, M. M.; Ferreira, E. C. Biological Process Optimal Design in a Wastewater

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ReceiVed for reView April 4, 2007 ReVised manuscript receiVed August 7, 2007 Accepted August 8, 2007 IE0704905