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Application to SMBC Processes: Case Study on Bi-Naphthol Enantiomers Separation

Application to SMBC Processes: Case Study on Bi-Naphthol Enantiomers Separation

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Wavelet Based Approaches and High Resolution Methods...

q *A, j =

q B* , j =

2.69C A, j

1 + 0.00336 C A, j + 0.0466 C B , j

3.73C B , j

1 + 0.00336C A, j + 0.0466C B , j



0.1C A, j

1 + C A, j + 3C B , j



0.3C B , j

1 + C A, j + 3C B , j

where, i = A, B ; j = 1" 8 .

To complete the dynamic modelling system, apart from the column model described in

Equation (26a) and (26b), initial conditions and boundary conditions are also essential.

C i[,kj] (0, x) = Ci[,kj−1] (t s , x) k: number of switching

The condition for x = 0 at t > 0 is:

The condition for x = L at t > 0 is:

∂Ci , j




∂C i , j



(Ci , j − C iin, j )





The Node Model for an SMB system

The flow and integral mass balance equations at each node are summarised in table 4

according to the sketch in figure 16, under the assumptions that the dead volume by the

switching devices, connecting tubes, and other parts is negligible. The operating conditions

and model parameters are given in table 5.

Figure 16. Flow diagram of 4 section SMB system.

Table 4. Node model

Flow Rate Balance

Composition Balance Eq (26g)

Desorbent node (eluent)


C iin, I QI = Ciout

, IV Q IV + C i , D Q D

Extract draw-off node


C i , E = C iin, II = Ciout


Feed node


Ciin, III QIII = Ciout

, II QII + Ci , F QF

Raffinate draw-off node


Ci , R = Ciin, IV = Ciout


Other nodes

Equal flow rates for the

columns in the same zone

C iin, j = C iout

, j −1


Hongmei Yao, Tonghua Zhang, Moses O. Tadé et al.

Table 5. System parameters and operating conditions for Bi-naphthol enantiomers




L (cm)


D (cm2/min)


k eff ,i


t switch (min)


Ci , feed





QF (ml/min)



QI (ml/min)



QII (ml/min)



QIII (ml/min)


QIV (ml/min)


Thus, the SMBC model is constructed by 16 PDEs from Equation (26a), 16 ODEs from

Equation (26b) and 20 AEs from Equations (26c) and (26g). All the single column models are

connected in series by boundary conditions. The dominating parameters of the interstitial

velocity and the inlet concentration of each column are restricted by node models. The

switching operation can be represented by a shifting of the initial or boundary conditions for

the single columns. This means that those conditions for each column change after the end of

each switch time interval. After the cyclic steady state is reached, the internal concentration

profiles vary during a given cycle, but they are identical at the same time for two successive


5.2. Numerical Simulations

As the Peclet number is close to 2000 in this application, the finite difference method will

not be adopted for solving the model equations numerically due to the reason given in Section

4. Therefore, numerical simulations have been performed using both the high resolution and

wavelet collocation methods for spatial discretization. The same integrator, the Alexander

semi-implicit method as described in Section 3.3, is used so that the results on the

effectiveness of different spatial discretization methods can be compared.

For the trials of the high resolution method, the number of mesh points along one

column length has been chosen to be Nz = 17 and 33, which are equivalent to the

collocation points generated by wavelet level of J = 4 and J = 5, respectively. Simulations

using wavelet collocation method are conducted on the level J = 4, 5, and 6, respectively.

The boundary conditions are treated using polynomial interpolation with the degree M = 1.

The number of mesh points along the time axis is 5 points each switching period for all the

trials. The reason for the less mesh points is that this semi-implicit integrator has built-in

Newton iteration mechanism for all its three stage equations, which improves the efficiency

of the scheme.

Figure 17 is the propagation of concentration profile with time.

Wavelet Based Approaches and High Resolution Methods...


Figure 17. Propagation of concentration wave at mid of each switching.

Figure 18 Concentration distributions at cyclic steady state.

Figure 19. Concentration distribution at cyclic steady state: HR 33 line (red); Wavelet-Collocation J=5

dash line (green).

Figure 18 and figure 19 illustrate the calculated concentration distribution against

experimental data, along the total columns length at the middle of 80th switching, which is


Hongmei Yao, Tonghua Zhang, Moses O. Tadé et al.

taken to be steady state. J = 4 of wavelet collocation or Nz = 17 of high resolution are not

good enough to predict the real value, and furthermore, wavelet collocation presents certain

degree of oscillation. However, wavelet collocation with J = 5 or high resolution with Nz =

33 produce better approximation, with high resolution has much closer results.






Relative error










HR 33












Number of switches

Figure 20. Comparison of relative error of prediction.

Figure 20 is the relative error defined by Minceva et al. (2003), which reflects the

algorithm convergence performance. Wavelet has an abrupt point at the end of first cycle or

the starting of next cycle (between 8th-9th switching). High resolution has consistent and better

convergence. As far as computational cost is concerned, standing on the same number of

spatial mesh points, wavelet takes less time for each switching period (16sec for J = 5)

because less iteration (2) is required in the solving of Jacobi matrices. The high resolution

needs 24.8sec for one switching calculation where 4 iterations are required. Nevertheless, the

results from the high resolution method are much closer to the reported experimental data.

6. Concluding Remarks

This chapter has explored some upfront discretization techniques for the solution of

complicated dynamic system models with sharp variations. Recently developed wavelet based

approaches and high resolution methods have been successfully used for solving models of

simulated moving bed chromatographic separation processes. To investigate the numerical

power of proposed methods, the solution of single column chromatographic process

represented by a Transport-Dispersive-Equilibrium linear model was firstly studied on the

prediction of transit behaviour of wave propagation. Comparisons of the numerical solutions

from finite difference, wavelet and high resolution methods with analytical solutions were

conducted for a range of Peclet numbers. It has revealed that all the proposed methods work

well when the PDEs system has low Peclet number, especially the upwind finite difference

method, which can offer good numerical solution with reasonable computing time. The high

resolution method provides an accurate numerical solution for the PDEs in question with

Wavelet Based Approaches and High Resolution Methods...


medium value of Pe. The wavelet collocation method is capable of catching up steep changes

in the solution, and thus can be used for numerically solving PDEs with high singularity.

The advantages of the wavelet based approaches and high resolution methods are further

demonstrated through applications to a dynamic SMB model for an enantiomers separation

process. It shows that both of the methods are good candidates for the numerical solution of

this complex model. They have provided encouraging results in terms of computation time

and prediction accuracy on steep front. However, high resolution methods would be more

preferable in this case because of better stability at achieving steady state and closer

approximation to experimental data.

Generally, in terms of the two approaches (wavelet collocation and high resolution)

mainly investigated here, wavelet based methods offers a better solution to PDEs with high

singularity, however, prior knowledge of wavelet is required in order to take advantage of this

kind of method. High resolution method is easy for implementation and can offer reasonable

result with reasonable computing time.

However, it is suggested that wavelet based methods should be used with cautious

because overestimation of wave peak is observed in our simulation results as well as in

previous work. Unlike finite difference and high resolution methods, it does not follow the

rule of the more points, the better approximation. The selection of wavelet resolution level is

highly related to specific problem, such as, the most accurate results are from J=6 for Pe=50,

J=7 for Pe=500, and J=9 for Pe=5000.

We have also noted that the selection of the value of the number of interpolating points,

M, is very important since it could affect the numerical solution significantly. No report has

been found on this issue so far in the open literature. Further investigation should be carried

out on this subject.

Although this is a preliminary study, the results are encouraging for their applications to

other complicated industrial systems. Further investigation is required into various aspects of

the discussed numerical computing method to improve their capability for numerically

solving difficult PDEs.

7. Nomenclature



i, j

C ,C

fluid phase concentration


i, j


D :

the concentrations of component i at the outlet or the inlet of

column j.

column diameter

Ti ,(1j) ; Ti ,( 2j ) :

the first and second derivative for the autocorrelation function of

scaling function

Dax :

axial dispersion coefficient of the bulk fluid phase

k eff :


effective fluid film mass transfer resistance



column length

concentration of component in the solid phase

equilibrium concentration in interface between two phases


Hongmei Yao, Tonghua Zhang, Moses O. Tadé et al.

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

t, x :

τ,z :

time and axial coordinates

u :

t switch :

interstitial velocity

switching time

εb :

Pe :

dimensionless time and length

void porosity of the mobile phase

Peclet number


The authors would like to acknowledge the support from Australian Research Council

(ARC) under Discovery Project Scheme (grant number DP0559111 to Tadé and Tian, grant

number DP0770420 to Zhang) and also the financial support from the Australian

Postgraduate Award to Yao.


[1] Bertoluzza, S. (1996). A wavelet collocation method for the numerical solution of

partial differential equations. Applied Computers and Harmonia Analysis, 3, 1-9.

[2] Bertoluzza, S., & Naldi, G. (1994). Some remarks on wavelet interpolation. Journal of

computational and applied mathematics, 13 (1), 13-32.

[3] Biegler, L. T., Jiang L., & Fox, V. G. (2004). Recent advances in simulation and optimal

design of pressure swing adsorption systems. Separation and Purification Reviews,

33(1), 1-39.

[4] Bindal, A., Khinast, J. G., & Ierapertritou, M. G. (2003). Adaptive multiscale solution of

dynamical systems in chemical processes using wavelets. Computers and Chemical

Engineering, 27, 131-142.

[5] Blom, J. G., & Zegeling, P. A. (1994). Algorith 731: a moving-grid interface for systems

of one-dimensional time-dependent partial differential equations. ACM Transactions on

Mathematical Software, 20, 194-214.

[6] Briesen, H., & Marquardt, W. (2005). Adaptive multigrid solution strategy for the

dynamic simulation of petroleum mixture processes. Computers and Chemical

Engineering, 29(1), 139-148.

[7] Ching, C. B. (1998). Parabolic intraparticle concentration profile assumption in

modeling and simulation of nonlinear simulated moving-bed separation processes.

Chemical Engineering Science, 53(6), 1311-1315.

Wavelet Based Approaches and High Resolution Methods...


[8] Cruz, P., Mendes, A., & Magalhães, F. D. (2001). Using wavelets for solving PDEs: an

adaptive collocation method. Chemical Engineering Science, 56, 3305-3309.

[9] Cruz, P., Mendes, A., & Magalhães, F. D. (2002). Wavelet-based adaptive grid method

for the resolution of nonlinear PDEs. American Institute of Chemical Engineering

Journal, 48(5), 774-785.

[10] Donoho, D. (1992). Interpolating wavelet transform. Technical Report, Department of

Statistics, Stanford University, 1992.

[11] Dorfi, E. A., & Drury, L. O. C. (1987). Simple adaptive grids for 1 - D initial value

problems. Journal of Computational Physics, 69, 175-195.

[12] Eigenberger, G., & Butt, J. B. (1976). A modified Crank-Nicolson technique with nonequidistant space steps. Chemical Engineering Science, 31, 681-691.

[13] Furzeland, R. M. (1990). A numerical study of three moving-grid methods fro onedimensional partial differential equations which are based on the method of lines.

Journal of Computational Physics, 89, 349-388.

[14] Gu, T. (1995). Mathematical modelling and scale up of liquid chromatography. New

York: Springer.

[15] Gunawan, R., Fusman, I., & Braatz, R. D. (2004). High resolution algorithms for

multidimensional population balance equations. AIChE Journal, 50, 2738-2749.

[16] Hu, S. S., & Schiesser, W. E. (1981). An adaptive grid method in the numerical method

of lines. In R. Vichnevetsky & R. S. Stepleman (Eds.), Advances in Computer Methods

for Partial Differential Equations (pp.305-311). IMACS: North-Holland.

[17] Huang, W., & Russell, R., D. (1996). A moving collocation method for solving time

dependent partial differential equations. Applied Numerical Mathematics, 20, 101.

[18] Jiang, G., & Shu, C. W. (1996). Efficient implementation of weighted ENO schemes.

Journal of Computational Physics, 126, 202-228.

[19] Koren, B. (1993) A robust upwind discretization method for advection, diffusion and

source terms, CWI Report NM-R9308 April, Department of Numerical Mathematics.

[20] Lapidus, L., & Amundson, N. R. (1952). Mathematics of adsorption in beds. VI. The

effect of longitudinal diffusion in ion exchange and chromatographic columns. Journal

of Physical Chemistry, 56, 984-988.

[21] Li, S., & Petzold, L. (1997). Moving mesh methods with upwinding schemes for timedependent PDEs. Journal of Computational Physics, 131, 368-377.

[22] Lim, Y., Lann, J. M., & Joulia, X. (2001). 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. Computers and

Chemical Engineering, 25, 1483-1492.

[23] Lim, Y., Lann, J. M., & Joulia, X. (2001). Moving mesh generation for tracking a shock

or steep moving front. Computers and Chemical Engineering, 25, 653-633.

[24] Liu, Y., Cameron, I. T., & Bhatia, S. K. (2001). The wavelet-collocation method for

adsorption problems involving steep gradients. Computers and Chemical Engineering,

25, 1611-1619.

[25] Minceva, M., Pais, L.S., & Rodrigues, A. (2003). Cyclic steady state of simulated

moving bed processes for enantiomers separation. Chemical Engineering and

Processing, 42, 93-104.

[26] Miller, K., & Miller, R. N. (1981). Moving Finite Elements. I. SIAM Journal of

numerical Analysis, 18, 1019-1032.


Hongmei Yao, Tonghua Zhang, Moses O. Tadé et al.

[27] Nikolaou, M., & You, Y. (1994). Chapter 7: Use of wavelets for numerical solution of

differential equations. In R. Motard & B. Joseph (Eds.), Wavelet applications in

chemical engineering (pp. 210-274). Kluwer: Academic Publisher.

[28] Pais, L. S., Loureiro, J. M., & Rodrigures, A. E. (1997). Separation of 1,1’-bi-2-naphthol

Enantiomers by continuous chromatography in simulated moving bed. Chemical

Engineering Science, 52, 245-257.

[29] Petzold, L. R. (1987). Observations on an adaptive moving grid method for onedimensional systems of partial differential equations. Applied Numerical Mathematics,

3, 347-360.

[30] Qamar, S., Elsner, M. P., Angelov, I. A., Warnecke, G., & Seidel-Morgenstern, A.

(2006). A comparative study of high resolution schemes for solving population balances

in crystallization. Computers and Chemical Engineering, 30, 1119-1131.

[31] Revilla, M. A. (1986). Simple time and space adaptation in one-dimensional

evolutionary partial differential equation. International Journal for Numerical Methods

in Engineering, 23, 2263-2275.

[32] Sanz-Serna, J. M., & Christie, I. (1986). A Simple adaptive technique for nonlinear

wave problems. Journal of Computational Physics, 67, 348-360.

[33] Vasilyev, O. V., Paolucci, S., & Sen, M. (1995). A multilevel wavelet collocation

method for solving partial differential equations in a finite domain. Journal of

Computational Physics, 120, 33-47.

[34] Wouwer, A. V., Saucez, P., & Schiesser, W. E. (1998). Some user-oriented comparisons

of adaptive grid methods for partial differential equations in one space dimension.

Applied Numerical Mathematics, 26, 49-62.

[35] Yao, H. M., Tian, Y. C., & Tade, M. O. (2008). Using Wavelets for Solving SMB

Separation Process Models. Industrial & Chemical Engineering Research, 47, 55855593.

[36] Zhang, T., Tade, M. O., Tian, Y. C., & Zang, H. (2008). High resolution method for

numerically solving PDEs in process engineering. Computers and Chemical

Engineering, 32, 2403-2408.

In: Handbook of Computational Chemistry Research

Editors: C.T. Collett and C.D. Robson, pp. 303-320

ISBN: 978-1-60741-047-8

© 2010 Nova Science Publishers, Inc.

Chapter 11



Zoltán Mucsi and György Keglevich

Department of Organic Chemistry and Technology,

Budapest University of Technology and Economics,

1521 Budapest, Hungary


Molecular structures are often influenced by aromatic stabilization and antiaromatic

destabilization effects. In spite of nearly a century’s effort – from Kekulé 1871 to Breslow and

Dewar 1965 – to synthesize cyclobutadiene, these attempts have proved to be unsuccessful

[1–6]. Only theoretical chemistry was able to explain this failure by introducing the concept of

antiaromaticity as a new phenomenon. The synthesis of these antiaromatic compounds has

long been considered as desirable target of preparative chemistry in order to examine

experimentally their species chemical properties, but only a few compounds could be prepared

and studied. One of the examples may be the family of phosphole oxides exhibiting a slightly

antiaromatic character [7–10]. At the same time, heterophosphetes, are of more considerable

antiaromatic character and they manifest only as high energy intermediate or transition state

(TS) [11–20]. In this paper, stability and thermodynamic, as well as kinetic properties of

heterophosphetes and phosphole oxides are discussed.

1. Heterophosphates

Heterophosphetes (1), such as oxaphosphetes, thiaphosphetes and azaphosphetes, a group

of four-membered β-heterocycles are structurally related to biologically active penicillin-type

β-lactams and belong also to the family of the antiaromatic molecules [11–20]. The

thermodynamic and kinetics aspects, as well as the electronic structures of heterophosphetes

containing three different ring heteroatom moieties (Y = NH, O, S) and twelve exocyclic

substituents (X = H, Me, Ph, OMe, NMe2, SMe, F, Cl, Br, CN, OCN, SCN) were studied

theoretically and experimentally [11,12].


Zoltán Mucsi and György Keglevich

1.1. Structure of Heterophosphates

In principle, heterophosphetes (1) may adopt two distinct conformations [11–20]. The

phosphorous-heteroatom bond (P–Y) can be either equatorial (1A) or apical (1B). The

situation is controlled by the apicofilicity of Y and X [21]. The interconversion of the two

forms falls into the domain of pseudorotation [21–23]. Heterophosphetes 1, usually prefer

undergoing a fast ring opening reaction to form the corresponding cis (3-I) or trans (3-II)

isomers of β-functionalized phosphoranes as isolable products [11–20]. This equilibrium may

be one of the reasons that earlier synthetic attempts to produce heterophosphetes have often

failed. From a theoretical point of view, one may consider the 1B ↔ 3-I reversible

transformation as a ring-chain valence tautomerism [11,12]. The saturated versions,

heterophoshetanes (2) existing also as two possible conformers (2A and 2B) are more stable

and in the case of Y=O are well-known as the intermediates of the Wittig reaction [24]. In

contrast to 1, compound 2 never undergoes ring-opening to form 4 [11]. The overall process

and structures are summarized in figure 1.

Figure 1. Structure of heterophosphetes (1A and 1B) and heterophosphetanes (2A and 2B) as well as

their ring opened forms (3 and 4).

Figure 2. Heterophosphates 5 and 6 prepared.

The Stability and Antiaromaticity of Heterophosphetes and Phosphole Oxides


In spite of the preparative efforts, only two compounds with the desired structure were

proved to be isolable and reported in the literature (5 [25] and 6, [26] figure 2) representing

the heterophosphete family. On the basis of the N⋅⋅⋅P distance of 2.170 Å, [25] compound 5

can only be regarded to have structure 1B in a broader sense. The relatively long bond

distance between the P and the Y atoms is rather an ionic interaction coming from strong

zwitter-ionic structure 3-I.

1.2. Synthetic Aspect of Heterophosphetes

There are many synthetic possibilities to prepare 1. From synthetic aspect, the synthesis

of 1B and 3-I is equivalent, because they are linked together by a ring-chain tautomerism,

therefore the successful synthesis of 1B requires a shifted equilibrium between 3-I and 1B

toward 1B, meaning that 1B lies at a lower energy level than 3-I. Two possible synthetic

approaches (inverse-Wittig route A and acylation C) were described for the preparation of 3-I

(Scheme 1) [13,18].

Scheme 1.

The route to produce 1B or 3-I involving an inverse-Wittig protocol was developed by

the Keglevich group [13–17,27] (Scheme 2) and parallelly by Japanese groups

[19,20,25,26,38]. In the former case, 2,4,6-trialkylphenyl cyclic phosphine oxides (7) were

treated with dialkyl acetylenedicarboxylate (8) at an elevated temperature to give first 9 as an

intermediate that underwent a ring-opening reaction to yield 10.

Scheme 2.

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