Minimizing the process time for ultrafiltration/diafiltration under gel polarization conditions R. Paulena,∗, G. Foleyb , M. Fikara , Z. Kov´acsc , P. Czermakc,d a Faculty
of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinskeho 9, Bratislava, Slovakia b School of Biotechnology, Dublin City University, Dublin, Ireland c Institute of Biopharmaceutical Technology, University of Applied Sciences Giessen-Friedberg, Wiesenstrasse 14, 35390 Giessen, Germany d Department of Chemical Engineering, Kansas State University, Manhattan, Kansas, USA
Abstract This study examines a generalized ultrafiltration/diafiltration process that is designed to reduce the initial volume of a given process liqueur and to eliminate impurities from the product solution. This theoretical investigation focuses on applications where the permeate flux is given by the gel polarization model. The goal of this paper is to use optimal control theory to determine optimal time-varying diluant addition that minimizes treatment time. We propose a diafiltration model in a dimensionless form with normalized model equations in order to determine general features of optimal diluant utilization strategy. Based on the model, we formulate the optimal control problem and apply the theory of optimal control exploiting the Pontryagin’s minimum principle. We confirm the analytical results by numerical computations using numerical methods of dynamic optimization. We prove that optimal control strategy is to perform a constant-volume diafiltration step at optimal macro-solute concentration that guarantees maximal removal of micro-solute at any time instant. This constant-volume diafiltration step is preceded and followed by optional ultrafiltration or pure dilution steps that force the concentrations at first to arrive to the optimal macro-solute concentration and at last to arrive to the desired final concentrations. Finally, we provide practical optimization diagrams that allow decision makers to determine the optimal diluant control of a given separation task. Keywords: ultrafiltration, diafiltration, gel polarization, Pontryagin’s minimum principle, dynamic optimization
1. Introduction Batch ultrafiltration (UF) and diafiltration (DF) are common processes used in the production of many food, chemical, biotech, and pharma products. The combination of UF and DF is known as a fast and effective technique especially for concentrating and desalting or buffer exchange of protein solutions. Some of the important applications in this field include antigen purification [1], fractionation of whey protein isolate [2], albumin production from human blood plasma for medical use [3] and recovery of animal blood proteins from slaughterhouse effluents [4], separation of protease from tuna spleen extract [5], recovery of β-galactosidase from PEG-rich top-phase of fermentation broth extract [6], production of recombinant DNA derived human protein pharmaceuticals [7] and antibody preparation [8], purification of soybean lecithin [9] and leaf proteins [10], or concentration and desalination of gelatin [11]. An effective separation, first and foremost, requires a permselective membrane which retains the highvalue product but allows the free passage of others while maintaining a high flux rate. A schematic diagram ∗ Corresponding
author. Tel.: +421 259 325 730; fax: +421 259 325 340. Email address:
[email protected] (R. Paulen)
Preprint submitted to Journal of Membrane Science
June 30, 2011
of the generalized UF/DF process, that is designed to reduce the processing volume and to eliminate the impurities from the product solution, is shown in Fig. 1, where α is a dimensionless variable that is defined as the ratio of diluant inflow to permeate outflow. Many efforts in both academia and industry have focused on diluant
retentate
αQp
permeate Qp membrane module
feed tank
Figure 1: Schematic representation of a generalized UF/DF process.
different aspects of ultrafiltration and diafiltration design. Process optimization of tangential-flow filtration systems is aimed at achieving the highest yield and purity taking into account the constraints on membrane area and process time [12]. These goals can be accomplished by employing several methods of process control such as constant flux, constant pressure, or constant wall concentration control. Description of these control modes can be found in the open literature (e.g., in [7, 13]). The work presented here examines the constant pressure approach. Choosing the right diluant utilization strategy is a critical aspect to consider in UF/DF process control. The standard way of performing such operation is to use a combination of ultrafiltration with constantvolume diafiltration (UFCVD) [14]. Here the UF step (α = 0) is done separately from the DF step which is done at constant retentate volume (α = 1). Another diafiltration technique proposed recently by Jaffrin and Charrier [3] is the so called variable-volume diafiltration (VVD). VVD utilizes the diluant at a rate that is less than the permeate flow rate. Thus, this process enables a simultaneous concentration of macro-solute and removal of micro-solute in one single step. A modification of VVD is the ultrafiltration with variablevolume diafiltration (UFVVD), i.e., a two step process in which the solution is first concentrated to an intermediate macro-solute concentration and then subjected to VVD at constant α to reach the final desired concentrations of both solutes [15]. For gel polarization conditions, Foley [16] has shown that UFVVD will nearly always require less diluant than UFCVD. In the case of UFCVD, carried out under gel polarization conditions, CVD should be performed when the macro-solute concentration is equal to cg /e where cg is the gel concentration and e is the base of the natural logarithm [17]. In recent work [18] it has been shown that UFVVD can never be done more rapidly than UFCVD. Whether there is any generalized UF/DF process that can be performed more rapidly than UFCVD is the main question addressed in this paper. The diluant utilization strategy of these processes can be represented with the α versus time profiles as illustrated in Fig. 2. A number of recent optimization works have been specifically devoted to determine the optimal time-varying α profile instead of comparing the above mentioned stepwise diluant utilization strategies. A notable contribution has been made by Lutz [19, 20]. The invention, assigned to Millipore Corp. (U.S. Pat. No. 5597486), introduces a control strategy that is based on maximizing the mass flux of the permeating species by continuously controlling the rate of addition of diluant in order to minimize treatment time. In recent years, novel numerical methods of dynamic optimization have been applied to determine 2
1
α
UFCVD UFVVD VVD 0 treatment time Figure 2: Representation of UFCVD, UFVVD and VVD in terms of the α function.
time-optimal control of diafiltration. Fikar et al. [21] have employed orthogonal collocation approach to compute optimal α profile for a nanofiltration application separating organic/inorganic substances. Paulen et al. [22] have demonstrated the power of control vector parameterization method to determine optimal diluant control of an UF process to purify and concentrate human albumin. The goal of this paper is to use optimal control theory to find a function α(t) that minimizes process time for the classic case where the membrane performance is given by the gel polarization model. In Sect. 2, we introduce the studied diafiltration problem and critically review the major technological factors that might limit the practical applicability of further mathematical analysis. In Sect. 3, we propose a dynamic model. We introduce dimensionless variables in the diafiltration model in order to obtain normalized model equations. The employment of these dimensionless variables enables us to determine some general features of optimal diluant utilization strategies. Based on the model, we formulate the optimal control problem and solve it with different approaches: exploiting the Pontryagin’s Minimum Principle (see Sect. 4.1) and using numerical dynamic optimization techniques (see Sect. 4.2). Finally, we provide practical optimization diagrams that allow decision makers to determine the optimal diluant control of a given UF/DF separation task. 2. Problem statement We consider an UF/DF process where a macro-solute is to be increased in concentration from c0 to cf and a micro-solute reduced in concentration from cs0 to csf . We examine a membrane filtration unit with a fixed membrane area which operates at constant applied pressure, temperature, and under fixed hydrodynamic conditions. We assume that no macro-solute passes through the membrane, i.e., the rejection coefficient of the macro-solute is unity. The rejection coefficient of the micro-solute is assumed to be zero. The targeted reduction in volume can be expressed with the volume concentration factor (VCF) that is the ratio of initial volume V0 to final volume Vf . Note that the assumption of complete macro-solute retention poses a hidden relation between volume and concentration such that VCF =
cf V0 = . Vf c0
(1)
Throughout this paper, the permeate flowrate is assumed to be given by the gel polarization model. This model has originally been introduced by Michaelis [23] and based on stagnant film theory. It is supposed that the rejected solutes accumulate on the membrane surface and form a concentration polarization layer. At steady state, the quantity of solutes conveyed by the solvent to the membrane is equal to those that diffuse back. The mass balance for the macro-solute leads to the following equation for the permeate flow Qp of a fully retentive membrane: Qp = kA ln 3
cg c
(2)
where k is the mass transfer coefficient, A is the membrane area, and cg is the macro-solute concentration at the membrane wall. The gel polarization model provides a simple and convenient procedure for interpreting experimental UF data. Under conditions of constant mass transfer coefficient and wall concentration, the permeate flux versus logarithm of bulk concentration plot gives a straight line. According to Eq. (2), cg can be obtained from the intercept of the extrapolated experimental data and the abscissa axis, and the slope of this line gives the mass transfer coefficient k [24]. In Eq. (2), cg represents the highest available concentration at the membrane wall where the permeate flux is equal to zero. In most UF applications, the permeate flux becomes essentially independent of pressure at high pressures. Historically, the term gel concentration was attributed to gelling of macromolecules under filtration conditions. It should be noted that limiting flux occurs independently of any supposed gelation effects. In fact, fitting procedures often result in a physically unreasonable gelation value for cg [25]. It has been shown that cg is rather a phenomenological variable than a true physical property of the solution [26]. Despite of that, the term gel is commonly accepted and being used in the literature. The gel polarization model has been widely used in the analysis of experimental macromolecule UF data [6, 7, 10, 12, 13, 17, 20, 25, 26, 27, 28]. The model fits experimental data very well in many cases, and thus, appropriate for practical purposes. The findings of this study are generally valid for membrane applications where the filtrate flux varies inversely as the logarithm of the bulk concentration of macro-solute. There is a number of other technological factors that might contribute to the UF/DF system design and should be mentioned in this section. It must be kept in mind that any biomolecule has a specific sensitivity to the various operating conditions [29], and the time spent exposed to those conditions should be kept to a minimum. Reduced process duration may prevent microbial contamination and reduce yield losses that are attributed to treatment time such as membrane adsorption and product denaturation [29]. However, operating temperature and pressure, or hydrodynamic conditions may also play an important role in process development. Such aspects are out of the scope of the mathematical analysis discussed in Sect. 3. The most important restrictions on the here examined system are as follows: • We assume that the rate of fouling is negligible under the given operational conditions. • The operational and economic sustainability of the examined membrane process is ensured by low adsorption and solubility losses [7, 30]. The process operates under conditions where the product concentration at the membrane wall does not exceed its solubility limit. • A change in feed composition during DF (i.e., alteration of pH and ionic strength), that might be a consequence of removal of salts or stabilizing molecules, does not result in protein denaturation and subsequent product losses [28]. • The appropriate reservoir size is available. Particularly, as shown later in the text, e · c0 /cg does not exceed the lower volumetric limit of the system. • No significant micro-solute binding to product occurs (e.g., in [31]). • The impact of membrane area(length)-dependent changes and membrane lot-to-lot variations on the separation performance are not considered here (e.g., in [32]). The findings of this work have to be interpreted in the light of the above listed restrictions. 3. Model Development The macro-solute balance can be written d(Vc) = 0, c(0) = c0 (3) dt where V is the retentate volume at time t, and c is the macro-solute concentration in the feed tank. The micro-solute balance is given by 4
d(Vcs ) = −Qp cs , cs (0) = cs0 (4) dt where cs is the micro-solute concentration in the feed reservoir. Finally, the volume balance can be written dV = (α − 1)Qp , V(0) = V0 . (5) dt Combining and rearranging equations (1)–(5) gives the model where α(t) is a function of time as follows cg dV = (α − 1)kA ln , dt c cg αcs dcs =− kA ln , dt V c c0 V 0 c= , V
(6a)
V(0) = V0 cs (0) = cs0 ,
cs (tf ) = csf
(6b)
c(0) = c0 ,
c(tf ) = cf .
(6c)
For computational purposes, it is more convenient to define the following dimensionless variables V ∗ (t) =
V(t) , V0
c∗s (t) =
cs (t) , csf
t∗ = t
kA , V0
x(t) =
cg . c(t)
(7)
The model then becomes dV ∗ = (α − 1) ln x, dt∗ ∗ αc∗s dcs = − ln x, dt∗ V∗ x = x0 V ∗ ,
V ∗ (0) = 1 c∗s (0) =
(8a)
cs0 , csf
x(0) = x0 =
c∗s (tf∗ ) = 1 cg , c0
x(tf∗ ) = xf =
(8b) cg . cf
(8c)
3.1. Optimization Problem Considering the process model derived in the previous section we can formulate the objective which is to minimize process operation time with respect to time-varying function α(t). The formulation is then as follows min tf∗ ∗
(9a)
α(t )
s.t. dV ∗ = (α − 1) ln(x0 V ∗ ), dt∗ dc∗s αc∗s = − ln(x0 V ∗ ), dt∗ V∗ α ≥ 0.
V ∗ (0) = 1, c∗s (0) =
cs0 , csf
V ∗ (tf∗ ) =
xf x0
c∗s (tf∗ ) = 1
(9b) (9c) (9d)
4. Results and Discussion We focus on two ways of solving the problem (9), theoretical and computational. The optimal control theory can be employed to find optimal process operation both from the theoretical point of view (e.g., in [33]) and numerically as well (e.g., in [34]). 5
4.1. Theoretical Results To solve the optimal control problem, we will make use of Pontryagin’s minimum principle, which is in our case as follows [35, 36]. Let us consider a dynamical system with states z and control u such that ˙z = f (z, u),
z(0) = z0 , z(tf ) = zf ,
u(t) ∈ [umin , umax ], t ∈ [0, tf]
(10)
and the cost function (11)
J = tf .
In our case z takes form [V ∗ (t), c∗s (t)] and u is represented by α(t). Further, let us define the Hamiltonian H and the vector of adjoint variables λ(t) such that H(z, u, λ) = 1 + λT f (z, u).
(12)
Necessary conditions of optimality as derived in Pontryagin’s principle of minimum then are defined as u = arg min H(z, u, λ), u
∂H , z(0) = z0 , ∂λ ∂H λ˙ = − ∂z H = 0, ∀t ∈ [0, tf] ˙z =
u ∈ [umin , umax ] z(tf ) = zf
(13) (14) (15) (16)
The Hamiltonian function for the studied problem is of the form c∗ H = 1 + λ1 (α − 1) ln(x0 V ∗ ) − λ2 α s∗ ln(x0 V ∗ ) V ∗ ! c = α ln(x0 V ∗ ) λ1 − λ2 s∗ + 1 − λ1 ln(x0 V ∗ ) V
(17) (18)
where the adjoint variables are defined by the following differential equations αc∗ α−1 ∂H λ˙ 1 = − ∗ = −λ1 ∗ + λ2 ∗ s 2 1 − ln(x0 V ∗ ) ∂V V (V ) α ∂H ∗ λ˙ 2 = − ∗ = λ2 ∗ ln(x0 V ). ∂cs V
(19) (20)
The Hamiltonian is linear in α. Thus, its minimum will be attained with α on its boundaries (bang-bang control) as follows 0 if ∂H ∂α > 0, α= (21) ∞ if ∂H < 0 ∂α where
! c∗s ∂H ∗ = ln(x0 V ) λ1 − λ2 ∗ . ∂α V
(22)
If this term is equal to zero, the Hamiltonian is singular and does not depend on α. Control trajectory in such singular arc can be obtained by its differentiation with respect to time and requiring it to be zero as well 6
This condition can be satisfied if
d ∂H = λ1 1 − ln(x0 V ∗ ) = 0. dt ∂α V∗ =
cg e ⇔c= . x0 e
(23)
(24)
As the result still does not depend on α, further differentiation is needed λ1 d2 ∂H = ∗ 1 − 2α ln(x0 V ∗ ) + α ln2 (x0 V ∗ ) = 0. 2 dt ∂α V This condition can be satisfied if 1 − 2α ln(x0 V ∗ ) + α ln2 (x0 V ∗ ) = 0 ⇔ α = 1.
(25)
(26)
We can easily check that these two conditions satisfy all necessary conditions of optimality (13)–(16). Therefore, to conclude the theoretical analysis, optimal control trajectory will either be at its boundaries α = {0, ∞} or it will be equal to one if the macro-solute concentration is given by c = cg /e as originally derived in [17]. This finding is in agreement with the optimization procedure suggested in [19]. The optimal process will then consist of consecutive operational steps of three basic operational modes in a certain order. These operational modes can be technically characterized as concentration mode (α = 0), CVD mode (α = 1), and pure dilution (α = ∞). This latter case, α = ∞, corresponds to instantaneous addition of diluant. As mentioned earlier, α is the ratio of diluant flow to permeate flow. In practice, the diluant is added with an external pump. The true value of the upper boundary of α can then be estimated by taking into account the capacity of the diluant pump and its relation to the permeate flow. We note that many authors implicitly assume that α is bounded above by α ≤ 1. This implies that process volume is a nonincreasing function of time and it is motivated by issues with gel polarization model if the process volume is too small. In that case, optimal control can be either zero or one. 4.2. Numerical Results In this section we solve the same optimization problem (9) numerically by applying the control vector parameterization approach [37, 38, 39, 40, 41]. It translates the original infinite dimensional problem into finite dimensional problem of non-linear programming (NLP). This NLP problem can be then handled by any of variety of available NLP solvers (e.g., MATLAB NLP solver fmincon). We have investigated numerically the effect of three parameters given by following values: x0 = (20, 30, 40), cs0 /csf = (5, 10, 20), and xf =(1.5, 2, e, 4, 5, 6, 7, 8, 9, 10). We have performed dynamic optimization with all possible combinations of these parameters. These computations showed that the value of minimum final time increases if: • x0 decreases (c0 increases). This is obvious since any increase in value of x0 means increase of flux during the operation and vice versa. • cs0 /csf increases. This means that we need to remove more micro-solute from initial solution to reach the same final goal. • xf decreases (cf increases). Solution volume has to be reduced more. Fig. 3 shows actual optimal α(t) trajectories for x0 = 20, cs0 /csf = 5, and with varying value of xf . Optimization results show that the type of optimal trajectory depends only on xf and is the same for any combination of investigated values of x0 and cs0 /csf . Subplots (d), (e) and (f) illustrate a three-step process including a concentration, a CVD, and a dilution step. In these cases, the solution is over-concentrated in the first concentration step. The arrows in the figures represent instantaneous dilution of the feed that ensures the desired final volume. 7
0.8
0.8
0.8
0.6
0.6
0.6 α
1
α
1
α
1
0.4
0.4
0.4
0.2
0.2
0.2
0 0
0.2
0.4
0.6
0.8
0 0
1
0.2
0.4
t∗/t∗f
0.8
0 0
1
(b) xf = 2
0.8
0.8
0.6
0.6
0.6
α
0.8 α
1
0.4
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8
1
0 0
0.2
0.4
t∗/t∗f
(d) xf = 4
0.6
0.8
1
0.8
1
(c) xf = e
1
0.2
0.4 t∗/t∗f
1
0 0
0.2
t∗/t∗f
(a) xf = 1.5
α
0.6
0.6 t∗/t∗f
0.8
1
0 0
0.2
0.4
0.6 t∗/t∗f
(e) xf = 7
(f) xf = 10
Figure 3: Optimal trajectories for α(t) for x0 = 20, cs0 /csf = 5, and chosen values of xf .
The optimal operation can be represented by a state diagram in Fig. 4. Here, S I denotes a general starting point S I = [c∗s0 , V0∗ ] = [cs0 /csf , 1] satisfying inequality x0 > e. There are three possible final points represented by X I , X II , and X III . These points lie on the final constraint c∗s = 1 and differ in final condition on Vf∗ = xf /x0 which decides about consecutive steps of the optimal strategy X I – (xf < e) represents three step strategy α = {0, 1, 0}, i.e., concentration (ultrafiltration) step is performed first, then it is followed by CVD step, and finishes with another UF step. In the first part, volume decreases until equal to e/x0 . The second part with constant volume (α = 1) continues until c∗s = 1. The third step reduces volume until its final condition is satisfied. Note that xf < e corresponds to cf > cg /e, the classic scenario considered by Ng et al. [17]. X II – (xf = e) represents two step strategy with α = {0, 1} performing a complete UF step first and then a CVD step. In the first part, volume decreases until equal to e/x0 . The second part with constant volume (α = 1) continues until c∗s = 1 where both final conditions are satisfied. The final macro-solute concentration in this case is given as cf = cg /e. III X – (xf > e) represents three step strategy α = {0, 1, ∞}. The simulations show that in the first part, dimensionless volume decreases until equal to e/x0 . Here, V ∗ = e/x0 corresponds to c = cg /e. The second part with constant volume (α = 1) continues until the concentration ratio of macro-solute to micro-solute reaches the final desired ratio. The CVD operation is terminated when the dimensionless micro-solute concentration is equal to xf /e. In the third step, the volume is increased by adding the correct amount of diluant instantaneously. In [c∗s , V ∗ ]–plane, this represents a move along a hyperbolic curve that is given as V∗ =
p1 c∗s
(27)
where p1 is a dimensionless process parameter defined as xf /x0 . The instantaneous addition of diluant is an abstraction and in practice (especially on large scales), this step would take a finite time. In a more realistic case where the upper bound on α is given as, for 8
instance, 10 or 100, the hyperbolic curve representing the third step is slightly modified in the [c∗s , V ∗ ]– plane according to the true value of α. However, the optimal control strategy remains unchanged as far as the type and order of applied operational steps is concerned. If we assume that α(t) ≤ 1, then the optimal strategy would be α = {0, 1}. In this case, however, the constant volume process will not operate at the optimal value e/x0 but at higher value xf /x0 and the final time will be larger.
V∗ SI
1
α=0
X III α=∞ e x0
X
II
α=1 α=0
X
V∗ =
S II
p1 c∗s
I
1
cs0 /csf
c∗s
Figure 4: State diagram of process with general starting point S I (x0 > e) and three chosen endpoints.
The second case starts at the point S II , lies at line x(t) = e and finishes at X II . This represents a single CVD operation as originally described in [17]. Of course, final points can also be X I or X III which would mean that CVD step will be followed by UF α = 0 or pure dilution α = ∞. Let us now discuss the optimal operation if we start from a point S III satisfying inequality x0 < e. This scenario can be represented by a state diagram as illustrated in Fig. 5. Here, the first part of the optimal operation path is achieved with a α = ∞ dilution step. Such predilution might be advantageous in many practical processes (e.g. in [3, 42]) where high feed concentration would result in a very low permeate flux and thus, a slow process. As shown in Fig. 5, optimal predilution is terminated when the dimensionless volume V ∗ reaches the value of e/x0 . Remaining optimal steps are the same as stated in the case x0 > e. This optimal operation was confirmed numerically with values x0 = (1.5, 2). Two different hyperbolic curves can occur for a move in state plane if α = ∞. This is due to the fact that dilution step is carried out at different state of the process which yields different parameters pi such that p1 =
xf , x0
p2 =
cs0 . csf
(28)
4.3. Optimization protocol The state diagrams, Fig. 4 and 5, provide a practical procedure to evaluate optimal diluant-utilization strategy. The number and type of consecutive operational steps can be simply determined from the relative position of V ∗ (0), e/x0 and V ∗ (t f ). The dynamics of the entire process can be then computed using Eq. (8). To summarize the results in original state coordinates, the optimal minimum-time operation under gel polarization conditions can be stated as follows: 1. The first (optional) step is either pure dilution (α = ∞) or pure ultrafiltration (α = 0) until optimal macro-solute concentration c = cg /e is obtained. 9
V∗ X
V∗ =
III
p2 c∗s
α=∞ e x0
X II
α=1
α=0
V∗ =
X
I
p1 c∗s
α=∞ S III
1 1
cs0 /csf
c∗s
Figure 5: State diagram of process with general starting point S III (x0 < e) and three chosen endpoints.
2. The second step is CVD (α = 1) maintaining the optimal macro-solute concentration. This step finishes if either final concentration of micro-solute or final ratio of macro-solute to micro-solute concentration is obtained. 3. Finally, the third (optional) step is again either pure dilution (α = ∞) or pure ultrafiltration (α = 0) until final concentration of both components are obtained. 5. Conclusions In this study we formulated the problem of optimal process operation in filtration if the flux is modeled by a gel polarization model. We have derived a normalized model differential equations, applied theory of optimal control, and derived necessary conditions of optimality. These analytical results were then confirmed by numerical computations, using numerical methods of dynamic optimization. We have investigated three parameters and showed their effect on minimum time control strategy. Any change in ratio cs0 /csf results in changing minimum process time, however, it has no impact on the optimal path. In contrary with that, values of parameters xf and x0 are decisive about the type of the optimal control strategy. We have proved that optimal control strategy is to operate on the optimal macro-solute concentration that guarantees maximal removal of the micro-solute at any time instant and thus maximum reduction of operational time. The CVD step at optimal macro-solute concentration is preceded and followed by optional ultrafiltration or pure dilution steps that force the concentrations at first to arrive to the optimal macro-solute concentration and at last to arrive to desired final concentrations. Although the outcomes of this study are restricted to gel polarization conditions-type problems, the optimization techniques that we applied are general. Therefore, we will focus in our future work on more complex cases dealing with arbitrary flux models, variable retention coefficients, and different types of cost functions. List of Symbols A c cg
membrane area (m2 ) macro-solute concentration in feed tank (mol m−3 ) limiting macro-solute concentration at membrane wall (mol m−3 ) 10
cs micro-solute concentration in feed tank (mol m−3 ) ∗ cs dimensionless micro-solute concentration as defined in Eq. (7) H Hamiltonian function J objective function k mass transfer coefficient (m s−1 ) p dimensionless process parameter as defined in Eq. (28) Qp permeate flow (m3 s−1 ) S starting coordinates representing process liqueur properties t operation time (s) t∗ dimensionless time as defined in Eq. (7) u vector of control variables V volume in feed tank (m3 ) ∗ V dimensionless volume as defined in Eq. (7) x dimensionless macro-solute concentration as defined in Eq. (7) X target coordinates representing desired product properties z vector of state variables Greek symbols α proportionality factor of diluant flow to permeate flow λ vector of adjoint variables Abbreviations VCF volume concentration factor UF ultrafiltration DF diafiltration CVD constant-volume diafiltration VVD variable-volume diafiltration UFVVD ultrafiltration combined with variable-volume diafiltration UFCVD ultrafiltration combined with constant-volume diafiltration NLP nonlinear programming Acknowledgments This research is a cooperative effort. The first and the third author acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grants 1/0071/09, 1/0537/10, 1/0095/11, and the Slovak Research and Development Agency under the project APVV-0029-07. The fourth author would like to thank the Hessen State Ministry of Higher Education, Research and the Arts for the financial support within the Hessen initiative for scientific and economic excellence (LOEWE-Program). [1] P. Schu, G. Mitra, Ultrafiltration membranes in the vaccine industry, in: W. K. Wang (Ed.), Membrane Separations in Biotechnology, 2nd Edition, Membrane Separations in Biotechnology, Marcel Dekker, Inc., New York, 2001, pp. 225–243. [2] B. Cheang, A. L. Zydney, A two-stage ultrafiltration process for fractionation of whey protein isolate, Journal of Membrane Science 231 (1-2) (2004) 159–167. [3] M. Jaffrin, J. Charrier, Optimization of ultrafiltration and diafiltration processes for albumin production, Journal of Membrane Science 97 (1994) 71–81. [4] D. Belhocine, H. Grib, D. Abdessmed, Y. Comeau, N. Mameri, Optimization of plasma proteins concentration by ultrafiltration, Journal of Membrane Science 142 (2) (1998) 159–171. [5] Z. Li, W. Youravong, A. H-Kittikun, Separation of proteases from yellowfin tuna spleen by ultrafiltration, Bioresource Technology 97 (18) (2006) 2364–2370. [6] A. Veide, T. Lindb¨ ack, S.-O. Enfors, Recovery of β-galactosidase from a poly (ethylene glycol) solution by diafiltration, Enzyme and Microbial Technology 11 (11) (1989) 744–751. [7] R. van Reis, E. M. Goodrich, C. L. Yson, L. N. Frautschy, R. Whiteley, A. L. Zydney, Constant Cwall ultrafiltration process control, Journal of Membrane Science 130 (1-2) (1997) 123–140. [8] R. Luo, R. Waghmare, M. Krishnan, C. Adams, E. Poon, D. Kahn, Process optimization for the ultrafiltration/diafiltration of abthrax antibody to very high final concentrations, Tech. Rep. PS1240EN00, Millipore Technical Publications, Millipore Corporation, Billerica, MA 01821 USA (2004). [9] R. C. Basso, L. A. G. Gon¸calves, R. Grimaldi, L. A. Viotto, Degumming and production of soy lecithin, and the cleaning of a ceramic membrane used in the ultrafiltration and diafiltration of crude soybean oil, Journal of Membrane Science 330 (1-2) (2009) 127–134.
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