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Journal of Chromatography A, 1217 (2010) 5337–5348

Contents lists available at ScienceDirect

Journal of Chromatography A

journa l homepage: www.e lsev ier .com/ locate /chroma

ptimization of simulated moving bed chromatography with fractionation andeedback: Part I. Fractionation of one outlet

uzhou Lia, Yoshiaki Kawajirib, Jörg Raischa,c, Andreas Seidel-Morgensterna,d,∗

Max-Planck-Institut für Dynamik komplexer technischer Systeme, Sandtorstraße 1, D-39106 Magdeburg, GermanySchool of Chemical & Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, GA 30332, USAFachgebiet Regelungssysteme, Technische Universität Berlin, Einsteinufer 17, D-10587 Berlin, GermanyLehrstuhl für Chemische Verfahrenstechnik, Otto-von-Guericke Universität, Universitätsplatz 2, D-39106 Magdeburg, Germany

r t i c l e i n f o

rticle history:eceived 17 December 2009eceived in revised form 4 June 2010ccepted 11 June 2010vailable online 19 June 2010

a b s t r a c t

A novel modification of simulated moving bed (SMB) technology, referred to as fractionation and feedbackSMB (FF-SMB), has been introduced recently. This concept is based on fractionating one or both outletstreams and feeding the off-spec fractions back into the unit alternatingly with the original feed mixture.In this paper, the optimization problem of FF-SMB realizing one outlet fractionation is considered. Amathematical optimization framework based on a detailed process model is presented which allows to

eywords:imulated moving bed chromatographyMBractionationeedback

evaluate quantitatively the potential of this operating scheme. Detailed optimization studies have beencarried out for a difficult separation characterized by small selectivity and low column efficiency. Theresults reveal that the proposed fractionation and feedback regime can be significantly superior to theclassical SMB chromatography, in terms of both feed throughput and desorbent consumption. The effectof the feeding sequence on the performance of FF-SMB is also examined.

F-SMBptimization

. Introduction

Simulated moving bed (SMB) chromatography, is a continu-us and effective multi-column chromatographic process [1], andas obtained considerable attention for the separation of multi-omponent mixtures. Since it circumvents technical restrictionsaused by the true moving bed (TMB) concept and limitations dueo the discontinuous batch operation, SMB has been widely usedver the last several decades in the petrochemical, fine chemicalnd sugar industries. Recently, SMB chromatography has been alsopplied increasingly in the pharmaceutical industry [2]. In partic-lar, SMB has become an important tool for enantioseparationssing chiral stationary phases [3]. Currently, many modificationsave been suggested exploiting various additional degrees of free-om offered by the classical process implementation. Overviews ofhe latest developments and future challenges in SMB technology

an be found in [4,5]. Some advanced SMB variants, for example,ariCol [7], PowerFeed [8] and ModiCon [6], have in common thatpecific operating parameters are modulated periodically duringperation. However, in these operating modes, the classical feature

∗ Corresponding author at: Max-Planck-Institut für Dynamik komplexer technis-her Systeme, Sandtorstraße 1, D-39106 Magdeburg, Germany.el.: +49 391 67 18644; fax: +49 391 67 12028.

E-mail address: anseidel@vst.uni-magdeburg.de (A. Seidel-Morgenstern).

021-9673/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.chroma.2010.06.031

© 2010 Elsevier B.V. All rights reserved.

of permanent withdrawal of products and constant feeding of unre-solved materials is still maintained. During the last years, attemptshave been made to exploit more sophisticated on–off character-istics at either the inlet or the outlet. For example, a simplifiedversion of PowerFeed, called “Partial Feed” [9], realizes a discon-tinuous pulse feed flow and can improve the separation efficiencyin terms of purity and recovery. The “Partial Discard” strategy [10]takes into account a non-permanent withdrawal of one or bothoutlet streams, and the remaining off-spec fractions are either dis-carded or remixed with the feed mixture. This scheme can enhancethe product purity, but discarding the off-spec fractions leads tolosses in recovery yields. Further, remixing of partially resolvedfractions with fresh feed is not attractive.

More recently, a novel SMB concept (FF-SMB) combining frac-tionation of one or both outlets and internal feedback has beenpresented [11]. In this newly developed process, the off-spec frac-tion is collected in a separate buffer vessel and fed back intothe unit alternatingly with the original fresh feed. In [11], somepreliminary results based on computationally expensive simula-tion studies identified advantages over both the conventional SMBand the “Partial Discard” strategy. However, the simulation-based

approach adopted in [11] cannot guarantee to determine opti-mum operating conditions. In order to exploit the full potentialof FF-SMB and assess its relative superiority over other operat-ing alternatives, efficient model-based optimization strategies areneeded.

5 gr. A 1217 (2010) 5337–5348

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338 S. Li et al. / J. Chromato

In this contribution, an optimization framework is developedor the FF-SMB process fractionating one outlet stream. This frame-ork is coupled with a detailed dynamic model of FF-SMB and able

o take advantage of the flexibility of the fractionation and feedbackegime. Using a difficult separation problem characterized by smallelectivity and low column efficiency as reference, systematic opti-ization studies are performed for FF-SMB and the conventional

oncept. The optimum performances achievable with both operat-ng schemes are quantitatively compared in terms of maximizationf feed throughput and minimization of desorbent consumption.he influence of the feeding sequence on the potential of FF-SMBs also evaluated. The paper is structured as follows. In the nextection, the principle of fractionation and feedback will be intro-uced. A mathematical model governing the dynamics of FF-SMBill be described in Section 3. In Section 4, the optimization prob-

em considering the additional degrees of freedom and constraintss formulated and the solution strategy is explained. The optimiza-ion results and performance comparison are detailed in Section 5ollowed by concluding remarks and future work.

. Principle of FF-SMB

.1. Basics of SMB

A typical SMB process, consisting of multiple identical chro-atographic columns connected in series, is illustrated in Fig. 1.ccording to the position of the columns relative to the feed and

he withdrawal ports, the unit can be divided into four zones withifferent functions allowing the separation of a binary mixture ofand B (see Fig. 1). The mixture is introduced between the central

eparation zones II and III. Due to different adsorption affinities,he less adsorbed species A is transported upstream to the raffi-

ate outlet with the liquid phase, and the more retained species B

s conveyed in the opposite direction to the extract outlet with theolid phase. In zone I, component B is desorbed to regenerate thedsorbent, while in zone IV, component A is adsorbed in order toecycle the desorbent. The counter-current movement between the

ig. 2. Illustration of outlet concentration profiles and integral purity profiles of convoncentration profiles for the less retained (solid line: A) and the more retained compononcentration profiles for the more retained (dotted line: B) and the less retained compo

Fig. 1. Schematic representation of a conventional SMB unit (4 zones and 4columns).

liquid and the solid phase is “simulated” by synchronously shiftingthe inlet and outlet ports one column ahead in the direction of thefluid flow after a certain cycle time tS .

A cyclic steady state, characterized by identical transient con-centrations inside the columns and at both outlets during eachcycle, is reached after a start-up period. Dimensionless outletconcentration profiles at the cyclic steady state are illustratedexemplarily in Fig. 2a and c for the raffinate and the extract ports,respectively. The product purities can be calculated by integrating

the concentration profiles over time (see Fig. 2b and d). In Fig. 2,only one cycle is considered and the dimensionless outlet concen-trations are obtained by relating the outlet concentration of eachcomponent with respect to its feed concentration. A dimensionless

entional SMB chromatography at cyclic steady state. (a) Dimensionless raffinateent (dotted line: B). (b) Raffinate integral purity profile. (c) Dimensionless extract

nent (solid line: A). (d) Extract integral purity profile.

S. Li et al. / J. Chromatogr. A 1217 (2010) 5337–5348 5339

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ocs

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ig. 3. Illustration of principle of outlet fractionation and resulting integral purity prurity profiles (thin solid line: without outlet fractionation; thick solid line: outlet

ntegral purity profiles (thin dotted line: without outlet fractionation; thick dotted5%.

ime � is defined as:

= t

tS, � ∈ [0, 1] (1)

Note that, in this classical concept, (a) both raffinate and extractutlet streams are continuously withdrawn over one completeycle (i.e., from � = 0 to � = 1), and (b) there is a permanent feedupply.

.2. Outlet fractionation

The outlet concentration profiles illustrated in Fig. 2a and c are ofime-dependent nature. In case of successful operation, the concen-ration of component A eluting from the raffinate outlet increasesypically over the course of one cycle time. At the same raffinateutlet, component B also shows a similar increasing elution profileut appears later (see Fig. 2a). At the extract outlet, both compo-ents elute typically with higher concentrations at the beginningf each cycle. As time increases, the concentration of A is character-zed by a faster drop than that of B (see Fig. 2c). Thus, the elution ofhe “unwanted” components from two outlets is most pronouncedt the end (raffinate outlet, component B) or the beginning (extractutlet, component A) of one cycle. If only certain fractions of theutlet streams are collected, the so-called “outlet fractionation”cheme results. The basic principle is demonstrated in Fig. 3, wherehe example shown above is used for illustrative purposes. A hypo-hetical purity threshold of 95% for both products is assumed (seeig. 3b and d, dashed lines). In the case without fractionation, col-ecting the outlet streams over one complete cycle leads to anntegral purity of 80% for the raffinate and of 90% for the extractsee Fig. 3b and d, thin solid and dotted lines), i.e., both below the

equired purity. In contrast, adopting the fractionation concept andithdrawing from � = 0 to � = 0.70 at the raffinate outlet and from= 0.30 to � = 1 at the extract outlet, allows to reach the desiredroduct purity of 95%. The two dimensionless times (i.e., 0.70 and.30) introduce new degrees of freedom defining “production peri-

. (a) Raffinate outlet fractionation (solid line: A; dotted line: B). (b) Raffinate integralionation). (c) Extract outlet fractionation (solid line: A; dotted line: B). (d) Extractoutlet fractionation). Dashed lines in (b) and (d): hypothetical purity threshold of

ods”, �Production, for each outlet. We will discuss them in more detailin Sections 3 and 4.

2.3. Feedback regime

Fractionation of one or both outlet streams can improve productpurity. However, it produces off-spec fractions in each cycle, whichcontain considerable amounts of components of high concentra-tion. Effective treatment of valuable off-spec fractions is of criticalimportance for process economics. The “Partial Discard” strategyadvocated in [10], tolerates reduced recovery yields. This appearsnot to be acceptable for the purification of highly valuable products.Alternatively, if the amount of the recycled fractions is small, [10]also suggested to remix these with the feed mixture. However, as[11] pointed out, the recycled fractions are already partially sep-arated and thus remixing with fresh feed destroys the achievedresolution.

In FF-SMB, a separate buffer vessel is introduced containing off-spec fractions, and the two mixtures in the buffer vessel and in thefeed tank are fed into the unit within each cycle in an alternatingmanner (see Fig. 4). Further discussion of the additional degrees offreedom associated with the distinct feeding scheme (e.g., feedbackperiods and feeding sequences) will be given in Sections 3 and 4.

The new concept utilizing the fractionation and feedback canbe easily implemented to the conventional SMB and other modifi-cations, by just adding a buffer tank and two standard three-wayvalves installed at the feed inlet and one of the outlet lines. Thus, theadditional hardware investment is very small. A schematic illus-tration of an FF-SMB unit with raffinate fractionation is given inFig. 5.

3. Process modeling

The basic principle outlined in the previous section reveals thatthe overall process model governing the dynamics of FF-SMB con-sists of two parts: an SMB model taking into account the continuous

5340 S. Li et al. / J. Chromatogr. A 1

Fig. 4. Non-constant feeding regime of FF-SMB (solid line: dimensionless feedconcentration of component A; dotted line: dimensionless feed concentration ofcomponent B). cFeed

iand cBuffer

idenote the dimensionless concentration of compo-

nrip

csatscadFDcwto

3

t

Fa

ent i in the feed tank and the buffer vessel, respectively. Both are normalized withespect to the concentration of component i in the feed tank CFeed

i, i = A, B. A feed-

ng sequence of buffer vessel followed by original feed tank is used for illustrativeurposes.

hromatographic separation in each column and the cyclic portwitching, and the buffer vessel dynamics. The two sub-processesre closely coupled through the fractionation and feedback opera-ion, leading to a rather complex hybrid dynamic system. In thisection, after a brief review regarding modeling classical SMBhromatography, the new degrees of freedom due to the fraction-tion and feedback strategy will be introduced. Then a simplifiedynamic model for the additional buffer tank will be described.inally, it will be demonstrated that classical SMB and the “Partialiscard” strategy are just special cases of the more general pro-ess investigated here. It should be noted that in this paper, weill consider only the fractionation of one outlet. An extension of

he modeling framework to the simultaneous fractionation of bothutlets is discussed in a joint paper.

.1. Modeling SMB

The conventional SMB process model can be assembled fromhe node balances describing the column interconnections and the

ig. 5. Illustration of an FF-SMB process with raffinate outlet fractionation. Twodditional valves and a buffer vessel are required.

217 (2010) 5337–5348

dynamic model of a single chromatographic column. Based on themass balances at the characteristic nodes, the inlet flow-rates andconcentrations of the four zones can be obtained:

Desorbent node:

QIV + QD = QI (2)

Couti,IV QIV = Cin

i,IQI (3)

Extract node:

QI − QE = QII (4)

Couti,I = Cin

i,II = CEi (5)

Feed node:

QII + QF = QIII (6)

Couti,II QII + CF

i QF = Cini,IIIQIII (7)

Raffinate node:

QIII − QR = QIV (8)

Couti,III = Cin

i,IV = CRi (9)

where Qj(j = I, II, III, IV) are the four internal flow-rates, QD the des-orbent flow-rate, QE the extract flow-rate, QF the feed flow-rate, QR

the raffinate flow-rate, Couti,j

and Cini,j

the concentrations of compo-

nent i in the liquid phase leaving or entering zone j, CEi

and CRi

theconcentrations of component i at the extract and the raffinate outletand CF

ithe feed concentration of component i, i = A, B.

The well established equilibrium dispersive model [13] wasused to calculate the concentration profiles in each column:

∂Ci

∂t+ 1 − �

�∂qi

∂t+ u

∂Ci

∂z= Dap,i

∂2Ci

∂z2, i = A, B (10)

where Ci and qi are the concentrations of component i in the liq-uid and the solid phase, respectively, u the interstitial velocity ofthe liquid phase in the column, z the spatial coordinate, t the time,and � the total column porosity. In this model, a local equilibriumbetween the solid and the liquid phase is assumed. An apparent dis-persion coefficient Dap,i, lumping axial dispersion and mass transferresistances, is used to consider all non-ideal effects which con-tribute to band broadening. In this work, for the sake of simplicity,the same coefficient calculated by Eq. (11) was used for both com-ponents [13]:

Dap,i = uL

2N(11)

where L is the column length and N the number of theoretical platesper column.

The following initial and boundary conditions were used:

Ci(t, z)|t=0 = 0 (12)

Dap,i∂Ci

∂z

∣∣∣∣z=0

− u(Ci|z=0 − Cini ) = 0,

Dap,i∂Ci

∂z

∣∣∣∣z=L

= 0 (13)

Additionally, the competitive Langmuir isotherm with Henry

constants Hi and thermodynamic coefficients Ki was used todescribe the adsorption equilibrium for the two components:

qi(CA, CB) = HiCi

1 + KACA + KBCB, i = A, B (14)

gr. A 1217 (2010) 5337–5348 5341

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Fig. 6. Illustration of development of axial concentration profiles and selection ofoptimal feeding sequence for FF-SMB. The raffinate outlet is used for fractiona-tion. (a) For � ∈ [0, �Feedback], the buffer vessel is supplied as inlet source. (b) For

S. Li et al. / J. Chromato

.2. Fractionation and feedback

As pointed out above, the novel concept offers more degrees ofreedom than the conventional SMB process. In this section, theefinition of these additional degrees of freedom and the resultingalance equations will be given.

.2.1. Production and recycle periodsThe outlet fractionation strategy described in Section 2.2 divides

ach cycle into two parts, defined by a production period and aecycle period (see Fig. 3a and c). Within the production period, thetream from the outlet used for fractionation satisfies the desiredroduct purity and is collected in a product tank. During the recycleeriod, the product collection is interrupted and the outlet stream

s recycled into the buffer vessel. Such a discontinuous collectionegime can be implemented by switching a three-way valve peri-dically. The length of the two periods can be denoted by �Production

nd �Recycle. Obviously, the following relation holds:

Production + �Recycle = 1 (15)

here �Production ∈ [0, 1]. Accordingly, the total volume leaving theractionated outlet can also be split into two portions:

Production = tS�ProductionQFrac (16)

Recycle = tS�RecycleQFrac (17)

here QFrac can be either QR or QE depending on which port isractionated.

It should be pointed out that a more general collection regimehich allows multiple transitions between production and recycleithin each cycle, could eventually lead to further improvements

n process performance. However, such a regime would increasehe complexity of FF-SMB, and is not taken into account here.

oreover, it is worth noting that the optimal sequence of thebove two periods depends on which outlet is fractionated ands relatively straightforward to determine. However, for schemesealizing multiple production and recycle operations, finding theptimum sequence is not trivial.

.2.2. Feeding and feedback periodsFeeding the mixture from the buffer vessel and the feed tank in

ne cycle in an alternating manner introduces two other periodssee Fig. 4). During a feedback period of length �Feedback, the off-pec fraction collected in the buffer vessel is fed back into the unit.

ithin the remainder of the cycle, the original feed tank is used asnlet source. Such a feeding regime can be easily realized by oper-ting periodically a three-way feed valve. The following relationsold:

Feedback + �Feeding = 1 (18)

Feedback = tS�FeedbackQF (19)

Feeding = tS�FeedingQF (20)

here �Feedback ∈ [0, 1], VFeedback and VFeeding are the liquid volumesrocessed within the respective periods.

In contrast to the fixed sequence of production and recycleor an outlet port, the inlet feeding sequence is variable so thatwo possible options (buffer vessel followed by feed tank, or fresheed followed by buffer vessel) are available. For example, for theequence of buffer vessel prior to fresh feed (see Fig. 4), the feedoncentration of component i at the SMB feed inlet can be defineds:

Fi (t) =

⎧⎪⎨⎪⎩

CBufferi

(t)t

tS∈ [0, �Feedback]

CFeedi

t

tS∈ (�Feedback, 1]

i = A, B (21)

� ∈ (�Feedback, 1], the original feed tank is used as inlet source.

where CFeedi

and CBufferi

represent the concentrations of compo-nent i in the feed tank and the buffer vessel, respectively. For theother feeding sequence, the corresponding definition is straight-forward. More complex feeding regimes incorporating multiplefeedbacks and various feeding sequences are not consideredhere.

It is not obvious to determine which feeding sequence ispreferable, even for the simplest case considered here. Fig. 6schematically presents the development of axial concentrationprofiles helping to identify the best feeding sequence. In theexample illustrated, the raffinate outlet is assumed to be frac-tionated. The buffer vessel is thus already enriched with the lessretained component (A). In this case, the recyclate should befed back at the beginning of each cycle, when the concentra-tion of A is higher than that of the more retained component B(see Fig. 6a). After the corresponding concentration fronts havemoved already downstream away from the feeding port, thefresh feed mixture containing a relatively higher fraction of B,should be supplied (see Fig. 6b). Inverting this order would reducethe separation performance. Alternatively, if the extract port isused for fractionation, the recyclate is enriched in componentB, and the reversed sequence is expected to be more favor-able.

It should be emphasized that for a given separation problem,the best sequence may depend in a more sophisticated mannerupon adsorption isotherms and feed concentrations. For non-linear adsorption isotherms of the Langmuir type, the slope ofthe isotherm decreases as the concentration increases, leading toincreased migration speeds. For FF-SMB, this effect could be alsoexploited as in the ModiCon concept [6], which manipulates the

concentration profiles by deliberately modulating the feed concen-trations.

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342 S. Li et al. / J. Chromato

.3. Buffer vessel dynamics

The liquid volume and concentrations in the buffer vessel cane described by the following simple mass balance equations of aell-mixed tank:

dVBuffer

dt= Qin(t) − Qout(t) (22)

d(VBufferCBufferi

)

dt= Qin(t)CFrac

i (t) − Qout(t)CBufferi

(t),

= A, B, Frac = E or R (23)

ith VBuffer being the liquid volume, Qin and Qout the inlet and out-

et flow-rates, CBufferi

and CFraci

the concentrations of component in the buffer vessel and at the SMB outlet used for fractionation,espectively.

Due to the discontinuous product withdrawal and non-constanteeding, Qin and Qout are characterized by two piecewise constantunctions. For the example of raffinate fractionation and feedinghe recycled fraction first, they can be defined as:

in(t) =

⎧⎪⎨⎪⎩

0t

tS∈ [0, �Production]

QRt

tS∈ (�Production, 1]

(24)

out(t) =

⎧⎪⎨⎪⎩

QFt

tS∈ [0, �Feedback]

0t

tS∈ (�Feedback, 1]

(25)

or the other feeding sequence and extract fractionation, the defi-itions are straightforward.

Initial conditions should be specified in order to predict theuffer vessel behavior:

Buffer(t = 0) = VBuffer,0 (26)

Bufferi

(t = 0) = CBufferi,0 , i = A, B (27)

uring the operation of FF-SMB, three different states for the liquidevel of the buffer vessel exist. A dimensionless factor �, defined asvolume ratio between VRecycle and VFeedback, was introduced in [11]

o characterize these scenarios:

= VRecycle

VFeedback= (1 − �Production)QFrac

�FeedbackQF, Frac = E or R (28)

f � = 1, the volume accumulated in the buffer vessel within eacheriod is zero and the liquid level remains cyclically constant. If> 1, the liquid accumulation becomes positive, leading to a con-

inuous increase in the level and the potential of overflow. If � < 1,gradual decrease occurs and the buffer vessel eventually “runs

ry”.Furthermore, the buffer vessel may have an impact on the

ynamics of the process. For example, the simulation studies in [11]howed that the initial conditions of the buffer vessel (Eqs. (26) and27)) could affect the transient behavior and thus the resulting timeequired to reach the cyclic steady state. However, a detailed anal-sis of the effect of the buffer tank on the dynamics is beyond thecope of this work.

.4. Special cases of FF-SMB

By fixing one or more degrees of freedom, the conventional SMBnd the “Partial Discard” operation can be regarded as two specialases of the FF-SMB process. The “Partial Discard” concept can be

217 (2010) 5337–5348

interpreted as the case where the “unwanted” fraction is discardedto the buffer vessel which has no feedback outlet to the unit:

�Feedback = 0, Qout(t) = 0, ∀t ∈ [0, tS] (29)

For the standard SMB mode realizing continuous product collec-tion and permanent feeding of fresh feed, obviously the followingconditions hold:

�Feedback = 0, �Production = 1 (30)

4. Optimization

The framework developed for the optimization of FF-SMB withone outlet fractionation will be presented below. The additionaldegrees of freedom and constraints will be underlined during for-mulating the optimization problem. An efficient solution strategyfor the optimization will be described briefly. The framework givenhere can also deal with the classical SMB and the “Partial Dis-card” operation. Thus, the optimization approach allows systematicinvestigation and comparison of these regimes.

4.1. Problem formulation

4.1.1. Objective functionFor the optimization of SMB and its various modifications, the

objective functions of interest typically differ from case to case.In [15], the authors suggested the objective function should bespecified based on detailed cost information. If such data are notavailable, only those dominant factors, such as the feed through-put, or the desorbent consumption for a given throughput, areoptimized. This approach using a single objective was adopted fre-quently to optimize SMB [14] and VariCol processes [16,17]. Onthe other hand, the multi-objective optimization has been compre-hensively studied in [19–22,25,32,33], where multiple competingperformance criteria were optimized simultaneously under givenconstraints. Both approaches reviewed above can be applied also tothe optimization of FF-SMB. It is worth noting that in the context ofthe fractionation and feedback concept, some of the performanceparameters used to evaluate the classical SMB operation must beappropriately adapted, in order to correctly reflect the discontin-uous product collection and non-constant feeding scheme. Suchadjusted parameters can be found in [11].

In the present work, the single objective optimization problemwith two different objective functions will be examined. For thefeed throughput maximization problem, the modified feed flow-rate introduced in [11] was chosen as the cost function:

Q ∗F = (1 − �Feedback)QF (31)

where Q ∗F represents the average rate of introducing fresh feed.

Obviously, for an FF-SMB process to be more productive, Q ∗F must be

larger than QF of the conventional SMB process. In the second casestudy, the desorbent flow-rate QD is minimized at a given Q ∗

F . If FF-SMB is more economical, a smaller QD should be achievable. It mustbe emphasized that such comparison is particularly instructive onlyif Q ∗

F takes the same value as QF in the classical SMB process.

4.1.2. Degrees of freedomFor the classical SMB process, the dimensionless flow-rate

ratios, the so-called m-values [23], were chosen as the optimizationvariables:

mj = QjtS − VCol�VCol(1 − �)

, j = I, II, III, IV (32)

with VCol being the column volume. The same selection has beendone in several previous optimization studies [21,24,25,27]. In

gr. A 1217 (2010) 5337–5348 5343

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Q

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ocairbuaml[

Table 1Summary of nonlinear optimization problem.

Objective function:max Q ∗

For min QD at a given Q ∗

F

Optimization variables:mI, mII , mIII , mIV , QF , �Production

Purity requirements:

Raffinate fractionation: PurE =∫ 1

0CE

B(�) d�∫ 1

0(CE

A(�)+CE

B(�)) d�

≥ PurE,min

PurR =∫ �Production

0CR

A(�) d�∫ �Production

0(CR

A(�)+CR

B(�)) d�

≥ PurR,min

Extract fractionation: PurE =

∫ 1

1−�Production

CEB

(�) d�∫ 1

1−�Production

(CEA

(�)+CEB

(�)) d�

≥ PurE,min

PurR =∫ 1

0CR

A(�) d�∫ 1

0(CR

A(�)+CR

B(�)) d�

≥ PurR,min

Maximum flow-rate constraint:QI ≤ Qmax

Feasibility constraint on �Feedback:0 ≤ �Feedback ≤ 1

S. Li et al. / J. Chromato

ddition to the four m-values, one additional operating parame-er must be specified. The external feed flow-rate QF was chosenn this study. In the case of FF-SMB, the dimensionless parametersProduction and �Feedback become two additional degrees of freedom.s shown in Section 3.3, VRecycle and VFeedback do not necessarilyatch: if VRecycle < VFeedback, the buffer tank level decreases, and if

Recycle > VFeedback, the level increases. In practical implementation,are must be taken to prevent the buffer vessel from “running dry”r overflowing. This can be realized, for example, by a cyclic exter-al compensation [11], or a regular removal of the recycled fraction

rom the buffer vessel. In this study, we restrict our discussion to thease where VRecycle = VFeedback, leaving other possibilities to futureork. Then the two variables are not independent any more and

elated by the following equality:

1 − �Production)QFrac = �FeedbackQF (33)

n the optimizations carried out, the dimensionless productioneriod �Production was specified as the independent variable, andFeedback was determined through one of the following two equa-ions, depending on which outlet is used for fractionation:

Feedback=

⎧⎪⎨⎪⎩

(mIII−mIV )(1−�Production)mIII−mII

raffinate fractionation

(mI−mII)(1−�Production)mIII−mII

extract fractionation(34)

In addition, the feeding sequence (i.e., the order of fresh feedupply and recyclate feedback, see Fig. 4) was also considered as aegree of freedom.

.1.3. ConstraintsIn this work, the purity constraints for both extract and raffinate

roducts were simultaneously taken into account:

urE ≥ PurE,min (35)

urR ≥ PurR,min (36)

here PurE,min and PurR,min are the pre-specified minimum purityequirements.

In order to respect the maximum allowable internal flow-ratemax caused by the limited pressure drop in the unit or the finiteapacity of the pump, an additional inequality constraint is imposedn the highest flow-rate occurring typically in zone I:

I ≤ Qmax (37)

n addition, the following obvious constraint regarding the feed-ack period is also required:

≤ �Feedback ≤ 1 (38)

he resulting nonlinear programming (NLP) problem is summa-ized in Table 1.

.2. Solution strategy

Depending upon how to deal with the cyclic steady state (CSS)f the process, generally two mainstream optimization strategiesan be found in the open literature. The first is the sequentialpproach, where the partial differential equations (PDEs) describ-ng SMB or its derivatives are only discretized in space and theesulting ordinary differential equations (ODEs) or differential alge-raic equations (DAEs) are integrated for a certain number of cycles

ntil the CSS is obtained. Then the values of the objective functionnd constraints are calculated and given back to an external opti-izer. The entire procedure is repeated until the optimal solution is

ocated. Applications of this method include optimization of SMBs14], SMB reactors [15] as well as VariCol processes [16,17].

Feasibility constraints on m-values:mI − mII > 0, mI − mIV > 0, mIII − mII > 0, mIII − mIV > 0

In order to reduce the computational effort associated withthe temporal integration required in the sequential method, thetailored simultaneous approach was proposed, where the CSScondition is treated as additional constraints of the optimiza-tion problem. Furthermore, two types of simultaneous methodshave been investigated in the literature. In the single discretiza-tion approach, the optimizer searches for the optimal operatingparameters and concentration profiles simultaneously. Here, forthe evaluation of the derivatives with respect to the decisionvariables, the sensitivity equations are integrated along with thedifferential equations. This scheme was proposed for the optimiza-tion of pressure swing adsorption (PSA) processes [34], as well as forSMB and its variants in the framework of the direct multiple shoot-ing [18]. On the other hand, in the full discretization method, themodel equations are fully discretized both in the spatial and tem-poral domain, and the PDE constrained optimization problem canbe transcribed into a large-scale problem constrained by nonlin-ear algebraic equations. Successful applications of this large-scaleoptimization strategy have been reported recently in [28–32].

For the FF-SMB process, it is not straightforward for thesimultaneous approach to deal with the discontinuous on–offcharacteristics exhibited at both the feed inlet and the fraction-ated outlet. Instead, the sequential method discussed above wasemployed. In particular, the PDE model described in Section 3 wasdiscretized by the orthogonal collocation on finite elements (OCFE)method, and the resulting system of DAEs was integrated usingthe package DASPK3.1 [35]. In recent years, some derivative-freestochastic optimization techniques, such as simulated annealingand genetic algorithms, etc., have also found many applications[19–22,25,27,26]. However, these techniques cannot check theoptimality conditions, and may require more computational effort.Alternatively, Newton-based approaches make use of gradient

information and have greater potential to solve optimization prob-lems efficiently. In this work, subroutine E04UCF from the NAGFortran Library [36], a solver based on sequential quadratic pro-gramming (SQP) technique, was chosen as the optimizer. In each

5344 S. Li et al. / J. Chromatogr. A 1217 (2010) 5337–5348

Table 2Model parameters and operating conditions for the example process.

SMB configuration 1-1-1-1Column dimensions (cm) 1 × 10Fractionation outlet Raffinate� 0.667N per column 40CFeed

i, i = A, B (g/L) 1

iCttC5tc

fotifiifnstuiwnttg

5

5

ppptzNwbaffiewo

TA

Qmax (mL/min) 10CBuffer

i,0, i = A, B (g/L) 0

VBuffer,0 (mL) 4

teration, the direct dynamic simulation was used to determine theSS solution. A tolerance was enforced on the difference betweenhe concentration profiles at the end of two successive cycles. Oncehe tolerance was fulfilled for both components simultaneously, theSS was considered to be attained. For the case studies in Section, the tolerance was specified to be 1 × 10−4 with the concentra-ion profile of each component normalized with respect to its feedoncentration.

In a Newton-based approach, the gradients of the objectiveunction and constraints are required by the solver. They can bebtained analytically except for those of the purity constraints. Inhis work, these gradients are obtained by finite difference approx-mation. Care must be taken in choosing appropriate intervals ofnite differences; poor approximation leads to a large number of

terations in optimization. Nevertheless, finding appropriate dif-erence intervals is not a difficult task in our problem, since theumber of decision variables is small (up to seven) and a small-cale nonlinear optimization problem results. We also investigatedhe sensitivity equation approach to evaluate the gradients exactly,tilizing an automatic differentiation (AD) technique. This was

mplemented using DASPK3.1 and an AD tool (Tapenade [37]). Itas noticed that although the number of iterations decreases sig-ificantly in this approach, the total computational time far exceedshat required by the finite difference approximation. This is due tohe large number of sensitivity equations which need to be inte-rated simultaneously with the model equations.

. Results and discussion

.1. Example process

In [11], a binary separation problem characterized by com-etitive Langmuir isotherms was used to investigate the FF-SMBrocess. The same illustrating example was also considered in theresent study. The model parameters and operating conditions forhis laboratory-scale unit are summarized in Table 2. In the four-one unit, four columns with low efficiencies characterized by an

value of 40 were used. For FF-SMB, only the raffinate outletas fractionated. Furthermore, it was assumed that initially the

uffer vessel contains only fresh solvent. The isotherm coefficientsssumed (Eq.(14)) are summarized in Table 3, which were adaptedrom experimental data given in [12]. For linear isotherms, the coef-

cients Ki(i = A, B) were set to zero. The selection of low columnfficiencies (N = 40) and small adsorption selectivity (˛ = 1.126)as motivated by our primary interest of evaluating the potential

f the new concept for difficult separations.

able 3dsorption isotherm parameters.

Component H K (L/g)

A 5.078 0.089B 5.718 0.105

Fig. 7. Comparison of maximum feed throughput for conventional SMB (circles)and FF-SMB with feeding sequence “FR” (diamonds) and “RF” (squares) (linearisotherms).

5.2. Case study 1: maximization of feed throughput

In this case, the desired extract product purity was fixed at90%, while a set of purity specifications (i.e., from 80% to 95%) wasimposed on the raffinate product. In order to identify the optimumfeeding regime for the FF-SMB process, the performance was opti-mized for each of the two feeding sequences (recyclate followed byfresh feed or fresh feed followed by recyclate). For simplicity, thesefeeding options are abbreviated as “RF” and “FR”, respectively.

5.2.1. Linear isothermFor specified raffinate purity requirements, the maximum feed

throughputs delivered by the classical SMB operation and the FF-SMB with two feeding sequences are shown in Fig. 7. The newprocess with the feeding sequence “RF” is capable of fulfilling allthe purity constraints examined here. The same process using theopposite feeding sequence “FR”, can provide purities up to 93%.In contrast, with conventional SMB, the highest attainable purityis only 92%. The results demonstrate that the fractionation andfeedback concept has potential for improving product purities.

On the other hand, it can be also observed that for the samepurity, the optimal Q ∗

F obtained with either feeding sequence ishigher than QF for SMB. Furthermore, this degree of improvementdepends on the raffinate purity requirement. The stricter the purityexpectations, the larger the advantages gained by the new conceptover the classical mode. For example, when the feeding sequence“RF” is applied, the processable feed can be improved by 94%, 152%and 365% for the desired purity of 90%, 91% and 92%, respectively. Incontrast, for the same set of purity specifications, an improvementof 36%, 62% and 159% can be achieved with the reversed feedingsequence. Both cases indicate that more fresh feed can be treatedby the FF-SMB process during the feeding period (i.e., �Feeding) thanthat by the conventional SMB over the complete cycle.

The optimal feeding sequence “RF” identified here to be moreattractive coincides well with the prediction made in Section 3.2.2.In the following, we restrict the further discussion to only thissequence and neglect the results for the other one.

In order to provide more rational explanations for the perfor-

mance improvement achieved by FF-SMB, the optimization resultswere also analyzed in the context of equilibrium theory [23]. Withinthis framework, a modified formulation of the original dimension-less factor mIII is introduced for the FF-SMB concept to reflect the

S. Li et al. / J. Chromatogr. A 1217 (2010) 5337–5348 5345

Ftp

r

m

sscawotoottStvmmco9bTwrn

�otaaftt

found again that FF-SMB with both feeding sequences (“RF” and“FR”) achieves higher purities. Furthermore, as in the case of linearisotherms, the optimized value of Q ∗

F is higher than that of QF . Forexample, with the feeding sequence “RF”, Q ∗

F is 1.70, 2.17 and 3.82

ig. 8. Comparison of classical SMB (open squares) and FF-SMB (solid squares) inhe combined (mII, mIII) or (mII, m∗

III) planes for linear adsorption isotherms. For each

rocess, raffinate purity requirements are also marked near the operating points.

educed feeding time [11]:

∗III = mII + Q ∗

F tS

VCol(1 − �)(39)

The optimal values of mII and m∗III for FF-SMB with the feeding

equence “RF” are illustrated in the (mII, m∗III) plane in Fig. 8. Corre-

ponding optimal (mII, mIII) pairs obtained from the conventionaloncept are also given in the (mII, mIII) plane in the same figure. Inddition, the raffinate purity requirements are also shown togetherith the corresponding optimal operating points. The following

bservations can be made. First, as the purity demand increases,he operating points for both processes move towards the diagonalf the respective operating plane, leading to a gradual reductionf the size of the feasible region. For stricter purity requirements,he optimization scheme failed to find a solution, which indicateshat such high purities cannot be fulfilled and are infeasible. For FF-MB, the highest achievable purity observed in the case study is upo 95%, while it is only 92% for the classical SMB. Second, the optimalalue of mII for FF-SMB is very similar to that of the conventionalode. This accordance is due to the fact that the extract purity isainly influenced by the flow-rate in zone II, and for both pro-

esses the extract outlet stream, which is continuously withdrawnver the complete cycle, needs to satisfy the same purity demand of0%. Compared to mIII for the SMB concept, a higher value of m∗

III cane delivered by the FF-SMB approach for the same raffinate purity.he difference between m∗

III and mIII tends to be more pronouncedhen the purity becomes more stringent. This indicates that using

affinate fractionation has a significant effect on m∗III , allowing the

ew process to exploit more productive operation points.The additional dimensionless parameters of the FF-SMB process,

Production and �Feedback, are shown in Fig. 9a and b, respectively. Inrder to better illustrate the development of these two variables,he optimization results for two lower purities (i.e., 70% and 75%)

re also given there. For the purity of 70%, the optimal �Production

pproaches 1, and the resulting �Feedback, calculated with Eq. (34)or the case of raffinate fractionation, is close to 0. This fact implieshat both outlet fractionation and feedback become negligible. Inhis case, the FF-SMB has almost degenerated into one of its special

Fig. 9. Development of (a) optimal production period �Production and (b) feedbackperiod �Feedback for FF-SMB (linear isotherms).

cases, the conventional SMB (see Eq. (30)). Thereafter, �Production

decreases rapidly in order to fulfill the increasing purity require-ments. For the purity of 90%, for example, the optimal value isnearly 0.89. For stricter purity specifications (e.g., from 91% to 95%),�Production begins to increase towards 1. For example, �Production ismore than 0.98 for the raffinate purity of 95%. This observationshows that for the separation under examination here, the func-tion of recycling the off-spec fraction tends to be most pronouncedfor the purity requirements around 90%. In contrast, a monotonicincrease up to about 0.41 can be observed for �Feedback, which indi-cates that within each cycle longer time is spent to feed backrecyclate.

5.2.2. Nonlinear isothermIn the case of nonlinear isotherms, the maximum feed through-

put for different purity specifications is illustrated in Fig. 10. It is

Fig. 10. Comparison of maximum feed throughput for conventional SMB (circles)and FF-SMB with feeding sequence “FR” (diamonds) and “RF” (squares) (nonlinearisotherms).

5346 S. Li et al. / J. Chromatogr. A 1217 (2010) 5337–5348

FtFp

ttfcttsc

sfsmttpvtit

iFeFt

5

ocitotw

For the fractionation and feedback approach with either feed-ing sequence, the potential desorbent savings (shown in bracketsin Table 4a and b) can be clearly identified for the same set of purityspecifications (i.e., 80%, 85%, and 90%). For increased purity expecta-

Table 4Results of minimization of desorbent consumption for SMB and FF-SMB with twofeeding sequences (“FR” and “RF”).

PurR,min (%) QD (mL/min)

SMB “FR” “RF”

(a) Linear isotherm80 0.384 0.375 (−2.2%) 0.347 (−9.5%)85 0.593 0.549 (−7.4%) 0.470 (−20.7%)90 1.482 1.070 (−27.9%) 0.760 (−48.8%)91 – 1.351 0.88292 – – 1.06693 – – 1.390

(b) Nonlinear isotherm80 0.609 0.592 (−2.8%) 0.571 (−6.3%)

ig. 11. Comparison of classical SMB (open squares) and FF-SMB (solid squares) inhe combined (mII, mIII) and (mII, m∗

III) planes for nonlinear adsorption isotherms.

or each process, raffinate purity requirements are also marked near the operatingoints.

imes higher for the raffinate purity of 90%, 91% and 92%, respec-ively. The improvement tends to decrease when using the oppositeeeding sequence. These results show again that the FF-SMB pro-ess not only allows to provide higher raffinate purities comparedo the conventional concept, but also improves productivities forhe same purity requirement. As in the linear case, the feedingequence “RF” proves to be better. Thus, only this sequence wasonsidered further.

The optimum operating conditions for different purities arehown in the combined (mII, mIII) and (mII, m∗

III) planes in Fig. 11or both processes. It can be seen that in order to satisfy moretringent purity requirements, the optimal operating point of eachode enters the complete separation region, and moves towards

he diagonal of the corresponding operating plane. This is similaro the behavior observed in the linear case. Moreover, for the sameurity requirement, m∗

III is similar to or smaller than mIII in the con-entional concept, and mII always shows lower values compared tohat of SMB. Thus, the larger value of m∗

III − mII achieved by FF-SMBs due to a smaller mII , rather than a higher m∗

III , which differs fromhe observation made for the linear case.

The dimensionless parameters �Production and �Feedback are shownn Fig. 12a and b, respectively. Compared to the linear case (seeig. 9), similar tendencies can be observed for these two param-ters. Note that in this case, the optimal value of �Production (seeig. 12a) does not begin to increase towards 1 until purities higherhan 91% are requested.

.3. Case study 2: minimization of desorbent consumption

In many SMB applications, the desorbent consumption is onef the dominant factors that contribute to overall separationosts. Reducing desorbent consumption can therefore considerably

mprove the process economics. In the second case study, condi-ions allowing the minimization of desorbent consumption wereptimized for a given feed throughput. The primary purpose waso evaluate the potential of the fractionation and feedback approachith respect to desorbent savings.

Fig. 12. Development of (a) optimal production period �Production and (b) feedbackperiod �Feedback for FF-SMB (nonlinear isotherms).

In our optimization study, the feed flow-rate QF was assumed tobe 0.12 mL/min considering the small dimensions of the laboratory-scale unit. For FF-SMB, the same value was specified to the modifiedfeed throughput Q ∗

F . In order to determine which feeding sequenceenables a more economic desorbent usage, both sequences (“RF”and “FR”) were optimized. The results obtained for both linear andnonlinear isotherms are summarized in Table 4a and b, respec-tively. Again, a minimum extract purity of 90% was required for alloptimization runs and different purity constraints were imposedon the raffinate product (see Table 4).

For the classical SMB mode, the minimum desorbent flow-rateincreases dramatically as the raffinate purity becomes higher. Forexample, in the case of linear isotherms (see Table 4a, SMB pro-cess), the desorbent consumed for delivering a purity of 90%, is upto 2.5 times higher than that for a lower purity of 85%. For the non-linear case, a more than twofold increase in QD can be observed.Such a high sensitivity of desorbent consumption with respect tothe purity requirement is mainly caused by the difficulty of the sep-aration under study. When a purity of 91% or higher was specified,the optimizer failed for both types of isotherms.

85 0.863 0.813 (−5.8%) 0.724 (−16.1%)90 1.899 1.444 (−23.9%) 1.053 (−44.5%)91 – 1.779 1.18392 – – 1.37493 – – 1.695

gr. A 1

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S. Li et al. / J. Chromato

ions, the advantage over the conventional process tends to be moreignificant. For example, in the case of linear isotherms, a desorbentow-rate reduced by about 50% can be achieved by FF-SMB withhe sequence “RF” for the purity of 90%. For nonlinear isotherms, theonsumption can be reduced by nearly 45%. The same concept usinghe sequence “FR” allows a reduction of 27.9% and 23.9% for lin-ar and nonlinear conditions, respectively. In contrast, the benefitsecome small for lower purity demands (e.g., 80%). Furthermore,

t is interesting to note that for stricter purity requirements (e.g.,1% or higher), only the fractionation and feedback regime allowso deliver both products while maintaining relatively low desor-ent consumptions. From Table 4, it can be observed that for both

sotherms, the highest achievable purity is 93% for the sequenceRF”, and 91% for the opposite case. Thus, the feeding sequenceRF” (raffinate recyclate followed by fresh feed) again provides theest performance and was found to be optimal in this case study.

. Conclusions and future work

The FF-SMB concept with one outlet fractionation has beennvestigated by a systematic optimization strategy. A model-basedptimization framework has been presented which allows dealingith the flexibility and exploiting the full potential of the fraction-

tion and feedback approach. Two typical optimization problemssing single objective functions and solved by the sequentialethod have been examined. On the basis of the extensive opti-ization studies performed for a difficult separation problem, the

ptimum performance of FF-SMB realizing raffinate outlet frac-ionation has been quantitatively evaluated with respect to that ofhe conventional concept. The results clearly demonstrate that the

ore flexible FF-SMB operating policy consistently outperformsMB chromatography in terms of the maximum throughput andinimum desorbent consumption for both linear and nonlinear

sotherms. For high purity requirements, the advantages becomearticularly pronounced. Moreover, the influence of the feedingequence on the optimal performance of FF-SMB has been alsoxamined. For the example considered in this paper, the sequencef raffinate recyclate followed by original fresh feed achieves supe-ior performance in both case studies.

Future work will focus on the extension of the optimizationramework to the process with simultaneous double fractionationf both outlets, as well as on the experimental validation of the the-retical results. For the theoretical study, particular emphasis wille concentrated on problems related to the practical applicabilityf the proposed new concept.

omenclature

liquid phase concentration (g/L)dimensionless liquid phase concentration

ap,i apparent axial dispersion coefficient of component i(cm2/s)Henry coefficientadsorption equilibrium constant (L/g)column length (cm)flow-rate rationumber of theoretical plates

ur purityliquid phase flow-rate (mL/min)

solid phase concentration (g/L)time (min)

S cycle time (min)interstitial velocity (cm/s)liquid phase volume (mL)

[[

[[

217 (2010) 5337–5348 5347

VCol column volume (cm3)z axial coordinate (cm)

Greek letters˛ selectivity� column porosity� dimensionless buffer vessel liquid factor in Eq.(28)� dimensionless time in Eq. (1)

Roman lettersI, II, III, IV zone index

Subscripts and superscripts∗ adapted parameters taking into account shortened feed-

ing timeA, B componentsBuffer buffer vesselD desorbentE extractF feedFeed original feed tankFeedback feedback from buffer vesselFeeding feed from original feed tankFrac SMB outlet used for fractionation, Frac = E or Ri component index, i = A, Bin inlet of zone or column for SMB, inlet of buffer vesselj zone index, j = I, II, III, IVout outlet of zone or column for SMB, outlet of buffer vesselProduction SMB outlet stream is withdrawn as productR raffinateRecycle SMB outlet stream is recycled to buffer vessel

Acknowledgements

The authors are indebted to Lars Christian Keßler for numer-ous fruitful discussions about the principle of FF-SMB. Y. Kawajiriacknowledges the financial support from the Alexander vonHumboldt Foundation. Support from Knauer WissenschaftlicheGerätebau GmbH (Berlin), Fonds der Chemischen Industrie (Köln)and the European Union project (“INTENANT”: FP7-NMP2-SL2008-214129) is gratefully acknowledged.

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