2013 IEEE Multi-conference on Systems and Control (Conference on Control Applications) Hyderabad, India, August 28-30, 2013
Economically Optimal Control of Batch Diafiltration Processes Martin Jelemensk´y1, Radoslav Paulen,1,2, Miroslav Fikar1 , Zolt´an Kov´acs3,4
optimal operation of utrafiltration/diafiltration processes with ideal membrane response. This optimization problem combines minimum time and minimum diluant problems and accounts for differences in prices of a unit of processing time (which comprises costs for running pumps and cooling costs) and a unit of consumed diluant. This work deals with more complex setups of diafiltration processes with complex membrane transport models that account for non-ideal membrane characteristics. As in our previous works, we apply Potryagin’s minimum principle. We derive an optimal solution to the economically optimal control problem such that it is parametrized with prices of a unit of processing time and a unit of consumed diluant. This form allows for responding to changes in these prices and subsequent restoring of economically optimal operation and, at the same time, it provides a tool for the analysis of long-term economics of the process. In the following, we describe a batch diafiltration process and studied problem of optimal control. Next, we provide an analytical form of a solution to presented problem of finding an economically optimal operation of the studied process. Finally, we show the optimal operation for two selected case studies.
Abstract— An economically optimal operation of batch diafiltration processes is studied. The processes are characterized by complex membrane transport models that adequately describe non-ideal membrane characteristics of real-life systems. We consider the optimization criterion which comprises two conflicting objectives, a minimization of processing time and a minimization of solvent consumption. The problem is treated in the multi-objective form in order to investigate the impact of operational cost factors on optimal operation policy. Thanks to the application of Pontryagin’s minimum principle (PMP), we find the economically optimal operation of studied process in the analytical form. Developed methodology is applied to two case studies taken from literature.
I. I NTRODUCTION Diafiltration (DF) is an effective and unique membrane process for separating two (or more) solutes present in a solution. It has found many applications in the fields of food and chemical (e.g. pharma) processing and biotechnology [5]. The process under consideration is designed to achieve the twin-objective of (i) fractionating two solutes, i. e. purifying a valuable component from a contaminant, and (ii) concentrating the valuable component up to a given value. Its control is realized by influencing of the species’ concentrations with addition of a solute-free solvent (diluant) into the feed tank. The desired dual goal can be achieved by a suitable choice of time-varying diluant flowrate. Additionally, this can be chosen in a way to minimize the processing time, the amount of consumed diluant, or the cost of the production. Several scientific works and industrial patents have been devoted for optimization of diluant addition during the DF process. The common feature of the earliest works relies on the optimization of switching times between arbitrarily predefined operational modes [13] such as concentration with constant-volume diafiltration (C-CVD). For optimization of processing time, the optimal concentration to start the CVD step was derived by [7] who considered the limiting flux conditions for transmembrane flow and an ideal membrane resistance. Following studies considered numerical [9] or analytical [6] approaches for identifying of time-optimal or minimum diluant control strategies. Recently, in [2] and [10], we presented a methodology for treating an economically
II. P ROBLEM A. Process description
We consider here a membrane filtration plant with a given membrane area that operates under fixed operating conditions such as constant cross-flow, temperature, and applied pressure. The schematic representation of diafiltration process is shown in Fig. 1. Filtered solution consists of solvent (water), micro-solute (lower molecular weight impurities) and macrosolute (higher molecular weight component, product) which is passed from the feed tank to the membrane (filtration) module. The membrane module is designed to retain macrosolute and to allow the passage of micro-solute. Permeate stream leaves the system at a flowrate q which is a function of concentrations of solutes in the feed q = q(c1 , c2 ). The retentate stream is recycled into the feed tank. The process control is achieved by adjusting the flowrate of water to the feed tank. Control variable α is defined as a dimensionless ratio between the inflow of diluant to the feed tank and the outflow of permeate q. We consider the processing of a solution with initial volume V0 and two species with concentrations c1 and c2 . A macro-solute is to be increased in concentration from c1,0 to c1,f and a micro-solute is to be reduced in concentration from c2,0 to c2,f . We assume that the utilized membrane is absolutely impermeable to product (which is the usual
1 Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Slovakia (e-mail: {martin.jelemensky, radoslav.paulen, miroslav.fikar}@stuba.sk) 2 Process Dynamics and Operations Group, Technische Universit¨ at Dortmund, Germany (
[email protected]) 3 Department of Food Engineering, Corvinus University of Budapest, Hungary (e-mail:
[email protected]) 4 Institute of Bioprocess Engineering and Pharmaceutical Technology, University of Applied Sciences Mittelhessen, Giessen, Germany (e-mail:
[email protected])
978-1-4799-1557-6/13/$31.00 ©2013 IEEE
FORMULATION
734
diluant
retentate
meα(t)q(c1 , c2 )
permeate q(c1 , c2 )
According to PMP, an optimal control minimizes the Hamiltonian function. Since Hamiltonian is affine in α, its minimum will be attained with α on its boundaries as ( 0 if Hα > 0, α= (6) αmax if Hα < 0.
membrane module feed tank
Fig. 1.
If Hα = 0, the Hamiltonian is singular and does not depend on α. In this case, it may be possible to construct switching surface S(x) = 0 corresponding to singular arc that depends on state variables only [3], [12]. We use the fact that the condition Hα = 0 implies that time derivatives of Hα are equal to zero as well. Therefore, we will make use of
Schematic representation of a batch diafiltration process.
c2 q c˙1 = 1 (1 − α), c1,0 V0 c1 c2 q c˙2 = (R2 − α), c1,0 V0
c1 (0) = c1,0 ,
(1)
c2 (0) = c2,0 .
(2)
III. R ESULTS Based on the statements of previous section, we eliminated adjoint variables from conditions (7) by simple algebraic manipulations. The switching surface was then found in the following form S(c1 , c2 , wT , wD ) = wT ST (c1 , c2 ) + wD SD (c1 , c2 ) = 0 (8) where ! 2 2 X X ci R2i , (9) ci qi + q ST (c1 , c2 ) = (R2 − 1) q +
The objective is to find the time dependent function α(t) which minimizes the economical objective function and drives the process from initial to final conditions. Mathematical formulation of this dynamic optimization problem reads as follows Z tf min (wT + wD αq) dt, (3a)
SD (c1 , c2 ) = q 2
R2 − 1 +
2 X
ci R2i
i=1
0
qi =
c21 q (1 − α), c1 (0) = c1,0 , c1 (tf ) = c1,f , (3b) c1,0 V0 c1 c2 q c˙2 = (R2 − α), c2 (0) = c2,0 , c2 (tf ) = c2,f , (3c) c1,0 V0 α ∈ [0, αmax ], (3d)
∂q , ∂ci
R2i =
!
∂R2 . ∂ci
,
(10) (11)
The conditions for limit cases, minimum time operation (wD = 0) and minimum diluant operation (wT = 0) are given by singular surfaces ST (c1 , c2 ) = 0 and SD (c1 , c2 ) = 0, respectively. The expression for singular control α that will keep the process at the optimal surface S(·) = 0 can be obtained by differentiating (8) with respect to time. Using process differential equations, this then yields
c˙1 =
where the coefficients wT ≥ 0 and wD ≥ 0 represent weights of final (processing) time (tf ) and total diluant consumption (Vw ), respectively, both being expressed in their integral forms. These weights can also be interpreted as unit prices of a unit of processing time (e.g. price for running the pumps) and a unit of consumed diluant. We make use of Pontryagin’s minimum principle [11] to R tf solve this problem. Hence that considered functional F (x, α)dt and the right-hand sides of process differential 0 equations x˙ = f (x, α) are affine in control. Due to this, Hamiltonian function takes control-affine form as well
α=
∂S ∂S ∂c1 c1 + ∂c2 c2 R2 . ∂S ∂S ∂c1 c1 + ∂c2 c2
(12)
Once the weights wT and wD are fixed, the optimal operation can be stated as follows: 1) In the first step, we use αmax or pure filtration (α = 0) until the optimal surface is reached S(c1 , c2 ) = 0. 2) The second step is characterized as diafiltration on singular surface (8). Thus we use singular control (12). 3) The last step is either pure filtration (α = 0) or operation with αmax until the final concentrations are reached. Any of these three steps can be missing from the optimal operation. This depends on particular process initial and final conditions. For instance, there might not exist a three-step
(4)
with x = (c1 , c2 )T and vector of adjoint variables λ = (λ1 , λ2 )T defined as ∂H . λ˙ = − ∂x
i=1
i=1
s.t.
H(x, α, λ) = F + λf = H0 (x, λ) + Hα (x, λ)α,
(7)
to eliminate the adjoint variables λ.
B. Process optimization
α(t)
H˙ α (x, λ) = 0,
Hα (x, λ) = 0,
industrial case) however its permeability of micro-solute varies with changing concentrations which can be expressed using a retention coefficient R2 = R2 (c1 , c2 ). Under these conditions, the process model can be found to read as
(5) 735
TABLE I T WO CASES OF FINAL
albumin, c1,f ethanol, c2,f
and albumin concentration does not change in CVD mode (α = 1). It follows from this analysis that the last step is such that ( 0 if c2 (t) = c2,f , α= (15) 1 if c1 (t) = c1,f ,
CONDITIONS CONSIDERED .
Case A 80 kg/m3 0.1 kg/m3
Case B 240 kg/m3 0.1 kg/m3
and the process is kept under this control until the prescribed final concentrations are met. With decreasing ratio wT /wD the switching surface can be observed to be pushed into the regions of higher concentrations of both solutes. This eventually causes the inadmissibility of reaching of this surface and the condition S(c1 , c2 ) = 0 cannot be satisfied. Using some insight, we may identify the limiting value of wT /wD which causes the optimal control structure to skip the singular arc for given initial conditions. This can be done by finding a value wT /wD such that the singular surface S(c1 , c2 , wD , wT ) is reached exactly at the point where a switching to the last step occurs. This value was identified to be
operation which fulfills final conditions on concentrations. In such case, the second step is usually skipped from optimal operation and optimal control is saturated on constraints. Note that PMP gives necessary conditions of optimality only. The simulation studies below and preliminary numerical optimizations confirm that the suggested operation is optimal. IV. C ASE S TUDIES A. Albumin/Ethanol Separation We study the separation of albumin and ethanol with a batch diafiltration process employing an ultrafiltration membrane with 1 m2 of membrane area. The membrane transport model was experimentally verified in [1]. Timeoptimal and minimum diluant operations of this process were studied in [8]. The goal is to process 0.0666 m3 of solution with 15 kg/m3 of albumin and 194.3 kg/m3 of ethanol. We consider two possible endpoints which are shown in Table I. Due to the properties of utilized membrane and a nature of the treated system (hence the big difference in effective sizes of molecules of albumin and ethanol), the membrane does not retain ethanol, i.e. R2 = 0. The model of the membrane flux was obtained experimentally and can be written in the following form q(c1 , c2 ) =
wT /wD =
(4b5 b6 −
4b5 2 b4 )c22,0
− 4b1 b5
= 0.074
(16)
Note that further decrease of wT /wD does not change the respective optimal operation and thus it coincides with minimum diluant operation (wT /wD = 0). 1) Case A: The above described procedure for finding an optimal control strategy was applied. Fig. 2 shows resulting transitions in the state diagram (top plot) and optimizing control (bottom plot) for the economically optimal (red curve), the minimum time (blue curve), and the minimum diluant (green curve) operations. The minimum time and the minimum diluant control strategies are shown for comparison and are in accord with [8]. The overall control strategy for cases of minimum time and economically optimal operation consists of three steps. Minimum diluant control strategy consist only of two steps, the reasons for this were discussed previously. In all cases, the process starts at initial concentrations (green circle) where either pure ultrafiltration or CVD step is applied according to Eq. (14). If singular arc (8) (plotted using dots in Fig. 2) is reached, control from (12) is applied. The last step is represented by the CVD operation where final concentration of albumin is maintained and both prescribed final concentrations (denoted by the red cross) are met at the end. In the case of minimum diluant operation, the middle step is skipped and pure ultrafiltration followed with CVD step is used. This behavior occurs since the singular surface is not reachable by any admissible control. Table II compares the different control strategies in terms of processing time, consumption of diluant and the cost of producing 10 kg of albumin. We observe that the economically optimal operation reduces the production cost by 3 % compared to minimum time operation. The minimum diluant and the traditionally used operation (C-CVD) operation are practically optimal. These results suggest that advanced optimal operation would be overly complicated and that the classical C-CVD practice would be sufficient.
1 , (13) b1 + b2 c1 + b3 c2 + b4 c1 c2 + b5 c21 + b6 c22
where constants b1 to b6 are taken from [1]. The upper limit on control variable value is αmax = 1, which represents an operation with constant-volume diafiltration. The price coefficients are set to wT = 2 e/h and wD = 10 e/m3 . After the values of wT and wD are fixed, the singular surface is found according to Eq. (8). We note that this represents a nonlinear function whose roots (singular state transitions c1 = c1 (c2 ) and c2 = c2 (c1 )) can be explicitly expressed. This feature mitigates the further analysis. Economically optimal operation comprises either two or three steps. The first step depends on the actual location of initial point x0 = [c1,0 , c2,0 ] in the state diagram. The control can be defined as ( 0 if S(x0 ) < 0, α= (14) 1 if S(x0 ) > 0. It is obvious that if S(x0 ) = 0, the first step is skipped and the operation starts directly on singular arc. After the singular surface is reached, the control for the next step is found from Eq. (12). In the last step, the control takes one of its extremal values. Note that the concentration of ethanol is constant during pure filtration mode (α = 0) 736
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c2 [kg/m3 ]
c2 [kg/m3 ]
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120 100 80 60
20 0 10
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S(c1,c2) = 0 20
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minimum diluant economically optimal minimum time ST(c1,c2) = 0
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minimum diluant economically optimal minimum time ST(c1,c2) = 0 S(c1,c2) = 0 50
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c1 [kg/m3 ]
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0 0
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minimum diluant economically optimal minimum time
0
2.5
0
Fig. 2. Economically optimal, minimum time and minimum diluant strategies for albumin/ethanol separation for Case A.
0.5
1
1.5
time [h]
2
2.5
Fig. 3. Economically optimal, minimum time and minimum diluant strategies for albumin/ethanol separation for Case B.
TABLE II C OMPARISON OF CONTROL STRATEGIES OF BATCH DF PROCESS FOR ALBUMIN / ETHANOL SEPARATION (C ASE A). control strategy
tf [h]
Vw [m3 ]
product cost [e/10 kg]
minimum time minimum diluant economically optimal C-CVD
2.54 2.59 2.58 2.59
0.123 0.095 0.096 0.095
63.12 61.26 61.24 61.26
We can also observe that, independently of the utilized control strategy, the processing times are lower for case B (when higher final concentration of albumin is desired; Tab. II) as compared to Case A (Tab. III). This seemingly strange behavior is, however, well explained by existence of higher permeation of the ethanol during the operations which precede the last CVD step in case B. The higher outlet of ethanol then shortens the time to reach the desired final concentrations for both solutes in the last step. Similar observations were reported in [9]. In Fig. 4, we show the optimal values for individual parts of objective function. These were obtained by changing the weight coefficients and evaluating respective optimal operation. For the minimum time case, optimal processing time and diluant consumption values are denoted by a square and the same quantities are marked using a circle for minimum diluant operation. The economically optimal operation is denoted by diamond. As it was stated above, the optimal operation starts either with α = 0 or α = 1. From the control point of view, it might be interesting to identify a ratio of price coefficients wT /wD that gives a switch in the optimal control structure for the first control arc. This can be identified from the condition
2) Case B: In comparison with the preceding case, similar behavior of resulting optimal control schemes was identified with one interesting difference. As the required final concentration of albumin is higher, a diafiltration at singular surface should last longer. It is however observed that the singular control is predicted to take physically inadmissible (negative) values due to the specified experimental model of the flowrate q. As a remedy, we use ultrafiltration mode (α = 0) in such cases (see also [4] for similar observations on behavior of singular control). In Fig. 3, resulting transitions in the state diagram (top plot) and optimizing control (bottom plot) are shown for the economically optimal, the minimum time, and the minimum diluant operations using the same color code as in Case A. 737
0.07
60 minimum time minimum diluant economically optimal optimal control structure switching point utopia point
0.065
50
0.055
c∗1 [g/dL]
Vw∗ [m3 ]
0.06
55
0.05 0.045
45 40
SD(c1) = 0 c
* 1
ST(c1) = 0 35
0.04
30 0.035
25 0.03 2.34
2.35
2.36
2.37
2.38
2.39
2.4
2.41
2.42
2.43
20 −6
t∗f [h]
−4
−2
0
2
4
6
log10 (wT /wD )
Fig. 4. Pareto front of optimal values of processing time and of diluant consumption for albumin and ethanol separation (Case B).
Fig. 5. Optimal macro-solute concentration during CVD step for different values of wT /wD for the separation of radiopaque and ethylene glycol.
TABLE III C OMPARISON OF CONTROL STRATEGIES OF BATCH DF PROCESS FOR ALBUMIN / ETHANOL SEPARATION (C ASE B). control strategy
tf [h]
Vw [m3 ]
product cost [e/10 kg]
minimum time minimum diluant economically optimal C-CVD
2.34 2.42 2.39 2.42
0.068 0.031 0.035 0.031
53.71 51.69 51.53 51.69
where c1 represents the concentration of radiopaque. The upper limit on control variable is αmax = ∞ which physically represents a pure dilution step, i.e. pouring of a certain amount of diluant into the feed tank in one time instant. We note that such operation maintains constant ratio of concentrations c1 (t)/c2 (t). The singular surface (8) depends only on the concentration of radiopaque. This predicts constant concentration of radiopaque on the singular arc which is maintained by using CVD operation, α = 1. Opposite to previous case, it is not possible to derive an explicit formula for the singular concentration of radiopaque, c∗1 , so its actual value (for given wT and wD ) has to be found numerically. Fig. 5 shows resulting values of c∗1 obtained for different values of ratio wT /wD . We can observe that c∗1 converges rather quickly to its limit values (dashed and dashed-dotted lines) represented by the optimal c∗1 for minimum time and minimum diluant operation. The solid curve can be approximated by three lines: almost linear increase in the middle interval (10−2 , 102 ) and approximately constant value equal to the limiting cases outside. It demonstrates that limiting cases of minimum time and minimum diluant can be approximated well by wT /wD = 10 and wT /wD = 0.01, respectively. Taking into account the initial conditions and based on the analysis similar to previous example, the optimal control sequence is either α = {0, 1, 0} or α = {0, 1, ∞}. The same price coefficients wT and wD as in previous example are used for evaluation of economically optimal control strategy of considered batch diafiltration process. Fig. 6 and Table IV show individual parts of the objective function obtained by changing the ratio wT /wD and by evaluating the respective optimal control strategy. The economically optimal operation takes 15.04 h by which the consumption of water is 29.93 dL. The minimum diluant operation with q = 0 is practically infeasible as it predicts infinite processing time. Therefore, we use a more realistic control strategy with the weights wT /wD = 0.01. Further, we also present results obtained for minimum time problem
S(x0 , wT , wD ) = 0 which represents the situation when the first control arc is skipped. The value of wT /wD = 0.443 was identified as a switching condition of the optimal control structure. The corresponding optimal values of t∗f and Vw∗ are denoted by hexagram in Fig. 4. The utopia point denoted by star lies at the intersection of minimal values of t∗f and Vw∗ . In Table III, we compare processing time, consumption of diluant, and the production cost of 10 kg of albumin. Again, we can observe that application of advanced optimal operations would not be beneficial and that the classical strategy should be used instead. The small differences between classical and optimal operations are, in both studied cases, caused by the fact that the control variable α is constrained by α ≤ 1. If this constraint was removed, much larger savings would be gained. B. Separation of Radiopaque and Ethylene Glycol An example of separation of radiopaque and ethylene glycol taken from [6] is studied. The membrane filtration system is represented by reverse osmosis plant with 0.28 m2 of membrane area which is employed together with diafiltration in order to increase the concentration of radiopaque (macro-solute) from 12 g/dL to 40 g/dL. The concentration of ethylene glycol (micro-solute) should be decreased from 0.5 g/dL to 0.01 g/dL in 4 L of processed solution. Experimentally obtained membrane characteristics are as follows q(c1 ) = −29.19 ln c1 + 118.1 R2 (c1 ) = 0.187 − 0.0073c1
(17) (18) 738
80
control. The middle step is presented by singular arc where various control trajectories can occur potentially. Economically optimal operation was derived and compared with traditionally used control approaches for two selected case studies. The effect of employing economically optimal control strategy appears to be more significant for membrane systems with imperfect micro-solute rejection. This provides a motivation for future research on more complex case studies, e.g. systems with imperfect membrane rejection of macro-solutes or ternary systems.
minimum time realistic minimum diluant economically optimal almost minimum time utopia point
75 70 65
Vw∗ [dL]
60 55 50 45 40 35
VI. ACKNOWLEDGMENTS
30 25 0
10
20
30
40
50
The authors acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grant 1/0053/13 and the Slovak Research and Development Agency under the project APVV-0551-11. ZK is grateful to the Hessen State Ministry of Higher Education, Research and Arts (LOEWE-Program), the MEMFIDA2 program (EUREKA HU 08-1-2010-0010), and the European Commission for the Marie Curie FP7 Integration Grant of the 7th European Union Framework Programme (PCIG11-GA-2012322219).
60
t∗f [h] Fig. 6. Optimal values of processing time and diluant consumption for different values of weight coefficients wT , wD in case of radiopaque and ethylene glycol separation. TABLE IV C OMPARISON OF CONTROL STRATEGIES OF BATCH DF
PROCESS FOR
RADIOPAQUE AND ETHYLENE GLYCOL SEPARATION .
control strategy
tf [h]
Vw [dL]
product cost [e/10 kg]
minimum time almost minimum time economically optimal realistic min. diluant minimum diluant traditional
4.18 4.99 15.04 55.33 ∞ 4.22
78.11 43.85 29.93 28.13 27.61 93.31
16,446 9,344 6,862 8,165 ∞ 19,615
R EFERENCES [1] M.Y. Jaffrin and J.Ph. Charrier. Optimization of ultrafiltration and diafiltration processes for albumin production. Journal of Membrane Science, 97:71–81, 1994. [2] M. Jelemensk´y, R. Paulen, M. Fikar, and Z. Kov´acs. Multi-objective optimal control of ultrafiltration/diafiltration processes. In European Control Conference, 2013. accepted. [3] C. D. Johnson and J. E. Gibson. Singular solutions in problems of optimal control. IEEE Trans. Automatic Control, 8(1):4–15, 1963. [4] U. Ledzewicz and H. Schattler. Optimal controlled trajectories for a mathematical model of anti-angiogenic therapy in cancer. In Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference., pages 1201–1206, 2009. [5] F. Lipnizki, J. Boelsmand, and R. F. Madsen. Concepts of industrialscale diafiltration systems. Desalination, 144(1-3):179 – 184, 2002. [6] H. Lutz. Membrane filtration with optimized addition of second liquid to maximize flux, January 1997. United States Patent 5597486, assignee: Millipore Investment Holdings Limited (Wilmington, DE, US). [7] P. Ng, J. Lundblad, and G. Mitra. Optimization of solute separation by diafiltration. Separation Science and Technology, 11(5):499–502, 1976. [8] R. Paulen, M. Fikar, G. Foley, Z. Kov´acs, and P. Czermak. Optimal feeding strategy of diafiltration buffer in batch membrane processes. Journal of Membrane Science, 411-412:160–172, 2012. [9] R. Paulen, M. Fikar, Z. Kov´acs, and P. Czermak. Process optimization of diafiltration with time-dependent water adding for albumin production. Chemical Engineering and Processing: Process Intensification, 50(8):815–821, August 2011. [10] R. Paulen, M. Jelemensk´y, M. Fikar, and Z. Kov´acs. Optimal balancing of temporal and buffer costs for ultrafiltration/diafiltration processes under limiting flux conditions. Journal of Membrane Science, 2013. In Press, DOI: 10.1016/j.memsci.2013.05.009. [11] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko. The Mathematical Theory of Optimal Processes. John Wiley & Sons, Inc., New York, 1962. [12] B. Srinivasan, S. Palanki, and D. Bonvin. Dynamic optimization of batch processes: I. Characterization of the nominal solution. Computers & Chemical Engineering, 27(1):1–26, 2003. [13] M. Yazdanshenas, A.R. Tabatabaeenezhad, R. Roostaazad, and A.B. Khoshfetrat. Full scale analysis of apple juice ultrafiltration and optimization of diafiltration. Separation and Purification Technology, 47(1–2):52–57, 2005.
and almost minimum time with the weights wT /wD = 10. The almost minimum time increases the processing time by only 20 % but decreases the water consumption almost by 80 %. Table IV indicates price of producing 10 kg of product by proposed control strategies. We also evaluate traditionally applied operation where three step strategy α = {0, 1, 0} is used and c∗1 is based on [7]. In this case, we can conclude that using of advanced control strategies can significantly reduce the production costs compared to the traditionally considered operations. V. C ONCLUSIONS In this paper, we studied the problem of economically optimal control of batch diafiltration processes where the economical criterion stands for the combination of minimum time and minimum diluant problems, traditionally considered for optimization of performance of membrane filtration systems. Cases with non-classical transmembrane flux models and imperfect micro-solute rejection were investigated. Necessary conditions for optimality were derived using Pontryagin’s minimum principle. These were then used to identify the economically optimal operation with respect to possible variations of prices of a unit of processing time and of consumed diluant. The optimal operation typically takes three steps, except for some cases which were discussed. The first and the last steps are represented by arcs with saturated 739
