Simultaneous modeling and optimization of nonlinear simulated moving bed chromatography by the prediction–correction method

Journal of Chromatography A, 1280 (2013) 51–63

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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma

Simultaneous modeling and optimization of nonlinear simulated moving bed chromatography by the prediction–correction method Jason Bentley, Charlotte Sloan, Yoshiaki Kawajiri ∗ School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

a r t i c l e

i n f o

Article history: Received 28 September 2012 Received in revised form 3 January 2013 Accepted 5 January 2013 Available online 16 January 2013 Keywords: Simulated moving bed chromatography Process development Optimization Parameter estimation Nonlinear isotherm Model selection

a b s t r a c t This work demonstrates a systematic prediction–correction (PC) method for simultaneously modeling and optimizing nonlinear simulated moving bed (SMB) chromatography. The PC method uses modelbased optimization, SMB startup data, isotherm model selection, and parameter estimation to iteratively refine model parameters and find optimal operating conditions in a matter of hours to ensure high purity constraints and achieve optimal productivity. The PC algorithm proceeds until the SMB process is optimized without manual tuning. In case studies, it is shown that a nonlinear isotherm model and parameter values are determined reliably using SMB startup data. In one case study, a nonlinear SMB system is optimized after only two changes of operating conditions following the PC algorithm. The refined isotherm models are validated by frontal analysis and perturbation analysis. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Simulated moving bed (SMB) chromatography is a continuous separation technology that can be used to resolve complex mixtures. Selective separation of a binary mixture can be achieved using SMB even if the separation factor is close to one. The SMB process has advantages over batch chromatography such as increased productivity and decreased solvent consumption. It is currently being used in a number of applications in the life sciences including sugars, enantiomers, and proteins. More details on SMB processes can be found in numerous references [1–3]. The attractive feature of SMB is that it simulates the counter-current flow of the stationary phase by rotating the positions of the inlet and outlet ports in the direction of fluid flow. This process allows for both continuous feeding and withdrawal of purified extract and raffinate products. Although SMB technology has been in use for about 50 years, finding optimal operating conditions for most systems is not yet a straightforward task due to the discontinuous motion of the inlet/outlet ports and complex kinetics and equilibria. For a new separation problem, if there is only a small amount of feed mixture available for lab and mini-plant tests, it may be impossible to systematically optimize the SMB process consuming a large amount of the feed mixture. Therefore the SMB is usually operated under conservative conditions to ensure high purity constraints are satisfied

∗ Corresponding author. Tel.: +1 404 894 2856. E-mail address: [email protected] (Y. Kawajiri). 0021-9673/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2013.01.026

with some safety margins; this is achieved, however, at the cost of reduced productivity and increased desorbent consumption. The typical process development scheme for a new separation problem relies on modeling with results from batch experiments, steady state SMB experiments, and manual tuning. In Fig. 1 there is an illustration of the typical steps of SMB process development. A new separation problem poses many challenges for the process designer including the estimation of reliable SMB model parameters for each component of the mixture. Once model parameters are determined, usually by batch experiments, operating conditions are selected based on design criteria that approximate SMB dynamics [4]. By observing the steady state SMB performance, the operating conditions are tuned manually until process specifications are met. Sometimes mini-plant experiments and computer simulation of a process model are used to aid in the design [5]. Once the desired operation is satisfied, the process may be controlled to maintain the target performance. Perhaps the most important aspect of SMB process design is the selection of a process model, including adsorption isotherm, and the determination of model parameters. Some detailed process models for liquid chromatography have been developed and there are a number of techniques that chromatographers have used for obtaining adsorption isotherm and kinetic parameters in the literature [2,6]. Details on moment analysis of pulse injections, frontal analysis, elution by characteristic points, perturbation analysis and inverse methods can be found [2,3,7,8]. Grosfils et al. [9] compared different single-column models for SMB modeling, such as the equilibrium-dispersion model and the kinetic model,

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J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

New sample

Batch experiments

SMB experiments

Desired producon

Product

CONTROL OPTIMIZATION MODELING Convenonal process development

MODELING + OPTIMIZATION Predicon-correcon method Fig. 1. Illustration of steps of SMB process development starting with a new mixture. The prediction–correction method is proposed to streamline SMB process development by simultaneous modeling and optimization.

and studied the identifiability of model parameters using pulse injections. These researchers used design of experiments to reduce the effort required to obtain reliable model parameters for SMB, although they do not consider isotherm model selection. In most case studies of nonlinear SMB process development in the literature, the adsorption isotherms of each component are characterized by batch experiments and then an isotherm model is assumed and used to fit the data. For example, Heuer et al. [10] used a careful perturbation analysis to estimate modified competitive Langmuir isotherm parameters for the design of enantiomer separation by SMB. Even though the estimated parameters were reliable for single-column tests, there was still significant model mismatch in the prediction of product concentrations and purity values. Indeed, there is no guarantee that the SMB modeling based on results of batch experiments will accurately predict SMB performance. Grosfils et al. [11] used a inverse method to determine competitive Langmuir isotherm parameters from two pulse injections. Using an SMB model with careful measurements of dead volumes in a real SMB plant, the process simulation was compared with plant data via UV detector signals. The inverse method has been shown to be a reliable parameter estimation technique in predicting batch experimental results for enantiomer separations [8], and is useful for isotherm model. Some other work has been done to attempt parameter estimation using SMB data. Küpper et al. [12] proposed and demonstrated a concept for SMB model parameter estimation using online measurements and computational simulation. They conclude that the model parameters can be estimated online by measurements of the extract, raffinate and recycle lines and can be adapted to physical changes in the column properties such as porosity. Yet their work requires online detection of entire internal concentration profiles over a step, and no optimization is performed using the estimated parameters. Araújo et al. [13] used a hybrid inverse method to determine competitive Langmuir isotherm parameters for an SMB model using numerous batch experiments and a periodic state represented on single-column set-up. The model parameters are refined by parameter estimation using the single-column cyclic steady state data, but these are only an approximation of real SMB dynamics and no optimization of the SMB process is performed. There are other researchers who have worked on optimizing control to maintain the desired production of the SMB process (see Fig. 1), by correcting operating conditions in the presence of disturbances. Klatt et al. [14] described a control strategy where the operating conditions and the assembled elution profile are measured during the periodic SMB operation and adjustments are made automatically to correct deviations from the target product purities. Further work on optimizing control for SMB has been done at ETH Zurich by Grossmann et al. [15] where average product concentrations are measured during a cycle as feedback information for a controller. This measurement strategy uses HPLC to

analyze the product concentrations, which is a more accurate measurement than online UV spectra. Their methodology assumes that a reliable set of process model parameters and optimal operating conditions for the SMB are known a priori. They do not consider updating the SMB model parameters using product concentration data. The goal of this work is to propose and demonstrate a systematic method for SMB process development, the prediction–correction (PC) algorithm, for nonlinear SMB optimization starting from a new separation problem and ending with optimal operating conditions. This algorithm uses a surrogate model (see Biegler et al. [16]), infrequent sampling of SMB outlet streams, parameter estimation, and dynamic optimization to reduce the time and effort to obtain reliable model parameters and simultaneously optimize nonlinear SMB processes in a systematic manner. It should be emphasized that the PC method is not designed as a technique for online control, but is an integrated modeling and optimization technique which can be used with a controller in a complementary manner. This paper is organized as follows: The detailed steps of the prediction–correction algorithm are discussed in Section 2, our laboratory equipment is detailed in Section 3, and case studies with a nonlinear isotherm system are presented in Section 4. The case studies show that the model selection and parameter estimation steps are successfully performed and the PC algorithm converges efficiently and robustly to optimize the SMB operation for a nonlinear system.

2. Methodology The PC method is an iterative scheme that uses SMB startup experiments, parameter estimation, and model-based optimization to rapidly achieve optimal operating conditions for a new separation problem. This method is described and demonstrated in our previous work for a linear isotherm system [17], which is modified in this work to allow isotherm model selection. Here, we assume that SMB column configuration, operating scheme, mobile phase, feed composition and operating temperature are decided a priori. The PC algorithm with model selection is shown in Fig. 2. During process development for a new separation problem the adsorption isotherm is unknown. Instead of spending valuable time and resources performing batch experiments to explore the adsorption isotherm for each component, we use a set of SMB startup data to perform model selection and parameter estimation. Here is a step-by-step walkthrough of the updated PC algorithm: 1. Initialize k = 0. A set of batch experiments are performed on a single column to estimate isotherm and kinetic parameters for the SMB model.

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

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2.1. Step 1: batch experiments (SMB modeling) 2.1.1. Pulse tests Pulse tests are performed to measure the overall porosity of the packed bed in a column and determine initial isotherm and kinetic parameters for each adsorbed component in the feed mixture. The following pulse tests should be done: i. Small-volume pulse injection of tracer compound to estimate the overall porosity of the column. ii. Small volume pulse injection of feed mixture (low concentration) to estimate the Henry’s constants and mass transfer coefficients by retention times and the number of theoretical plates (NTP) for each species. iii. Increased volume pulse injection of feed mixture (increased concentration) for parameter estimation by the inverse method. These simple pulse tests yield a set of initial model parameters that are not measured at various conditions, and they do not account for any dead volumes or mixing behaviors in the real SMB unit. In our approach we do not perform laborious batch experiments to rigorously characterize the adsorption isotherm or account for parameter dependencies on the flowrate. For example, we do not use loading tests such as frontal analysis or perturbation analysis, van Deemter relation [3] which shows the effect of flowrate on mass transfer coefficients, or extra-column dead volume models to account for real dead volumes in the SMB unit [18]. The initial parameter estimates for a relatively simple model obtained by pulse tests are used only to calculate initial SMB operating conditions, and they are later corrected iteratively.

Fig. 2. prediction–correction (PC) algorithm with model selection. Model selection [in brackets] is performed during parameter estimation in Step 4.

2.1.2. SMB model The SMB model presented below is described in more detail in our previous work [17]. The SMB consists of columns in series that are assumed to be at isothermal conditions with a homogeneous stationary phase. The one-dimensional component mass balance in the liquid phase is, j

2. Model-based optimization of the SMB is performed using nonlinear programming (NLP) to obtain an initial set of optimal operating conditions. 3. An SMB experiment is performed using the kth optimal operating conditions and a set of concentration data and product purities are obtained. 4. Let k = k + 1. Model selection is performed by solving a parameter estimation problem with different isotherm models. Then, the isotherm model is fixed and the estimated parameter values for the SMB model are used (correction step). 5. Model-based optimization of the SMB is repeated with the selected isotherm model and refined parameters to obtain updated operating conditions (prediction step). 6. The termination criteria are checked: if the product purity constraints were met by the kth SMB experiment and the k + 1th optimal objective function value is not significantly different from the kth optimal objective function value, then the algorithm is terminated. If not, then return to Step 3 and repeat. By adding model selection in Step 4, it is possible to optimize the SMB with a given isotherm model and then discover if a different isotherm model can significantly improve prediction of real SMB performance. This is a useful step if the initial guess of isotherm model is inadequate to describe the real system. For example, it may be assumed that the system is linear at first and then a nonlinear isotherm model can be selected, following systematic criteria, which fits the SMB experimental data.

∂ci (z, t) ∂t i = A, B

j

+

j

∂c (z, t) 1 − εb ∂qi (z, t) = −vj (t) i εb ∂t ∂z

j = 1, 2, . . . , ncol

(1)

j

where ci (z, t) is the liquid concentration of component i in column j

j at axial position z at time t, εb is the overall bed porosity, qi (z, t) is the adsorbed concentration of component i in column j, vj (t) is the linear mobile phase velocity in column j, component A is weakly retained, component B is strongly retained, and ncol is the number of columns in the SMB. The component mass balance in the adsorbed phase is approximated by the linear driving force (LDF) model, j

∂qi (z, t) ∂t

eq,j

= ki (qi

j

(z, t) − qi (z, t))

(2)

where ki is the overall mass transfer coefficient of component i, and eq,j qi (z, t) is the theoretical equilibrium adsorbed concentration of component i in column j. This equilibrium adsorbed concentration is given by the isotherm model, eq,j

qi

(z, t) = f (cA , cB )

(3)

where f is some function of the liquid concentrations of each component. In this work, we consider three isotherm models. The first is the linear isotherm, eq,j

qi

j

(z, t) = Hi ci (z, t)

(4)

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J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

where Hi is the Henry’s constant for component i. Eq. (4) usually applies when the adsorbing components are dilute in the mobile phase. The second is the single-component Langmuir (sLangmuir) isotherm, j

eq,j

qi

(z, t) =

Hi ci (z, t)

(5)

j

1 + bi ci (z, t)

where bi is the equilibrium constant for component i. Eq. (5) is a nonlinear isotherm that considers the adsorption of each component independently. The third is the competitive Langmuir (cLangmuir) isotherm, j

eq,j

qi

(z, t) =

Hi ci (z, t) j

(6)

j

1 + bA cA (z, t) + bB cB (z, t) eq

where the same bi values are used in both equations for qA and eq qB . Eq. (6) is a nonlinear isotherm that assumes both components compete for the same adsorption sites in the stationary phase. Most physical systems with monotonically increasing adsorption isotherms may be adequately described by one or more of these three isotherm models. More examples of adsorption isotherm models can be found [19]. 2.1.3. Inverse method Initial parameter values may be estimated by the inverse method. In this work, a single-column model is written in gPROMS using Eqs. (1)–(3). The chromatograms obtained from pulse tests described in Section 2.1.1 are used to solve a maximum likelihood estimation problem in gPROMS. The methods involved in this calculation are the same as those described in Section 2.4. A similar approach is also used by Grosfils et al. [11]. 2.2. Steps 2 and 5: model-based SMB optimization (prediction step) 2.2.1. Optimization problem formulation The following model-based SMB optimization is performed initially in Step 2 of the PC algorithm as shown in Fig. 1 and then again in Step 5 of each iteration. As in our previous work [17], the SMB optimization objective function, ϕSMB (u) is formulated as maximization of the feed flowrate, Ff which is a measure of productivity. The optimization problem is formulated as, max ϕSMB (u) = Ff Raf

≥ PurityA,min ,

Raf

j+1

(z, tS ),

s.t. PurityA j

(7)

ci (z, 0) = ci

Ext PurityExt B ≥ PurityB,min

j = 1, 2, . . . , ncol − 1

(8) (9)

cincol (z, 0) = ci1 (z, tS ) j

j+1

qi (z, 0) = qi

(z, tS ),

j = 1, 2, . . . , ncol − 1

(10)

PurityRaf A

is the purity of A in the raffinate prodEqs. (1)–(3), where uct, PurityExt is the purity of B in the extract product, PurityRaf is B A,min

is the the minimum purity of A in the raffinate product, PurityExt B,min minimum purity of B in the extract product, u is the control vector of SMB operating conditions, u = [tS , m1 ,. . ., m4 ], tS is the step time, mx is the liquid to solid flowrate ratio in zone x, mx =

Fx tS − Vcol εb Vcol (1 − εb )

2.2.2. Numerical optimization The SMB optimization problem given by Eqs. (1)–(3) and (7)–(10) is formulated as an NLP and solved by the IPOPT solver in AMPL [20,22]. The solution method requires that the PDE system given by Eqs. (1)–(3) be fully discretized in time and space yielding a large-scale optimization problem with algebraic constraints. For initialization of the NLP in Step 2 of the PC algorithm, the operating conditions are selected so that the triangle theory constraints set forth by Mazzotti et al. [4] are satisfied, but these constraints are not needed in subsequent iterations. 2.3. Step 3: SMB experiment An SMB experiment is performed by implementing the kth optimal operating conditions, uk and measuring the concentrations in the extract, raffinate and recycle lines during operation. The composition of each sample is determined using HPLC analysis as described by [15]. The sampling strategy used in this work is described in Section 3.2. The sampling is continued following a set schedule until the following condition is met,

    i=A,B ciExt (n − 1) − i=A,B ciExt (n)  k  ≤ εc     c Ext (n − 1)

(12)

i=A,B i

where ciExt (n) is the concentration of component i in extract sample

n, and εkc is the tolerance for the change in concentration of the extract product in the kth experiment. Once the product sampling is completed, the SMB operation continues using uk , and Step 4 of the PC algorithm begins. In order to determine the CSS performance of the SMB, average extract and raffinate products are collected for a single step. This purity measurement is done before the decision is made to either switch the SMB operating conditions or terminate the PC algorithm and continue operation with uk . 2.4. Step 4: parameter estimation and isotherm model selection

qni col (z, 0) = q1i (z, tS ) Fx ≤ Fmax , x = 1, 2, 3, 4

where Fx is the volumetric flowrate in zone x, and Vcol is the total column volume. With these m-values specified, the optimal flowrates can be calculated given the column volume and overall porosity. The purity constraints in Eq. (8) require that the purity of component A in the raffinate must be at least the minimum required and the purity of component B in the extract must be at least the minimum required. The cyclic steady state (CSS) constraints in Eq. (9) are defined for a single step of the SMB process. This is the same problem implemented for SMB optimization by Kawajiri and Biegler [20], which was first proposed for periodic adsorption processes by Nilchan and Pantiledes [21]. The CSS constraints mean that the internal concentration profiles are equal at the beginning and end of a single step only shifted one column in the direction of fluid flow. The flowrates in each zone are constrained in Eq. (10) by the maximum allowable flowrate based on the total pressure drop limits for the SMB unit.

(11)

2.4.1. Parameter estimation problem definition The SMB model parameters are corrected after the kth experiment by parameter estimation. Each experiment is like a loading test using the SMB unit. The parameter estimation is posed as a constrained maximum likelihood problem, max ϕPE () =

NVp NMpl NE   



2 ln(pln )+

(ˆcpln − cpln )

p=1 l=1 n=1 min max ≤ m ≤ m , s.t. m

m = 1, 2, . . . , npar

2 pln

2



(13)

(14)

where  is the set of model parameters to be estimated, NE is the number of experiments, NVp is the number of variables measured in

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

the pth experiment, NMpl is the number of measurements of the lth 2 is the variance of the nth meavariable in the pth experiment, pln surement of the lth variable in the pth experiment (determined by the variance model), cpln is the nth measured value of the lth variable in the pth experiment and ˆcpln is the nth model-predicted value of the lth variable in the pth experiment. In Eq. (14), the lower bound min is the minimum allowable value of the mth model paramem max is the maximum allowable value of the ter, the upper bound m mth model parameter, and npar is the number of model parameters. The set of model parameters,  includes both the isotherm parameters and mass transfer coefficients. This is the same parameter estimation method used by [17], and the numerical solution was also performed in gPROMS using the SRQPD solver. The parameter estimation routine in gPROMS includes statistical analysis and calculation of 95% confidence intervals. Further information can be found in the gPROMS documentation [23]. The definition of the variance model is needed to calculate the objective function in Eq. (13) and the 95% confidence intervals. The variance model is determined for each component by creating a calibration curve of HPLC data at varying concentration levels. From this regression, the standard deviation (dependent on cpln ) can be calculated. The following linear variance model was used, 2 pln

= (˛cpln + ˇ)

2

(15)

where ˛ and ˇ are the regression coefficients. Other variance models may be used, but the linear model is chosen to fit the trend of HPLC measurements.

55

The value of εMS is chosen to be a positive value less than 1.0, such as 0.5. This systematic approach will ensure that the selected model will simultaneously satisfy a small number of parameters and have sufficient fitting of the experimental data. 2.5. Step 6: termination of the PC algorithm The PC algorithm has systematic termination criteria and the following two criteria must both be satisfied before termination occurs. The termination criteria are: i. The purity constraints are satisfied in the kth SMB experiment. ii. In the k + 1th SMB optimization problem ϕSMB (uk+1 ) should not differ significantly from ϕSMB (uk ). The following condition should be satisfied, k ϕSMB

   ϕSMB (uk ) − ϕSMB (uk+1 )    ≤ εtol =  ϕSMB (uk )

(17)

where εtol is the tolerance for change in the objective function between successive SMB optimization problems. If both of the above conditions are satisfied, then the k + 1th SMB experiment is deemed to be unnecessary, the PC algorithm is terminated and the current operating conditions in the kth experiment, uk are considered optimal. 3. Experimental 3.1. SMB equipment

2.4.2. Isotherm model selection The isotherm model selection may be done systematically for any number of isotherm models. There is a trade-off in model selection between the number of independent model parameters and the maximum likelihood estimation. Some information criteria may be used in order to select from multiple models minimizing a factor that takes into account both of these objectives [24]. In this work, we use a simplified model selection procedure by comparing the number of parameters and the parameter estimation solutions in Eqs. (13) and (14) for three models. Using the data collected in an SMB experiment, the parameter estimation problem can be solved for multiple isotherm models in parallel. The models are ranked based on the number of parameters, and their optimal objective function values are compared. The selection is based on the following prioritization and comparison:

Experiments were performed using a laboratory scale SMB unit (CSEP C190, Knauer, Germany). A schematic diagram of the SMB unit is shown in Fig. 3. Four double-piston pumps (P1 through P4) were used along with two ultra-violet detectors (UV1 and UV2) at the outlet streams. A single rotary valve with 16 positions was used to achieve port switching. Four to eight HPLC columns (C1 through C4) were used (YMC-Pack ODS-A, YMC Inc., Japan). These columns are 25 cm in length and 1 cm in diameter. Each YMC column is packed with C18 stationary phase for reverse phase chromatography with an average particle size of 20 ␮m. While the UV detector data was observed during operation, the signals were not used for composition analysis or comparison with SMB model simulations.

i. The models are ranked in increasing order depending on the total number of parameters and the number of parameters in each equation. For example, the linear isotherm in Eq. (4) has first priority, the sLangmuir isotherm in Eq. (5) has second, and the cLangmuir isotherm in Eq. (6) has third. ii. The optimal objective function values are compared for each pair of isotherm models, and the model with lesser priority is only selected if here is a sufficient decrease in the objective function value. The following comparison is made for each pair of isotherm models,

 r,∗  s,∗   ϕPE − ϕPE    ϕr,∗  ≥ εMS ,

r = 1, 2, ..., nmod − 1

PE

s = 2, 3, . . . , nmod , s > r

(16)

r,∗ is the optimal objective function value for the rth where ϕPE model, nmod is the total number of isotherm models considered for selection, and εMS is the tolerance for model selection. If the condition in Eq. (16) is satisfied, then the sth model is selected instead of the rth model.

Fig. 3. Schematic diagram of SMB unit.

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J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

3.2. SMB experimental technique

3.3. Determination of variance model for parameter estimation Calibration curves for cyclopentanone, C5 (Alfa Aesar, CAS# 12092-3, USA) and cyclohexanone, C6 (Alfa Aesar, CAS# 108-94-1, USA) were obtained using Shimadzu HPLC data with the C18 analytical column. Using a solvent composition of 50% water and 50% methanol by volume, three injections of 10 ␮L each were made at various feed concentrations in the HPLC analyzer. Each sample of C5 and C6 was prepared in the same mobile phase used in the SMB unit; 60% water 40% methanol. The flowrate was 1.0 mL min−1 and the column temperature was 40 ◦ C. Standard deviations of the peak area measurements were calculated at each concentration level to establish a linear variance model for use in the parameter estimation routine in gPROMS. The data is shown in Fig. 4. Linear variance models as in Eq. (15), were determined for both C5 and C6. The coefficients for C5 were ˛ = 1.1 × 10−3 and ˇ = 9 × 10−4 . The coefficients for C6 were ˛ = 1.1 × 10−3 and ˇ = 4 × 10−4 . As can be observed in Fig. 4b, the standard deviations for both C5 and C6 increase with increasing feed concentration. This relationship is modeled most simply by a linear regression. The increasing variance with feed concentration may be due to error in the injection volume for the three samples. There may also be error in the baseline definition during the calculation of peak areas. 4. Results 4.1. Demonstration of isotherm model selection An important step in the PC algorithm is isotherm model selection. In order to demonstrate isotherm model selection for a nonlinear system the following case studies were done with cyclopentanone (C5) and cyclohexanone (C6). We consider three case studies, Cases A, B and C, as shown in Table 1. 4.1.1. Initial pulse tests In all case studies, the PC algorithm was initiated at k = 0 by pulse tests on a single SMB column for parameter estimation. The column porosity was measured by a 2 ␮L injection of uracil at a flowrate of 3.0 mL min−1 , and the value was εb = 0.678. This porosity value was fixed in all case studies. The Henry’s constants, mass transfer coefficients and nonlinear isotherm parameters for C5 and C6 were

Peak area (x1,000,000 mAU min)

3.0 2.5

y = 0.1345x - 0.004 R² = 1

2.0 1.5 1.0

y = 0.1077x - 0.0066 R² = 0.9998

0.5 0.0 0

5

10

(a)

15

20

25

30

Concentraon (g/L)

0.035 0.030

Standard deviaon (g/L)

At the onset of new SMB operating conditions, the extract and raffinate products were collected by placing the outlet tubing into empty glass beakers with stirring to maintain uniform composition. As the compositions of each outlet stream change with time, it is important to achieve sufficient mixing to eliminate concentration gradients in the product vessel. In this way, cumulative average extract and raffinate products were collected. Further details on the sampling strategy are in Section 2.3. Each sample was analyzed with Shimazdu HPLC equipment with an analytical C18 column (Daisopak, SP2-120-10-ODS-BP, Daiso, Japan). A flowmeter (Model 5025000, GJC Instruments) was used to ensure the consistency of the set flowrate of the HPLC equipment as well as extract, raffinate and recycle lines at the end of the sampling period. The recycle line was not collected in a cumulative way during the experiment, but was sampled once by collecting the average concentration over a step as discussed by [17]. This was done assuming that after a sufficient number of cycles the cumulative concentration measurement equals the average concentration measurement over a step. This single sample for the recycle line was included in the parameter estimation. The SMB flowrates were measured at the end of the sampling period in order to prevent disruption of the internal profiles that may occur by disconnecting the recycle line.

C5 C6 Linear (C5) Linear (C6)

3.5

y = 0.0011x + 0.0009 R² = 0.8067

0.025 0.020 0.015

y = 0.0011x + 0.0004 R² = 0.9024

0.010 0.005 0.000 0

(b)

5

10

15

20

25

30

Concentraon (g/L)

Fig. 4. (a) Calibration curve using Shimadzu HPLC with three injections made at each concentration. (b) Standard deviations of the measured concentration at each concentration level with linear regression equations and correlation.

estimated simultaneously using the inverse method on two smallvolume injections of the feed mixture with chromatograms shown in Fig. 5. The 310 nm UV signal from the detector was converted into concentration units for C5 and C6 so that the parameter estimation problem, Eqs. (13) and (14) could be solved in gPROMS. The simulated chromatograms are also shown in Fig. 5 using the estimated model parameters for the sLangmuir isotherm, Eq. (5). The inverse method problem was solved for three isotherm models, and the initial parameter values are shown in Table 2. The experimental chromatograms exhibit some tailing due to asymmetry of the injection pulse as well as nonlinearity of the isotherm. The concentrations of C5 and C6 in the column are probably not large enough to characterize the nonlinearity in the Table 1 Operation details of three case studies with C5 and C6. Variable

Case A

Case B

Case C

ncol Zone configuration [cf ,C5 , cf ,C6 ] (g L−1 ) Desorbent [water%, methanol%] Temperature (◦ C) Initial isotherm model Fmax (mL min−1 ) Raf PurityA,min (%) PurityExt (%) B,min Purity safety margin (%)

4 1-1-1-1 [34, 34] [60, 40] 40 sLangmuir 6.5 97 97 1.0

8 2-2-2-2 [34, 34] [60, 40] 40 Linear 6.5 97 97 1.0

4 1-1-1-1 [20, 20] [60, 40] 40 sLangmuir 5.0 97 97 1.0

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

Concentraon (g/L)

1.5

Experiment, C5 Experiment, C6 sLangmuir, C5 sLangmuir, C6

1.2

40 μL

0.9 0.6 0.3 0 0

3

(a)

6

9

12

1.5

Concentraon (g/L)

15

Time (min) 80 μL

1.2 0.9 0.6 0.3 0 0

3

(b)

6

9

12

15

Time (min)

Fig. 5. Chromatograms of C5 and C6 with (a) 40 ␮L and (b) 80 ␮L injection, with cf ,C5 = cf ,C6 = 34.0 g L−1 , F = 3.0 mL min−1 , T = 40 ◦ C, and inverse method fitting using sLangmuir model. Table 2 Comparison of initial parameter set  0 and refined parameter set  1 of three isotherm models after parameter estimation using SMB startup data from Case A. Parameter

Initial,  0

Linear isotherm 1.793 HC5 HC6 3.743 kC5 (min−1 ) 310 kC6 (min−1 ) 180 1,∗ ϕPE sLangmuir isotherm 1.793 HC5 HC6 3.743 bC5 (L g−1 ) 0.0225 bC6 (L g−1 ) 0.0405 −1 kC5 (min ) 310 −1 180 kC6 (min ) 2,∗ ϕPE cLangmuir isotherm 1.793 HC5 HC6 3.743 −1 bC5 (L g ) 0.0207 −1 bC6 (L g ) 0.0378 −1 kC5 (min ) 310 180 kC6 (min−1 ) 3,∗ ϕPE a

Refined,  1

95% confidence interval

1.593 2.771 186.5 108.1 1.0

5.3 × 10−4 6.4 × 10−4 4.5 88

1.766 3.413 0.0121 0.0337 372a 26.8 0.38

8.6 × 10−4 7.3 × 10−4 8.2 × 10−5 3.0 × 10−4 – 0.99

2.011 3.581 0.0115 0.0367 372a 130.7 0.080

1.6 × 10−3 4.6 × 10−3 9.4 × 10−5 2.4 × 10−4 – 11

Parameter value at active upper bound.

isotherm. However, these are only initial estimates of the Langmuir isotherm parameters which can be refined later by the PC algorithm. 4.1.2. Isotherm model selection from SMB experiment Isotherm model selection was performed during the first iteration of the PC algorithm at k = 1. The optimal operating conditions, u0 were implemented in the SMB and the cumulative average

57

extract and raffinate product concentrations were sampled for four cycles of operation. The recycle line was also sampled once to improve observability of the SMB internal concentration profiles. This data was used to solve the parameter estimation problems for three isotherm models. In Fig. 6 the fittings of the isotherm models to the experimental concentration data obtained in Case A are compared. Using the linear model (Eq. (4)) there are four model parameters estimated simultaneously. Accordingly, with the sLangmuir (Eq. (5)) and cLangmuir (Eq. (6)) models there are six estimated parameters. In Table 2 the initial parameters, refined parameters, 95% confidence intervals, and normalized objection function values for the paramr,∗ eter estimation problems are shown for Case A. The values of ϕPE in

1,∗ Table 2 are normalized by ϕPE , where 1 represents the linear model. From Fig. 6 it is clear that the linear isotherm model does not fit the experimental data, while the sLangmuir model has improved fitting, and the cLangmuir model exhibits the best fit. In Fig. 6a and b it can be seen that the C6 profiles are fit better than the C5 profiles. The C5 profiles are more difficult to fit because C5 is the less retained component in the SMB; nonlinear and competition effects may be observed in the raffinate data where the wave front is steepened. By comparison, the C6 profiles are easier to fit because C6 is the more retained component and the extract data describes the tailing end of the internal concentration profiles. In Fig. 6d and e note that the profiles for C5 are only fit well by the cLangmuir isotherm, indicating that significant competition exists in the SMB at the Case A conditions. In agreement with these experimental observations, it can be concluded from the objective function values in Table 2 that the cLangmuir model is the best choice to fit the data. The objective function value for the sLangmuir model is 61% less than that of the linear model, and the objective function value for the cLangmuir model is 79% less than that of the sLangmuir model. In Table 2 it can also be seen why the linear isotherm should not be selected in this case study. The values of the Henry’s constants are dramatically reduced in the refined parameter set  1 . Note that HC6 was reduced by about 26%, which indicates that the retention of C6 is much less than observed in single-column pulse tests. This is because the optimizer reduced this parameter value in order to account for the nonlinearity of the isotherm. It should be noted that the model selection step of the PC algorithm can be performed by an SMB operator without applying any prior knowledge of adsorption fundamentals. In this case study with cycloketones, the cLangmuir isotherm is selected following the systematic steps of the PC algorithm without any prior experience or understanding of the adsorption behavior of cycloketones on the reversed-phase column. Another observation from Table 2 is that systematic selection between sLangmuir and cLangmuir isotherms, which are both nonlinear, is enabled by the PC algorithm. The estimated parameter values for these models are similar, especially in the nonlinear bi values. However, there is more than 50% decrease in the objective function value for the cLangmuir isotherm compared to the sLangmuir isotherm. Therefore, based on the selection criteria explained in Section 2.4.2, the cLangmuir model is to be selected for the next iteration of the PC algorithm. From Table 2 it should be noted that the value of the mass transfer coefficient for the less retained component kC5 reached the upper bound in k = 1 using the sLangmuir and cLangmuir models. The mass transfer coefficient is related to the NTP of a column, and the sensitivity of this parameter may be insufficient from the first set of experimental data. This issue is more distinctive when there is a combination of nonlinearity and fast mass transfer, which are highly correlated effects. The former contributes to the steepness of wave fronts and the latter describes overall band broadening. This is what happened in our case studies when we used a nonlinear system and HPLC columns with high

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0.25

4

Experiment, C5 Experiment, C6 linear, C5 linear, C6 sLangmuir, C5 sLangmuir, C6 cLangmuir, C5 cLangmuir, C6

3 2 1

Extract product (g/L)

Extract product (g/L)

5

0

0.15 0.10 0.05 0.00

0

(a)

Zoom in of (a)

0.20

40

80

120

160

200

Time (min)

0

(d)

20

40

60

80

100

80

100

Time (min) 4

5

Zoom in of (b)

Raffinate (g/L)

Raffinate (g/L)

4 3 2 1 0

3 2 1 0

(b) 0

40

80

120

160

200

Time (min)

(e)

0

20

40

60

Time (min)

Recycle line (g/L)

1.00 0.75 0.50 0.25 0.00

(c)

0

40

80

120

160

200

Time (min)

Fig. 6. Experimental (a) extract, (b) raffinate and (c) recycle line data using u0 of Case A, simulated profiles using linear, sLangmuir and cLangmuir isotherms with refined parameter set  1 . Zoom-in of C5 profiles in (d) extract and (e) raffinate streams.

efficiency. However, the PC algorithm can be continued even with this over-estimation of the mass transfer coefficient for C5, which can be corrected later by additional experimental data.

We demonstrate the robustness of the PC algorithm in Case B by showing that even if a poor isotherm model is selected at first and the initial SMB performance is poor, isotherm model selection can correct the model structure and estimate parameter values for the re-optimization of the SMB. The SMB performance can be dramatically improved in the next iteration of the PC algorithm after model selection. In Case B, the first SMB experiment had very poor purity values with 84% C6 in the extract and 62% C5 in the raffinate, because a linear isotherm was assumed initially despite the true adsorption behavior being nonlinear. The experimental data in k = 1 was used to perform model selection and the results were consistent with those discussed in Section 4.1. The parameter estimation objective function values for each isotherm model are shown in Fig. 7 as Case B. Refined model parameters for the cLangmuir isotherm were used to obtain the next optimal operating conditions, u1 which were implemented in k = 2. The improvement in product purities is shown in Fig. 8 after the operating conditions are switched from u0 to u1 in the SMB. After the switch, the measured purities in the extract and raffinate products improved by 8% and 30% respectively. The operating conditions u0 were kept for 3 h after collecting samples in k = 1 before implementing the next conditions u1 . This was done to allow sufficient time for model selection

sLangmuir

cLangmuir

1

Normalized ϕPEr

4.2. SMB performance improvement after isotherm model selection

linear

0.8 0.6 0.4 0.2 0

Case A

Case B

Case C

1,∗ Fig. 7. Parameter estimation objective function values normalized by ϕPE in three case studies with C5 and C6.

and re-optimization of the process. The most difficult parameter estimation problem in gPROMS involves the simulation of multiple SMB experiments, and most problems take less than 60 min of CPU time. The SMB optimization problem usually takes less than 15 min of CPU time in AMPL. To show how the optimal operating conditions were updated from k = 1 to k = 2, Fig. 9 shows triangle diagrams with regions of complete separation [4], superimposed for the linear and

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

k=1, use u0 in SMB

k=2, use u1 in SMB 96% purity

Product purity (%)

100 90 80 70

Extract

60

Raffinate

Table 3 Refinement of SMB model parameters in Case C using cLangmuir isotherm in k = 1–3. Parameter

Initial,  0

Refined,  1

HC5 HC6 bC5 (L g−1 ) bC6 (L g−1 ) kC5 (min−1 ) kC6 (min−1 )

1.793 3.743 0.0207 0.0378 310 180

1.870 3.578 0.00302 0.0424 148 44.8

a

50 100

0

200

300

400

500

Fig. 8. Experimental product purities are plotted for k = 1 where u0 is obtained with linear isotherm, and k = 2 where u1 is obtained with cLangmuir isotherm in Case B. The target product purity is 96%.

4

3

u0

linear PurityBExt = 92% PurityARaf = 92% u1

m3 2

cLangmuir cf,C5 = cf,C6 = 34 g L-1 1

0 0

1

2

3

Refined,  2 2.202 3.814 0.0223 0.0360 372a 110

Refined,  3 2.134 3.819 0.017 0.042 18.7 71.5

Parameter value at active upper bound.

600

Time (min)

PurityBExt = 84% PurityARaf = 62%

59

4

m2 Fig. 9. Optimal m2 and m3 values from operating conditions u0 and u1 with linear and cLangmuir triangle regions determined by refined parameter sets  1 in Case B. Experimental product purities are included.

cLangmuir models with u0 and u1 and the experimental product purities. The separation regions are drawn using refined parameter values  1 . The move from u0 to u1 shows the reduction in feed throughput after re-optimization, which was about 60%. Based on the cLangmuir triangle diagram, the position of u1 is much closer than u0 to the theoretical optimal operating conditions, which explains the improvement in observed purity values. 4.3. Convergence of PC algorithm and parameter refinement The PC algorithm is shown to converge for a nonlinear system in Case C. For the termination criteria, the tolerance on change in feed throughput εtol was 1.5% because the nonlinear SMB performance is very sensitive to the feed flowrate. The initial model parameters for the sLangmuir model were used in k = 0, and after performing model selection on the first set of SMB data, the cLangmuir model was selected. This result is consistent with the model selection shown in Section 4.1, and the parameter estimation objective function values for each model are shown in Fig. 7 as Case C. Table 3 shows the iteration history of the cLangmuir model parameter values determined by the PC algorithm. It should be noted that the final values of the Henry’s constants are greater by a few percent than the initial estimates. This can be explained by the fact that there are dead volumes in the real SMB unit with column connections, rotary valve and pumps that are not described

explicitly in the SMB model. The values of the cLangmuir isotherm parameters bi fluctuate in each iteration with bC5 having the greatest change from  1 to  2 . In general the values of the mass transfer coefficients estimated by the PC method are reduced by more than 50% of the initial guess when ki ’s are obtained by the inverse method. This agrees with previous results in our case study with uridine and guanosine [17] where the apparent mass transfer rates in the real SMB unit were about 50% less than the initial guess from pulse tests. This reduction is probably due to mixing and various flowrates in the real SMB unit with column connections, rotary valve and pumps. The ki values are largely insensitive in the SMB model, thus it is not surprising that their estimated values vary greatly from step to step. Note that in  2 , it can be seen that kC5 has an active upper bound which is 15% higher than the initial value. This is an indication of the steep concentration profiles for C5, the less retained component. The large value of kC5 was unexpected because the apparent mass transfer rates in the SMB unit are usually less than the initial guess using pulse tests. Nonetheless, in the final iteration k = 3 the value of kC5 was corrected to be a small fraction of the initial guess. In Fig. 10 the convergence of the SMB experiments to optimal productivity with high purity constraints is shown. The SMB experiments are performed relatively quickly with a small number of samples. In k = 2 and k = 3 only six measurements were taken to observe the transition in operating conditions. It can be seen how the concentration profiles evolve after switching the operating conditions, and the improvement of the model fitting is shown by plotting simulated profiles using  0 and  3 . The most significant improvement in the model fitting is in the C5 profiles of the extract and raffinate products. Another observation is that product concentrations are increased in the final iteration with an increase of about 400% in the raffinate stream. The simulated C5 raffinate profile has some steep oscillations in each step of operation, as seen in Fig. 10. As observed by [17], these oscillations are not due to numerical instability of the solver, but instead are a result of internal concentration profiles during a step. The product concentrations that enter the extract and raffinate tanks have decreasing and increasing patterns respectively during a step. These patterns are observed distinctly at the beginning of each experiment where the accumulated volume of components in the tanks is still small. The simulated concentration profiles have discontinuities at the time points where the next operating conditions are initialized. This discontinuity is also observed in the experiments by collecting the extract and raffinate products in an empty container at the moment the operating conditions are switched as described in Section 3.2. Table 4 shows the operating conditions in each iteration of the PC algorithm in Case C, along with the product purities and the change in feed throughput for each experiment. Fig. 11 also shows how the change in feed throughput affects the experimental purity values in each iteration. It should be noted that the optimal solutions u0 through u3 do not differ greatly, yet the experimental purities and the product concentrations are significantly affected by even slight changes in the step time and feed throughput. The termination criteria are satisfied in the third iteration because both products have purity greater than 96% and the change in

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Experiment k = 1 Use u0

Experiment, C5 Experiment, C6 Model, k=0, C5 Model, k=0, C6 Model, k=3, C5 Model, k=3, C6

15

Extract product (g/L)

k=2 Empty product tanks Use u1

12 9

k=3 Empty product tanks Use u2

6 3 0 0

50

100

150

200

250

300

350

400

450

500

550

600

450

500

550

600

450

500

550

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Time (min) 15

Raffinate (g/L)

12 9 6 3 0 0

50

100

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Time (min) 15

Recycle line (g/L)

12

9

6

3

0 0

50

100

150

200

250

300

350

400

Time (min) Fig. 10. Experimental extract, raffinate, and recycle line data for three iterations of the PC algorithm in Case C. The simulated profiles are plotted using  0 and the final parameter set  3 .

the optimal feed throughput between u2 and u3 is less than 1.5%. Therefore u2 is considered optimal. In Fig. 12 it can be seen how the operating conditions change after each SMB optimization in order to improve both the product purities and feed throughput. The complete separation region is drawn using  3 and it is seen that the initial operating conditions Table 4 Optimal operating conditions and termination criteria for Case C. Operating conditions

u0

u1

u2

u3

tS (min) m1 m2 m3 m4

6.83 3.26 1.64 2.17 −0.724

6.61 3.01 1.23 1.85 0.84

6.83 3.18 1.36 1.93 1.10

6.83 3.25 1.44 2.01 0.921

Termination criteria

k=1

k=2

k=3

PurityExt (%) C6 Raf PurityC5 (%) k ϕSMB (%)

99.2 98.2 21.5

91.0 93.9 11.7

98.0 97.8 1.36

u0 were close to the tip of the triangle even though the sLangmuir model was used to solve for u0 . Indeed, the product purity values were high in the first experiment. The step from u0 to u1 appears to be overly aggressive because the feed flowrate increased by 22% and the product purities decreased below the minimum constraint of 96%. Nonetheless, the step from u1 to u2 was a good adjustment because the purities were a few percent greater than 96%, and the throughput was improved by 8% from k = 1 to k = 3. 4.4. Validity of model parameter values We show that the values of cLangmuir isotherm parameters estimated using the PC algorithm in our nonlinear case studies are reliable when compared with those measured by singlecomponent frontal analysis (FA) and perturbation analysis (PA). The detailed batch experiments described here are not required to implement the PC algorithm, yet they are shown to demonstrate the reliability of the parameter estimation using SMB data. The nonlinear isotherm parameters are more difficult to estimate, especially

100

0.65

98 96

0.60

94 0.55 92 0.50

90

0.45

Throughput

Extract

Raffinate

88

k=1

k=2

k=3

20

0

5

10

15

20

80

m3

35

40

45

50

60 40 Cases A and B

20 Case C 0

cLangmuir cf,C5 = cf,C6 = 20 g L-1

30

FA data PC model linear model

0

2.0

25

Liquid c, (g/L)

100

Adsorbed q, (g/L)

u0 u0 u1 u1 u2 u2 u3 u3

Case C

40

(a)

3.0

Cases A and B

60

0

Fig. 11. Experimental throughput values are shown on the left axis, and experimental extract and raffinate purity values are shown on the right axis for three iterations of the PC algorithm in Case C.

2.5

FA data PC model linear model

80

86

0.40

61

100

Adsorbed q, (g/L)

0.70

Product purity (%)

Throughput (mL min-1)

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

5

10

15

(b)

20

25

30

35

40

45

50

Liquid c, (g/L)

Fig. 13. Comparison of single-component FA data for (a) C5 and (b) C6 with cLangmuir model parameters obtained by PC method in Case C and linear model parameters obtained by pulse tests.

1.5

1.0 1.0

1.5

2.0

2.5

3.0

m2 Fig. 12. Optimal m2 and m3 values in Case C for u0 through u3 with the cLangmuir triangle region determined by refined parameter set  3 .

when competitive adsorption is involved, yet we show that the values obtained by the PC method are reliable in the SMB model. In Fig. 13 single-component FA results for C5 and C6 are compared with the model using cLangmuir isotherm parameters obtained by the PC method. The FA tests were performed by injecting step inputs of single-components up to 70 g L−1 into an SMB column with 60% water 40% methanol mobile phase, flow rate of 3 mL min−1 , and temperature of 40 ◦ C. The linear isotherm model is shown in Fig. 13a and b using the retention times of small-volume pulse injections of C5 and C6 on the SMB column. It can be seen that C6 has stronger nonlinearity than C5 on the column, and both components exhibit deviations from linear adsorption at 34 g L−1 , which was the feed concentration of each component used in Cases A and B. The cLangmuir model is plotted in Fig. 13 using refined parameter values  3 from Case C. In this comparison the cLangmuir model reduces to the sLangmuir model in Eq. (5), because the FA tests were performed with single components. Note that Fig. 13 shows that the PC model fits the C5 isotherm better than the C6 isotherm, especially over the concentration ranges of interest for Cases A and B. The PC model predicts both single-component isotherms relatively well at the low-tomoderate concentration range, although the deviation increases for C6 at higher concentrations. We find that a better estimate for bC5 than bC6 is obtained by the PC method because nonlinear effects are directly observed in the raffinate stream, which is upstream of the feed location, where C5 is pure. From Fig. 6 we noted that the cLangmuir isotherm fit the C5 profiles in the raffinate and extract while all isotherm models fit the C6 profiles reasonably well. This indicates that the nonlinearity of C6 is not completely observed at

the tailing end of the concentration profiles of the SMB which are collected in the extract product. In Table 5 the cLangmuir isotherm parameter values obtained by inverse method, frontal analysis, and PC method are compared. There, theoretical saturation capacities (qsat ) are obtained from the i following equation: qsat = i

Hi bi

(18)

The parameters Hi and bi are correlated by the theoretical saturation capacity of the stationary phase. The bi ’s estimated by the PC method are within a reasonable range when compared to those obtained by single-component FA, and it seems that in Case C the initial guess of cLangmuir bi ’s obtained by the inverse method were also fairly reliable. If the underlying assumptions of the fundamental Langmuir equation we use in Eq. (6) are reasonable in our practical application of multi-component adsorption, we can expect the values of bi obtained by single-component FA to match those of the competitive FA [19]. Since the PC experiment is essentially a competitive loading test on the SMB columns, we are able to observe non-ideal competitive adsorption behavior that is lost in the single-component FA experiment. For further validation of estimated cLangmuir parameters using the PC method we performed competitive PA. An SMB column was used with 60% water 40% methanol as mobile phase, flow rate of 2.5 mL min−1 , and temperature of 40 ◦ C. The column was Table 5 Comparison of cLangmuir isotherm parameter values estimated by three methods. The PC method parameters are from  3 in Case C. Parameter

Inverse method

Single-component frontal analysis

PC method

bC5 (L g−1 ) bC6 (L g−1 ) (g L−1 ) qsat C5 qsat (g L−1 ) C6

0.0207 0.0380 86.6 98.5

0.0107 0.0197 152 176

0.017 0.042 130 91

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J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

(a)

0.008

PC, C5

FA, C5

H,i Henry’s constants

Deviaon from experimental dq/dc (%)

150 120 90 60 30 0

C5 0.006 0.004 0.002 0.000

(a)

(b)

FA, C6

k=1

k=2

k=3

k=1

k=2

k=3

k=1

k=2

k=3

0.0004

120

b,i cLangmuir parameters (L g-1)

Deviaon from experimental dq/dc (%)

150

PC, C6

C6

90 60 30

0.0003 0.0002 0.0001

0

cC5 = 0 cC6 = 0

cC5 = 3.71 cC5 = 34.3 cC5 = 3.69 cC5 = 34.0 cC6 = 3.55 cC6 = 3.54 cC6 = 34.7 cC6 = 34.7

0.0000

(b)

Mobile phase soluons: (g/L)

equilibrated with five different solutions of C5 and C6 that were prepared to imitate various internal locations of the SMB unit. A 30 ␮L injection of cf ,C5 = cf ,C6 = 20 g L−1 was made at each equilibrium level, and the retention times of the C5 and C6 peaks were recorded. The results of the PA are summarized in Fig. 14 where each equilibrium state is shown as a category on the horizontal axis. The values of dqi /dci were calculated, based on the method described in [10], for the experimental results and the model predictions using both cLangmuir parameters obtained by the PC method and the single-component FA. The values of bC5 and bC6 obtained by FA were used in the competitive Langmuir isotherm, Eq. (6) to calculate dqi /dci at each equilibrium state [25]. The percent deviations shown in Fig. 13 were calculated by the following equation:

   (dq /dc )j − (dq /dc )j  i i Mod   i i Exp   × 100% j   (dqi /dci )Exp

(19)

where j is the index for the equilibrium condition, Exp is the experimental value, and Mod is the model-predicted value of the local derivative of the adsorption isotherm. Equal values of the Henry’s constants were used in the PC and FA model parameter sets in order to evaluate the estimates of bi for competitive adsorption. This result shows that the cLangmuir model parameters obtained by the PC method can predict the retention times of perturbation peaks more reliably than the cLangmuir parameters measured by single-component FA, especially around the feed concentration of 34 g L−1 each C5 and C6. Using the solution where cC5 = 34.0 g L−1 and cC6 = 34.7 g L−1 , the predicted dqi /dci values using the FA parameters were off by 144% and 114% for C5 and C6 respectively. This means that the assumptions of the fundamental Langmuir theory, such as one molecule–one site interaction, or no interaction among adsorbing components, may not hold in our case studies. Another way we test the reliability of the optimal parameter set obtained by the PC method is to analyze the confidence intervals for each estimated parameter at 95% statistical confidence. In

k,i mass transfer

Fig. 14. Comparison of experimental dqi /dci values to those predicted by cLangmuir model parameters obtained by PC method in Case C and single-component FA for (a) C5 and (b) C6.

coefficients (min-1)

80

(c)

60 40 20 0

Fig. 15. 95% confidence intervals for each SMB model parameter in each iteration of the PC algorithm in Case C.

Fig. 15 the computed confidence intervals for the Henry’s constants, cLangmuir parameters and mass transfer coefficients are shown in each iteration of the PC algorithm for Case C. These statistics were calculated by the parameter estimation routine in gPROMS, which depend on the specification of the variance model for experimental measurements discussed in Section 2.4. In general, the confidence intervals decreased from one parameter estimation result to the next. The values of the confidence intervals for HC5 and HC6 were tight relative to the parameter values of 2.134 and 3.819 respectively and slight improvement was observed. The values of the confidence intervals for bC5 and bC6 were also tight relative to the parameter values of 0.017 and 0.042 L g−1 respectively and slight improvement was observed. Comparatively, the values of the confidence intervals for kC5 and kC6 were loose relative to the parameter values of 18.7 and 71.5 min−1 respectively. Yet the confidence interval for kC5 improved significantly from k = 1 to k = 3. This is a similar trend to that observed in our previous work [17] and we believe the larger confidence intervals for the mass transfer coefficients are another indication of their insensitivity in the SMB model. 5. Conclusions and future work The prediction–correction algorithm for nonlinear SMB process development is extended with isotherm model selection and

J. Bentley et al. / J. Chromatogr. A 1280 (2013) 51–63

tested. It is found to be an efficient and robust algorithm in numerous laboratory experiments with a nonlinear isotherm system. In all case studies with cyclopentanone and cyclohexanone the competitive Langmuir isotherm is distinguished from the linear and single-component Langmuir models, and the isotherm parameters are determined from SMB startup data. The parameter values estimated using the PC method are comparable to those obtained using batch techniques such as moment analysis, inverse methods and frontal analysis. In one case study the robustness of the PC algorithm is demonstrated where the operating conditions are corrected from a poor initial point in a single step after isotherm model selection. The product purities were improved by 8% in the extract and 30% in the raffinate. In another case study the PC algorithm is shown to converge to optimal operating conditions in only three iterations while maintaining high purity constraints and improving the feed throughput of the process. The PC algorithm proceeds in an automatic and sequential manner until it terminates so that an SMB expert is not required to manually tune the SMB operating conditions. It should be noted that this algorithm is designed to function with any choice of isotherm model, and up to six model parameters have been simultaneously determined using the PC method in this study. In the near future we plan to extend the PC algorithm to optimize nonstandard SMB operations for binary and ternary separations. Work is currently underway to experimentally verify the optimal startup strategy for SMB using the reliable model parameters obtained by the PC method for dynamic optimization. Finally, the optimal design of SMB experimental conditions for reliable parameter estimation using the PC method will be explored. Acknowledgments The financial support by Semba Biosciences is gratefully acknowledged. We also received technical support from Daiso Co.,

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