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Some Dynamic Features of Fluidised Bed Reactors for Partial Oxidation of o-Xylene.

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Dev.Chem. Eng. Mineral Process., 11(3/4), pp. 349-361, 2003.
Some Dynamic Features of Fluidised
Bed Reactors for Partial Oxidation
of o-Xylene
M.E.E. Abashar
Department of Chemical Engineering, King Saud University,
P. 0. Box 800, Riyadh 11421, Saudi Arabia
A dynamic analysis offuidised bed reactors for the partial oxidation of o-qlene to
phthalic anhydride is presented. The theory of bfurcation, continuation and Poincare‘
techniques play key roles in the analysis. A new type of period-adding bifurcation of
Poincare‘ return points is observed and is shown to generate a complex 4namic
behaviour. The period adding is classified as a second kind of period adding
according to the classrfication of Holden and Fan [15J It is found that relaration
oscillations develop through distortion of limit cycles, which have fast and slow
motion in the different regions of the phase space.
Introduction
Chemical reactors are known to exhibit fascinating phenomena in their dynamic
behaviour, including steady-state multiplicity, periodic oscillations, quasi-periodicity,
strange chaotic and non-chaotic behaviour [ 1-31. Such phenomena are caused by the
inherent non-linearities of the reacting system, the most common of which are the
Arrhenius-type dependence of the reaction rate on temperature and also the nonmonotonic kinetics. The nonlinear phenomena can be either very useful or harmful to
chemical processes. For example chaos is known to develop through sequences of
bifurcations. This nonlinear phenomenon can be utilized in combustion applications
to enhance the mixing of air and fuel and thus leading to improved performance.
Chaotic regions can be used to offer greater flexibility in the operation of chemical
systems due to an infinite number of unstable attractors embedded in a chaotic
attractor. These unstable attractors can be stabilized, e.g. by the OGY (Ott, Grebogi
and York) method as required [4]. Unsteady-state operations of chemical reactors are
also known to improve performance with regard to the conversion, stability and
parameters sensitivity [ 5 ] . However, chaotic regions can be very harmful to the
stability and control of chemical processes. For example the “butterfly effect”, which
can be produced by small unintentional disturbances, can render our long-term
predictions of the performance of chemical processes invalid.
* Author for correspondence (tnuhnslinrir:
Xsii. edii
su).
349
M.E.E. Abashar
Many industrial gas-solid reactions or solid-catalysed gas-phase reactions are
carried out in fluidised bed reactors. The proper design of these reactors has always
been hampered by our inability to correctly incorporate the highly complex kinetics
and poorly understood hydrodynamics into mathematical models. However, recently
many researchers have shown that the implementation of the modem theory of
bifurcation and chaos gives a new tool for understanding, scaling, modelling, design
and operation of fluidised bed reactors [6, 71.Abashar and co-workers analysed the
phenomena of strange chaotic and non-chaotic behaviour of fluidised bed catalytic
reactors [8,9]. The conventional approach of Van Heerden was used by Elshishini et
al. [lo] to analyse the steady-state multiplicity in fluidised bed reactors for partial
oxidation of o-xylene.
The present investigation is an extension of the work by Elshishini et al. [101. The
major difference between the present study and the earlier ones is that our analysis is
based on the theory of bifurcation, continuation and Poincar6 techniques. The recent
concepts and modes of thought have greatly increased our understanding of these
types of complicated systems as shown in this preliminary investigation.
Kinetics and Rate Expression
For the partial oxidation reaction of o-xylene, the following simplified rdaction
scheme given by Froment [I 11 is used:
The corresponding rate expressions can be written in the form [ 101:
kgmol
kgcat s
RA = C, aA exp(--) Y A x I x 3
9
XI
R, = C , a, exp(--)Y w x 2 x t
9
x3
R, = C , a, exp(--)Yc x , x ,
9
XI
kgmol
kgcat s
kgmol
kgcat s
(3)
where C, is the reference concentration (kgmoVm3); a, is the pre-exponential factor
for reaction i (m3/kgcat.s); yi is the dimensionless activation energy for reaction i;
x I and x 2 are the dimensionless concentration of o-xylene and phthalic anhydride
respectively; and x is the dimensionlesstemperature.
350
Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene
Model Development
A fluidised bed reactor with a simple linear-proportional feedback control system is
considered in this investigation. The mathematical model is developed based on the
two-phase bubbling bed reactor model. The following simplifying assumptions are
used in the derivation of the conservation equations of the model [lo]:
1. The emulsion phase is assumed to be perfectly mixed.
2. The gas in the bubble phase is assumed as plug flow.
3. The bubble size is taken as uniform throughout the reactor.
4. The bubble phase is solids free.
5. The reactor is adiabatic.
6. Due to the low concentration of o-xylene in the feed ( a1%) and the large
excess of oxygen in the air, the partial pressure of oxygen is considered to
be constant.
The unsteady-state mass balance equations for the emulsion phase are thus
given by:
For o-xylene:
For phthalic anhydride:
The unsteady-state energy balance for the emulsion is given by:
(6)
F, 'T,
+K(T, - T )
(7)
351
M.E.E. Abmhar
The bubble-phase mass and energy balance equations are at pseudo steady-state
because of their negligible mass and heat capacities and are given by:
-
Equations (8) (10) can be solved analytically and the resulting solutions used to
evaluate analytically the integrals in Equations (4) - (6). Assuming (K,,), = (K,,)2
= H, and casting the equations in dimensionless forms gives:
dx
I d.t= G ( X , - X l ) - L
dB
dt
O-=G((8
(1 1)
-- @ + A
i
+acPcexp -k X ,
( 6 )
b e , + K ( e , -e)
,
whereX and X are the dimensionless emulsion-phase concentrations of o-xylene
and phthalic anhydride respectively, and 6 is the dimensionless ernulsion-phase
temperature. The data used for the present system is given in Table 1.
352
Fluidised Bed Reactors for Partial Oxidation of o-Xylene
Table 1. Data usedfor thejluidised-bed catalytic reactor [ I 01.
Clf
2.031 I X lo4 kgmol/m3
aE
CZf
0.00
kgmovm3
a,
D
0.618
m
YA
H
3
m
YE
Tf
600
K
u,
0.175
m/S
- M A
umf
0.079
m/S
m
B
Yc
23.484
0.488294
aA
I 11.832 X lo5 m3/kgcat.s I
8,
I
0.929
Results and Discussion
Figure 1(a) shows the dimensionless concentration of phthalic anhydride (X2) versus
the dimensionless emulsion-phase temperature (0) for the uncontrolled adiabatic case
(K=O). It is clear that the maximum concentration of phthalic anhydride corresponds
to a middle unstable static steady state (saddle-type) as shown in Figure l(b). The
feed and reactor temperatures at this optimal condition are 8
, =0.032806
(-253.47"C) and 8 = 0.929 (284.25"C) respectively. It is obvious that this feed
temperature is of no practical use. A practical feed temperature of about 20°C
(8 = 0.488294) which corresponds to 35.7% of the phthalic anhydride optimal value
was selected for this study to investigate some dynamical features of this important
industrial system. In order to operate the reactor at this unstable steady-state, a simple
linear proportional feedback control system is used.
The autonomous model of the fluidised bed reactor (Equations 1 1 - 13) includes a
large number of parameters (G,K,XIF,X~F,a, ,aE
PA PE PC 3 Y A Y E ~ Y ce1
8, , A ,0 ). It is a difficult task to investigate the entire static and dynamic behaviour
of this system over seventeen dimensional parameter space. Instead the dynamic
behaviour is investigated by considering two parameters of the control system (K, 0, )
with all other parameters kept constant. The richness and complexity of the behaviour
associated with the one parameter investigation reported in this study strongly
justifies this severe reduction in the parameter space dimensions.
Figure 2(a) shows a bifbrcation diagram for the controlled fluidised bed reactor.
The bifurcation diagram is obtained by using the software package AUTO86 of
Doedel and Kerntvez [12]. The bifurcation diagram is characterized by a region of
multiplicity of steady-state and three Hopf bifurcation points (HBI at K=6.070374,
HB2 at K=6.622175, and HB3at Kg.541273). The Hopf bifurcation point connects a
steady-state branch with a periodic branch (limit cycle); exchange of stability occurs
9
9
9
9
9
353
M.E.E. Abashar
I.so
I.20
0.80
0.40
2.00
0
-
1.oo
(b)
0.50
0.00
-1.50
-
----
Stable
Unstable
/'Sl
I
I
I
I
I
I
t
I
-
\
0.40
0.80
1.20
I.60
2.00
0
Figure 1. Uncontrolled adiabatic case (K=O): (a) dimensionless concentration of
phthalic anhydride (Xd vs the dimensionless emulsion-phase temperature
(0); (b) one-parameter bifurcation diagram of q v s 0for K=O.
354
Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene
0.60
0.40
x"
0.20
0.00
160.0
120.0
-(ID
U
.-o
80.0
tl
a
40.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
K
Figure 2. Controlledfluidised bed reactor: (a) one-parameter bifurcation diagram of
X , versus K for Of= 0.488294; (b) period of oscillation versus K.
355
M.E.E.Abashar
as shown in Figure 2(a). At the Hopf bifurcation point a complex conjugate pair of
eigenvalues crosses the imaginary axis transversally. The periodic branches
originating from these Hopf bifurcation points terminate homoclinically, i.e. forming
homoclinic orbits with infinite periods as shown in Figure 2(b). This global
phenomenon requires a saddle type of unstable steady-state. The eigenvalues for the
unstable saddle steady-state are pure and real of opposite sign (-,+,-).The negative
values correspond to two stable manifolds that enter the saddle point and are called
the separatrices; the positive eigenvalue corresponds to the unstable manifold. When
the stable and unstable manifolds of the same saddle coincide (connecting the saddle
to irself), we have the homoclinic orbit (saddle loop) that marks the end of the
periodic branch. The homoclinic orbits strongly influence the nature of other
trajectories passing near them.
The region between HB2 and HB3 forms a region between two homoclinic orbits
and contains three unstable attractors of steady states as shown in Figure 2(a). Even
though such global homoclinic bihcations involve the saddle-type steady state, they
cannot be detected through the local information (e.g. linearization) around this steady
state. Due to this difficulty, and the lack of the starting point for the computation, it is
not possible to use AUTO86 for this region. Therefore, we developed a Poincard
bifurcation diagram to study this complex region.
Figure 3 shows a one parameter Poincard bifurcation diagram for the region
between the two-homoclinic orbits. This technique is based on the reduction of the
dimension of the problem fiom three to two [8]. Instead of following the whole
trajectory, discrete points are taken when the trajectory crosses a certain hyperplane of
two dimensions transversally and in the same direction. These discrete points are
called return points. The plotting of one of the co-ordinate of the return points (in this
case X,) versus one of the free parameters (in this case K) is known as the Poincard
bikcation diagram as shown in Figure 3. Figure 3 shows clearly that with increasing
K the system bifurcates through period adding mechanism. A periodic branch with
period 3 (P3) originates fiom the homoclinic orbit and goes through a sequence of
period adding. The periodic sequence of periods are: P3,P4, P5, P6, P7etc, until
accumulation point with very high periodicity after which the system bifurcates in a
reverse way through period decreasing e.g. ...P16,PIS, Pi4,..., P4 up to the end of this
region. The stability of the periodic solutions is determined by the eigenvalues of
certain monodromy matrix called the characteristic or Floquet multipliers. The
multiplier with largest absolute value is called the principle Floquet multiplier (PFM).
When the PFM crosses a unit circle the periodic orbit loses stability and a dynamic
bifurcation occurs. In fact, the appearance of P3 represents a type of “bench-mark” in
this process because it is the first member of Sharkovsky’s T set, so that all types of
periodic solutions are possible [ 131. The behaviour of the system in this region is
similar to behaviour of the autonomous three-dimensional Rose-Hindmarsh model for
neuronal activity which exhibits a new type of period adding bifurcation [14]. Holden
and Fan in a series of papers [ 15- 181, investigated the complex dynamic behaviour of
the Rose-Hindmarsh model of action potential. This model was proposed as a
mathematical representation of the bursting behaviour of neurons and is expected to
simulate the repetitive, patterned and irregular activity in molluscan neurons. Using
this model, Holden and Fan discovered a new period-adding type of bifurcation on the
Poincard bifurcation diagram. They classified the bifbrcation behaviour into three
356
Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene
kinds of period adding. It is clear that the period-adding bifimation presented in
Figure 3 belongs to the second kind where the lines of periodicity on the Poincard
bifurcation diagram are discontinuous at the period-adding point. It is interesting to
observe that the sizes of the periodic regions (windows) are different and there is no
clear relationship between their sizes.
0.36
0.34
0.32
0.30
Qf
0.28
6.0
= 0.488294
I
I
I
7.0
8.0
9.0
10.0
K
Figure 3. Poincare' bifurcation diagram of X , versus K.
Figures 4(a - d) show the time traces and phase planes at different values of K. It
is clear that as K increases, the limit cycle is distorted (see Figures 4b - 4 4 and
becomes different from the circular form (Figure 4a). The distorted limit cycle
exhibits regions of fast and slow motion in the phase space and the initial smoothness
diminishes gradually and gives large and sharp oscillations of a saw-tooth shape. This
distortion of the limit cycle produces large amplitude oscillations called relaxation
oscillations [ 131.
Figure 5 shows various bifbrcation diagrams obtained by changing the feed
temperature (0, ). The nature of these diagrams depends upon the value of 0, and
whether it is smaller or larger than 0, =0.488294. It is clear that there exist complex
regions of unstable multiple steady states. The problem in these regions becomes too
stiff for the software AUTO86 to complete the periodic branches, as shown in Figures
5(b) and 5(c). Further investigation for these regions needs to be conducted.
357
M.E.E. Abashar
"
.
(
I42
!' ' '
. ,'
'
. ;-
I
1147L1
l161lJ
I -1.1
"'1"'
I-8.18
1.01,
1
1.11
- 4
1.11
lMm.0
lOI10.0
1 0 ~ ~ 0 llEla.0
1124a.P
0.20
0.40
a.81
XZ
I(4
1
Figure 4. Time traces andphase planes at different values of K (6.65, 6.9, 8.00, 9.5).
358
Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene
0.4
s
0.2
0.0
2.0
4.0
8.0
8.0
10.0
120
1.0
K
0.40
i
ef 0.6
0.00
0.0
4.0
8.0
K
120
8.0
Figure 5. Bijiurcation diagrams at different values of 0, (0.46, 0.SO, 0.64).
Conclusions
We have shown that for a specific parameter space the dynamic behaviour of the
fluidised bed reactor for partial oxidation of o-xylene is quite complex. The periodadding phenomenon is observed. It may be too early to comment on the practical
implications of this type of strange phenomenon on the design and control of such
reactors. The results presented in this investigation suggest that such strange dynamic
behaviour is to be avoided (or suppressed) when it is harmful to the system, and
exploited to the maximum when it is beneficial. Further study is needed to
characterize the period-adding bihcation &om a mathematical standpoint.
359
M.E.E. Abashar
Nomenclature
Cross-section areas of the bed occupied by bubble and dense phases
respectively (m2)
Concentrations of component i in bubble and emulsion phases
respectively (kmovm3)
Reference concentration (km0Vm3)
Specific heat of gas and catalyst respectively (k.J/kg K)
Bed diameter (m)
Reciprocal of the modified residence time of the bed (s-’)
Vertical coordinate measured from distributor plate (m)
Bed height (in)
Heat exchange coefficient between bubble and emulsion phases (s-’)
Heat of reaction i (kJkm01)
Dimensionless controller gain
Mass exchange coefficient of component i (s-I)
Bubble and emulsion phase flow rates respectively (m3/s)
Rate of reaction i (km0Vkgcat.s)
Time (s)
Bubble and emulsion phase temperatures respectively (K)
Desired temperature (set point) (K)
Superficial and minimum fluidisation velocities respectively ( d s )
Dimensionless concentration of o-xylene and phthalic anhydride
respectively
Dimensionless temperature.
Dimensionless emulsion phase concentrations of o-xylene and phthalic
anhydride respectively
Greek symbols
Pre-exponential factor for reaction i (m3/kgcats)
Dimensionless overall exothermicity factor for reaction i
Dimensionless activation energy for reaction i
Void fiaction in the bed at minimum fluidising conditions
Dimensionless feed and emulsion phase temperatures respectively
Set point for the controller
Dimensionless parameter (= (1 -Ern/ ) p, / E,, )
Gas and solid densities respectively (kg/m3)
Dimensionlesseffective mass capacity of the bed
Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene
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Received: 12 November 2001; Accepted afier revision: 1 April 2002.
361
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