Experimental Validation of Optimized Model-Based Startup

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Experimental validation of optimized model-based startup acceleration strategies for simulated moving bed chromatography Jason Bentley, Suzhou Li, and Yoshiaki Kawajiri Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 24 Jun 2014 Downloaded from http://pubs.acs.org on June 28, 2014

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Industrial & Engineering Chemistry Research

Experimental validation of optimized model-based startup acceleration strategies for simulated moving bed chromatography Jason Bentley,† Suzhou Li,‡ and Yoshiaki Kawajiri∗,† School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA. 30332, and Max Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany E-mail: [email protected]

Abstract It is demonstrated experimentally by this work that significant startup acceleration can be achieved for simulated moving bed (SMB) chromatography when the transient operation is optimized. The desorbent consumption can be reduced d average product concentrations can be increased during the startup period by systematically switching the SMB operating conditions during startup. In this work, systematic model development and subsequent model-based optimization are used to find control policies that are implemented experimentally to realize startup acceleration. The SMB model is reliably determined using the recently developed prediction-correction method (Bentley, J.; Kawajiri, Y. AIChE J. 2013, 59, 736-746), which is used to efficiently optimize the cyclic steady state operation. The startup acceleration problem is solved approximately following an algorithm proposed by Li et al. (Li, S.; Kawajiri, Y.; Raisch, ∗

To whom correspondence should be addressed Georgia Institute of Technology ‡ Max Planck Institute for Dynamics of Complex Technical Systems †

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ACS Paragon Plus Environment


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J.; Seidel-Morgenstern, A. J. Chromatogr. A 2011, 1218, 3876-3889). The tradeoff between the startup acceleration and product purity is investigated using the SMB model, and validated experimentally. It is shown that the startup time and desorbent consumption during the startup can be reduced significantly by systematically switching the operating conditions three times in a case study with a linear isotherm system.

1

Introduction

A simulated moving bed (SMB) process exploits multiple chromatographic columns connected in series where inlet/outlet port switching occurs between columns to simulate the counter-current movement between the liquid and solid phases. 1–3 SMB technology mimics the behavior of a true moving bed (TMB) unit, without having to convey the solid phase. During the last two decades, SMB has received increasing attention in the chemical industry, especially for sugar, petrochemical, and chiral separations. 4,5 For separation where preparative batch chromatography has been conventionally used, the SMB process offers significantly increased productivity and product concentrations. The typical SMB configuration for binary separation has four zones where the mostretained component is desorbed in Zone 1, and adsorbed in Zone 2, and the least-retained component is desorbed in Zone 3, and adsorbed in Zone 4. A schematic of the classical SMB process with four columns and one column in each zone, is shown in Fig. 1 to illustrate the concept of port switching and continuous binary separation. In Step 1, the feed is supplied between Zones 2 and 3 consisting of a mixture of components, A and B, usually dissolved in the desorbent, D. The extract product is withdrawn between Zones 1 and 2 where the purity of B is high, and the raffinate product is withdrawn between Zones 3 and 4 where the purity of A is high. Desorbent is supplied between Zones 4 and 1 to regenerate the adsorbent. After one step time, the positions of the inlet/outlet ports advance ahead one column in the direction of fluid flow, as shown in Fig. 1b. This motion maintains the product purities in 2


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the extract and raffinate streams throughout the process, following the concentration profiles of components A and B through the columns. The SMB process has a cyclic steady state (CSS) where the concentration profiles are equal at the beginning and end of a cycle. For step symmetric operations, where the operating conditions are the same in each step, the CSS condition exists at the beginning and end of each step of operation. In practice, the SMB unit is conventionally operated using constant operating conditions that are found to yield high purity products at the CSS. However, only little attention has been given to the optimization of the transient operation (startup and shutdown) in the literature. Yet there are some important dynamics that take place during startup while the concentration profiles of each component are not yet established in the SMB unit. If only constant operating conditions are used during startup, the system may take a long time to reach the CSS conditions, and some of the products collected during the startup operation may need to be discarded or reprocessed if the product concentration is too low or if the purity is off-specification. In addition, during shutdown the conventional operating strategy may not allow for any products to be collected if the purity constraints are violated. As discussed by Li et al. 6 , there is a common scenario in the pharmaceutical industry where the same SMB unit is sequentially used to purify products from small batches of different mixtures. In Fig. 2 there is a schematic illustration of this scenario with a three-stage production campaign, where the transient SMB operation requires a significant portion of the total operation time. For the two products, (2) and (3), the amount of the feed is small, and there is not sufficient time to reach the CSS using the conventional startup strategy, which makes application of SMB difficult. If the transient operation is optimized, the SMB can reach the CSS when processing (2) and (3). Furthermore, the overall feed throughput is increased, and this may significantly reduce the processing time of the campaign. Furthermore, in the sugar industry where SMB is used for large scale productions, evaporating water from the dilute products during the startup and shutdown periods requires substantial

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Zone 1

Extract, B+D

Zone 2

Zone 4

Desorbent, D

Fluid flow and port switching direction

Raffinate, A+D

Zone 3

Feed, A+B+D

Zone 4

Desorbent, D

Raffinate, A+D

Zone 1

(a) Step 1

Zone 3

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Fluid flow and port switching direction

Feed, A+B+D

Zone 2

Extract, B+D

(b) Step 2

Figure 1: Schematic of SMB process showing two steps of cyclic operation. Four-zone configuration with one column in each zone for (a) Step 1 and (b) Step 2

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cost. Some portion of the dilute products may need to be discarded if water evaporation cost exceeds the profit obtained by recovering the sugar. Normalized concentration Ci(t)/CCSS Normalized concentration Ci(t)/CCSS

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(1) 1.0

(2) (3)

(a)

Time

(1)

(2)

(3)

1.0

Time (b)

Figure 2: Schematic illustration of product concentrations for products (1), (2), and (3) over time for SMB operated with (a) conventional and (b) optimized transient strategies for a three-stage production campaign

Some researchers have applied heuristic startup and shutdown strategies such as preloading the columns with feed and having pre-elution, including recovery of adsorbed products during shutdown. Pre-loading is achieved by activating the feed pump only to build up the concentration profiles in the unit before the operating conditions for CSS production are implemented. Lim and Ching 7 worked on reducing the transient operation in a modeling study of SMB, and their simulation studies showed that pre-loading certain columns with the feed mixture reduces the transient operation time. Both numerical simulation and experimental demonstration of pre-loading, pre-elution and product recovery during shutdown were performed by Xie et al. 8 for insulin production. They found that the startup time 5



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of an SMB cascade system is reduced by pre-loading certain SMB columns with the feed mixture. Bae et al. 9 used an SMB model and experiments to study startup dynamics for different loadings of the less-retained feed component. In an extension of that work, Bae et al. 10 studied the effects of the flow rate ratios, or m-values, on both the CSS and startup performance of an SMB unit. Abunasser and Wankat 11 used one-column and two-column analogues to emulate the behavior of a 4-zone SMB unit for small-scale production campaigns, and they analyzed the transient operation using a similar heuristic approach as Xie et al. 8 They also showed that pre-loading the columns may significantly reduce the startup time which is critical for such small-scale campaigns. 11 Roderigues et al. 12 showed that using a one-column setup, the analogous SMB CSS concentration profiles may be obtained rapidly by manipulating the inlet composition during the initial feed injection. However, they noted that this approach can not be used for multi-column SMB operation. Yet, none of these previous works apply any model-based optimization methods to find the best control parameters for transient operation of SMB units. In recent work by Li et al. 6 , systematic startup and shutdown strategies are presented for an SMB unit using nonlinear programming. In that work the potential for significant startup and shutdown acceleration is shown using simulation of nonlinear adsorption experiments with optimal transient control strategies. The optimal operating conditions are solved offline by minimizing the difference between the transient and CSS concentration profiles for multiple stages of the startup time horizon. Yet it is unclear if the SMB model used in that work is sufficient to predict the transient SMB operation because the optimal transient control strategies obtained by Li et al. 6 were not validated experimentally. In this work, the prediction-correction (PC) method, recently developed by Bentley and Kawajiri 13 and Bentley et al., 14 is used to systematically obtain reliable model parameters that can be used to optimize the transient SMB control variables. Then optimal startup controls are calculated and implemented in a lab-scale SMB unit for a linear isotherm system, building upon the methodology used by Li et al. 6 . This work demonstrates experimentally

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the feasibility of using a model-based optimization approach to significantly reduce the SMB startup time. The benefits of an optimized startup acceleration strategy include increased overall productivity and reduced desorbent consumption. The paper is organized as follows. The SMB model, a description of the PC method which is used to obtain model parameters, and the formulation of the startup optimization problem are presented in Section 2. The experimental equipment and operation are described in Section 3. Results of the CSS optimization, calculated startup acceleration strategies, and experimental performances of startup acceleration strategies are reported in Section 4. Conclusions are drawn in Section 5.

2

Methodology

2.1

SMB model

A detailed SMB model that takes into account axial dispersion and mass transfer kinetics, or some combination of these, is critical for model-based optimization of both CSS and transient operation. 15 The linear driving force (LDF) model is shown here for a binary mixture. For each column the component mass balance in the fluid phase is: ∂cji (z, t) 1 − ǫb ∂qij (z, t) ∂cj (z, t) + = −v j (t) i ∂t ǫb ∂t ∂z i = A, B,

(1)

j = 1, 2, . . . , ncol

where cji is the fluid phase concentration of component i in column j, ǫb is the overall bed porosity, qij is the adsorbed phase concentration of component i in column j, v j is the linear mobile phase velocity in column j, z is the axial coordinate, t is the time, A and B are the less- and more-retained components respectively, and ncol is the number of columns.

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The component mass balance in the adsorbed phase is: ∂qij (z, t) = ki (qieq,j (z, t) − qij (z, t)) ∂t

(2)

where ki is the overall mass transfer coefficient for component i, and qieq,j is the adsorbed phase concentration of component i in equilibrium with the fluid phase in column j. The lumped parameter ki is used to describe band broadening effects including axial dispersion, and it has been shown that such a simplified model can predict experimental behavior well provided that a reliable value for ki is obtained. 4,16 The linear adsorption isotherm is: qieq,j = Hi cji

(3)

where Hi is the Henry’s constant for component i.

2.2

Determination of optimized CSS operation by the PC method

Reliable model parameters and CSS operating conditions are required in order to optimize the SMB startup operation. These are determined systematically by the prediction-correction (PC) method, developed recently by Bentley and Kawajiri 13 and Bentley et al. 14 The PC algorithm is shown in Fig. 3. The CSS operating conditions obtained using this method are optimized while satisfying specified purity constraints. The initial model mismatch that is inherent to SMB modeling using batch experiments is readily resolved using the PC method, which follows an iterative scheme to find model parameters that predict the SMB experimental data in the sequence. Since the details of the PC method can be found in our previous studies, 13,14 only a summary of this method is presented here. In Step 1, initial SMB model parameters are obtained by pulse tests on a single column. Optimized CSS operating conditions are determined by

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Batch experiments

Step 1.

Initial model parameters

k=0

Model-based SMB optimization

Step 2.

Initial operating conditions

SMB experiment

Step 3.

Concentration data, product purities

k=k+1

Step 4.

Parameter estimation (correction) Refined model parameters

Step 5.

Model-based SMB optimization (prediction) Updated operating conditions

Step 6.

Termination criteria satisfied? YES

NO Perform switch of operating conditions

Terminate

Figure 3: Prediction-correction (PC) algorithm for SMB process development

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solving the following nonlinear programming (NLP) problem in Steps 2 and 5:

max

Ffeed

uCSS

(4a)

raf s.t. Purityraf A ≥ PurityA,CSS,min + ∆safety

(4b)

ext Purityext B ≥ PurityB,CSS,min + ∆safety

(4c)

cji (z, 0) = cj+1 (z, tstep ), i

j = 1, 2, . . . , ncol − 1

(4d)

cni col (z, 0) = c1i (z, tstep ) qij (z, 0) = qij+1 (z, tstep ),

j = 1, 2, . . . , ncol − 1

(4e)

qincol (z, 0) = qi1 (z, tstep ) Fx ≤ Fmax ,

x = 1, 2, 3, 4

(4f)

Eqs. (1)-(3) where uCSS = [tstep , m1 , m2 , m3 , m4 ]T is the vector of SMB operating conditions, tstep is the step time, mx is the fluid-to-solid flow rate ratio in zone x, determined by:

mx =

Fx tstep − Vcol ǫb Vcol (1 − ǫb )

(5)

where Fx is the volumetric flow rate in zone x, and Vcol is the total column volume. Purityraf A,CSS,min and Purityext B,CSS,min are the lower bounds on the raffinate and extract purities, respectively, in the SMB experiment. A safety factor for the purities, ∆safety , is added for both the extract and raffinate in the optimization problem. This is done to guarantee the purity requirements under minor and acceptable model uncertainty that remains after the convergence of the PC algorithm.

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The purity of component A in the raffinate is defined for a step of SMB operation as: tstep R

craf A (t)dt

0

Purityraf A =

P

tstep R

(6a) craf i (t)dt

i=A,B 0

tstep R

cext B (t)dt

0

Purityext B =

P

tstep R

(6b) cext i (t)dt

i=A,B 0

ext where craf A (t) is the concentration of A in the raffinate product, and cB (t) is the concentration

of B in the extract product. The CSS constraints in Eqs. (4d) and (4e) are defined for a single step of the SMB process. These constraints show that the fluid and adsorbed phase concentration profiles are the same at the beginning and end of a step only shifted one column in the direction of fluid flow. The maximum flow rate constraint in Eq. (4f) is based on the total pressure drop limits for the SMB pumps. More details on the optimization problem are given by Bentley and Kawajiri 13 . Note that this optimization problem does not consider transient dynamics; it only optimizes the CSS performance. The optimized operating conditions, u∗CSS , are implemented in an SMB experiment in Step 3, where cumulative product concentrations are collected over time during startup. To resolve model mismatch, the isotherm and kinetic parameters are corrected by solving the following parameter estimation problem in Step 4:

min θ

N Vp N Mpl NE X X X p=1 l=1 d=1

"

2 ln(σpld )

(˜ ypld − ypld )2 + 2 σpld

s.t. θmin ≤ θ ≤ θmax Eqs. (1)-(3)

11


#

(7a)

(7b)


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where θ is the set of model parameters to be estimated, N E is the number of experiments, N Vp is the number of variables measured in experiment p, N Mpl is the number of mea2 surements of variable l in experiment p, σpld is the variance of measurement d of variable

l in experiment p, which is determined by the measured variable’s variance model, y˜pld is the value of measurement d of variable l in experiment p, ypld is model-predicted value of measurement d of variable l in experiment p, θmin and θmax are the lower and upper bounds on the set of model parameters to be estimated. More details on the parameter estimation problem are given by Bentley and Kawajiri 13 . The SMB is re-optimized using the refined model parameters with the same constraints. This sequence of SMB experiments, followed by parameter estimation and optimization, is repeated until the termination criteria are satisfied. The termination criteria are that the experimental product purities are satisfied, and the optimal feed throughput is converged. When the termination criteria are satisfied, the PC algorithm is terminated and the current operating conditions, u∗CSS , are considered as the final results.

2.3

Conventional startup strategy

To demonstrate the advantages of SMB startup acceleration, the conventional startup strategy is implemented and compared with the accelerated startup strategies. The conventional startup strategy uses constant operating conditions equal to the optimal CSS operating conditions for the duration of the experiment:

u(t) = u∗CSS ,

∀t ∈ [0, tCSS ]

where u(t) are the SMB operating conditions as a function of time, and tCSS is the time when the SMB plant reaches the CSS conditions. The conventional startup strategy is easy to implement in practice because there is only one set of transient operating conditions. However, conventional startup often results in long startup times and significant desorbent

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consumption. Furthermore, there is no guarantee that the products collected during startup will meet the desired purity specifications. 6 In some SMB operations, the evaporation cost of the product is so high that the dilute product in the startup period may need to be discarded.

2.4 2.4.1

Accelerated startup strategy Problem formulation

In this work the startup acceleration problem is proposed to be solved using a multistage approach. The startup time horizon is divided into P stages, as shown in Fig. 4a. Each stage is assumed to include nstep,n SMB steps in Stage n, during which piecewise constant operating conditions are used. Therefore, u(t) = un , t ∈ [tn , tn+1 ], n = 1, 2, . . . , P , where un is the vector of operating conditions for Stage n, and tn is the time that Stage n begins. In this way the operating conditions may be changed during startup in order to shorten the overall startup time. The piecewise constant control variables allow for a more practical implementation of the startup acceleration strategy. The goal of the startup acceleration problem is to solve for un , n = 1, 2, . . . , P , and determine nstep,n and P , the number of stages needed to reach the CSS performance starting from the initial condition of clean beds in order to minimize the startup time while respecting any imposed constraints. A straightforward formulation of the objective function can be made as follows: min

tCSS =

P X

nstep,n tstep,n

(8)

n=1

where tCSS is the startup time. As discussed in Li et al., 6 this objective function may lead to an ill-conditioned optimization problem. Therefore, an alternative objective function is employed in this work which uses a dimensionless time coordinate:

τn =

t − tn ∈ [0, 1], tn+1 − tn

13

n = 1, 2, . . . , P


(9)


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where τn is the normalized time for Stage n. This transformation of the time horizon is shown in Fig. 4(b). The objective function may be reformulated as:

min

J=

P X n=1

tstep,n

Z1

kC(τn ) − CCSS (τn )k2 dτn

(10)

0

where C(τn ) is a vector of the internal concentration profile for each component in each column: C = [cjA , cjB , qAj , qBj ]T ,

j = 1, 2, . . . , ncol

and CCSS (τn ) is a vector of the CSS total internal concentration profile for each component in each column. After discretization, this vector has a dimension of 4N F E, where N F E is the total number of spatial finite elements. The objective function in Eq. (10) considers minimizing the difference between the transient concentration profile and the CSS concentration profile. The difference between C(τn ) and CCSS (τn ) may be calculated by simultaneous simulations of the SMB model. The objective function J using the exact integral in Eq. (10) can be used to formulate the optimal startup problem as: min

un ,n=1,...,P

s.t.

(11a)

J

kC(τP = 1) − CCSS (τP = 1)k ≤ ǫconc

raf Purityraf A,accum (τP = 1) ≥ PurityA,accum,min

(11b)

(11c)

ext Purityext B,accum (τP = 1) ≥ PurityB,accum,min

mx,P = m∗x ,

tstep,P = t∗step ,

Fx,n ≤ Fmax , F1,n − F2,n ≥ 0,

F1,n − F4,n ≥ 0, 14

x = 1, 2, 3, 4

x = 1, 2, 3, 4 F3,n − F2,n ≥ 0,


(11d) (11e)

F3,n − F4,n ≥ 0

(11f)

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Stage 1

Stage n

Stage P

tstep,n

time, t 0 = t1

tn+1

tn

t2

tP

tP+1

(a)

Stage 1

Stage n

1

0

0

(b)

Stage P

0.25

0.25

0.25

1 0 normalized time, 2

1

CSS

Startup period u1 u*(t)

Operating conditions, u

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un Stage 1

0

(c)

Stage n

1

0

uCSS*

uP

Stage P

1 0 normalized time, 2

1

Figure 4: (a) Schematic of startup time horizon decomposed into stages. (b) Normalized time horizon with fixed 4 steps per stage. (c) Piece-wise constant operating conditions approximating the optimal control profile, u∗ (t)

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with n = 1, . . . , P . The CSS constraint in Eq. (11b) enforces the convergence of the transient concentration profile to the CSS concentration profile within a given tolerance, ǫconc . The cumulative purity constraints given by Eq. (11c) must be satisfied by the end of the startup period, when τP = 1. The cumulative purity values are defined later by Eq. (23). The equality constraints in Eq. (11d) enforce that the operating conditions converge to the CSS optimum values by the end of the startup period. The flow rates are constrained to maintain feasible and safe operation of the SMB unit throughout the startup period. This optimization problem may be solved to find the optimal piece-wise constant operating conditions that minimize the time it takes to reach the CSS concentration profile while satisfying given cumulative purity constraints for each product. It should be noted that there are many choices for the vector norm in the CSS constraint Eq. (11b), such as the L2 and L∞ norms. In this work, we consistently employ the L2 norm kC(τP = 1) − CCSS (τP = 1)k2 . The L2 norm does not involve nondifferentiability, which is convenient in the gradient-based optimization approach employed in this study. Possibilities of employing other norms are discussed later. Due to numerical difficulties in the simultaneous simulation of two SMB models, one to solve for C(τn ) and another to solve for CCSS (τn ), with potentially different time horizons, an alternative formulation, proposed by Li et al., 6 is used in this work. In their approach, the objective function in Eq. (10) is approximated stage-wise as:

J ≈ J˜ =

P X n=1

Jñ =

P X n=1

tstep,n kC(τn = 1) − CCSS (τn = 1)k22

(12)

where Jñ is the rectangular approximation of the exact integral in Eq. (10) for Stage n, C(τn = 1) is the transient concentration profile at the end of Stage n, and CCSS (τn = 1) is the optimized CSS concentration profile. In the proposed accelerated startup strategy, we always assume that the concentration profiles always converge to the same CSS regardless of the trajectory (history) of the operat-

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ing conditions. This assumption holds as long as there is only a unique cyclic steady state for a given set of CSS operating conditions uCSS ; i.e. the mapping [uCSS , t] 7→ cji (t), ∀t ≥ tstartup is unique. Although this cannot be proved rigorously using the mathematical model, we confirm in our numerical simulation that the above condition holds in all observations. We also observe in our experiments that the steady state product concentrations are always unique, as shown later. 2.4.2

Solution strategy

It is not yet straightforward to solve directly the problem Eq. (11) because the number of stages required to reach the CSS conditions is unknown a priori. To circumvent the numerical difficulties in this problem, the following sequential decomposition algorithm is used, as proposed by Li et al. 6 The objective function is to be evaluated in each stage, one at a time, and nstep,n is fixed in this work. Therefore, the startup acceleration problem is decomposed into a sequence of stage-wise optimization sub-problems, and each sub-problem is formulated as: min un

J˜startup,n = Jñ + αkun − u∗CSS k22

raf s.t. Purityraf A,n ≥ PurityA,n,min

Fx,n ≤ Fmax , F1,n − F2,n ≥ 0,

F1,n − F4,n ≥ 0,

(13a)

ext Purityext B,n ≥ PurityB,n,min

x = 1, 2, 3, 4 F3,n − F2,n ≥ 0,

(13b) (13c)

F3,n − F4,n ≥ 0

(13d)

where α is a coefficient for the penalty function that guides the optimizer to choose the CSS operating conditions, u∗CSS , as C(τn = 1) converges to CCSS (τn = 1) in subsequent stages. The CSS constraints cannot be evaluated explicitly in the above formulation. The purity of

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component A in the raffinate is defined for Stage n of transient SMB operation as:

Purityraf A,n =

R1

craf A (τ )dτ

0

P R1

(14) craf i (τ )dτ

i=A,B 0

and the purity of component B in the extract for Stage n is:

Purityext B,n =

R1

cext B (τ )dτ

0

P R1

(15) cext i (τ )dτ

i=A,B 0

The purity constraints in these optimization sub-problems may be implemented to preserve the product quality during startup in each stage. However, choosing the minimum puext rity requirements for Stage n, i.e., Purityraf A,n,min and PurityB,n,min , n = 1, 2, . . . , P is not a

straightforward decision. This is discussed further in Section 4.3. The approximated SMB startup acceleration problem is solved iteratively using an algorithm based on the one proposed by Li et al. 6 The steps of the solution algorithm used in this work are shown in Fig. 5 and are outlined below: 1. Initialize and set n = 1; 2. Set number of SMB steps in Stage n, initial guess of un , and update initial concentration profile, C(τn = 0); 3. Solve stage-wise sub-problem given by Eq. (13); 4. Check convergence of operating conditions given by kun − u∗CSS k2 ≤ ǫOC . If the condition is satisfied, then u(t) ≃ u∗CSS , ∀t ∈ (tn , tCSS ], and the algorithm is terminated in Stage n. If this condition is not satisfied, then let n = n + 1, return to Step 2, and repeat.

18


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For updates of the initial concentration profile in Stage n, the final concentration profile from Stage n − 1 is used. Therefore, C(τn = 0) = C(τn−1 = 1). It should be noted that the termination criteria given in Step 4 of this solution algorithm is changed from the CSS convergence constraint given by Eq. (11b). We check the convergence of the operating conditions to u∗CSS , so that the algorithm can be terminated once the operating conditions converge. Although Eq. (11b) may not be satisfied at the stage of algorithm termination, the converged operating conditions lead to the satisfaction of Eq. (11b) at a later stage. Step 1.

Step 2.

Initialize and set n = 1

Set nstep,n , initial guess of un , initial C(2n=0)

Solve Eq. (15)

Step 3.

Step 4.

Check:

un u *CSS

2

d H OC

NO

YES

n=n+1

Terminate

Figure 5: Algorithm for solving stage-wise SMB startup acceleration problem

In this work, each stage is defined to be a cycle of SMB operation, and this strategy is also used by Li et al. 6 . However, the decision about how many steps per stage, nstep,n , has not been fully explored in simulation or experiment, and remains an open question for further research. Each SMB startup sub-problem is solved using the nested, single-discretization approach, described by Kawajiri and Biegler, 17 and Jiang et al. 18 There are no explicit CSS conditions in the sub-problems, and thus the number of decision variables is much less than in the si19



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Page 21 of 49

multaneous optimization methods. The system of DAEs resulting from single-discretization of the SMB model using the method of lines is integrated over the time horizon of a singlestage, 19 and the objective function and constraints are evaluated. Each sub-problem is written in gPROMS and solved using SRQPD in an iterative scheme until the startup acceleration solution algorithm is terminated. 20 The CSS concentration profile at the end of a cycle, CCSS (τn = 1), is obtained a priori from simulation of the SMB model in gPROMS using the optimal CSS operating conditions, u∗CSS , obtained by the PC method.

3

Experiments

3.1

SMB equipment and HPLC analysis

For all the experiments in this work, a mixture of uridine (CAS 58-96-8, EMD Biosciences) and guanosine (CAS 118-00-3, Spectrum Chemical Mfg. Corp.) dissolved in the desorbent (90% water and 10% methanol) was used as the feed. The composition of each sample taken from the unit was analyzed by a Shimadzu HPLC system with an analytical C18 column (Daisopak, SP-120-10-ODS-BP, Daiso, Japan). The standard deviation of the measured concentration was reported previously in Bentley and Kawajiri. 13 Experiments were performed on a laboratory scale 4-zone SMB unit (CSEP C190, Knauer, Berlin, Germany) which is shown in Fig. 6 schematically. The unit includes four doublepiston pumps, two ultra-violet detectors at the extract and raffinate outlet streams, and a rotary valve with sixteen positions. The ports of the rotary valve are connected to each other by continuous channels. All unoccupied positions were filled with short capillary tubing. Four HPLC columns (YMC-Pack ODS-A, YMC Co., Ltd., Japan) were used in a 4-zone configuration. The HPLC column dimensions are 250 mm length and 10 mm inner diameter. These columns are packed with a C18 stationary phase for reverse phase chromatography with an average particle size of 20 µm. The average pump flow rates were measured periodically using a flow meter (Model 20


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5025000, GJC Instruments Ltd.) at the extract outlet, raffinate outlet, and recycle line between Zone 4 and Zone 1 (referring to Fig. 6) during each experiment to verify the operating conditions. The UV signals at the extract and raffinate points were monitored, but this data was not used for composition analysis or modeling because of poor baseline quality and the need for deconvolution of the signal. 21,22 The pulse injection test in the first step in the prediction-correction approach was performed using the same Shimadzu HPLC system. By carrying out multiple tests of different injection volumes, we confirmed that the peaks of both components are always nearly symmetric and retention times do not change, which indicates that the isotherms are linear. The columns used in the SMB unit were tested in the same HPLC system as the one used in the product analysis. The overall porosity of the column was determined by injecting a solution of uracil dissolved in the mobile phase. The chromatograms of the pulse injection tests are included in the Supporting Information.

3.2

SMB startup experiments

The CSS constraint in Eq. (11b) cannot be checked in an experiment, since the internal liquid concentrations in columns cannot be measured. The only observable state variables raf are the concentrations of the extract and raffinate products, cext i (t) and ci (t) . Thus, the

attainment of the CSS in an SMB experiment must be defined only from these measurable quantities. Furthermore, since online measurement of the individual component is often difficult, we carry out intermittent sampling which is more reliable. 21,22 In this sampling method, only the average concentrations of samples are available. In this work, we carry out product sampling over one step, and thus the concentration can be measured as the average value. The average concentrations of the raffinate and extract over a step after reaching the CSS can be given by:

c¯ext i,CSS

=

−1 t∗step

Z

tCSS +t∗step

cext i,CSS (t)dt, i = A, B

tCSS

21


(16a)


Desorbent

Sampling

Sampling Feed Raffinate

Extract P3

P2

P1

P4

Recycle sampling

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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C2

C1

C3

C4

Zone 1 Zone 2 Zone 3 Zone 4

Figure 6: Schematic of experimental SMB unit with four pumps (P1 through P4), and sampling points at the extract, raffinate and recycle lines. HPLC columns (C1 through C4) are in a 4-zone configuration, connected to a rotary valve

c¯raf i,CSS

=

−1 t∗step

Z

tCSS +t∗step

craf i,CSS (t)dt, i = A, B

(16b)

tCSS

Similarly, the average concentration of the extract and raffinate at the m-th step is given by:

c¯ext i,m

c¯raf i,m

=

=

t−1 step,m

t−1 step,m

Z

tm +tstep,m

cext i (t)dt, i = A, B

(17a)

craf i (t)dt, i = A, B

(17b)

tm

Z

tm +tstep,m tm

We further define the vector of product concentration deviation from the CSS at the m-th step using these cumulative concentrations, as follows:

em = c¯ext ¯ext ¯ext ¯ext ¯raf ¯raf ¯raf ¯raf A,CSS − c A,m , c B,CSS − c B,m , c A,CSS − c A,m , c B,CSS − c B,m 22


(18)

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Using this vector, the CSS condition Eq. (11b) can be approximated as follows:

||em || ≤ ǫ¯CSS

(19)

where ǫ¯CSS is a tolerance for the CSS. Note that the observability in the constraints is reduced substantially by this approximation; the dimension of the vector (C (τP = 1) − CCSS (τP = 1)) in Eq. (11b) is 4N F E, while that of em is only four. The startup time tCSS can be defined using this condition:

tCSS = min{tm : ||en || ≤ ǫ¯CSS , n = m, m + 1, m + 2, ...}

(20)

where tm is the final time of the m-th step. Note that there are many choices for the vector norm ||em ||. As discussed in Section 2.4, the CSS condition is checked by the L2 norm of the vector (C − CCSS ) when the accelerated strategies are obtained in our optimization approach (Eq. (12)). Accordingly, the L2 norm of ei can also be used in the approximated CSS condition Eq. (20) which gives the following condition: 2 raf 2 raf 2 ext 2 ≤ (¯ǫCSS )2 , n = m, m+1, m+2, ...} (21) tCSS = min{tm : eext A,n + eB,n + eA,n + eB,n In the above condition, the squared sum of both components in the raffinate and extract products must be sufficiently close to the value at the CSS. An alternative definition of the startup time using the L∞ norm is given in the Supporting Information. The tolerance ǫ¯CSS must be chosen carefully considering the accuracy of the measurement in the experiment. In our study, the product concentrations are measured by HPLC, as described in Section 3.1. From our analysis on the standard deviation of the HPLC measurement, we find 2.0 mg/L is sufficiently larger than the standard deviation 13 . From this observation, we choose the tolerance for the four-dimensional vector in Eq. (21) as follows:

23



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ǫ¯CSS =

√

2.02 × 4 = 4.0 mg/L.

The accelerated startup experiments are carried out as follows: At t = 0 the operating conditions are set to u1 , the optimal operating conditions determined for Stage 1 by solving Eq. (13). For the first cycle of operation there is no sampling because the concentrations are low. After Stage 1 is complete, the operating conditions are switched to u2 , and the first sample is taken from the extract and raffinate products during the first step of the second cycle. Similarly, the operating conditions are switched to un in Stage n, and the products are sampled once per cycle at the beginning of each cycle. After Stage P is completed (see Fig. 4), the optimal CSS operating conditions are used, and the product sampling continues to check the consistent and robust operation of the SMB at CSS.

4

Results and Discussion

4.1

Model development by the PC method

It is crucial to have a reliable SMB model for prediction of transient startup behavior. In this study, the SMB model parameters and optimized CSS operating conditions are determined systematically by following the PC method. 13,14 In our previous work, the same Knauer SMB unit was optimized for the separation of uridine and guanosine using the PC method by Bentley and Kawajiri. 13 In the present work, two additional iterations of the PC method are performed on an eight-column SMB unit to obtain the refined parameters used for modeling transient SMB operation. Due to trouble with our equipment, the number of columns was reduced from eight to four, and the validity of the mathematical model was examined later. After two brief SMB experiments, the SMB model parameters were obtained by solving Eq. (7) and the CSS operating conditions were optimized by solving Eq. (4) to obtain u∗CSS . The SMB design and model parameters used in all the startup acceleration experiments are given in Table 1. The optimized CSS operating conditions, u∗CSS , are shown in Table 2. The entire iteration history of the prediction24


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correction method is given in the Supporting Information. Table 1: System parameters for separation of uridine (A) and guanosine (B) Parameter Overall porosity, ǫb Henry’s constant for uridine, HA Henry’s constant for guanosine, HB Mass transfer coefficient for uridine, kA [min−1 ] Mass transfer coefficient for guanosine, kB [min−1 ] Column diameter, d [m] Column length, L [m] Number of columns, ncol Operating temperature, T [◦ C] Feed concentration of uridine, cA,feed [mg L−1 ] Feed concentration of guanosine, cB,feed [mg L−1 ]

Value 0.82 0.911 2.59 51.7 26.1 0.01 0.25 4 40.0 42.05 40.75

Table 2: Optimal CSS operating conditions, u∗CSS Operating conditions Step time, t∗step [min] m∗1 m∗2 m∗3 m∗4 Zone 1 flow rate, F1∗ [mL min−1 ] ∗ Feed flow rate, Ffeed [mL min−1 ] Extract flow rate, F ext,∗ [mL min−1 ] Raffinate flow rate, F raf,∗ [mL min−1 ] +

4.2

Value 3.79 2.99 0.913 2.48 0.779 7.00+ 1.46 1.91 1.58

indicates active upper bound in constrained optimization problem

Conventional startup performance

To validate the SMB model, an experiment was carried out using u∗CSS in a four-column SMB unit. The minimum purity set in the SMB optimization problem was 96% for extract and raffinate products with a 1% safety margin (see Eqs. (4b) and (4c) with ∆safety = 0.01). The maximum allowable flow rate in the optimization problem was set at 7.0 mL min−1 due to overall pressure drop considerations. 25



40

Extract concentration ciext(t) (mg/L)

35 30 25 20 15 10 5 0 0

30

60

90

120

150

180

210

Time (min) 40 35

Raffinate concentration ciraf(t) (mg/L)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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30 25 20 Experiment, compound A 15

Experiment, compound B

10

Model

5 0 0

30

60

90

120

150

180

210

Time (min) Figure 7: Cumulative extract, raffinate, and recycle line concentrations using u∗CSS in the SMB unit. Model prediction using refined parameters given in Table 1

26


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In Fig.7 the experimental data and model-predicted product concentrations (average concentrations over each step of operation) for the conventional startup strategy are shown. The operating conditions used in the conventional startup strategy are listed in Table 2. It can be seen that the model prediction matches well the experimental data. The SMB model is shown to predict well the experimental behavior, although there is still minor model mismatch in the raffinate profile for component A. This model mismatch is probably due to the difference in dead volume between the eight-column and four-column SMB configurations. Nevertheless, the model is adequately reliable to predict the transient dynamics and both product purities exceed 96% at CSS for the maximized feed throughput in this unit.

4.3

Theoretical optimal startup acceleration strategies

An important issue in the startup acceleration problem is that there exists a trade-off between the overall startup time and the cumulative product purities. For example, if the product purity constraints are too strict in the initial cycles of SMB operation, the feed throughput is reduced and the startup time cannot be reduced sufficiently. On the other hand, if the purity constraints are relaxed, the feed throughput can be increased and the startup time can be significantly reduced, although the product may need to be discarded in the initial cycles of operation. In this work, the startup acceleration problem given by Eq. (13) is solved for multiple values of the minimum purity constraints imposed in each stage, yielding a set of optimized startup strategies. However, it is still unclear how to systematically choose these minimum ext purity constraints, Purityraf A,n,min and PurityB,n,min , in each stage that results in the optimal

startup performance. To investigate this trade-off between the overall startup time and cumulative product purities, we perform some case studies, some of which are validated experimentally. Our strategy in these case studies is to use a stage-wise constant purity decrement in the initial

27



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stages according to a parameter, ∆P , as follows: raf raf P urityA,n,min = P urityA,CSS,min − (P − n)∆P

ext ext P urityB,n,min = P urityB,CSS,min − (P − n)∆P ,

n = 1, . . . , P

(22a)

(22b)

where P is the maximum number of stages calculated by solving the stage-wise startup acceleration problem without purity constraints in Eq. (13). Several values of ∆P were considered to calculate optimized startup operating conditions. Initially, the stage-wise startup acceleration problem is solved without imposing any purity constraints in order to determine how many stages are required for startup. In this case study, the number of stages required for startup using the stage-wise approximation without purity constraints was found to be P = 3. We confirm that three stages are always sufficient in all cases considered in this study. The cumulative purity values are calculated using the following equations.

Purityraf A,accum (tu ) =

Rtu

craf A (t)dt

0

P Rtu

(23a) craf i (t)dt

i=A,B 0

Purityext B,accum (tu ) =

Rtu

cext B (t)dt

0

P Rtu

(23b) cext i (t)dt

i=A,B 0

where tu is an arbitrary time of SMB operation where the cumulative product purity is evaluated. For all of the points shown in Fig. 8a, the upper limit of integration for the cumulative product purities is set at tu = 180 (i.e., 3.0 h). The average throughput values, Throughputave , are calculated for each startup strategy:

Throughputave =

ΣPn=1 Ff eed,n tstep,n ΣPn=1 tstep,n

28


(24)

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The average throughput and startup time can be improved by increasing ∆P , which leads to lower purity. Therefore, it appears advantageous to increase ∆P as much as possible before the extract purity requirement is violated. The cumulative product purities, the objective function J˜ given in Eq. (12), and average throughput are shown in Fig. 8. As can be seen in Fig. 8(a), the cumulative extract purity, Purityext B,accum (tu ), decreases more significantly than the cumulative raffinate purity, Purityraf A,accum (tu ) when ∆P is increased. This is because the optimizer gives priority to increasing the internal concentration profile of the less-retained component A, which builds up faster and contaminates the extract product. It should also be noted that J˜ reaches a plateau at around ∆P = 10 (%). Table 3: Predicted cumulative product purities for optimized startup strategies using the purity decrement method Case (a) Case (b)

Purity decrement ∆P = 5% ∆P = ∞

Cumulative purity (%) Purityraf A,accum (t = 180) = 96.4 Purityext B,accum (t = 180) = 96.4 Purityraf A,accum (t = 180) = 97.0 ext PurityB,accum (t = 180) = 93.5

It should be noted that using the purity decrement method with ∆P = ∞ is the same as to ignore the purity constraints in the first two stages. This strategy was employed by Li et al. 6 , where the purity constraints were ignored in Stages 1 and 2, and the product was discarded during the first two stages. We also consider this method as an experimental case study. Using the purity decrement method with ∆P = 5%, some contamination is allowed at the beginning of the operation, and thus the feed flow rate in Stage 1 is increased. As expected, the cumulative purity constraints are satisfied for this case, which we verify experimentally. It should also be noted that if ∆P > 22.5%, the purity constraints are inactive in the optimal solutions to Eq. (13) and the product purities does not get any worse. In this work we verify two case studies experimentally. Case (a) employs a purity decre29



Cumulative purity (%)

100

Extract, t=180 min Raffinate, t=180 min

98

conventional

96 94 92 90 0

5

10

15

20

25

30

Purity constraint decrement, ûp (%)

Objective Function, ´ ×10-6 (mg2 min /L2)

3.0 2.5

conventional

2.0 1.5 1.0 0.5 0.0 0

5

10

15

20

25

30

Purity constraint decrement, ûp (%) Average throughput (mL min-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2.5 2.0 1.5 conventional 1.0 0.5 0.0 0

5

10

15

20

25

30

Purity constraint decrement, ûp (%)

Figure 8: Simulated (a) cumulative purities after 180 min, (b) overall startup times, and (c) average throughput values for various ∆P . Conventional startup strategy values are shown with the dotted line 30 ACS Paragon Plus Environment

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ment of ∆P = ∞, which is expected to achieve the maximum startup acceleration at the sacrifice of violating purity constraints. Similarly, Case (b) employs ∆P =5%, which is also expected to achieve significant startup acceleration while preserving the cumulative product purities during startup. In Fig. 9 the operating conditions for Cases (a) and (b), are shown. It is apparent that the operating conditions for Case (a), shown in Fig. 9(a), have sharp changes from one stage to the next. The feed flow rate in Stage 1 is 126% higher than its CSS value, and then it is reduced substantially in Stage 2. Such sharp changes in operation may lead to model mismatch as confirmed in the experimental results discussed in Section 4.4. By contrast, the operating conditions for Case (b), shown in Fig. 9(b), have smoother changes, and this may result in more predictable SMB operation compared to Case (a).

4.4

Experimental startup acceleration performance

The experimental results for Cases (a) and (b) are compared in terms of overall startup time, average product concentration, desorbent consumption, and cumulative product purities. In both case studies, it is shown that significant performance enhancements are achieved over the conventional startup strategy. First the experimental data and model predictions for the two startup strategies are compared. In Fig. 10 the experimental and model-predicted product concentrations for Case (a) are superimposed on the conventional startup results. The control profiles for Case (a) are shown on the left in Fig. 9. The model mismatch is apparent in Fig. 10 in the first few cycles of operation. This mismatch is probably because there are sharp changes in the operating conditions during Stage 1 and Stage 2. The model prediction fits the CSS performance and the product concentrations are built up rapidly in the SMB unit. The operating conditions used in this strategy cause a significant contamination in the extract during the first two stages of operation. The extract product in the first two cycles can be discarded in order to improve the extract purity. 31


Case (a) Step time (min)

Extract flow rate (mL min-1) Feed flow rate (mL min-1)

0 3.5 3 2.5 2 1.5 1 0.5 3.5 3 2.5 2 1.5 1 0.5 3 2.5 2 1.5 1 0.5 0

0

4

4

4

12

8

8

8

16

20

24

12 16 20 24

0

4

8

3.5 3 2.5 2 1.5 1 0.5

12 16 20 24

12 16 20 24

12 16 20 24

Operation steps

0 7.5 7 6.5 6 5.5 5 4.5 3.5 3 2.5 2 1.5 1 0.5

Raffinate flow rate (mL min-1)

0

8

Zone 1 flow rate (mL min-1)

4

Extract flow rate (mL min-1)

0 7.5 7 6.5 6 5.5 5 4.5

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Case (b)

6 5.5 5 4.5 4 3.5 3

Feed flow rate (mL min-1)

Zone 1 flow rate (mL min-1)

6 5.5 5 4.5 4 3.5 3

Raffinate flow rate (mL min-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Step time (min)


4

8

12

16

20

24

0

4

8

12 16 20 24

0

4

8

12 16 20 24

0

4

8

12 16 20 24

0

4

8

12 16 20 24

3 2.5 2 1.5 1 0.5 0

Operation steps

Figure 9: Optimized operating conditions for Case (a) and Case (b) startup strategies

32


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40 35

Conventional experiment, compund A

30

Conventional experiment, compound B

25 20

Case (a) experiment, compound A

15

Case (a) experiment, compound B

10

Conventional model

5

Case(a) model

0 0

30

60

90

120

Time (min) 40 35


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


30 25 20 15 10 5 0 0

30

60

90

120

Time (min) Figure 10: Experimental and model-predicted product concentrations for Case (a) superimposed on conventional startup results

33




40 35

Conventional experiment, compound A

30

Conventional experiment, compound B

25 20

Case (b) experiment, compound A

15

Case (b) experiment, compound B

10

Conventional model

5

Case (b), model

0 0

30

60

90

120

Time (min)

40 35


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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30 25 20 15 10 5 0 0

30

60

90

120

Time (min) Figure 11: Experimental and model-predicted product concentrations for Case (b) superimposed on conventional startup results

34


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In Fig. 11 the experimental and model-predicted product concentrations are shown for Case (b), which are also superimposed on the conventional startup results. The operating conditions for Case (b) are shown on the right of Fig. 9. It can be seen that the model prediction fits well the experimental behavior, even in the first few cycles, and this is probably because there are relatively smooth changes in the operating conditions during startup. Nevertheless, the model prediction fits well the CSS performance and the product concentrations are built up rapidly in the SMB unit. The operating conditions used in this strategy cause some contamination in the extract and raffinate during the first two stages of operation, yet the cumulative product purities are both greater than 96% after 180 min of operation as shown later. Thus, there is no need to discard any of the products collected in the initial stages of operation. Figure 12 shows the deviation in the concentration from the CSS. As can be seen in this figure, both of the accelerated startup strategies, Case (a) and Case (b), reduce the deviation from the CSS more quickly than the conventional startup strategy. The startup time for the tolerance of ǫ¯CSS =4.0 (mg/L) is given in Table 5. The startup times which were observed experimentally were obtained by linearly interpolating the discrete values. Although all startup times in the experiment are longer than the model error due to the model mismatch, the accelerated strategies reduce the startup time by approximately 30%. It should be noted that the startup time depends on the choice of the tolerance ǫ¯CSS . An alternative analysis using the L∞ norm of em is given in the Supporting Information. Table 4: Comparison of startup times for ǫ¯CSS =4.0 (mg/L) Model (min) Experiment, obtained by linear interpolation (min)

Conventional 52.98 61.19

Case (a) 34.11 42.59

Case (b) 37.66 38.44

In Fig. 13 experimental cumulative product concentrations are compared for the three

35



2500 80 Concentration deviation from CSS ||em||22 (mg2/L2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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60 2000

40 ‹CSS2

20

16.0 (mg 2 /L2 )

0

1500

30

60

90

Conventional experiment Conventional model Case (a) experiment Case (a) model Case (b) experiment Case (b) model

1000

500

0 0

30

60

90

120

Time (min)

Figure 12: Comparison of deviation from the CSS ||em ||22 . The horizontal line shows the tolerance ǫ¯2CSS .

36


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startup strategies. The cumulative product concentrations are: 1 Cumulative extract concentration of B = t

Zt

cext B (t) dt

(25a)

Zt

craf A (t) dt

(25b)

0

1 Cumulative raffinate concentration of A = t

0

which are approximated using the trapezoidal rule with the experimental data points. The product concentrations increase rapidly in Cases (a) and (b) during the first few cycles of operation due to the average increase in throughput and decrease in desorbent consumption. The concentrations are about 20% greater in Case (a), and about 15% greater in Case (b) than the conventional startup strategy after 100 min of operation. This can lead to significant cost savings in reduced processing time of a fixed amount of feed, and reduced evaporation cost of solvent recovery from the products after SMB processing. The relative desorbent consumption (normalized by the value of the conventional startup strategy) is compared in Figure 14, which is defined as:

Relative desorbent consumption =

Rt

accel Fdes (t) dt

0

Rt

(26) conv Fdes (t) dt

0

accel conv where Fdes (t) and Fdes (t) are the desorbent flow rate of the accelerated and conventional

strategies, respectively. It should be noted from Fig. 14 that both accelerated startup strategies, Cases (a) and (b), require nearly the same desorbent consumption in this case study. Employing an accelerated startup strategy achieves a reduction of about 30% in desorbent consumption in three hours. This can result in significant cost savings if a non-volatile solvent is used in the mobile phase or if there is a high turnover frequency with the SMB equipment. In Figure 15, there is a comparison of three startup strategies in terms of product purities 37


Page 39 of 49

35 30 25 20

Conventional

15

Case (a)

10

Case (b) 5 0 0

30

60

90

120

150

120

150

180

Time (min) Cumulative raffinate concentration (mg/L)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Cumulative extract concentration (mg/L)


35 30 25 20 15 10 5 0 0

30

60

90

180

Time (min) Figure 13: Comparison of cumulative product concentrations over time for three startup strategies

38


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1.0 Relative desorbent usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8 0.6 Conventional 0.4

Case (a) Case (b)

0.2 0.0 0

50

100 Time (min)

150

200

Figure 14: Comparison of desorbent consumption over time for different startup strategies

observed experimentally. The target purities were 96% in both raffinate and extract products. As can be seen in this figure, the raffinate purity in Case (a) was significantly lower than the target value, while in Case (b) the drop was less significant. In contrast, the purity drop in the extract product was less pronounced in both cases. These observations are in agreement with the prediction by the model. The purity drop in the extract product both in Case (a) and (b) at around 50 minutes was probably caused by the error in implementing the operating condition; we believe the timing of the switching from the second stage to the third stage (8th step) shown in Figure 9 was not exact. The cumulative purity measured experimentally in 180 min is summarized in Table 5. The product purities were measured in the collected product tanks after 3.0 hrs of operation assuming that none of the product is discarded. The target purities were 96% in both raffinate and extract products with a 1% safety margin. The target purity was not reached in the extract product for Case (a) after 3.0 hrs of startup operation as predicted by the model. Therefore, in this startup strategy the products in the first two cycles should be discarded in order to maintain the cumulative product purity, which could be unacceptable if the feed is expensive. On the other hand, Case (b) does not have the same drawback, and

39



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the purity constraints are satisfied within the safety margin after 3.0 hrs of operation. As predicted by the model, even though the products collected in the first one or two cycles are off-spec, after a few more cycles the purity is re-established in the product tank and thus no products need to be discarded during startup. In this case study with a linear isotherm system, the conventional startup strategy maintains higher purity during startup but at the sacrifice of significantly longer startup time. In a nonlinear isotherm system, however, there may be significant violations in the product purities even using the conventional startup operation, as reported by Li et al. 6 In such a case, the proposed startup strategy with purity constraints could resolve this issue. Table 5: Cumulative product purities measured experimentally for three startup strategies Strategy Conventional Case (a) Case (b)

5

Purityraf A,accum (t = 180 min) (%) 97.8 98.0 97.6

Purityext B,accum (t = 180 min) (%) 97.2 94.5 96.1

Conclusions and Future Work

It is shown experimentally that the startup time for an SMB unit can be significantly reduced by solving model-based startup optimization problems using the SMB model developed by following the PC method. The optimized startup strategy using the purity decrement method with ∆P = ∞ was shown to be successful at increasing the product concentration more rapidly although there was model mismatch in the first few cycles of operation. On the other hand, the optimized startup strategy using the purity decrement method with ∆P = 5% was also successful, and the model matched the experimental data well while some model mismatch remains for the impurity components. Using a relatively simple three-stage startup acceleration strategy can yield significant benefits in terms of decreased processing time, increased average product concentrations, and decreased desorbent consumption.

40


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ext ext ext Extract purity cB ,m / (c A,m cB ,m ) u100 (%)

100

Raffinate purity c Araf,m / (c Araf,m cBraf,m ) u100 (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


95 90 85 Conventional

80

Conventional model

75

Case (a)

70

Case (a) model

65

Case (b)

60

Case (b) model

55 50 0

30

60 90 Time (min)

120

150

180

0

30

60 90 Time (min)

120

150

180

100 95 90 85 80 75 70 65 60 55 50

Figure 15: Comparison of purity over time for different startup strategies

41



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Page 43 of 49

Although the startup acceleration strategies were shown to be successful in this work, there remain some opportunities for further improvements. For example, it may be possible to gain further startup acceleration if more stages are employed with more frequent switching of the operating conditions (i.e., P > 3). However, implementation of such a startup strategy would introduce more complexity and operational uncertainty to the process due to frequent switching of the flow rates and step time. Hence, the number of operating steps per stage, nstep,n , should be further investigated. There remain many questions which are subject to future work. In addition to startup, shutdown of SMB can be accelerated, as demonstrated by Li et al. 6 By accelerating the shutdown, water evaporation cost can be reduced further in sugar applications, and the shortened campaign as shown in Figure 2 can be realized which allows processing small batches of different feeds in pharmaceutical applications. The previous work by Li et al. considered a nonlinear isotherm in a computational study, which will be also validated experimentally in the future. Finally, the effect of isotherm and mass transfer properties as well as the system configurations such as the number and length of columns on the overall startup time should be investigated. We found in this work that the mathematical model developed by the prediction-correction approach predicted the experimental performance well in general, although some model mismatch remains. When there was a sharp change in the operating condition, we observed deviation of our model from the experimentally observed values. The model accuracy may need to be improved in applications of high target purities. Furthermore, in such applications, the standard deviation in the product concentration analysis may need to be be quantified accurately to determine the tolerance in the CSS condition carefully.

42


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6

Nomenclature

A

less-retained component

B

more-retained component

c

fluid phase concentration

c¯

average concentration over a step vector of internal concentration profile

C

vector of optimized CSS internal concentration profile

CCSS d

column diameter vector of concentration deviation from CSS at m-th step

em F

volumetric flow rate

H

Henry’s constant

J

startup acceleration objective function

J˜

approximated objective function

J˜startup

approximated stage-wise objective function

k

overall mass transfer coefficient

L

column length

m

fluid-to-solid flow rate ratio number of columns

ncol nstep

number of operation steps

NE

number of experiments total number of spatial finite elements for discretization

NF E NM

number of measurements

NV

number of variables

P

number of stages for startup

q

adsorbed phase concentration

q eq t

adsorbed phase concentration in equilibrium with fluid phase time 43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

time that Stage n begins

tn tCSS

time to reach CSS conditions

tstep

time of operation step

tu

arbitrary time to evaluate cumulative purity

T

temperature

u

vector of operating conditions

v

linear mobile phase velocity total column volume

Vcol y

model-predicted value

y˜

experimentally measured value

z

axial coordinate in column

Greek letters α

penalty function coefficient

γ

binary variable

∆P

purity decrement parameter

∆safety

safety factor for purity

overall bed porosity

ǫb ǫconc

tolerance for concentration profile convergence to CSS

ǫOC

tolerance for operating conditions convergence to CSS

ǫ¯CSS

tolerance for product concentration convergence to CSS

θ σ2 τ

set of model parameters variance dimensionless time coordinate

Subscripts and superscripts ∗

optimized value

44


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


accum ave CSS d

cumulative value average value cyclic steady state value

measurement index

des

value for desorbent stream

ext

value for extract stream

feed

value for feed stream

i in

adsorbable component index value for inlet stream

j

column index

k

iteration counter

l

variable index

m

alternative stage index

n

stage index

out p raf x

value for outlet stream experiment index value for raffinate stream zone index

Acknowledgement Special thanks to Andrew Tatum for his assistance in the laboratory experiments. The technical support from Asahi Kasei Technikrom, Daiso, and YMC America is gratefully acknowledged.

Supporting Information Available This information is available free of charge via the Internet at http://pubs.acs.org/.

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References (1) Broughton, D. Continuous Sorption Process Employing Fixed Bed. 1961; US Patent 2,985,589. (2) Ruthven, D. Principles of adsorption and adsorption processes; Wiley New York, 1984. (3) Ruthven, D. M.; Ching, C. B. Counter-Current and Simulated Counter-Current Adsorption Separation Processes. Chem. Eng. Sci. 1989, 44, 1011–1038. (4) Guiochon, G.; Felinger, A.; Shirazi, D. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: San Diego, 2006. (5) Schmidt-Traub, H. Preparative Chromatography: Of Fine Chemicals and Pharmaceutical Agents; Wiley VCH: Weinheim, 2005. (6) Li, S.; Kawajiri, Y.; Raisch, J.; Seidel-Morgenstern, A. Optimization of startup and shutdown operation of simulated moving bed chromatographic processes. J. Chromatogr. A 2011, 1218, 3876–3889. (7) Lim, B.-G.; Ching, C.-B. Modelling studies on the transient and steady state behaviour of a simulated counter-current chromatographic system. Separ. Technol. 1996, 6, 29 – 41. (8) Xie, Y.; Mun, S.-Y.; Wang, N.-H. L. Startup and Shutdown Strategies of Simulated Moving Bed for Insulin Purification. Ind. Eng. Chem. Res. 2003, 42, 1414–1425. (9) Bae, Y.-S.; Moon, J.-H.; Lee, C.-H. Effects of Feed Concentration on the Startup and Performance Behaviors of Simulated Moving Bed Chromatography. Ind. Eng. Chem. Res. 2006, 45, 777–790. (10) Bae, Y.-S.; Kim, K.-M.; Moon, J.-H.; Byeon, S.-H.; Ahn, I.-S.; Lee, C.-H. Effects of flow rate ratio on startup and cyclic steady state behaviors of simulated moving bed under linear conditions. Sep. Purif. Technol. 2008, 62, 148–159. 46


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www.psenterprise.com.

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Experimental Validation of Optimized Model-Based Startup

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