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
295
(26c)
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
∂x
uj
=
∂C i , j
∂x
Dax
(Ci , j − C iin, j )
(26d)
(26e)
(26f)
=0
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)
QI = QIV + QD
C iin, I QI = Ciout
, IV Q IV + C i , D Q D
Extract draw-off node
QII = QI − QE
C i , E = C iin, II = Ciout
,I
Feed node
QIII = QII + QF
Ciin, III QIII = Ciout
, II QII + Ci , F QF
Raffinate draw-off node
QIV = QIII − QR
Ci , R = Ciin, IV = Ciout
, III
Other nodes
Equal flow rates for the
columns in the same zone
C iin, j = C iout
, j −1
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Hongmei Yao, Tonghua Zhang, Moses O. Tadé et al.
Table 5. System parameters and operating conditions for Bi-naphthol enantiomers
separation
Symbol
Value
L (cm)
10.5
D (cm2/min)
εb
k eff ,i
(min-1)
t switch (min)
Symbol
Ci , feed
Value
(g/L)
2.9
0.00525u
QF (ml/min)
3.64
0.4
QI (ml/min)
56.83
6.0
QII (ml/min)
38.85
2.87
QIII (ml/min)
42.49
QIV (ml/min)
35.38
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
cycles.
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...
297
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
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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.
5
0.005
0.004
0.003
4
Relative error
0.002
0.001
3
0
60
70
80
2
J=5/1
HR 33
1
0
0
10
20
30
40
50
60
70
80
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...
299
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
C:
in
i, j
C ,C
fluid phase concentration
out
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 :
L:
effective fluid film mass transfer resistance
q:
q∗:
column length
concentration of component in the solid phase
equilibrium concentration in interface between two phases
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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
Acknowledgment
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.
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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
THE STABILITY AND ANTIAROMATICITY
OF HETEROPHOSPHETES AND PHOSPHOLE OXIDES
Zoltán Mucsi and György Keglevich
Department of Organic Chemistry and Technology,
Budapest University of Technology and Economics,
1521 Budapest, Hungary
Abstract
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].
304
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
305
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.