Ind. Eng. Chem. Res. 2008, 47, 5585–5593
5585
Using Wavelets for Solving SMB Separation Process Models Hongmei Yao,† Yu-Chu Tian,*,‡ and Moses O. Tade´† Department of Chemical Engineering, Curtin UniVersity of Technology, GPO Box U1987, Perth, Western Australia 6845, Australia, and Faculty of Information Technology, Queensland UniVersity of Technology, GPO Box 2434, Brisbane, Queensland 4001, Australia
With the expanding applications of the simulated moving bed (SMB) chromatography technology in various industries, it becomes increasingly significant to systematically investigate SMB process modeling, simulation, and model-based control design. How to control an SMB process effectively is a significant and challenging problem at the frontier of process systems engineering research as well as industrial operation. Addressing the challenges in this area, this paper reviews recent advances in modeling and simulation of SMB separation processes and also clearly identifies the limitation of the existing modeling techniques in industrial applications. To facilitate SMB process modeling, the paper analyses the structure of the existing models for a better understanding of the functionality and suitability of each model. It is our belief that the effort on model development for process control can be made in two aspects: obtaining the sufficiently accurate model with simple representation and good robustness, and development of computationally efficient algorithms for solving the model equations which ultimately capture the process dynamics. Case studies are carried out to demonstrate the key concepts of process modeling and numerical solution of the models. The wavelet-based methods that we recently investigated are highlighted. 1. Background of SMB Chromatographic Separation Separation processes are significant in various industries. They have been estimated as accounting for 40%-70% of the capital and operating costs in chemical industries.1 Energy-intensive separation processes like distillation are facing increasing challenges from other alternative technologies. Just like membrane technologies are offering a viable alternative, so is the simulated moving bed (SMB) chromatography. Chromatography was initially developed for extraction and purification of complex mixtures of vegetal origin,2 and the technique achieved rapid growth later when it became a ubiquitous analytical method. The SMB chromatography is the technical realization of a countercurrent adsorption process, approximating the countercurrent flow by a cyclic port switching. It consists of a certain number of chromatographic columns connected in series, and the countercurrent movement is achieved by sequentially switching the inlet and outlet ports of one column downward in the direction of the liquid flow after a certain period of time. This mechanism is illustrated in the diagram of Figure 1. Compared with the conventional fixed-bed operation, the SMB technology has many advantages. Countercurrent flow enhances the potential for separation and continuous feeding improves the throughput of the equipment. By reducing the required volumes of stationary and mobile phase, SMB processes can achieve higher productivities with lower solvent consumption, implying that the products are less diluted and consequently the product recovery step is easier and cheaper. Another great advantage of the SMB chromatography, compared to other processes, is its capacity to scale up linearly. A pilot scale process can be reproduced quickly at production scale without sacrificing the purity and production rate. The chromatographic separation technique used in industry for purification and separation of materials can be traced back * To whom correspondence should be addressed. Phone: +61-73138 2177. Fax: +61-7-3138 1801. E-mail:
[email protected]). † Curtin University of Technology. ‡ Queensland University of Technology.
to as early as in the 1950s, when it was evaluated for manufacturing of chemicals. Industrial scale preparative separations took a giant step in the 1960s when UOP introduced and commercialized the Sorbex family of SMB processes for petroleum refining and petrochemical. Since then, major commodity applications have been found in the petroleum industry, e.g., separation of xylene isomers,3–7 food industries, e.g., separation of sugars or amino acid,8–12 fine chemicals and others.13–15 Figure 2 is a statistical analysis of the SMB separation related publications during the past 30 years. The figure clearly shows the increasing interest in this area. In 1990s, the use of chromatography in biotechnology, pharmaceutical and fine chemical industries has a significant explosion fuelled by the market trend toward chirally pure therapeutic compounds.16–20 During the past decade, chromatography was recognized by the pharmaceutical industry as the key general separation method for the purification of the drug intermediates and the pharmaceuticals that it produces.2,21 Figure 3 indicates that, among three major applications, chiral separation is growing the most rapidly. The motivation for these advances is that the SMB technology reduces the cost of packing materials with high loading capacity and can also provide high purity and high recovery in a very short time. Furthermore, the pharmaceutical industry is also attracted by the SMB technology for its ease of scaling up which reduces the time from laboratory to market. 2. Challenges in Industrial Operation The industrial applications of the SMB technology require a good design of the SMB systems and effective control of the operation. However, there are two difficulties. The system we are dealing with has complex process dynamics with significant uncertainties and time delay. It also presents high sensitivity to various disturbances, especially the SMB process operated close to its optimum.22 Usually, system analysis will be first conducted and tested at laboratory scale. Then, in real applications, the operating
10.1021/ie071246g CCC: $40.75 2008 American Chemical Society Published on Web 07/10/2008
5586 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 1. An SMB unit with five columns in four sections configuration of 1/2/1/1 (RF ) direction of fluid flow and port switching).
Figure 2. Total number of publications related to SMB separation (extracted from EngineeringVillage2 database on May 1, 2008).
Figure 3. Distribution of the research interest on SMB separation (extracted from ngineeringVillage2 database on May 1, 2008).
conditions are manually tuned to their suboptimum values. The optimization of the SMB process operation can be achieved through optimizing a performance index, or multiple indices, with respect to the manipulated process variables. It requires accurate process models as constraints. Process modeling is also necessary for scale-up from laboratory to industrial scale, for prediction of process dynamics, and for online process control. Mathematical modeling and simulation of SMB chromatographic processes is widely documented, yet scattered, in the open literature. As a consequence of interests in applications, a great amount of work has been done involving researchers from various disciplines of chemical engineering, process engineering,
chemistry, and even biomolecular engineering. So far, design and optimization have accounted for the majority of the work within these potential applications of the modeling. In the following, we will summarize the advances in SMB modeling by reviewing some of the research activities in terms of their significant impact to this area. 2.1. Some Advances in SMB Modeling. Earlier contributions to the development of first-principles models for countercurrent and simulated countercurrent separation systems have been made by Ruthven and Ching.23–25 Most of their publications have become the guide for later SMB model development. The proposed node model is still adopted for modeling SMB systems nowadays. These researchers and colleagues have also explored industrial applications of those systems. Since the 1980s, a large number of papers have been published on chromatographic separation from an interdisciplinary collaborated group of Mazzotti, Morbidelli, and Storti, etc. Their investigations in the optimization of the operation parameters are conducted mainly on the plane of operational region. They derived a graphical short-cut design methodology, the so-called Triangle Theory, and extended the theory to systems with nonlinear adsorption isotherms.26–31 This methodology has once been widely used for initial guesses of a feasible operating point of the process. It is also used as a tool for manually tuning the suboptimal operating conditions. More recently, work has been conducted on process control.32–36 Other than the optimization of traditional SMB systems (fixed feed rate and simultaneous port switching), Varicol processes (nonsimultaneous and unequal shifts of ports) and PowerFeed operation (changing flow rates during the switching interval) have also been studied.37,38 The online optimization based SMB control scheme using repetitive model predictive control method has been applied to different cases through simulations and experimental implementation. Some fundamental issues such as column dynamics and thermodynamics, retention mechanisms, and isotherm effects have been addressed by Guiochon, Zhong, and their collaborators.39–44 Especially, Guiochon’s work has great impact on the separation science through development of new theory and showing how theory can guide the practical application of separations.45 Undoubtedly, through investigating the effect of the different parameters on the band profiles and the effect of the concentration histories on its two outputs, their work will help understand the SMB behavior in detail. Since the late 1990s, Du¨nnebier and Klatt have also studied SMB processes from process control engineering point of view with the focus on process modeling for optimal design and model-based control purpose.22,46–48 Earlier work was on the comparison of different modeling approaches. This led to their later proposal on process optimization and design. From a thorough review of previous design methods, they put forward a new model-based optimization strategy from the selection objective function, the degrees of freedom, constrains, and the optimization problem description. They proposed a two-layer control architecture in which the optimal operating trajectory is calculated off-line by dynamic optimization based on a rigorous process model. The low-level control task is to keep the process on the optimal trajectory despite disturbances and plant model mismatch. Effort was also made to use neural networks for process identification.49 Enantiomers and p-xylene separation processes have been studied by the Research of Laboratory of Separation and Reaction Engineering (LSRE) at the University of Porto, Portugal, for the investigation of SMB transient and steady-
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5587 50–54
state behavior. Different strategies for numerical solution of model and approaches on the determination of cyclic steady state were compared. The developed simulation software will be useful in the future numerical algorithm development for a efficient solution of SMB model as well as for the fast prediction of CSS. 2.2. Problems under Consideration. From the process control point of view, to ultimately use a model for control purpose encounters at least the following problems. 1. The first challenge deals with the development of algorithms for solving SMB models. Thanks to the many researchers with many years’ effort, the mathematical representation of SMB separation process is becoming mature. The SMB processes are distributed parameter systems in which the dependent variables vary with axial position and time. The models consist of a set of partial differential equations (PDEs) for mass balance over the column, ordinary differential equations (ODEs) for parabolic intraparticle concentration profile, and algebraic equations (AEs) for equilibrium isotherms and node mass balances. These equations are highly coupled, making it almost impossible to be solved analytically. The methods of finite difference, finite element, or orthogonal collocation are traditionally used for solving PDEs.55–58 Finite discretization of the PDEs gives rise to dynamic systems of a very high order. Problems with sharp variations of solutions require even larger discretization models. The CPU time required for the simulation largely depends on how many ODEs need to be solved. Systems with more components and stiff concentration profiles require more ODEs. Due to its intensive computation, the large size is a problem when online optimization and real-time control are necessary.22,59 Effort has been made to investigate alternative and more effective numerical solvers, e.g., the iterative space-time conservation element/solution element (CE/SE) method used by Lim and Jorgensen60 in the SMB application significantly reduced the computational time. It remains an important area worthy to be explored. 2. The second challenge relates to the simulation of the SMB cycles to fully capture the process dynamics. SMB unit operation exhibits complex dynamics because an SMB process never reaches a steady state, implying that all process variables do not reach time-invariant profiles. The stationary regime of this process is a cyclic steady state (CSS), in which the state of the spatially distributed concentration at the certain time of last switching is identical to the state at that time instant of the next switch. It is also indicated by the convergent time profiles of concentration at the withdrawing ports. Process metrics for design and operation are determined only after the system reaches a CSS. Generally speaking, the PDE models will be solved cycle after cycle until it converges to the CSS. Since this procedure is repeated sufficiently for a large number of cycles (more than 10 cycles), the calculations are expensive and an efficient numerical scheme is therefore essential. Over the past decade, effective simulation strategies have been investigated for the SMB chromatographic systems.53,61,62 The difference among those methods lies in the iteration cycle by cycle or by one switching time, starting from given initial conditions or using the CSS profile from previous simulation. Depending on the complexities of the adsorption model and the time constants of the system, this direct substitution approach mimics the startup of the cycle accurately, but can also be time-consuming and expensive. As a result, the optimization of SMB chromatographic process cycles with detailed models still remains a challenging problem.
In our research, we have been making efforts to promote wavelet-based approaches for solving SMBC models. In chromatographic separation, the concentration fronts experience steep changes. Such sharp transitions are typically moving with time along a spatial coordinate. The computational complexity typically increases with the separation of scales, which need to be resolved in order to obtain a physically meaningful solution. Additional complexity arises due to the fact that time scale differences occur at dynamically changing spatial domains. Wavelets play a role for efficient and full adaptive solution of this kind of problems. In the case of moving steep front, using the wavelet transform, one can track its position and increase the local resolution of the grid by adding higher resolution wavelets in that region. On the other hand, the resolution level in the smoother regions can be appropriately decreased to avoid an unnecessarily dense grid.63 Applications of wavelet transformation to the solution of partial differential equation arising in the chemical engineering are limited.64–71 Among those, Cruz, Lim, and Liu exemplified the application on a fixed-bed adsorption model to simulate the propagation of a concentration peak along a chromatographic column with axial dispersion. Application of wavelet technique to the SMB systems has not been found in the open literature. In the following, we will give a brief introduction of the SMB model system and then discuss a case study for fructose-glucose separation to demonstrate the key concepts of SMB process modeling and numerical solution of the models. 3. Dynamic SMB Model The SMB system modeling approaches can be classified as (1) true moving bed model (E-TMB), in which the system is represented in terms of an equivalent true moving bed; and (2) dynamic SMB (D-SMB), in which the system is assembled by the models of single chromatographic columns under explicit consideration of the cyclic switching operation. The main difference between the E-TMB and D-SMB models is their stationary regime. The E-TMB represents the traditional concept of steady state operation while modeling using D-SMB deals with its cyclic steady state. Practically, when increasing the number of columns and decreasing column length, an SMB system will approach an equivalent TMB system. Because the cyclic port switching is neglected, the E-TMB model is simplified and can be solved efficiently. It has been shown that the steady-state solution of a detailed TMB model reproduces the solution of an SMB model reasonably well under certain circumstances.30,54,72 However, E-TMB is a rather severe idealization of the simulated countercurrent process and is restricted to the cases of three or more columns per zone with linear isotherms and no reaction involved.46 Research has shown that, in three-section processes, the TMB approach is of limited quantitative value55 and consequently needs to be treated with caution. Comparative simulations73 also suggest that since E-TMB models do not include the transient regime of the quasistationary and periodic steady state of SMB processes, and do not consider the upper and lower bounds of the retention time, an SMB optimization based on E-TMB model will lead to a miscalculated operating condition. Earlier applications of SMB were large separation units in food and petrochemical industries and were operated under nearly linear behavior of the equilibrium isotherm. Most of the recent applications are in the areas of fine chemicals and pharmaceutical products and are operated with nonlinear adsorption behavior and in a high-purity range. SMB modeling through dynamic simulation gives essential information for a
5588 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 4. (a) Adsorbent particle; (b) mass transfer and mass exchange in separation process.
Figure 5. Column model development chart and illustration.
better understanding of chromatographic processes. It is a crucial aid to develop control strategies and to analyze the stability of SMB processes concerning malfunctions. Dynamic SMB models consist of two main parts: a single chromatographic column model, and a node balance model. Since the behavior of the equilibrium isotherm determines the characteristics of chromatography being linear or nonlinear, the models of the chromatography column are not specific. They are the combination of column models with any isotherm models. The level of mathematical difficulty encountered depends on the nature of the equilibrium relationship, the concentration level, and the choice of flow models.23 The node balance model describes the connection of the columns combined with the cyclic switching.58
3.1. Column Model. In a separation column, the mobile phase containing the solvent and components flows through the solid phase, the package of fine, globular and porous particles (adsorbent). As shown in Figure 4a, the particle consists of a solid part and a liquid pore phase and is surrounded by a film. The affinity of the component to the adsorbent leads to the transport of the component from the solvent to the particles. The driving force of the transport is the tendency of equilibrating the molecular load between the solvent and the surface of the particles. The mass transfer between the two phases is dominated by three phenomena: convection, diffusion, and dispersion. Convective and dispersive mass transfers are presented in the mobile phase, while the mass exchange between the mobile and solid
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5589
phase takes place by diffusion through the film of solvent surrounding the particle. In principle, adsorptive exchange takes place all over the particle surface; however, for ease of modeling, only the pore phase is assumed to be in exchange with the particle surface, and the exchange between the pore phase and the mobile phase occurs through the film. The mechanism is illustrated in Figure 4b. From fluid dynamics and mass transfer, mathematical descriptions are available for the above three phenomenon. The ways of describing mass transfer resistance between the mobile phase and the solid phase contribute significantly to the complexity of the models. According to the effect of those mass transfer phenomena, column models can be roughly classified as the general model,46,59,73,74 pore model,75 transport-dispersive linear driving force (LDF), transport-dispersive equilibrium,76–78 and ideal model.47,48 The derivation of each model is illustrated logically in Figure 5, where assumptions are outlined to help the development of the next stage model. More details can be found from the review paper of Guiochon.2 The transport-dispersive LDF model is the application of the first Fick’s law of mass transport and is an equation of equilibrium isotherm, and also may be called lumped solid diffusion model35 or nonequilibrium model.79 It offers a realistic representation of industrial processes and proves to be a good compromise between accurate and efficient solutions of these models.80 LDF models have been widely used in modeling many adsorption processes due to their remarkable simplicity and good agreement with experimental results.25 In this study, we will use the following typical expression of transport-dispersive LDF model to describe the kinetics of each column. ∂C ∂2C 1 - εb ∂C +u ) Dax 2 k (q/ - q) ∂t ∂x εb eff ∂x
(1)
(4)
The condition for x ) L at t > 0 i ∂Ci,j )0 ∂x
desorbent node (eluent) extract draw-off node feed node raffinate draw-off node other nodes
flow rate balance
composition balance
QI ) QIV + QD QII ) QI - QE QIII ) QII + QF QIV ) QIII - QR equal flow rates for the columns in the same zone
out Ci,IQI ) Ci,IV QIV + Ci,DQD in out Ci,E ) Ci,II ) Ci,I in out Ci,III QIII ) Ci,II QII + Ci,FQF in out Ci,R ) Ci,IV ) Ci,III in out Ci,j ) Ci,j-1 in
Jorgensen60 is taken as our case study system in this work. The SMBCP has 8 columns with the configuration of 2:2:2:2. The schematic system is similar to that in Figure 1 except with two columns in each section. The operating conditions and model parameters are summarized in Table 2. The transport-dispersive LDF model is selected to represent the column dynamics. Thus, the model consists of 16 PDEs, 16 ODEs, and 20 AEs connecting all the variables together.
symbol
value
symbol
L (cm) D (cm) εb τ (min) QF (mL/min)
52.07 2.6 0.41448 16.39 1.67 zone I 15.89 1.105
keff,i (min-1) Ci,feed (g/L)
Q (mL/min) Dax (cm2/min)
value (A, fructose)
value (B, glucose)
0.72 0.9 363 322 qA/ ) 0.675CA qB/ ) 0.32CB + 0.000457CACB
zone II 11.0 0.765
zone III 12.67 0.881
zone IV 9.1 0.633
(3)
Boundary Conditions. There are two boundary conditions, one at the column inlet and the other at the column outlet (i ) component A or B, j ) column number). The condition for x ) 0 at t > 0 is uj ∂Ci,j ) (C - Cin i,j) ∂x Dax i,j
Table 1. Node Model
Table 2. Parameters of the Fructose-Glucose Separation
∂q ) keff(q/ - q) (2) ∂t 3.2. Node Model. To complete the dynamic modeling system, apart from the column model described in eqs 1 and 2, initial conditions and boundary conditions are also essential. These are summarized below. Initial conditions describe the status of the column at the beginning of the switching [k-1] C[k] (ts, x) k: number of switching i,j (0, x) ) Ci,j
Figure 6. SMB dynamic modeling system.
(5)
in where Ci,j are subject to the material balances at the column conjunctions as listed in Table 1. A schematic description of the modeling system is depicted in Figure 6.
4. Case Study: SMB Modeling and Wavelet-Based Numerical Computing 4.1. System Description. The separation of fructose-glucose in deionized water used in Beste et al.81 and Lim and
4.2. Numerical Computing. Since it is almost impossible to solve the system model analytically, numerical computing of the system model becomes crucial. Various numerical computing techniques have been investigated for the SMB model computation, e.g., finite difference method and finite element method. However, these methods are computationally intensive and unsuitable for real-time control study. Moreover, they may not retain the global model dynamics and the computed results may differ significantly from the real solutions. We have recently developed new wavelet collocation methods for solving SMB models numerically. The effectiveness of these methods will be shown below through the case study. Numerical simulations have been performed in this work using both finite difference and wavelet collocation methods for spatial discretization. The same integrator, the Alexander semi-implicit method (third-order Runge-Kutta), is adopted so that the results can be compared on the effectiveness of different spatial discretization methods. For the trials of FD, the number of mesh along one column length has been chosen as Nx ) 33 and 65, which are equivalent to the collocation points generated by wavelet level of J ) 5 and J ) 6. Simulations using wavelet
5590 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
collocation were conducted on the level J ) 4, 5, and 6. The boundary conditions are treated using polynomial interpolation proposed by Liu71 with the degree M of 1, 2, and 3. The number of mesh along the time axis is 5 points each switching period for all the trials. The reason for fewer mesh points is that this semi-implicit integrator has built-in Newton iteration mechanism for all its three-stage equations. A brief introduction into the numerical computing equations will be given in the following section. 4.3. Numerical Computing Equations. The spatial discretization of eq (1) and boundary conditions for column model are performed in this paper using either finite difference or wavelet method to transform partial differential equations to ordinary equations for each spatial mesh point (or collocation point). 1. Finite Difference Discretization Using Upwind-1 Method
(
)
(
2Dax Dax dC(xi, t) Dax u ) 2 C(xi+1, t) + C(xi, t) + + 2 dt ∆x ∆x ∆x ∆x2 1 - εb u C(xi-1, t) k [q*(xi, t) - q(xi, t)] ∆x εb eff
)
where i ) 2,..., N - 1,∆x ) xi+1 - xi, and N is the number of mesh points. Boundary conditions are treated using the false boundary method, which gives
(
)
(
Dax dC(x1, t) Dax u2 u2 u ) 2 C(x3, t) + + C(x2, t) + + 2 dt Dax ∆x Dax ∆x ∆x 1 - εb u in C (x1, t) k [q*(x1, t) - q(x1, t)] ∆x εb eff
)
and
(
)
Dax dC(xN, t) u ) + [C(xN-1, t) - C(xN, t)] 2 dt ∆x ∆x 1 - εb k [q*(xN, t) - q(xN, t)] εb eff
3. Time Integratorsthe Alexander Method. This is a thirdorder semi-implicit Runge-Kutta method, which requires several implicit equations to be solved in series. The Alexander method has three-stage equations with the order of 3 and is L-stable. Its Butcher block looks like
where γ ) 0.435 866 5, c ) (1 + γ)/2, b1 ) -(6γ2 - 16γ + 1)/4, b2 ) (6γ2 - 20γ + 5)/4. The good stability of the method makes it particularly well suited for stiff problems. 4.4. Results and Analysis. Figure 7 illustrates the concentration distribution along the total columns length at the middle of 80th switching. Although simulation using wavelet approach has been conducted under different levels, the results from lower levels can fit equivalently well with the experimental data, especially at the feeding port, where the concentration front experiences a sudden change. Higher level (J ) 6) wavelet demands much more computational time while giving a similar prediction performance. The two methods produce almost identical profile, however, as far as computational cost is concerned, wavelet takes remarkably less time for each switching period (5 s for J ) 4; 16 s for J ) 5; 285 s for Nx ) 65 of the FD method in this study). From the time profile of average concentration in Figure 8, numerical simulation from wavelet reaches steady state faster than the solution from finite difference. Furthermore, the product purity and yield are used as criterion for the evaluation of separation quality. They were calculated as average during a switching period. Table 3 lists the result from our simulation, reported experimental data, and other published simulation results based on the same case study. The numerical solution using wavelet is very close to the results reported in ref 81 which was carried out on Sun Ultra Spark I platform using the method of line (MOL).
2. Wavelet Collocation Discretization. The compactly supported orthonormal wavelets71 is constructed to approximate the function to the interval of [0,1]. J
J
2 dC(xi, t) Dax 2 u )C(xj, t)D(2) C(xj, t)D(1) + i,j i,j dt L j)0 L2 j)0
∑
∑
1 - εb k [q*(xi, t) - q(xi, t)] εb eff where i ) 2,..., N - 1, N ) 2J + 1 is the number of collocation (1) (2) points; J is the resolution level; and Di,j and Di,j are the first and second derivatives, respectively, for the autocorrelation function of the scaling function. The left boundary condition is 2J
∑ C(x , t)D j
(1) 1,j )
j)0
uL [C(x1, t) - Cin(x1, t)] Dax
And the right boundary condition is 2J
∑ C(x , t)D j
j)0
(1) 2J,j ) 0
Figure 7. Concentration distribution at middle of 80th switch.
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5591 Table 3. Separation Quality Analysis (Average at 73th Switching unless Ns Specified) purity (%)
exptl (ref 81)a finite difference Nx ) 65 wavelet J ) 4 (M ) 1) wavelet J ) 5 (M ) 2) simulation (MOL) (ref 81)a CE/AE at Ns ) 40 (ref 60)a a
yield (%)
extract
raffinate
extract
raffinate
81.6 88.0 88.0 89.2 89.4 97.8
92.9 96.2 97.8 98.0 97.9 98.3
96.4 97.6 98.9 99.3 98.3 89.5
80.4 85.5 85.1 86.9 86.9 97.3
elapsed time (for 80 switches) 6 h (P4 3.00 GHz) 6 min (P4 3.00 GHz) 21 min (P4 3.00 GHz) 4-5 h (Sun Spark I)
Denotes the results from references.
LDF column model has been studied in this work. Using wavelets for spatial discretization combined with a third-order semi-implicit integrator has led to encouraging results in terms of computation time and prediction accuracy on steep front. Moreover, it is faster at achieving steady state. Further research can be conducted from the following aspects using the wavelet technique. 1. Development of control-relevant model reduction techniques for nonlinear SMB chromatographic processes. Limited work has been reported in the open literature on systematic investigation of model reduction for SMB control. 2. Development of nonlinear model predictive control on the reduced model in order to optimize SMB operation. Though model predictive control is now a fairly standard technique for model-based control, it is still challenging to design a nonlinear version for optimal SMB operation with sufficient stability and robustness. Nomenclature
Figure 8. Time profile of average concentration at the extract and raffinate ports.
As for the computing demand and efficiency, the reported calculation time for the method of line (MOL) on Sun Ultra Spark I platform is 4-5 h. The numerical simulations of this paper have been conducted on a personal computer with Intel’s Pentium IV 3.00 GHz processor, and thus the computation times cannot be compared with that from the Sun Ultra Spark I platform. However, we have simulated both the finite difference method and wavelet method on the same personal computer, and therefore the computation times can be directly compared with each other. It is seen from Table 3 that the finite difference method consumes a few hours to get the results, while the wavelet method requires only 6 to 21 min. This indicates that the computing performance of the wavelet method is encouraging. 5. Concluding Remarks With the recognition of the SMB chromatographic process technology by various industries, systematic investigations become increasingly significant into the modeling and modelbased control design. Application of wavelets to the solution of dynamic SMB modeling system with a transport-dispersive
C ) fluid phase concentration Cp ) concentration in the pores of the particle j p ) average concentration in the pores of the particle C in out Ci,j , Ci,j ) the concentrations of component i at the outlet or the inlet of column j D ) column diameter (1) (2) Di,j ; Di,j ) the first and second derivative for the autocorrelation function of scaling function Dax ) axial dispersion coefficient of the bulk fluid phase J ) wavelet resolution level keff ) effective fluid film mass transfer resistance L ) column length N ) number of mesh points (FD) or collocation points (wavelets) along spatial discretization q ) concentration of component in the solid phase q* ) equilibrium concentration in interface between two phases QI, QII, QIII, QIV ) volumetric flow rate through the corresponding sections QD ) desorbent flow rate QE ) extract flow rate QF ) feed flow rate QR ) raffinate flow rate r ) radial coordinate for particle Rp ) particle radius t,x ) time and axial coordinates u ) interstitial velocity τ ) switching time εb ) void porosity of the mobile phase
Acknowledgment The authors acknowledge the anonymous reviewers for their constructive comments and suggestions, which were invaluable
5592 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
for the significant improvement of the quality and readability of this paper. Literature Cited (1) Spear, M. Separations in Flux. http://www.chemicalprocessing.com(accessed Oct 2006). (2) Guiochon, G. Review: Preparative Liquid Chromatography. J. Chromatogr., A 2002, 965, 129. (3) Jin, S. H.; Wankat, P. C. New Design of Simulated Moving Bed for Ternary Separations. Ind. Eng. Chem. Res. 2005, 441, 1906. (4) Broughton, D. B. Production-Scale Adsorptive Separations of Liquid Mixtures By Simulated Moving-Bed Technology. Sep. Sci. Technol. 1984, 19, 723. (5) Snyder, L. R.; Kirkland, J. J. Introduction to Modern Liquid Chromatography; Wiley: New York, 2nd ed., 1979. (6) deRosset, A. J.; Neuzil, R. W.; Korous; D. J. Liquid Column Chromatography as a Predictive Tool for Continuous Countercurrent. Ind. Eng. Chem. Proc. Des. DeV. 1976, 15, 261. (7) Minceva, M.; Rodrigues, A. E. Understanding and Revamping of Industrial Acale SMB units for p-xylene Separation. AIChE J. 2007, 53, 138. (8) Azevedo, D. C. S.; Rodrigues, A. S. Obtainment of High-Fructose Solutions from Cashew (Anacardium Occidentale) Apple Juice by Simulated Moving-Bed Chromatography. Sep. Sci. Technol. 2000, 35, 2561. (9) Lu, Y.; Wei, F.; Shen, B.; Ren, Q.; Wu, P. Modeling, Simulation of a Simulated Moving Bed for Separation of Phosphatidylcholine from Soybean Phospholipids. Chin. J. Chem. Eng. 2006, 14, 171. (10) Molnar, Z.; Nagy, M.; Aranyi, A.; Hanak, L.; Argyelan, J.; Pencz, I.; Szanya, T. Separation of Amino Acids with Simulated Moving Bed Chromatography. J. Chromatogr., A 2005, 1075, 77. (11) Xie, Y.; Chin, C. Y.; Phelps, D. S. C.; Lee, C. H.; Lee, K. B.; Mun, S.; Wang, L. A. Five-Zone Simulated Moving Bed for Isolation of Six Sugars From Biomass Hydrolyzate. 2005 AIChE Annual Meeting and Fall Showcase, Conference Proceedings; 2005; p 1675. (12) Lancrenon, X.; Herve, D. Sugar Technol. ReV. 1988, 14, 207. (13) Pope, J. 211th American Chemical Society National Meeting, Symposium on Industrial Scale Process Chromatographic Separations, New Orleans, 27-28 March, 1996. (14) Borges da Silva, E. A.; Souza, D. P.; Ulson de Souza, A. A.; Souza, S. M.; Rodrigues, A. E. Analysis of the behaviour of the simulated moving bed reactor in the sucrose inversion process. Sep. Sci. Technol. 2005, 40, 2373. (15) Bauer, J. E.; Chandhok, A. K.; Scanlon, B. W.; Wilcher, S. A. A ComprehesiVe Look at Scaling-Up SMB Chiral Separations. Proceedings of Chiratech’97; Philadelphia Catalyst Consultants Publishing, Inc.: Philadelphia, PA, 1997. (16) Pais, L. S.; Loureiro, J. M.; Rodrigures, A. E. Chiral Separation by SMB Chromatography. Sep. Purif. Technol. 2000, 20, 67. (17) Pedeferri, M.; Zenoni, G.; Mazzotti, M.; Morbidelli, M. Experimental Analysis of a Chiral Separation through Simulated Moving Bed Chromatography. Chem. Eng. Sci. 1999, 54, 3735. (18) Schulte, M. Preparative Enantioseparation by Simulated Moving Bed Chromatography. J. Chromatogr., A 2001, 906, 399. (19) Wang, X.; Ching, C. B. Chiral Separation and Modelling of the Three-Chiral-Center β-Blocker Drug Nadolol by Simulated Moving Bed Chromatography. J. Chromatogr., A 2004, 1035, 167. (20) Wongso, F.; Hidajat, K.; Ray, A. K. Optimal Operating Mode for Enantioseparation Of SB-553261 Racemate based on Simulated Moving Bed Technology. Biotechnol. Bioeng. 2004, 87, 704. (21) Rodrigues, A. E.; Minceva, M. Modelling and Simulation in Chemical Engineering: Tools for Process Innovation. Comput. Chem. Eng. 2005, 29, 1167. (22) Klatt, K-U.; Hanisch, F.; Du¨nnebier, G. Model-Based Control of a Simulated Moving Bed Chromatographic Process for the Separation of Fructose and Glucose. J. Process Control 2002, 12, 203. (23) Ruthven, D. M. Principles of Adsorption and Adsorption Process; John Wiley & Sons: New York, 1984. (24) Ruthven, D. M.; Ching, C. B. Counter-Current and Simulated Counter-Current Adsorption Separation Processes. Chem. Eng. Sci. 1989, 44, 1011. (25) Ching, C. B. Parabolic Intraparticle Concentration Profile Assumption in Modeling and Simulation of Nonlinear Simulated Moving-Bed Separation Processes. Chem. Eng. Sci. 1998, 53, 1311. (26) Mazzotti, M.; Storti, G.; Morbidelli, M. Robust Design of Countercurrent Adsorption Separation Processes: 2. Multicomponent Systems. AIChE J. 1994, 40, 1825.
(27) Mazzotti, M.; Storti, G.; Morbidelli, M. Robust Design of Countercurrent Adsorption Separation Processes: 3. Nonstoichiometric Systems. AIChE J. 1996, 42, 2784. (28) Mazzotti, M.; Storti, G.; Morbidelli, M. Robust Design of Countercurrent Adsorption Separation Processes: 4. Desorbent In The Feed. AIChE J. 1997, 43, 64. (29) Storti, G.; Baciocchi, R.; Mazzotti, M.; Morbidelli, M. Design of Optimal Operating Conditions of Simulated Moving Bed Adsorptive Separation Units. Ind. Eng. Chem. Res. 1995, 34, 288. (30) Storti, G.; Masi, M.; Paludetto, R.; Morbidelli, M.; Carra, S. Adsorption Separation Processes: Countercurrent and Simulated Countercurrent Operations. Comput. Chem. Eng. 1988, 12, 475. (31) Storti, G.; Mazzotti, M.; Morbidelli, M.; Carra, S. Robust Design of Binary Countercurrent Adsorption Separation Processes. AIChE J. 1993, 39, 471. (32) Abel, S.; Erdem, G.; Mazzotti, M.; Morari, M.; Morbidelli, M. Optimizing Control of Simulated Moving Beds - Linear Isotherm. J. Chromatogr., A 2004, 1033, 229. (33) Biressi, G.; Ludemann-Hombourger, O.; Mazzotti, M.; Nicoud, R. M.; Morbidelli, M. Design and Optimisation of a Simulated Moving Bed Unit: Role of Deviations from Equilibrium Theory. J. Chromatogr. A 2000, 876, 3. (34) Erdem, G.; Morari, M.; Amanullah, M.; Mazzotti, M.; Morbidelli, M. Optimizing Control of an Experimental Simulated Moving Bed Unit. AIChE J. 2006, 52, 1481. (35) Song, I. H.; Lee, S. B.; Rhee, H. K.; Mazzotti, M. Identification and Predictive Control of a Simulated Moving Bed Process. Purity Control Chem. Eng. Sci. 2006, 61, 1973. (36) Strohlein, G.; Mazzotti, M.; Morbidelli, M. Optimal Operation of Simulated-Moving-Bed Reactors for Nonlinear Adsorption Isotherms and Equilibrium Reactions. Chem. Eng. Sci. 2005, 60, 1525. (37) Zhang, Z. Y.; Mazzotti, M.; Morbidelli, M. PowerFeed Operation of Simulated Moving Bed Units: Changing Flow-Rates during the Switching Interval. J. Chromatogr., A 2003a, 1006, 87. (38) Zhang, Z. Y.; Mazzotti, M.; Morbidelli, M. Multiobjective Optimization of Simulated Moving Bed and Varicol Processes using a Genetic Algorithm. J. Chromatogr., A 2003b, 989, 95. (39) Guiochon, G.; Ghodbane, S. Computer Simulation of the Separation of a Two-Component Mixture in Preparative Scale Liquid Chromatography. J. Phys. Chem. 1988, 92, 3682. (40) Zhong, G. M.; Guiochon, G. Theoretical Analysis of Band Profiles in Non-Linear Ideal Countercurrent Chromatography. J. Chromatogr., A 1994, 688, 1. (41) Zhong, G. M.; Guiochon, G. Analytical Solution for the Linear Ideal Model of Simulated Moving Bed Chromatography. Chem. Eng. Sci. 1996, 51, 4307. (42) Zhong, G. M.; Guiochon, G. Simulated Moving Bed Chromatography. Effects of an Axial Dispersion and Mass Transfer under Linear Conditions. Chem. Eng. Sci. 1997, 52, 3117. (43) Zhong, G. M.; Guiochon, G. Simulated Moving Bed Chromatography: Comparison between the Behaviours under Linear and Nonlinear Conditions. Chem. Eng. Sci. 1997, 52, 4403. (44) Zhong, G. M.; Smith, M. S.; Guiochon, G. Effect of the Flow Rates in Linear, Ideal, Simulated Moving-Bed Chromatography. AIChE J. 1997, 43, 2960. (45) Schure, M. R.; Katti, A. M. Commemoration: Professor Georges A. Guiochon’s 75th Birthday. J. Chromatogr., A 2006, 1126, 2. (46) Du¨nnebier, G.; Fricke, J.; Klatt, K. U. Optimal Design and Operation of Simulated Moving Bed Chromatographic Reactors. Ind. Eng. Chem. Res. 2000, 39, 2290. (47) Du¨nnebier, G.; Weirich, I.; Klatt, K-U. Computationally Efficient Dynamic Modelling and Simulation of SMB Chromatographic Processes with Linear Isotherms. Chem. Eng. Sci. 1998, 3, 2537. (48) Du¨nnerbier, G.; Klatt, K-U. Modelling and Simulation of Nonlinear Chromatographic Separation Processes: A Comparison of Different Modelling Approaches. Chem. Eng. Sci. 2000, 55, 373. (49) Wang, C.; Klatt, K-U.; Du¨nnerbier, G.; Engell, S.; Hanisch, F. Neural Network-Based Identification of SMB Chromatographic Processes. Control Eng. Pract. 2003, 11, 949. (50) Lea˜o, C. P.; Rodrigues, A. E. Transient and Steady-State Models for Simulated Moving Bed Processes: Numerical Solutions. Comput. Chem. Eng. 2004, 28, 1725. (51) Minceva, M.; Rodrigues, A. Modeling And Simulation of a Simulated Moving Bed for the Separation of P-Xylene. Ind. Eng. Chem. Res. 2002, 41, 3454. (52) Minceva, M.; Rodrigues, A. Two-Level Optimization of an Existing SMB For P-Xylene. Comput. Chem. Eng. 2005, 29, 2215.
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5593 (53) Minceva, M.; Pais, L. S.; Rodrigues, A. Cyclic Steady State of Simulated Moving Bed Processes for Enantiomers Separation. Chem. Eng. Process. 2003, 42, 93. (54) Pais, L. S.; Loureiro, J. M.; Rodrigures, A. E. Modelling Strategies For Enantiomers Separation by SMB Chromatography. AIChE J. 1998, 44, 561. (55) Chu, K. H.; Hashim, M. A. Simulated Countercurrent Adsorption Processes: A Comparison of Modelling Strategies. Chem. Eng. J. 1995, 56, 59. (56) Kaczmarski, K.; Mazzotti, M.; Storti, G.; Morbidelli, M. Modeling Fixed-Bed Adsorption Columns through Orthogonal Collocations on Moving Finite Elements. Comput. Chem. Eng. 1997, 21, 641. (57) Cruz, P.; Santos, J. C.; Magalha˜es, F. D.; Mendes, A. Simulation of Separation Processes using Finite Volume Method. Comput. Chem. Eng. 2005, 30, 83. (58) Arau´jo, J. M. M.; Rodrigues, R. C. R.; Mota, J. P. B. Use of SingleColumn Models for Efficient Computation of the Periodic State of a Simulated Moving-Bed Process. Ind. Eng. Chem. Res. 2006, 45, 5314. (59) Toumi, A.; Engell, S. Optimization-Based Control of A Reactive Simulated Moving Bed Process for Glucose Isomerisation. Chem. Eng. Sci. 2004, 59, 3777. (60) Lim, Y.; Jorgensen, S. B. A Fast And Accurate Numerical Method for Solving Simulated Moving Bed (SMB) Chromatographic Separation Problems. Chem. Eng. Sci. 2004, 59, 1931. (61) Kloppenburg, E.; Gilles, E. D. A New Concept for Operating Simulated Moving-Bed Processes. Chem. Eng. Technol. 1999, 22, 813. (62) Nilchan, S.; Pantelides, C. On Optimization Of Periodic Adsorption Processes. Adsorption 1998, 4, 113. (63) Cruz, P.; Mendes, A.; Magalha˜es, F. D. Using Wavelets for Solving PDEs: an Adaptive Collocation Method. Chem. Eng. Sci. 2001, 56, 3305. (64) Bertoluzza, S. A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations. Appl. Comput. Harmonia Anal. 1996, 3, 1. (65) Bertoluzza, S.; Canuto, C.; Urban, K. On the Adaptive Computation of Integral Wavelets. Appl. Numer. Math. 2000, 34, 13. (66) Lim, Y.; Lann, J. M.; Joulia, X. Moving Mesh Generation for Tracking a Shock or Steep Moving Front. Comput. Chem. Eng. 2001, 25, 653. (67) Lim, Y.; Lann, J. M.; Joulia, X. Accuracy, Temporal Performance and Stability Comparisons of Discretization Methods for the Numerical Solution of Partial Differential Equations (PDEs) in the Presence of Steep Moving Fronts. Comput. Chem. Eng. 2001, 25, 1483. (68) Cruz, P.; Mendes, A.; Magalha˜es, F. D. Wavelet-Based Adaptive Grid Method for the Resolution of Nonlinear PDEs. AIChE J. 2002, 48, 774.
(69) Bindal, A.; Khinast, J. G.; Ierapertritou, M. G. Adaptive Multiscale Solution of Dynamical Systems in Chemical Processes Using Wavelets. Comput. Chem. Eng. 2003, 27, 131. (70) Briesen, H.; Marquardt, W. Adaptive Multigrid Solution Strategy for the Dynamic Simulation of Petroleum Mixture Processes. Comput. Chem. Eng. 2005, 29, 139. (71) Liu, Y.; Cameron, I. T.; Wang, F. Y. The Wavelet-Collocation Method for Transient Problems with Steep Gradients. Chem. Eng. Sci. 2000, 55, 1729. (72) Hashimoto, K.; Adachi, S.; Noujima, H.; Maruyama, A. Models for the Separation of Glucose/Fructose Mixture Using a Simulated MovingBed Adsorber. J. Chem. Eng. Jpn. 1983, 16, 400. (73) Strube, J.; Schmidt-Traub, H. Dynamic Simulation of SimulatedMoving-Bed Chromatographic Processes. Comput. Chem. Eng. 1998, 22, 1309. (74) Lu, Z. P.; Ching, C. B. Dynamics of Simulated Moving-Bed Adsorption Separation Processes. Sep. Sci. Technol. 1997, 32, 1993. (75) Natarajan, S.; Lee, J. H. Repetitive Model Predictive Control Applied to a Simulated Moving Bed Chromatograph System. Comput. Chem. Eng. 2000, 24, 1127. (76) Migliorini, C.; Gentilini, A.; Mazzotti, M.; Morbidelli, M. Design of Simulated Moving Bed Units under Nonideal Conditions. Ind. Eng. Chem. Res. 1999, 38, 2400. (77) Erdem, G.; Abel, S.; Marari, M.; Mazzotti, M.; Morbidelli, M. Automatic Control of Simulated Moving Beds II: Nonlinear Isotherm. Ind. Eng. Chem. Res. 2004, 43, 3895. (78) Amanullah, M.; Grossmann, C.; Mazzotti, M.; Morari, M.; Morbidelli, M. Experimental Implementation Of Automatic ‘Cycle to Cycle’ Control of a Chiral Simulated Moving Bed Separation. J. Chromatogr., A 2007, 1165, 100. (79) Lim, Y. An Optimization Strategy For Nonlinear Simulated Moving Bed Chromatography: Multi-Level Optimization Procedure (MLOP). Korean J. Chem. Eng. 2004, 21, 836. (80) Biegler, L. T.; Jiang, L.; Fox, V. G. Recent Advances in Simulation and Optimal Design of Pressure Swing Adsorption Systems. Sep. Purif. Technol. 2004, 33, 1. (81) Beste, Y. A.; Lisso, M.; Wozny, G.; Arlt, W. Optimization of Simulated Moving Bed Plants With Low Efficient Stationary Phases: Separation of Fructose and Glucose. J. Chromatogr., A 2000, 868, 169.
ReceiVed for reView September 13, 2007 ReVised manuscript receiVed May 6, 2008 Accepted May 15, 2008 IE071246G
