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Optimal Operation Modes for a Fixed-Bed Bioreactor Used in Wastewater Treatment
7e-045
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
OPTIMAL OPERATION MODES FOR A FIXED-BED BIOREACTOR USED IN WASTEWATER TREATMENT C. Benthack, B. Srinivasan, and D. Bonvin Institut d'Automatique, Ecole Polytechnique Federale de Lausanne, CH-l015 Lausanne, Switzerland, e-mail: name~ia.epf1.ch
Abstract. The problem of optimizing the efficiency of a fixed-bed bioreactor by manipulating the feed flowrate and the inlet concentration is addressed. A simple macroscopic model of the bio-filter is developed for this purpose. Two criteria, one maximizing the space efficiency and the other the time efficiency of wastewater treatment, are proposed. It is shown that, optimal operation is obtained when one of the input variables is set to its upper limit and the other adjusted such that the effluent quality is met. The optimal input is calculated using a feedback structure which also helps tackle large perturbations inherent to biological systems. Finally, an improved operational scheme, termed the pseudo-batch mode, is developed for which both the time and the space criteria are maximized simultaneously. Keywords. Feedback Optimal Control, Maximum Principle, Distributed-Parameter Systems, Biotechnology 1. INTRODUCTION
Bioreactors have been used increasingly in wastewater treatment due to their capability of consuming various chemicals present therein. Recently, the plug-flow fixedbed bioreactor has emerged as an important treatment method (Pujol et al. 1993). However, due to the nonavailability of dynamic models, fixed-bed bioreactors for wastewater treatment have normally been operated with constant inlet concentration and constant feed flowrate. Yet, one is often interested in using the reactor in a more efficient manner. Two criteria which reflect possible economic objectives (one maximizing the space efficiency and the other the time efficiency) will be proposed. The inlet concentration and feed fiowrate are considered as variables that can be manipulated for the purpose of this optimization. The inlet concentration is varied by recycling the effluent. To optimize the efficiency of wastewater treatment, a dynamic process model that adequately describes the main phenomena and/or trends of the system (macroscopic model) is required. Currently available models are mostly microscopic models that include models by Wanner and Gujer (1986) and Grasmick et al. (1979). Macroscopic models are also available (Grady and Lim 1980) but they neglect shear and filtering effects. Hence, the first part of the article is devoted to developing a simplified macroscopic model of the bio-filter.
Though optimization with differential and algebraic constraints is a well researched field, little work has been reported on the dynamic optimization of fixed-bed bioreactors. Work on dynamic optimization is, in general, based on the Pontryagin's Maximum Principle (PMP) (Bryson and Ho 1969). This requires the solution of a two-point boundary value problem which is computationally expensive. However, for the bioreactor investigated in this study, the proposed objective functions are monotonic in the decision variables, and the optimal solution is on the input/state constraints of the system (Bryson and Ho 1969). The input constraints can be easily met, while to satisfy the state constraints, a feedback structure is used (Choi and Butala 1989). This has the disadvantage that the input so calculated is suboptimal, while the advantages include robustness and computational efficiency. However, only one of the objectives can be maximized at a time with these schemes. For maximizing both optimization goals simultaneously, a new scheme, termed the pseudo-batch mode is proposed. The paper is organized as follows. In Section 2 the dynamic model of the process is introduced, followed by simulation and analysis results in Section 3. The goals of optimization for this process are specified in Section 4. The feedback modes of operation are explained in Section 5, and the pseudo-batch mode is developed in Section 6. Finally, conclusions are drawn in Section 7.
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2. MODELING The bioreactor, shown schematically in Figure 1, is an aerobic cocurrent up-flow fixed-bed reactor for wastewater treatment (Grady and Lim 1980). The reactor is filled with a stationary support (viz., fixed bed) on which the micro-organisms are attached. They grow by consuming oxygen and the carbonaeeous (organic) ingredients, thereby forming biofilms around the support. As the biofilms grow, the reactor volume available for liquid and air flows is reduced. To avoid clogging the normal operation has to be stopped and the reactor backwashed. Hence, such a reactor is considered as a semi-continuous process. In order to model the bioreactor, the following assumptions are made: (i) The reactor is operated in isothermal and aerobic conditions, (ii) It is characterized by plug-flow, i.e., axial and radial dispersion are neglected, (iii) Microbial growth in the liquid phase is negligible, (iv) External mass transfer limitations are negligible, (v) Growth kinetics are substrate limited and of the Monod type (Grady and Lim 1980) and (vi) The influent wastewater does not contain any particulate matter.
volume, M the concentration of the suspended solids in the bulk liquid, Q the volumetric feed flowrate, A the cross-sectional area of the reactor and L its length. Ef and El represent the bed void fractions occupied by the biofilm and the liquid phase, respectively. rs represents the substrate removal rate, rx the biomass growth rate and rM the transfer rate of biomass from the immobilized phase to the liquid phase. The substrate balance contains a convection term and a substrate consumption term. The substrate removal rate rs is assumed to be dependent only on microbial growth kinetics and mass-transfer limitations within the biofilm. Since analytical expressions for rs for Monod kinetics combined with internal diffusion are not available (van't Riet and Tramper 1991), a numerical approximation is used (Benthack and Samb 1995): rs =
SDS) [SeLf + l n (1 _ S+Ks' SeL,)] ( 2f1PK YL} Ks
h . the were JL IS
maximum specific growth rate, p the biofilm density, Ks the Monod saturation constant, Y the yield co-
efficient, Ds the internal diffusion coefficient, L f the biofilm thickness and c a numerical constant. The dynamics of the biomass is described by equation (2). The biomass growth rate rx is proportional to the substrate removal rate: rx = rSYEf. The suspended solids evolve in accordance to the equation (3). rM represents the detachment and attachment effects and is given by: rM = (kd X - ka 1;f). Particulate matter will be detached from the biofilm by shear forces and will be attached to the immobilized phase due to the inherent filtering property of the column.
t
3. SIMULATION AND ANALYSIS Figure 1.
Schematic view of the fixed-bed bioreactor
Based on material balances over infinitesimal volume elements, a model expressed as a set of partial and ordinary differential equations has been developed:
as
Q
as
Ej
-=----r"-
at
S(z, t
AEI
az
C
El '
(1)
= 0) = 0, S(z= 0, t) = Sin(t)
ax
at =
rX -
El rM
(2)
X(z, t = 0) = Xo
aM Q aM at Aft az ' M(z, t = 0) = 0, M(z = 0, t) = 0 -=---+rM
(3)
where S is the substrate concentration in the bulk liquid, X the biomass concentration with respect to reactor
The dynamic model described in Section 2 was discretized in space using finite differences and integrated in time using the Adams-Bashford method. A cycle is considered complete when at least 85 % (say) of the bed void fraction is occupied by the biomass at any height in the column. The variables that can be manipulated during the operation are (i) the input concentration Sin(t) and (ii) the wastewater flowrate Q(t). Typical concentration profiles along the column at time intervals of 2 hours are shown in Figures 2 and 3 for S and X respectively, for constant operating conditions of Q == 6 m 3 / h and Sin == 0.3 kg COD/m 3 . The substrate concentration decreases with the vertical position in the column due to consumption by the biomass. It can also be seen from Figure 2 that, at a given height, S decreases with time as the biomass present in the reactor increases. Since more substrate is available at the bottom (entrance) than at the top
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of the reactor, the growth rate and hence the biomass concentration are higher at the bottom. The biomass spatial profiles corresponding to a higher substrate inlet concentration but the same feed flowrate are shown in Figure 4. Comparing Figures 3 and 4, certain remarks are in order: (i) Faster growth rate which can be visualized by more widely spread profiles in time. (ii) The operation time, tf, is reduced as faster growth rate leads to earlier clogging.
shows that (i) X at the reactor exit is higher for larger flowrates. A larger flowrate leads to a lower residence time, a less steep concentration profile, and a more uniform distribution of the biomass. (ii) The operation time t f is less sensitive to variations in feed flowrate than to those in Sin. The operation time depends on the rate of growth at the bottom of the reactor, which is almost independent of Q. 4. GOALS OF OPTIMIZATION
0.2
S [kg~?D] 0.1
O.OOL--~l-z-[-m=]~2~~~3 Substrate S profiles
Figure 2.
4 X
[~]
28±=~~ °0~--~-~~-~3 1 z [rn] 2
Figure 3.
Biomass X profiles
6~--r--~---.
The main goal in all wastewater treatment processes is to treat the given wastewater in a space- and timeefficient manner such that the effluent meets the specified quality requirements. For a given bio-filter, the problem of selecting the optimal operating conditions (Le., choosing Q(t) and Sin(t)) is addressed here. The two quantities that will be maximized are the amount of substrate treated, (a) during one cycle (Ja) and (b) per unit of cycle time (Jb). In both cases, the same operational constraints apply. These include constraints on the feed flowrate, feed composition and effluent concentrations. The first criterion, J a , looks for a space-efficient procedure. This is of particular interest when the backwashing procedure is energy intensive and expensive. On the contrary, with Jb, a time-efficient operation is preferred so that more wastewater can be treated per unit time. The time efficiency is defined with respect to the cycle time, t e , which not only includes operation time tf but also the time required for backwashing tw: te = tf + two The free terminal time optimization problems can be formulated as follows:
4 X
[~]
s.t.maxX(z,tf) = X max , Sout(t)::; z
00
Figure 4.
(4)
max J a or Jb Q(t),Sin(t)
2
1 z [m] 2
Slim
Qlb ::; Q(t) ::; Qub, Sib::; Sin(t) ::; Sub
3
tf
X profiles for Sin == 0.5 kg COD 1m3
Ja
= jQSw(Sin - Sout) dt o
Sw - Sout
(5)
(6)
00
Figure 5.
1 z [m] 2
3
X profiles for Q == 8 m 3 /h
The biomass profiles corresponding to larger feed flowrate are shown in Figure 5. A comparison of Figures 3 and 5
where [Qlb, QubJ and [Sib, Sub] are the ranges of admissible values for the feed flowrate and the inlet substrate concentration, respectively, and Slim the maximum acceptable effluent concentration. Xmax is the maximum permissible biomass concentration, attaining which at any height, the column has to be backwashed. Sw is the concentration of the wastewater that needs to be cleaned and ('?,:=,~':,~t' is the fraction of the original wastewater
)
6834
stream that actually enters the reactor, taking recycling into account.
direct and stronger influence on TS than Q, at its upper bound.
The developments that follow are based on monotonicity arguments, i.e., the derivative of the costs with respect to certain variables being always positive. Then by invoking the Maximum Principle, it will be shown that some of the variables should be at their constraint limits. Though these arguments can be established analytically, for the sake of brevity we will only state the results and provide intuitive arguments substantiating them.
The feasible region and the sensitivities at a given point of time are summarized in the Figure 6. The constraint surface and the optima move towards the top right corner as time increases.
Monotonicity with respect to Sout: Both costs increase when the limit on Sout is relaxed i.e., &~::, > 0 and a~:UI > o. When Sout is allowed to increase, Q can be increased for that purpose. Compared to the increase in Q, the decrease in ~i"~%Q"t is only marginal and hence {Jut J a increases. Also, since t f is insensitive to changes in Q, Jb increases with Sout. From PMP, the optimum is achieved on the constraints which, in this case, means Sout == Slim for maximizing both J a and Jb.
Sin
must be
lV
Optimization of the cost J a : Having set Sout == Slim, it has to be decided how the two input variables can be manipulated along this constraint surface so as to maximize Ja.. A sensitivity analysis leads to : (7)
From PMP, one can conclude that the optimal choice is == Qub. However, since we cannot control the effluent concentration using Q, the inlet concentration needs to be adjusted to meet Sout == Slim. Intuitively, the reason why Q has to be at its maximum for maximizing J a can be explained as follows : Seeking a set of input variables that result in space-efficient wastewater treatment is 'equivalent' to having a more uniform biomass distribution. As can be seen in Fig. 5, such a situation can be achieved with a larger Q. Hence, increasing Q is beneficial as far as maximizing J a is concerned. Q*(t)
Optimization of the cost Jb : In this case, a sensitivity analysis along the constraint surface, Sout == Slim, results in : (8)
From PMP, the optimal choice is Sin(t) == Sub. Hence, the feed flowrate should be adjusted to meet the limit on Sout. The reason for Sin to be at its maximum for time efficiency can be explained as follows : Time-efficient wastewater treatment is 'equivalent' to maximizing the substrate consumption rate TS. Since Sin has a more
Feasible Region
Figure 6.
Feasible region at time t
5. OPTIMAL MODES OF OPERATION For both the optimization criteria proposed in the previous section, it was found that one of the input variables (Sin or Q) has to be adjusted such that the effluent quality is met exactly. This can be achieved either by explicitly solving the corresponding dynamic optimization problem or by imposing a feedback structure that regulate Sout at Slim. An explicit solution requires a good dynamic model and a considerable amount of computation time. On the other hand, solving the problem implicitly by imposing a feedback structure is computationally efficient and can be easily implemented on-line. The set-point for the feedback system is set to Slim and the controller so designed that the output tracks the reference. However, there is no guarantee that the output will be equal to Slim at all times and, hence, the scheme should be considered suboptimal. Yet, the advantages are considerable : (i) There is no necessity to obtain an exact model, as long as the observed profiles follow the trends indicated in the previous sections. (ii) The feedback control provides a certain amount of robustness. This is extremely important, as biological systems are inherently non-deterministic and are characterized by large perturbations and parameter variations. Considering the system at hand, especially its biological nature, the best choice would be to use a suboptimal feedback scheme rather than a 'true' optimal feedforward action. The price paid, however, is that the effluent substrate concentration needs to be measured. The following feedback structures have been investigated in this study:
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6837
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
OPTIMAL OPERATION MODES FOR A FIXED-BED BIOREACTOR USED IN WASTEWATER TREATMENT C. Benthack, B. Srinivasan, and D. Bonvin Institut d'Automatique, Ecole Polytechnique Federale de Lausanne, CH-l015 Lausanne, Switzerland, e-mail: name~ia.epf1.ch
Abstract. The problem of optimizing the efficiency of a fixed-bed bioreactor by manipulating the feed flowrate and the inlet concentration is addressed. A simple macroscopic model of the bio-filter is developed for this purpose. Two criteria, one maximizing the space efficiency and the other the time efficiency of wastewater treatment, are proposed. It is shown that, optimal operation is obtained when one of the input variables is set to its upper limit and the other adjusted such that the effluent quality is met. The optimal input is calculated using a feedback structure which also helps tackle large perturbations inherent to biological systems. Finally, an improved operational scheme, termed the pseudo-batch mode, is developed for which both the time and the space criteria are maximized simultaneously. Keywords. Feedback Optimal Control, Maximum Principle, Distributed-Parameter Systems, Biotechnology 1. INTRODUCTION
Bioreactors have been used increasingly in wastewater treatment due to their capability of consuming various chemicals present therein. Recently, the plug-flow fixedbed bioreactor has emerged as an important treatment method (Pujol et al. 1993). However, due to the nonavailability of dynamic models, fixed-bed bioreactors for wastewater treatment have normally been operated with constant inlet concentration and constant feed flowrate. Yet, one is often interested in using the reactor in a more efficient manner. Two criteria which reflect possible economic objectives (one maximizing the space efficiency and the other the time efficiency) will be proposed. The inlet concentration and feed fiowrate are considered as variables that can be manipulated for the purpose of this optimization. The inlet concentration is varied by recycling the effluent. To optimize the efficiency of wastewater treatment, a dynamic process model that adequately describes the main phenomena and/or trends of the system (macroscopic model) is required. Currently available models are mostly microscopic models that include models by Wanner and Gujer (1986) and Grasmick et al. (1979). Macroscopic models are also available (Grady and Lim 1980) but they neglect shear and filtering effects. Hence, the first part of the article is devoted to developing a simplified macroscopic model of the bio-filter.
Though optimization with differential and algebraic constraints is a well researched field, little work has been reported on the dynamic optimization of fixed-bed bioreactors. Work on dynamic optimization is, in general, based on the Pontryagin's Maximum Principle (PMP) (Bryson and Ho 1969). This requires the solution of a two-point boundary value problem which is computationally expensive. However, for the bioreactor investigated in this study, the proposed objective functions are monotonic in the decision variables, and the optimal solution is on the input/state constraints of the system (Bryson and Ho 1969). The input constraints can be easily met, while to satisfy the state constraints, a feedback structure is used (Choi and Butala 1989). This has the disadvantage that the input so calculated is suboptimal, while the advantages include robustness and computational efficiency. However, only one of the objectives can be maximized at a time with these schemes. For maximizing both optimization goals simultaneously, a new scheme, termed the pseudo-batch mode is proposed. The paper is organized as follows. In Section 2 the dynamic model of the process is introduced, followed by simulation and analysis results in Section 3. The goals of optimization for this process are specified in Section 4. The feedback modes of operation are explained in Section 5, and the pseudo-batch mode is developed in Section 6. Finally, conclusions are drawn in Section 7.
6832
2. MODELING The bioreactor, shown schematically in Figure 1, is an aerobic cocurrent up-flow fixed-bed reactor for wastewater treatment (Grady and Lim 1980). The reactor is filled with a stationary support (viz., fixed bed) on which the micro-organisms are attached. They grow by consuming oxygen and the carbonaeeous (organic) ingredients, thereby forming biofilms around the support. As the biofilms grow, the reactor volume available for liquid and air flows is reduced. To avoid clogging the normal operation has to be stopped and the reactor backwashed. Hence, such a reactor is considered as a semi-continuous process. In order to model the bioreactor, the following assumptions are made: (i) The reactor is operated in isothermal and aerobic conditions, (ii) It is characterized by plug-flow, i.e., axial and radial dispersion are neglected, (iii) Microbial growth in the liquid phase is negligible, (iv) External mass transfer limitations are negligible, (v) Growth kinetics are substrate limited and of the Monod type (Grady and Lim 1980) and (vi) The influent wastewater does not contain any particulate matter.
volume, M the concentration of the suspended solids in the bulk liquid, Q the volumetric feed flowrate, A the cross-sectional area of the reactor and L its length. Ef and El represent the bed void fractions occupied by the biofilm and the liquid phase, respectively. rs represents the substrate removal rate, rx the biomass growth rate and rM the transfer rate of biomass from the immobilized phase to the liquid phase. The substrate balance contains a convection term and a substrate consumption term. The substrate removal rate rs is assumed to be dependent only on microbial growth kinetics and mass-transfer limitations within the biofilm. Since analytical expressions for rs for Monod kinetics combined with internal diffusion are not available (van't Riet and Tramper 1991), a numerical approximation is used (Benthack and Samb 1995): rs =
SDS) [SeLf + l n (1 _ S+Ks' SeL,)] ( 2f1PK YL} Ks
h . the were JL IS
maximum specific growth rate, p the biofilm density, Ks the Monod saturation constant, Y the yield co-
efficient, Ds the internal diffusion coefficient, L f the biofilm thickness and c a numerical constant. The dynamics of the biomass is described by equation (2). The biomass growth rate rx is proportional to the substrate removal rate: rx = rSYEf. The suspended solids evolve in accordance to the equation (3). rM represents the detachment and attachment effects and is given by: rM = (kd X - ka 1;f). Particulate matter will be detached from the biofilm by shear forces and will be attached to the immobilized phase due to the inherent filtering property of the column.
t
3. SIMULATION AND ANALYSIS Figure 1.
Schematic view of the fixed-bed bioreactor
Based on material balances over infinitesimal volume elements, a model expressed as a set of partial and ordinary differential equations has been developed:
as
Q
as
Ej
-=----r"-
at
S(z, t
AEI
az
C
El '
(1)
= 0) = 0, S(z= 0, t) = Sin(t)
ax
at =
rX -
El rM
(2)
X(z, t = 0) = Xo
aM Q aM at Aft az ' M(z, t = 0) = 0, M(z = 0, t) = 0 -=---+rM
(3)
where S is the substrate concentration in the bulk liquid, X the biomass concentration with respect to reactor
The dynamic model described in Section 2 was discretized in space using finite differences and integrated in time using the Adams-Bashford method. A cycle is considered complete when at least 85 % (say) of the bed void fraction is occupied by the biomass at any height in the column. The variables that can be manipulated during the operation are (i) the input concentration Sin(t) and (ii) the wastewater flowrate Q(t). Typical concentration profiles along the column at time intervals of 2 hours are shown in Figures 2 and 3 for S and X respectively, for constant operating conditions of Q == 6 m 3 / h and Sin == 0.3 kg COD/m 3 . The substrate concentration decreases with the vertical position in the column due to consumption by the biomass. It can also be seen from Figure 2 that, at a given height, S decreases with time as the biomass present in the reactor increases. Since more substrate is available at the bottom (entrance) than at the top
6833
of the reactor, the growth rate and hence the biomass concentration are higher at the bottom. The biomass spatial profiles corresponding to a higher substrate inlet concentration but the same feed flowrate are shown in Figure 4. Comparing Figures 3 and 4, certain remarks are in order: (i) Faster growth rate which can be visualized by more widely spread profiles in time. (ii) The operation time, tf, is reduced as faster growth rate leads to earlier clogging.
shows that (i) X at the reactor exit is higher for larger flowrates. A larger flowrate leads to a lower residence time, a less steep concentration profile, and a more uniform distribution of the biomass. (ii) The operation time t f is less sensitive to variations in feed flowrate than to those in Sin. The operation time depends on the rate of growth at the bottom of the reactor, which is almost independent of Q. 4. GOALS OF OPTIMIZATION
0.2
S [kg~?D] 0.1
O.OOL--~l-z-[-m=]~2~~~3 Substrate S profiles
Figure 2.
4 X
[~]
28±=~~ °0~--~-~~-~3 1 z [rn] 2
Figure 3.
Biomass X profiles
6~--r--~---.
The main goal in all wastewater treatment processes is to treat the given wastewater in a space- and timeefficient manner such that the effluent meets the specified quality requirements. For a given bio-filter, the problem of selecting the optimal operating conditions (Le., choosing Q(t) and Sin(t)) is addressed here. The two quantities that will be maximized are the amount of substrate treated, (a) during one cycle (Ja) and (b) per unit of cycle time (Jb). In both cases, the same operational constraints apply. These include constraints on the feed flowrate, feed composition and effluent concentrations. The first criterion, J a , looks for a space-efficient procedure. This is of particular interest when the backwashing procedure is energy intensive and expensive. On the contrary, with Jb, a time-efficient operation is preferred so that more wastewater can be treated per unit time. The time efficiency is defined with respect to the cycle time, t e , which not only includes operation time tf but also the time required for backwashing tw: te = tf + two The free terminal time optimization problems can be formulated as follows:
4 X
[~]
s.t.maxX(z,tf) = X max , Sout(t)::; z
00
Figure 4.
(4)
max J a or Jb Q(t),Sin(t)
2
1 z [m] 2
Slim
Qlb ::; Q(t) ::; Qub, Sib::; Sin(t) ::; Sub
3
tf
X profiles for Sin == 0.5 kg COD 1m3
Ja
= jQSw(Sin - Sout) dt o
Sw - Sout
(5)
(6)
00
Figure 5.
1 z [m] 2
3
X profiles for Q == 8 m 3 /h
The biomass profiles corresponding to larger feed flowrate are shown in Figure 5. A comparison of Figures 3 and 5
where [Qlb, QubJ and [Sib, Sub] are the ranges of admissible values for the feed flowrate and the inlet substrate concentration, respectively, and Slim the maximum acceptable effluent concentration. Xmax is the maximum permissible biomass concentration, attaining which at any height, the column has to be backwashed. Sw is the concentration of the wastewater that needs to be cleaned and ('?,:=,~':,~t' is the fraction of the original wastewater
)
6834
stream that actually enters the reactor, taking recycling into account.
direct and stronger influence on TS than Q, at its upper bound.
The developments that follow are based on monotonicity arguments, i.e., the derivative of the costs with respect to certain variables being always positive. Then by invoking the Maximum Principle, it will be shown that some of the variables should be at their constraint limits. Though these arguments can be established analytically, for the sake of brevity we will only state the results and provide intuitive arguments substantiating them.
The feasible region and the sensitivities at a given point of time are summarized in the Figure 6. The constraint surface and the optima move towards the top right corner as time increases.
Monotonicity with respect to Sout: Both costs increase when the limit on Sout is relaxed i.e., &~::, > 0 and a~:UI > o. When Sout is allowed to increase, Q can be increased for that purpose. Compared to the increase in Q, the decrease in ~i"~%Q"t is only marginal and hence {Jut J a increases. Also, since t f is insensitive to changes in Q, Jb increases with Sout. From PMP, the optimum is achieved on the constraints which, in this case, means Sout == Slim for maximizing both J a and Jb.
Sin
must be
lV
Optimization of the cost J a : Having set Sout == Slim, it has to be decided how the two input variables can be manipulated along this constraint surface so as to maximize Ja.. A sensitivity analysis leads to : (7)
From PMP, one can conclude that the optimal choice is == Qub. However, since we cannot control the effluent concentration using Q, the inlet concentration needs to be adjusted to meet Sout == Slim. Intuitively, the reason why Q has to be at its maximum for maximizing J a can be explained as follows : Seeking a set of input variables that result in space-efficient wastewater treatment is 'equivalent' to having a more uniform biomass distribution. As can be seen in Fig. 5, such a situation can be achieved with a larger Q. Hence, increasing Q is beneficial as far as maximizing J a is concerned. Q*(t)
Optimization of the cost Jb : In this case, a sensitivity analysis along the constraint surface, Sout == Slim, results in : (8)
From PMP, the optimal choice is Sin(t) == Sub. Hence, the feed flowrate should be adjusted to meet the limit on Sout. The reason for Sin to be at its maximum for time efficiency can be explained as follows : Time-efficient wastewater treatment is 'equivalent' to maximizing the substrate consumption rate TS. Since Sin has a more
Feasible Region
Figure 6.
Feasible region at time t
5. OPTIMAL MODES OF OPERATION For both the optimization criteria proposed in the previous section, it was found that one of the input variables (Sin or Q) has to be adjusted such that the effluent quality is met exactly. This can be achieved either by explicitly solving the corresponding dynamic optimization problem or by imposing a feedback structure that regulate Sout at Slim. An explicit solution requires a good dynamic model and a considerable amount of computation time. On the other hand, solving the problem implicitly by imposing a feedback structure is computationally efficient and can be easily implemented on-line. The set-point for the feedback system is set to Slim and the controller so designed that the output tracks the reference. However, there is no guarantee that the output will be equal to Slim at all times and, hence, the scheme should be considered suboptimal. Yet, the advantages are considerable : (i) There is no necessity to obtain an exact model, as long as the observed profiles follow the trends indicated in the previous sections. (ii) The feedback control provides a certain amount of robustness. This is extremely important, as biological systems are inherently non-deterministic and are characterized by large perturbations and parameter variations. Considering the system at hand, especially its biological nature, the best choice would be to use a suboptimal feedback scheme rather than a 'true' optimal feedforward action. The price paid, however, is that the effluent substrate concentration needs to be measured. The following feedback structures have been investigated in this study:
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6837