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On the chaotic behavior of externally forced industrial fluid catalytic cracking units
Chaos, So/irons & Fracmls, Vol. 9, No. 3. pp. 455-470, 199X 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0964.0779/98 $19.00 + 0.00
Pergamon
PII: !30960-0779(97)ooo88-x
On the Chaotic Behavior of Externally Forced Industrial Fluid Catalytic Cracking Units A. E. ABASAEED Chemical Engineering Department, KSU, P.O. Box 800, Riyadh 11421, Saudi Arabia
and S. S. E. H. ELNASHAIE Chemical Engineering Department, Cairo University, Cairo, Egypt (Accepted 27 March 1997)
good insight into the effect of external disturbances on industrial units can be obtained by the investigation of the effect of periodic forcing. Therefore, the effect of periodic forcing of the air feed temperature on the bifurcation behavior, gasoline yield, and amplitude modulation of a type IV industrial fluid catalytic cracking (FCC) unit is investigated. The investigation concentrates on a case in which the autonomous system shows bistability between a high temperature static attractor and a periodic attractor (the emphasis being on the periodic attractor). The center of forcing considered in this case is moderately removed from the homoclinic termination point of the autonomous periodic branch. The one-parameter stroboscopic Poincart bifurcation diagram reveals the richness of the dynamic behavior of the forced system (periodicity, aperiodicity, bifurcations, inverse bifurcations, intermittency, boundary crisis, shrinking size torus and chaos). The investigation reveals that period doubling, type 1 intermittency and boundary crisis are the dominant routes to chaos. Lyapunov exponents are computed to distinguish between chaotic and non-chaotic complex attractors. The average gasoline yield obtained with the attractor resulting from the forcing of the autonomous static attractor increases slightly with increase in forcing amplitude. All chaotic attractors resulting from the forcing of the autonomous periodic attractor give slightly higher gasoline yields than their autonomous counterparts. 0 1998 Elsevier Science Ltd. All rights reserved
Abstract-A
NOMENCLATURE dimensionless groups ” forcing amplitude modified space velocities for the reactor and regenerator respectively (l/s) coke burning rate [kg coke bumed/(kg solid s)] dimensionless group connected with the coke balance in the reactor and regenerator respectively bubble flow rate through any horizontal section for reactor and regenerator respectively (m3/s) height of reactor and regenerator respectively (m) heat losses in reactor and regenerator respectively dimensionless rate of heat absorption due to endothermic cracking reactions in the reactor
HV Kc RC xsb X sbf x@ x&w X gob X gof Yf Yfa
YGB
455
dimensionless heat of vaporization for gas oil (kJ/kg) controller gain coke make rate dimensionless bubble phase gasoline concentration dimensionless gasoline concentration in feed dimensionless dense phase gasoline concentration dimensionless gas oil concentration dimensionless bubble phase gas oil concentration dimensionless gas oil concentration in feed dimensionless feed temperature dimensionless air feed temperature dimensionless regenerator bubble phase temperature
,;*I, t;:,, ~‘I?,, ):,,
dimensionless regenerator dense phase temperature dimensionless reactor bubble phase temperature dimensionless reactor dense phase temperaturedimensionless set point reactor
t., i , !4 i-.
temperature I'!,
:l (ireCk lmm Cl,. CY1,ru, c-!
dimensionless gas oil vaporization temperature forcing frequench
/A
dimensionless pre-exponential factors for the cracking reactions dynamic parameters in heat balance equation for regenerator
I&~ l/f,; $5. PI* /‘I
c Y.
dynamic sorption dynamic sorption dynamic equation dynamic
parameters for catalyst adcapacity for gas oil parameters for catalyst atlcapacity for gasoline parameters in coke balance for reactor parameters in coke balance equation for regenerator exothermicity factor for coke comhustion reaction bed voidagc dimensionless activation energies for cracking reactions catalyst activity within reactor catalyst activity within regenerator density of air, liquid gas oil. vaporized qas oil respectively (kg/m’)
1. INTRODUCTION
Fluid catalytic cracking is one of the most important processes in the petroleum refining industry. These units have been shown to operate at a controllable unstable steady state, saddle-type [ I-S]. A number of investigations concentrated on studying the interaction of feed disturbances (intentional or unintentional) with the non-line&ties in chemically reactive systems [C-9]. Very few results are presented for the cases with the center of forcing being relatively close to the homoclinic orbit [9, lo]. In a previous investigation. Abasaeed and Elnashaie [Ill reported the bifurcation behavior of an industrial type IV FCC’ unit subject to periodic forcing of air feed temperature with the centerof forcing being extremely close to the homoclinic orbit. Period doubling and ~type I intermittency have been identified as the routes to chaotic behavior. The chaotic behavior occurred at very low values of forcing amplitudes. Boundary crisis terminated the first chaotic regime before a P4 attractor emerged by a possibIe blue-sky catastrophe. The P4 also underwent period doubling bifurcations leading to chaos before it completely disappeared. In this investigation, the center of forcing has been moderately removed from the homodinic termination point of the autonomous periodic branch. The investigation concentrates on the effect of forcing on the chaotic behavior as well as on the gasoline yield and amplitude of temperature osciljations of the forced system at selected forcing amplitudes. Various techniques are used to characterize the periodicity and aperiodicity of Ihe forced system. Lyapunov exponents are also computed to characterize chaotic behavior.
2. THE
MODEL
The model used here is based on the two-phase theory of Auidized bed reactors, which was developed earlier and compared successfully against data from two industrial units [4]. This model has been extended by Elnashaie et ul. [12] to include the dynamics of gas oil and gasoline. The details of the model development and its associated assumptions have been given earlier 14,s. 12,131 and will not be repeated here. The normalized dynamic model equations are given as follows,
Chaotic behaviour
of forced FCCUs
457
The reactor dense phase normalized heat balance equation: -d&u, = B(Yu - I-,,)
dr
+ A2(Yf - Yu) - Hu + HRCB + A,(YG,
- Y,,) - HL.
(1)
The regenerator dense phase normalized heat balance equation:
dycm = &{(BD(Y,, - Km)) + Pc(CB) + AOk, dr
- Km> - HL,).
The gas oil normalized mass balance in the reactor dense phase:
$$’ =[2(B{Xgof -
X,,} - +R[ale-YI’YRD+ a3e-y~‘y~]x&,).
(3)
The gasoline normalized mass balance in the reactor dense phase: 2
= &{B(X+
- Xgd) - t,bR(a2Xgde-.y2’YRD - a1(Xgo)2e-Y1’Y~D)~.
(4)
The carbon balance (catalyst activity) in the reactor:
2 = &(CRD{~- Icrd- (Ir&),
(9
where R, = a2e- y2’Y~~Xgd+ ff3e - wmx* go-
The carbon balance (catalyst activity) in the regenerator:
(6) The air feed temperature: Yfa= Y,, + Am sin(w,z) + Kc( Y,, - Y,,).
(7
The normalized mass and heat balance equations for the bubble phase which is assumed to be at pseudo-steady state are given as follows. The reactor bubble phase normalized heat balance equation: Y,, = YRD= (Yu - YRD)eeURHR.
@I
The regenerator bubble phase normalized heat balance equation: YoB = Y,, + (Yu - YGD)e-a~H~.
(9)
The gas oil normalized mass balance in the reactor bubble phase: X gob= X,, + (Xgof - Xgo)eeaRHR.
(10)
J<\
A
I,. Al3ASAEF.I)
:I&
S. 5. t:.
H. T:I.NASHAIF
The gasoline normalized mass balance in the reactor bubble phase:
x,, = x,,, -i ix,,,
X,Jc “;,“J:.
(11)
ln eqn (7). Am =O.O implies the autonomous system. The various parameters and variables used in the above equations are defined in the Nomenclature.
3. RESULTS AND DISCUSSION
Ihe bifurcation diagrams are constructed using AUTO86 developed by Doedel and Kernevez [14]. The Lyapunov exponents are computed using the algorithm developed by Wolf et (I/. [15]. To ensure accuracy of the simulation results, the bound on the allowable error was maintained at lo-.“. with automatic step size integration routines. 3. 1
C‘losed-loop
wtonom0u.s
system
Figure I shows the bifurcation diagram for the reactor dense phase gasoline yield for the autonomous system in the histability region (2.6863 5 Kc 5 2.8235) with the air feed temperature yt, as the bifurcation parameter. The solid lines in the figure indicate stable steady states static branches, whereas the dashed lines shows the unstable one. The stable periodic branch is given in the figure as the symbol (0). and the Hopf bifurcation points, denoted HB, as the symbol (m). The enlargement of the periodic branch emanating from the Hopf bifurcation point (denoted HJ31 in Fig. l(a)) is shown in Fig. l(b). This periodic branch terminates homoclinically at Y,,,= I.505187 as shown in Fig. l(b). It is evident from Figs l(a) and (b) that the unstable static (dashed lines) and the stable periodic branches (0) give a much higher gasoline yield (>0.4) compared to the stable static branch (solid lines). A detailed discussion of this figure is given in Abasaeed and Elnashaie [I 11. 3.2
The f;mwi
.sy.stm
The case in which periodic forcing of this system with the center of forcing (Y,;, = 1.505188) being very close to the homoclinic orbit and the forcing frequency five times the frequency of the autonomous system has been discussed in a previous study [ 1I]. The one-parameter stroboscopic PoincarC bifurcation diagram for that case revealed that the
(b) ---__44*m t HBI
--j----
Chaotic behaviour of forced FCCUs
459
attractor resulting from the forcing of the autonomous periodic attractor goes through period doubling bifurcations leading to chaos (PGPlO-P20 chaos, based on strobing every forcing period, this is equivalent to Pl-P2-P4 chaos when strobing every autonomous period). At extremely small values of the forcing amplitude (Am < 0.94 X 10p5), the resulting chaotic attractor lost its stability due to a boundary crisis [16]. This situation continued until Am = 0.245 X 10e3, when a P4 attractor (based on strobing every forcing period) suddenly appeared. The only attractor existing in the range 0.94 X 10V55 Am 5 0.245 X 10m3was the Pl (period 1) attractor resulting from the forcing of the high temperature static attractor. The resulting P4 attractor went through a period doubling sequence cascading to chaos. A boundary crisis occurs at Am = 0.48 X lop3 and the chaotic attractor resulting from the forcing of the autonomous periodic attractor lost completely its stability to the attractor resulting from the forcing of the autonomous static high temperature attractor. In the present investigation, the bifurcation behavior of the forced system is analysed for the case when the center of forcing (Yf, = 1.505583) is moved moderately from the homoclinic orbit. The natural period of the autonomous periodic attractor is 259.6571 s at Y, = 1.505583. The one-parameter stroboscopic Poincare bifurcation diagram is shown in Fig. 2 for the reactor dense phase temperature against the dimensionless forcing amplitude. Since the forcing is for a case with the autonomous system in a region of bistability, it is to 2.17 2.16 y1 2.15 2 5*
2.14 2.13 2.12:,,
, , , , ) , , , , ( , , , , , , , , , , , , , , , , , , , , , ,
1.26
1.18,, 0.0
, I / , , , , , , , , , I ,, 0.1 0.2 0.3
, , , , , , , , , , , , , ,, 0.4 0.5 0.6
, , , ( 7
Forcing amplitude Fig. 2. One-parameter stroboscopic Poincart bifurcation diagram in the bistability region for the reactor dense phase temperature against the forcing amplitude (strobing every forcing period). The attractor resulting from the forcing of the (a) autonomous static attractor and (b) autonomous periodic attractor. Center of forcing is at Am = 1.505583.
be expected that, depending on initial conditions, the forcing will affect either of the two attractors. For this reason Fig. 2 is composed of two portions: the upper portion (Fig. 2(a)) of the figure is for the attractor resulting from the forcing of the static high temperature attractor (of course nothing exciting takes place with this attractor because periodic forcing of a static attractor results in a periodic attractor having period one), while the lower portion (Fig. 2(b)) is for the attractor resulting from the forcing of the autonomous periodic attractor (this attractor is expected to give quite a rich behavior). The following analysis concentrates on the attraetor resulting from the forcing of the autonomous periodic attractor (i.c.~the attractor shown in Fig. 2(b)). 3.3
T/w forced autonom0u.s periodic nttructor
The amplitude of temperature oscillations for the autonomous periodic attractor has been determined to be 17.7OK and the average gasoline yield obtained with this attractor is 0.43844. It is clear from Fig. 2(b) that five branches emerge-(based-on strobing every forcing period, in fact these five branches would show as one branch if the autonomous period is used for strobing) as the forcing amplitude is increased. These five branches undergo period doubhng bifurcation (P5 -4 PlO) and inverse period doubling bifurcation (period halving, PIO-+PS) with increase in forcing-amplitude to Am - 0.19875, where this attractor loses stabitity. Figure 3(a) and (b) shows the time trace and phase planes for the attractor resulting from the forcing of the autonomous periodic attractor (a Pl attractor based on strobing every natural period) respectively at A?rz-- 0. I. This attractor gives an average gasoline yield of 0.438395 (slightly less than the yield obtained with the autonomous periodic attractor). The amplitude of temperature oscillations increases to XX.84K. As the forcing amplitude is increased to At/7 = (Li5. the Pl attractor undergoes period doubling bifurcation (i.e. a PI? attractor is ~born based on strobing every natural period). as is clearly seen from Fig. .3(c) and (d) for time trace and phase plane respectively. The resulting P2 attractor -.-__1
-I-
(~a) Amp = 0.1 n
n
n
!dj
Chaotic behaviour
of forced FCCUs
461
oscillates with higher amplitudes (20.02 K) and gives an average gasoline yield of 0.438403 (still lower than the autonomous periodic attractor yield, 0.43844). Depending on the set of initial conditions used, it has been found that it is possible to locate two different attractors (one periodic and the other chaotic) at a forcing amplitude of 0.19874. Figure 4(a) and (b) shows the time trace and phase portrait for the periodic attractor (this is a Pl attractor) at Am = 0.19874. The average gasoline yield for this attractor is 0.438382 and the amplitude of oscillations is 19.76 K. It is postulated here that a blue-sky catastrophe [17] has occurred and this attractor loses its stability at Am = 0.19875. The time trace and the phase plane for the chaotic attractor at Am = 0.19874 are shown in Fig. 4(c) and (d) respectively. The chaotic nature of this attractor has been confirmed by computing the Lyapunov exponents (the leading Lyapunov exponent is positive, +0.0014). The maximum amplitude of temperature oscillations for this attractor is 18.55 K (lower than the other coexisting Pl attractor). The average gasoline yield obtained with the resulting chaotic attractor is 0.438471 (slightly higher than the average yield obtained with the autonomous periodic attractor, 0.43844). An enlargement of Fig. 2(b) for a forcing amplitude range of 0.19 < Am < 0.23 is shown in Fig. 5(a). It is clear from the figure that a P5 attractor (based on strobing every forcing period) emerges from the left of the diagram followed by a thin strip of chaos followed by a P9 attractor (at Am = 0.2045) which undergoes period doubling bifurcations. The time trace. and the phase plane for this attractor at Am = 0.2043 (slightly into the chaotic strip) are shown in Fig. 5(b) and (c) respectively. Intermittency is a term used to describe the presence of laminar channels (periodic oscillations) which are sandwiched inside chaotic bursts [18]. The intermittency is not clear from Fig. 5(b) and (c), however, the existence of nine laminar phases within the chaotic regime is clear from the stroboscopic points histogram for the reactor dense phase temperature against the forcing period (Fig. 5(d)). The two-dimensional stroboscopic map for the gasoline yield against the reactor dense phase temperature is shown in Fig. 5(e). The spread of this diagram over a relatively wide range of reactor dense phase temperatures indicates the hard to dissipate nature of this system. Type 1 intermittency (which occurs when the Floquet multiplier exits the unit circle through +l) can be determined by plotting the ith iterate map [18,19]. The 9th iterate map for the reactor dense phase temperature is plotted in Fig. 5(f). It is clear from this figure that the curve approaches the bisectrix tangentially at nine distinct points which confirms the type 1 intermittency for this case. The average gasoline yield for this attractor is 0.438474 (slightly higher than the average yield of the autonomous periodic attractor) and the amplitude of temperature oscillations is 18.33 K at Am = 0.2043. The time trace and phase plane diagrams for the P9 attractor which emerges from the chaotic strip by type 1 intermittency are shown in Fig. 6(a) and (b) respectively for Am = 0.2092. For the resulting P9 attractor, the amplitude of oscillations and the average gasoline yield are respectively 17.71 K (slightly higher than the amplitude of oscillations for the autonomous periodic attractor) and 0.438524 (slightly higher than the average yield obtained with the autonomous periodic attractor). This P9 attractor undergoes period doubling bifurcations leading to banded chaos and eventually fully developed chaos as the forcing amplitude is increased (shown in Fig. 7 which is an enlargement of Fig. 2(b) for 0.2 5 Am I 0.3 and 1.23 % Ya,, 5 1.26). The time trace and the phase portrait for the chaotic attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.232 are shown in Fig. 8(a) and (b) respectively. Computation of the Lyapunov exponents gave a positive leading Lyapunov exponent (+0.00122) which confirms the chaotic nature of this attractor. This attractor gives an average gasoline yield of 0.438499 (slightly higher than that obtained with the autonomous periodic attractor) and has an amplitude of temperature oscillations of 18.12 K (slightly higher than the autonomous
0 442
I
I
(a) Amp = 0.19874
0.442 0.440 UP x
0.438 0 436 0 434
25000
30000
~5000 Time
J0000
: lfi
I.18
1.20
1.22
1.24
1.26
t 28
YRI3
Fir. 3. Time trace and phase planes al Am = 0.19874 for the two possible resulting attractors from forcing rhe antonomous periodic attractor: 1a.h) periodic, Pl and (c.d) chaotic.
periodic attractorj. The chaotic attractor then undergoes period halving bifurcation leading rn a PI8 attractor. The P3 P attractor resulting from the forcing of the autonomous periodic attractor undergoes another sequence of period doubling bifurcations leading to chaos and subsequent inverse period doubling (period halving) bifurcation to a P9 attractor before it finally loses stability due to a probable blue-sky catastrophe 1171at a forcing amplitude of about 0.2746. <%t-a forcing amplitude of about 0.2665 a PI3 attractor suddenly appears (by a possible blue-sky catastrophe) 1171,as is clearly shown in Fig. 9 which is an enlargement of Fig. 7 for 0.260 5 Am “= 0.275. (The coexistence of the PI 8 and PI3 attractors is clearly manifested in Fig. 9.) The time trace and phase plane at Awl = 0.26% are shown in Fig. 10(a) and (b) for the P18 attractor, This PI 8 attractor gives an average gasoline yield of 0.43849 and amplitude of temperature oscillations of 17.90K (slightly higher average gasoline yield and amplitude of oscillations compared to the autonomous periodicattractor). The other attractor (the PI3 attractor) gives an average gasoline yield of 0.438521 (slightly higher than the gasoline yield obtained with the P18 attractor) and an amplitude of temperature oscillations of 17.76K (riightly lower than the P1S attractor). As the forcing amplitude is increased the P18 attractor undergoes period halving bifurcation to P9 and eventually loses stability at Am = 0.2746 as discussed above. The P13 attractor undergoes period doubling bifurcations leading to chaos which eventually gives a 1~4attractor by type 1 intermittency at a forcing amplitude of about 0.32928. Figure 21(a) shows an enlargement of Fig. 2(b) for 0.30 5 Am 5 0.35. The time trace and the phase piane for this attractor at Am = 0.32927 (slightly into the chaotic region) are shown in Fig. 11(b) and (c) respectively. The presence of four laminar channels inside the chaotic bursts is clear from the stroboscopic points histogram (strobing every forcing period) shown in Fig. 11(d). The 4th iterate map shown in Fig. 11(f) clearly shows that the curve approaches (becomes
Chaotic behaviour of forced FCCUs 1.26
463
1.29 i
j
1.24
I
1
1
!
(b)
I :
,..::I!
Amp = 0.2043
i .
.
:I.
.
j
j
:,
j
1
:
:
:
..,I .
: :
I
I
:
j
:
:
:
, ,
! !
: !
! :
: :
.
: :
x
2
1.24
1.18
(a> 0.20
0.21
0.22
1.19 1 12000
0
13000
14000
15000
16000
17000
Time
Amp 0.444 -,
1.29
(d)
(c) Amp = 0.2043
Amp = 0.2043
0.442 0.440 *
8 0.438 0.436 0.434 $ 1.16
1.18
1.20
1.22
1.24
1.26
1.28
200000
250000
Forcing period
‘RD 0.444
1.28 (e) Amp = 0.2043 1.26
0.442 ,^ + E x’
*% 0.440 0.438
1.24 1.22 1.20
0.436 1.18
1.20
1.22
1.24
1.26
1.28
1.18 1.20 1.22
1.24
1.26
1.28
Fig. 5. (a) Enlargement of Fig. 3 for 0.19 5Am % 0.23. Type 1 intermittency at Am = 0.2043: (b) time trace; (c) phase plane; (d) stroboscopic points histogram showing nine laminar channels; (e) two-dimensional stroboscopic map; (f) iterate map showing nine distinct approach points.
1.280
0.444
1.260
0.442
(b)
1.240 0.440 p
1.220
8 R
x
0.438
1.200 1.180 1
I
(a)
Amp = 0.2092 ,,I,,,,,I1,,I,,,,,,,,r,,,,,,,,,,,,,,,, 27000 26000 26500
Time
0.436 0.434
27500
28000
1
1.18
1.20
1.22
1.24
1.26
1.28
‘RD
Fig. 6. (a) Time trace and (b) phase planes for the attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.2092.
-464
\.
0.20
F,. AHASAEED
0 22
and
S. S. F.
H. ELNASHAIE
0 26
0.24
0.30
0.28
Forcing amplitude Fig. 7. Enlargement of Fig. 3. 0.2 c Am 5 0.3 and 1.23c YRD-~0.23 for the attractor resulting from the forcing IJF the autontimous periodic attractor showing periodicity-banded chaos-chaos periodicitp also showing the appearance of anolher attractor.
almost tangent to) the diagonal at four distinct points which confirms the type 1 intermittency. The dissipative nature of the forced system can be measured by constructing the two-dimensional stroboscopic map (gasoline yield against the reactor dense phase temperature) as shown in Fig. 11(e). I1 is clear from the figure that the forced system shows hard-to-dissipate characteristics as indicated by the spread of the diagram. Conclusive evidence for the existence of chaos can be obtained by computing the Lyapunov exponents. The hugest Lyapunov exponent for this system has been found to he positive (+0.0012) which confirms the chaotic nature of the attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.32927. The average gasoline yield obtained with this chaotic attractor is 0.438542 and-the ampIitude of temperature oscillations is 18.14 K. Figure 12 is an enlargement of Fig. 3(b) for 0.48 or Am 5 0.64. It is clear from the enlarged figure (Fig. 12) that as the forcing amplitude is increased the P4 attractor resulting from the forcing of the autonomous periodic attractor (and emerging by type 1 intermittency from i W
0 444
-__--_-
I LX
0.442
1.26 c *”
0.440
!.24
* 2% 0.438
1.22 1.20
ii.436 I.18
I
0.434 I2000
18000
24000
‘10000
1 16
1 I8
1 20
I 22
1.24
1.26
I 2x
klc. 8. ia) Time trace and (h) phase planes for the chaotic attractor resulting from the forcing of the autonomous periodic attractor at All? ==(1.311.
465
Chaotic behaviour of forced FCCUs ,
1.26
:;i;
j !::::,.....’
x2 1.24-
x
2 ,:,a:
.24
,
; : , : : ! ..:. ! :
.’
_
:I!::;;, : /
:,: _.:,;:.... : I :
,,I
0.260
: ! 1 /
:
I(
:.
..
. ,
,,,,,,’
I8
0.265
,I
8
,_
I
,,I
0.270
11
I
I
I
I
I
IL
0.275
Forcing amplitude Fig. 9. Enlargement of Fig. 8 for 0.26 5 Am 5 0.275 showing the two possible periodic attractors resulting from forcing the autonomous periodic attractor (initial conditions lead to either of the two attractors).
the previous chaotic region) continues as a P4 attractor until Am = 0.537, where it loses stability. Depending on the set of initial conditions, it is possible to have two attractors resulting from the forcing of the autonomous periodic attractor in the region 0.52 5 Am I 0.537. The time trace and phase planes for these possible attractors are shown in Fig. 13 for Am = 0.532. The two possible attractors are a P4 attractor (Fig. 13(a) and (b)) and a strange attractor (Fig. 13(c) and (d)). The P4 attractor (which loses stability at Am = 0.537) showed the highest amplitude of oscillations (20.13 K) compared to previously found attractors. The average gasoline yield obtained with this attractor is 0.438421, which is quite close to the gasoline yield obtained with the autonomous periodic attractor. The strange attractor has the least amplitude of temperature oscillations of 16.73 K (lower than the autonomous periodic attractor) with an average gasoline yield of 0.438502 (slightly higher than the autonomous periodic attractor). Figure 14 shows different projections for this attractor at Am = 0.532. Computation of the Lyapunov exponent has shown that none of the Lyapunov exponents are positive, with the two largest Lyapunov exponents being very close to zero (L, = -7.65 X 10-4, L2 = -1.7 X 10p4). From Fig. 14 and the analysis of Lyapunov exponents, it is believed that this attractor is a strange non-chaotic attractor, higher-dimensional torus [20,21]. The shrinking size strange non-chaotic attractor resembles the shrinking size chaotic attractor obtained by Holden and Fan [22] in their study of the Rose-Hindmarch model for
(3) I MOO
.41np = 0 2685 12500
13noo
l~sno
I4ono
I Ih
! IX
I.20
I 22
I 24
I 26
I 2x
neuronnl activity. J’hc higher-dimension~ll torus continues to shrink in size and cventuah!; hccomcs a Pl attractor as the forcing amplitude is increased. J‘ahlc I contains a summary 01 the effect of periodic forcing of the autonomous periodic attractor on the amplitude of temperature oscillations and the average gasoline yield at selected forcing amplitude values. It is clear from the table that for the attractor resulting from the forcing of the autonomous periodic attractor the amplitude of temperature oscillations is gcncrally greater than that of the autonomous system (except for the strange non-chaotic attractor. the higher-dimensional torus. which showed lower amplitude of oscillattons). The average gasoline yield is slightly affected by forcing.
4. CONCI,USION In this paper the model developed earlier 1121for gas oil conversion to gasoline in a type 11: industrial lluid catalyttc cracking unit is used to investigate the effect of periodic forcing of air feed temperature on the system performance with special emphasis on gasoline yield and amplitude of temperature oscillations of the forced system. The autonomous case reveals a histability m the region of investigation. The bifurcation behavior of the forced svstem f’or a case in which the center of forcing is moderatelv removed from the homoclinic termination point of the autonomous periodic attractor revealed the rich dynamic behavior ()I’ the system. ,4s the forcing amplitude is increased the attractor resulting from the forcing of the autonomous periodic attractor undergoes period doubling and inverse period doubling bifurcations and eventually loses stability due to a blue-sky catastrophe. Another chaotic attractor has heen found before the first attractor loses stabilitv. A P9 attractor is born by
Chaotic behaviour of forced FCCUs
467
1.29
1.26
(a) Amp = 0.32927
x
(4
2
1.24
1.191
1.16 C
0.32
0.31
0.33
0.34
0
I
12000
13000
Amp
14000 Time
15000
16000
17000
130000
140000
150000
(d) Amp = 0.32927
0.433: 0.43om 1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.191 100000
110000
120600
Forcing period
YRD 1.28
(e> 1.26
0.442 =:
1.24
2 g
1.22
0.438 0.436 1
1 1.21
1.22
1.23
1.24
Y RD
1.25
1.26
1.:
1.18 1
1.20
1.22
1.24
1.26
1.28
Y(n)
Fig. 11. (a) Enlargement of Fig. 3 for 0.30 earn 5 0.35. Type 1 intermittency at Am = 0.32927: (b) time trace; (cl phase plane; (d) stroboscopic points histogram showing four laminar channels; (e) two-dimensional stroboscopic map; (f) iterate map showing four distinct approach points.
type 1 intermittency from the previous chaotic attractor and it undergoes period doubling bifurcation leading to banded and fully developed chaos before it regains periodicity by period halving and eventually loses stability. A P13 attractor is also present which undergoes period doubling bifurcations leading to chaos and returns to periodicity (P4) by intermittency. The P4 attractor loses stability and a non-chaotic attractor is born which shrinks in size to give a Pl attractor. The average gasoline yield obtained from the attractor resulting from the forcing of the autonomous periodic attractor is slightly affected by forcing amplitude. As expected, the amplitude of oscillations increases slightly with increase in forcing amplitude (the only exception is the shrinking size torus which oscillates with lower amplitude and higher gasoline yield). It is worth noting that the above analysis shows the robustness of operating in the oscillatory region under significantly high forcing amplitudes on the average gasoline yield,
.A. F.
m&rr,, 0 4x
and S. S. E. f-i. ELNASHAIF
AHASAEED
,
,
,
,
:
0.5?
,
, , , 0.56
,
/
I
!
I,,
& 1 I
I
I
I,
i
0. 64
0.60
Forcing amplitude Fig. 12. Enlargement of Fig. 3 for 0.48 5 Am 5 U.64 showing the possibility of having two attractors, a P4 and a shrinkins size stranpc attractor (depending on initial conditions) upon forcing the autonomous periodic attractor.
which is comparable to the optimum gasoline yield (only attainable at extremely high and impractical- values of controller gain) with minimum increase in the amplitude of temperature oscillati6ns.
I.15 1 18 I 20 I 22
1.24
1.26
I.28
0.432 12000
12500
17000
IV00
t4nQn
1
8
YRD
Time Fig. 13 Time trace and phase planes at .4nz = 0.532 ior the two possible resulting attractors from forcing adtonomrrus periodic-attractor: (ah) periodic, P4 and (c.d) strange non-chaotic attractor.
the
Chaotic behaviour of forced FCCUs
‘go
“’
VG ”
‘RD
469
VR -is ‘RD
‘RD
VR “’
‘gd
Fig. 14. Projections of the state variables for the shrinking size strange non-chaotic attractor at Am = 0.532 Table 1. Resulting amplitudes and gasoline yield for the attractor resulting from forcing the autonomous periodic attractor at various forcing amplitudes Am Am
0.00000 0.10000 0.15000 0.19874 0.19874 0.20430 0.20920 0.23200 0.26850 0.26850 0.32927 0.53200 0.53200
Amplitude of oscillations (K)
Gasoline yield
Type of attractor
17.65 18.84 20.02 19.76 18.55 18.33 17.71 18.12 17.90 17.76 18.14 20.13 16.73
0.438440 0.438395 0.438403 0.438382 0.438471 0.438478 0.438524 0.438499 0.438491 0.438521 0.438542 0.438412 0.438502
Periodic Pl P2 Pl Chaotic Chaotic P9 Chaotic P18 P13 Chaotic P4 Torus
REFERENCES 1. Iscol, L., The dynamics and stability of a fluid catalytic cracker. In Proceedings of the Automatic Control Conference, Atlanta, GA, 1970, pp. 602-607. 2. Elnashaie, S. S. E. H. and El-Hennawi, I. M., Multiplicity of the steady states in fluidized bed reactors IV. Fluid catalytic cracking. Chem. Eng. Sci., 1979, 34, 1113.
1. f- ABASAEhl)
i-i!
.Ind S. S. F. II. ELNAStlAff-
1 F.dwards. W. M. and Kim. Ii. N.. Mulhple rtcadv stale< in Huidiled bed FC‘(.‘ unit operation. Ch~tn f’rty .Scr.. 1988.37, 161 I. -1. Elshishini. S. S. and Elnashaie, S. S. I:. H.. Digital simulation of mdustrial titud catalytic cracking unltu I. Bifurcation and its implications. Che,n. Eq. Sci., 1990, 45, 553. 5. Elnashaie, S. S. E. H. and Elshishini, S. S.. Digital simulation of industrial flmd catalytic cracking units IV. Dynamic behavior. Chum. Eng. Sci., 1993, 48, 567. 0. Bailey. .I. E.. Periodic operation of chemical reactors: a review. C’hern. f?ng. Cbm,~ut~.. 1973, 1. 111. 7. Al-Taie. A. S. and Kershenbaum. L. S.. Effect of periodic operation cm the selectivity of catalytic rcacticms. /Ir?l. Chrnl. sot.. sv,np. ser.. 1978. (is, 5 12. t(. Sllveslon. P. L., Hudtins, R. R.. Adesina. A. A.. Ross. G. S. and Feimer. J. L.. Activity and selectivitv control through period comp&ion forcing over Fischer-Tropsch catalysts. Gem. Eng. Sci.. 1986. 41. 923. 0. I’lnashaie. S. S. E. H. and Abashar. M. E.. C‘haotic behavior of oeriodicallv forced fluidized-bed catalvtic reactors with consecutive exothermic chr.mical reactions. C’h~m. 6ng.’ Sci., 49, lb93. 2183. II). Can&a, C;. T. and Crossey. M. J.. Chaos in a periodically perturbed CSTR. Uzern. Eng. Cornrnu~., lY87. 51. I. i 1. Abasaeed, A. E. and Elnashaie, S. S. E. H.. The effects of cxtemal disturbances on the performnncc :lnd chaotic behavtour of industrial FCC units. C/XI~S, Solifons & I;ntcruls, lW7, 8, 1941-1955. 1’‘. Elnashaie. S. S. E. H.. Abasaeed, A. E. and Elshishini. S. S.. Digital simulation of industrial fluid catalytic cracking units V. Static and dynamic bifurcation. Chen?. Eng. Sci., 1995, 50, 1615. Bifurcarion md Chuck Hehmuw of CAw Solid I?. Elnashaie, S. S. E. H. and Elshishini, S. S.. Dynnmic, Mndvling. Cafalyfic Reuc~ors. Gordon & Breach. CK, 1996. l-l. Doedel. E. J. and Kernevcz. J. P.. Auto: Sofrwaw fir C’onfinrcar~o~r und iYifurcarro!l !‘rohlems irl Ordlnurj, Differenfiaf Equafions. California Institute of Technology. Pasadena, CA. 1086. i T. Wolf. A., Swift, J. B.. Swinnev. H. L.. and Vastano. J. A.. Determining I.yapunov exponents from a ttmc \crics. Ph?;.kYJ n. 1985. 16, 285. 16. Marek. M. and Schreiber, I.. C.‘haom~ Brhcnlt~~~ rjf I)efermm~.~fic I)r.t.sipnrive .T,.~c*~lr. (‘amhridgc I !ntvcrzit! Press. Cambridge, 1991, p. 70. 17. Jackson, E. A., Perspecrives o/Nonlinear Dynujnics. Cambrtdgc I iniversity Press, Cambridge, 1989, p. I. IK. Pomeau, Y. and Mannevillc, I’.. Intermittent transition to turhulcnce in dissipative dvnamical systems. Conmun.
Math.
Phys.,
1980. 74, 189.
IY. Tanibe. S. S. and Kulkami. B. D.. Intermittency route to chaos 111a periodically forced model reaction system. Chem. Eng Sci.. 1093, 48. 2X I7. XJ. Ding, M., Grebogi, c‘. and Ott, tl.. E vo I u t’ton of attracton in quast-periodically forced systems. from quasi-periodic to strange non-chaotic to chaotic. Pllys. Rev. A. lY8Y. 39, 2593. .! I. Kapitaniak. T.. Ponce, F. and Wojewoda, J.. Route to chao\ via non-chaotic attractor\. 1. t%ys. A: Mafh.. IYYO. 23, 1,387. 22. fiolden. A. V. and Fan, Y.. C‘rlsls-induced chaos in the Rose- Hindmarsh model tor nruronal activity. (.‘hti~. .Solironc
Pergamon
PII: !30960-0779(97)ooo88-x
On the Chaotic Behavior of Externally Forced Industrial Fluid Catalytic Cracking Units A. E. ABASAEED Chemical Engineering Department, KSU, P.O. Box 800, Riyadh 11421, Saudi Arabia
and S. S. E. H. ELNASHAIE Chemical Engineering Department, Cairo University, Cairo, Egypt (Accepted 27 March 1997)
good insight into the effect of external disturbances on industrial units can be obtained by the investigation of the effect of periodic forcing. Therefore, the effect of periodic forcing of the air feed temperature on the bifurcation behavior, gasoline yield, and amplitude modulation of a type IV industrial fluid catalytic cracking (FCC) unit is investigated. The investigation concentrates on a case in which the autonomous system shows bistability between a high temperature static attractor and a periodic attractor (the emphasis being on the periodic attractor). The center of forcing considered in this case is moderately removed from the homoclinic termination point of the autonomous periodic branch. The one-parameter stroboscopic Poincart bifurcation diagram reveals the richness of the dynamic behavior of the forced system (periodicity, aperiodicity, bifurcations, inverse bifurcations, intermittency, boundary crisis, shrinking size torus and chaos). The investigation reveals that period doubling, type 1 intermittency and boundary crisis are the dominant routes to chaos. Lyapunov exponents are computed to distinguish between chaotic and non-chaotic complex attractors. The average gasoline yield obtained with the attractor resulting from the forcing of the autonomous static attractor increases slightly with increase in forcing amplitude. All chaotic attractors resulting from the forcing of the autonomous periodic attractor give slightly higher gasoline yields than their autonomous counterparts. 0 1998 Elsevier Science Ltd. All rights reserved
Abstract-A
NOMENCLATURE dimensionless groups ” forcing amplitude modified space velocities for the reactor and regenerator respectively (l/s) coke burning rate [kg coke bumed/(kg solid s)] dimensionless group connected with the coke balance in the reactor and regenerator respectively bubble flow rate through any horizontal section for reactor and regenerator respectively (m3/s) height of reactor and regenerator respectively (m) heat losses in reactor and regenerator respectively dimensionless rate of heat absorption due to endothermic cracking reactions in the reactor
HV Kc RC xsb X sbf x@ x&w X gob X gof Yf Yfa
YGB
455
dimensionless heat of vaporization for gas oil (kJ/kg) controller gain coke make rate dimensionless bubble phase gasoline concentration dimensionless gasoline concentration in feed dimensionless dense phase gasoline concentration dimensionless gas oil concentration dimensionless bubble phase gas oil concentration dimensionless gas oil concentration in feed dimensionless feed temperature dimensionless air feed temperature dimensionless regenerator bubble phase temperature
,;*I, t;:,, ~‘I?,, ):,,
dimensionless regenerator dense phase temperature dimensionless reactor bubble phase temperature dimensionless reactor dense phase temperaturedimensionless set point reactor
t., i , !4 i-.
temperature I'!,
:l (ireCk lmm Cl,. CY1,ru, c-!
dimensionless gas oil vaporization temperature forcing frequench
/A
dimensionless pre-exponential factors for the cracking reactions dynamic parameters in heat balance equation for regenerator
I&~ l/f,; $5. PI* /‘I
c Y.
dynamic sorption dynamic sorption dynamic equation dynamic
parameters for catalyst adcapacity for gas oil parameters for catalyst atlcapacity for gasoline parameters in coke balance for reactor parameters in coke balance equation for regenerator exothermicity factor for coke comhustion reaction bed voidagc dimensionless activation energies for cracking reactions catalyst activity within reactor catalyst activity within regenerator density of air, liquid gas oil. vaporized qas oil respectively (kg/m’)
1. INTRODUCTION
Fluid catalytic cracking is one of the most important processes in the petroleum refining industry. These units have been shown to operate at a controllable unstable steady state, saddle-type [ I-S]. A number of investigations concentrated on studying the interaction of feed disturbances (intentional or unintentional) with the non-line&ties in chemically reactive systems [C-9]. Very few results are presented for the cases with the center of forcing being relatively close to the homoclinic orbit [9, lo]. In a previous investigation. Abasaeed and Elnashaie [Ill reported the bifurcation behavior of an industrial type IV FCC’ unit subject to periodic forcing of air feed temperature with the centerof forcing being extremely close to the homoclinic orbit. Period doubling and ~type I intermittency have been identified as the routes to chaotic behavior. The chaotic behavior occurred at very low values of forcing amplitudes. Boundary crisis terminated the first chaotic regime before a P4 attractor emerged by a possibIe blue-sky catastrophe. The P4 also underwent period doubling bifurcations leading to chaos before it completely disappeared. In this investigation, the center of forcing has been moderately removed from the homodinic termination point of the autonomous periodic branch. The investigation concentrates on the effect of forcing on the chaotic behavior as well as on the gasoline yield and amplitude of temperature osciljations of the forced system at selected forcing amplitudes. Various techniques are used to characterize the periodicity and aperiodicity of Ihe forced system. Lyapunov exponents are also computed to characterize chaotic behavior.
2. THE
MODEL
The model used here is based on the two-phase theory of Auidized bed reactors, which was developed earlier and compared successfully against data from two industrial units [4]. This model has been extended by Elnashaie et ul. [12] to include the dynamics of gas oil and gasoline. The details of the model development and its associated assumptions have been given earlier 14,s. 12,131 and will not be repeated here. The normalized dynamic model equations are given as follows,
Chaotic behaviour
of forced FCCUs
457
The reactor dense phase normalized heat balance equation: -d&u, = B(Yu - I-,,)
dr
+ A2(Yf - Yu) - Hu + HRCB + A,(YG,
- Y,,) - HL.
(1)
The regenerator dense phase normalized heat balance equation:
dycm = &{(BD(Y,, - Km)) + Pc(CB) + AOk, dr
- Km> - HL,).
The gas oil normalized mass balance in the reactor dense phase:
$$’ =[2(B{Xgof -
X,,} - +R[ale-YI’YRD+ a3e-y~‘y~]x&,).
(3)
The gasoline normalized mass balance in the reactor dense phase: 2
= &{B(X+
- Xgd) - t,bR(a2Xgde-.y2’YRD - a1(Xgo)2e-Y1’Y~D)~.
(4)
The carbon balance (catalyst activity) in the reactor:
2 = &(CRD{~- Icrd- (Ir&),
(9
where R, = a2e- y2’Y~~Xgd+ ff3e - wmx* go-
The carbon balance (catalyst activity) in the regenerator:
(6) The air feed temperature: Yfa= Y,, + Am sin(w,z) + Kc( Y,, - Y,,).
(7
The normalized mass and heat balance equations for the bubble phase which is assumed to be at pseudo-steady state are given as follows. The reactor bubble phase normalized heat balance equation: Y,, = YRD= (Yu - YRD)eeURHR.
@I
The regenerator bubble phase normalized heat balance equation: YoB = Y,, + (Yu - YGD)e-a~H~.
(9)
The gas oil normalized mass balance in the reactor bubble phase: X gob= X,, + (Xgof - Xgo)eeaRHR.
(10)
J<\
A
I,. Al3ASAEF.I)
:I&
S. 5. t:.
H. T:I.NASHAIF
The gasoline normalized mass balance in the reactor bubble phase:
x,, = x,,, -i ix,,,
X,Jc “;,“J:.
(11)
ln eqn (7). Am =O.O implies the autonomous system. The various parameters and variables used in the above equations are defined in the Nomenclature.
3. RESULTS AND DISCUSSION
Ihe bifurcation diagrams are constructed using AUTO86 developed by Doedel and Kernevez [14]. The Lyapunov exponents are computed using the algorithm developed by Wolf et (I/. [15]. To ensure accuracy of the simulation results, the bound on the allowable error was maintained at lo-.“. with automatic step size integration routines. 3. 1
C‘losed-loop
wtonom0u.s
system
Figure I shows the bifurcation diagram for the reactor dense phase gasoline yield for the autonomous system in the histability region (2.6863 5 Kc 5 2.8235) with the air feed temperature yt, as the bifurcation parameter. The solid lines in the figure indicate stable steady states static branches, whereas the dashed lines shows the unstable one. The stable periodic branch is given in the figure as the symbol (0). and the Hopf bifurcation points, denoted HB, as the symbol (m). The enlargement of the periodic branch emanating from the Hopf bifurcation point (denoted HJ31 in Fig. l(a)) is shown in Fig. l(b). This periodic branch terminates homoclinically at Y,,,= I.505187 as shown in Fig. l(b). It is evident from Figs l(a) and (b) that the unstable static (dashed lines) and the stable periodic branches (0) give a much higher gasoline yield (>0.4) compared to the stable static branch (solid lines). A detailed discussion of this figure is given in Abasaeed and Elnashaie [I 11. 3.2
The f;mwi
.sy.stm
The case in which periodic forcing of this system with the center of forcing (Y,;, = 1.505188) being very close to the homoclinic orbit and the forcing frequency five times the frequency of the autonomous system has been discussed in a previous study [ 1I]. The one-parameter stroboscopic PoincarC bifurcation diagram for that case revealed that the
(b) ---__44*m t HBI
--j----
Chaotic behaviour of forced FCCUs
459
attractor resulting from the forcing of the autonomous periodic attractor goes through period doubling bifurcations leading to chaos (PGPlO-P20 chaos, based on strobing every forcing period, this is equivalent to Pl-P2-P4 chaos when strobing every autonomous period). At extremely small values of the forcing amplitude (Am < 0.94 X 10p5), the resulting chaotic attractor lost its stability due to a boundary crisis [16]. This situation continued until Am = 0.245 X 10e3, when a P4 attractor (based on strobing every forcing period) suddenly appeared. The only attractor existing in the range 0.94 X 10V55 Am 5 0.245 X 10m3was the Pl (period 1) attractor resulting from the forcing of the high temperature static attractor. The resulting P4 attractor went through a period doubling sequence cascading to chaos. A boundary crisis occurs at Am = 0.48 X lop3 and the chaotic attractor resulting from the forcing of the autonomous periodic attractor lost completely its stability to the attractor resulting from the forcing of the autonomous static high temperature attractor. In the present investigation, the bifurcation behavior of the forced system is analysed for the case when the center of forcing (Yf, = 1.505583) is moved moderately from the homoclinic orbit. The natural period of the autonomous periodic attractor is 259.6571 s at Y, = 1.505583. The one-parameter stroboscopic Poincare bifurcation diagram is shown in Fig. 2 for the reactor dense phase temperature against the dimensionless forcing amplitude. Since the forcing is for a case with the autonomous system in a region of bistability, it is to 2.17 2.16 y1 2.15 2 5*
2.14 2.13 2.12:,,
, , , , ) , , , , ( , , , , , , , , , , , , , , , , , , , , , ,
1.26
1.18,, 0.0
, I / , , , , , , , , , I ,, 0.1 0.2 0.3
, , , , , , , , , , , , , ,, 0.4 0.5 0.6
, , , ( 7
Forcing amplitude Fig. 2. One-parameter stroboscopic Poincart bifurcation diagram in the bistability region for the reactor dense phase temperature against the forcing amplitude (strobing every forcing period). The attractor resulting from the forcing of the (a) autonomous static attractor and (b) autonomous periodic attractor. Center of forcing is at Am = 1.505583.
be expected that, depending on initial conditions, the forcing will affect either of the two attractors. For this reason Fig. 2 is composed of two portions: the upper portion (Fig. 2(a)) of the figure is for the attractor resulting from the forcing of the static high temperature attractor (of course nothing exciting takes place with this attractor because periodic forcing of a static attractor results in a periodic attractor having period one), while the lower portion (Fig. 2(b)) is for the attractor resulting from the forcing of the autonomous periodic attractor (this attractor is expected to give quite a rich behavior). The following analysis concentrates on the attraetor resulting from the forcing of the autonomous periodic attractor (i.c.~the attractor shown in Fig. 2(b)). 3.3
T/w forced autonom0u.s periodic nttructor
The amplitude of temperature oscillations for the autonomous periodic attractor has been determined to be 17.7OK and the average gasoline yield obtained with this attractor is 0.43844. It is clear from Fig. 2(b) that five branches emerge-(based-on strobing every forcing period, in fact these five branches would show as one branch if the autonomous period is used for strobing) as the forcing amplitude is increased. These five branches undergo period doubhng bifurcation (P5 -4 PlO) and inverse period doubling bifurcation (period halving, PIO-+PS) with increase in forcing-amplitude to Am - 0.19875, where this attractor loses stabitity. Figure 3(a) and (b) shows the time trace and phase planes for the attractor resulting from the forcing of the autonomous periodic attractor (a Pl attractor based on strobing every natural period) respectively at A?rz-- 0. I. This attractor gives an average gasoline yield of 0.438395 (slightly less than the yield obtained with the autonomous periodic attractor). The amplitude of temperature oscillations increases to XX.84K. As the forcing amplitude is increased to At/7 = (Li5. the Pl attractor undergoes period doubling bifurcation (i.e. a PI? attractor is ~born based on strobing every natural period). as is clearly seen from Fig. .3(c) and (d) for time trace and phase plane respectively. The resulting P2 attractor -.-__1
-I-
(~a) Amp = 0.1 n
n
n
!dj
Chaotic behaviour
of forced FCCUs
461
oscillates with higher amplitudes (20.02 K) and gives an average gasoline yield of 0.438403 (still lower than the autonomous periodic attractor yield, 0.43844). Depending on the set of initial conditions used, it has been found that it is possible to locate two different attractors (one periodic and the other chaotic) at a forcing amplitude of 0.19874. Figure 4(a) and (b) shows the time trace and phase portrait for the periodic attractor (this is a Pl attractor) at Am = 0.19874. The average gasoline yield for this attractor is 0.438382 and the amplitude of oscillations is 19.76 K. It is postulated here that a blue-sky catastrophe [17] has occurred and this attractor loses its stability at Am = 0.19875. The time trace and the phase plane for the chaotic attractor at Am = 0.19874 are shown in Fig. 4(c) and (d) respectively. The chaotic nature of this attractor has been confirmed by computing the Lyapunov exponents (the leading Lyapunov exponent is positive, +0.0014). The maximum amplitude of temperature oscillations for this attractor is 18.55 K (lower than the other coexisting Pl attractor). The average gasoline yield obtained with the resulting chaotic attractor is 0.438471 (slightly higher than the average yield obtained with the autonomous periodic attractor, 0.43844). An enlargement of Fig. 2(b) for a forcing amplitude range of 0.19 < Am < 0.23 is shown in Fig. 5(a). It is clear from the figure that a P5 attractor (based on strobing every forcing period) emerges from the left of the diagram followed by a thin strip of chaos followed by a P9 attractor (at Am = 0.2045) which undergoes period doubling bifurcations. The time trace. and the phase plane for this attractor at Am = 0.2043 (slightly into the chaotic strip) are shown in Fig. 5(b) and (c) respectively. Intermittency is a term used to describe the presence of laminar channels (periodic oscillations) which are sandwiched inside chaotic bursts [18]. The intermittency is not clear from Fig. 5(b) and (c), however, the existence of nine laminar phases within the chaotic regime is clear from the stroboscopic points histogram for the reactor dense phase temperature against the forcing period (Fig. 5(d)). The two-dimensional stroboscopic map for the gasoline yield against the reactor dense phase temperature is shown in Fig. 5(e). The spread of this diagram over a relatively wide range of reactor dense phase temperatures indicates the hard to dissipate nature of this system. Type 1 intermittency (which occurs when the Floquet multiplier exits the unit circle through +l) can be determined by plotting the ith iterate map [18,19]. The 9th iterate map for the reactor dense phase temperature is plotted in Fig. 5(f). It is clear from this figure that the curve approaches the bisectrix tangentially at nine distinct points which confirms the type 1 intermittency for this case. The average gasoline yield for this attractor is 0.438474 (slightly higher than the average yield of the autonomous periodic attractor) and the amplitude of temperature oscillations is 18.33 K at Am = 0.2043. The time trace and phase plane diagrams for the P9 attractor which emerges from the chaotic strip by type 1 intermittency are shown in Fig. 6(a) and (b) respectively for Am = 0.2092. For the resulting P9 attractor, the amplitude of oscillations and the average gasoline yield are respectively 17.71 K (slightly higher than the amplitude of oscillations for the autonomous periodic attractor) and 0.438524 (slightly higher than the average yield obtained with the autonomous periodic attractor). This P9 attractor undergoes period doubling bifurcations leading to banded chaos and eventually fully developed chaos as the forcing amplitude is increased (shown in Fig. 7 which is an enlargement of Fig. 2(b) for 0.2 5 Am I 0.3 and 1.23 % Ya,, 5 1.26). The time trace and the phase portrait for the chaotic attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.232 are shown in Fig. 8(a) and (b) respectively. Computation of the Lyapunov exponents gave a positive leading Lyapunov exponent (+0.00122) which confirms the chaotic nature of this attractor. This attractor gives an average gasoline yield of 0.438499 (slightly higher than that obtained with the autonomous periodic attractor) and has an amplitude of temperature oscillations of 18.12 K (slightly higher than the autonomous
0 442
I
I
(a) Amp = 0.19874
0.442 0.440 UP x
0.438 0 436 0 434
25000
30000
~5000 Time
J0000
: lfi
I.18
1.20
1.22
1.24
1.26
t 28
YRI3
Fir. 3. Time trace and phase planes al Am = 0.19874 for the two possible resulting attractors from forcing rhe antonomous periodic attractor: 1a.h) periodic, Pl and (c.d) chaotic.
periodic attractorj. The chaotic attractor then undergoes period halving bifurcation leading rn a PI8 attractor. The P3 P attractor resulting from the forcing of the autonomous periodic attractor undergoes another sequence of period doubling bifurcations leading to chaos and subsequent inverse period doubling (period halving) bifurcation to a P9 attractor before it finally loses stability due to a probable blue-sky catastrophe 1171at a forcing amplitude of about 0.2746. <%t-a forcing amplitude of about 0.2665 a PI3 attractor suddenly appears (by a possible blue-sky catastrophe) 1171,as is clearly shown in Fig. 9 which is an enlargement of Fig. 7 for 0.260 5 Am “= 0.275. (The coexistence of the PI 8 and PI3 attractors is clearly manifested in Fig. 9.) The time trace and phase plane at Awl = 0.26% are shown in Fig. 10(a) and (b) for the P18 attractor, This PI 8 attractor gives an average gasoline yield of 0.43849 and amplitude of temperature oscillations of 17.90K (slightly higher average gasoline yield and amplitude of oscillations compared to the autonomous periodicattractor). The other attractor (the PI3 attractor) gives an average gasoline yield of 0.438521 (slightly higher than the gasoline yield obtained with the P18 attractor) and an amplitude of temperature oscillations of 17.76K (riightly lower than the P1S attractor). As the forcing amplitude is increased the P18 attractor undergoes period halving bifurcation to P9 and eventually loses stability at Am = 0.2746 as discussed above. The P13 attractor undergoes period doubling bifurcations leading to chaos which eventually gives a 1~4attractor by type 1 intermittency at a forcing amplitude of about 0.32928. Figure 21(a) shows an enlargement of Fig. 2(b) for 0.30 5 Am 5 0.35. The time trace and the phase piane for this attractor at Am = 0.32927 (slightly into the chaotic region) are shown in Fig. 11(b) and (c) respectively. The presence of four laminar channels inside the chaotic bursts is clear from the stroboscopic points histogram (strobing every forcing period) shown in Fig. 11(d). The 4th iterate map shown in Fig. 11(f) clearly shows that the curve approaches (becomes
Chaotic behaviour of forced FCCUs 1.26
463
1.29 i
j
1.24
I
1
1
!
(b)
I :
,..::I!
Amp = 0.2043
i .
.
:I.
.
j
j
:,
j
1
:
:
:
..,I .
: :
I
I
:
j
:
:
:
, ,
! !
: !
! :
: :
.
: :
x
2
1.24
1.18
(a> 0.20
0.21
0.22
1.19 1 12000
0
13000
14000
15000
16000
17000
Time
Amp 0.444 -,
1.29
(d)
(c) Amp = 0.2043
Amp = 0.2043
0.442 0.440 *
8 0.438 0.436 0.434 $ 1.16
1.18
1.20
1.22
1.24
1.26
1.28
200000
250000
Forcing period
‘RD 0.444
1.28 (e) Amp = 0.2043 1.26
0.442 ,^ + E x’
*% 0.440 0.438
1.24 1.22 1.20
0.436 1.18
1.20
1.22
1.24
1.26
1.28
1.18 1.20 1.22
1.24
1.26
1.28
Fig. 5. (a) Enlargement of Fig. 3 for 0.19 5Am % 0.23. Type 1 intermittency at Am = 0.2043: (b) time trace; (c) phase plane; (d) stroboscopic points histogram showing nine laminar channels; (e) two-dimensional stroboscopic map; (f) iterate map showing nine distinct approach points.
1.280
0.444
1.260
0.442
(b)
1.240 0.440 p
1.220
8 R
x
0.438
1.200 1.180 1
I
(a)
Amp = 0.2092 ,,I,,,,,I1,,I,,,,,,,,r,,,,,,,,,,,,,,,, 27000 26000 26500
Time
0.436 0.434
27500
28000
1
1.18
1.20
1.22
1.24
1.26
1.28
‘RD
Fig. 6. (a) Time trace and (b) phase planes for the attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.2092.
-464
\.
0.20
F,. AHASAEED
0 22
and
S. S. F.
H. ELNASHAIE
0 26
0.24
0.30
0.28
Forcing amplitude Fig. 7. Enlargement of Fig. 3. 0.2 c Am 5 0.3 and 1.23c YRD-~0.23 for the attractor resulting from the forcing IJF the autontimous periodic attractor showing periodicity-banded chaos-chaos periodicitp also showing the appearance of anolher attractor.
almost tangent to) the diagonal at four distinct points which confirms the type 1 intermittency. The dissipative nature of the forced system can be measured by constructing the two-dimensional stroboscopic map (gasoline yield against the reactor dense phase temperature) as shown in Fig. 11(e). I1 is clear from the figure that the forced system shows hard-to-dissipate characteristics as indicated by the spread of the diagram. Conclusive evidence for the existence of chaos can be obtained by computing the Lyapunov exponents. The hugest Lyapunov exponent for this system has been found to he positive (+0.0012) which confirms the chaotic nature of the attractor resulting from the forcing of the autonomous periodic attractor at Am = 0.32927. The average gasoline yield obtained with this chaotic attractor is 0.438542 and-the ampIitude of temperature oscillations is 18.14 K. Figure 12 is an enlargement of Fig. 3(b) for 0.48 or Am 5 0.64. It is clear from the enlarged figure (Fig. 12) that as the forcing amplitude is increased the P4 attractor resulting from the forcing of the autonomous periodic attractor (and emerging by type 1 intermittency from i W
0 444
-__--_-
I LX
0.442
1.26 c *”
0.440
!.24
* 2% 0.438
1.22 1.20
ii.436 I.18
I
0.434 I2000
18000
24000
‘10000
1 16
1 I8
1 20
I 22
1.24
1.26
I 2x
klc. 8. ia) Time trace and (h) phase planes for the chaotic attractor resulting from the forcing of the autonomous periodic attractor at All? ==(1.311.
465
Chaotic behaviour of forced FCCUs ,
1.26
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j !::::,.....’
x2 1.24-
x
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.24
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8
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11
I
I
I
I
I
IL
0.275
Forcing amplitude Fig. 9. Enlargement of Fig. 8 for 0.26 5 Am 5 0.275 showing the two possible periodic attractors resulting from forcing the autonomous periodic attractor (initial conditions lead to either of the two attractors).
the previous chaotic region) continues as a P4 attractor until Am = 0.537, where it loses stability. Depending on the set of initial conditions, it is possible to have two attractors resulting from the forcing of the autonomous periodic attractor in the region 0.52 5 Am I 0.537. The time trace and phase planes for these possible attractors are shown in Fig. 13 for Am = 0.532. The two possible attractors are a P4 attractor (Fig. 13(a) and (b)) and a strange attractor (Fig. 13(c) and (d)). The P4 attractor (which loses stability at Am = 0.537) showed the highest amplitude of oscillations (20.13 K) compared to previously found attractors. The average gasoline yield obtained with this attractor is 0.438421, which is quite close to the gasoline yield obtained with the autonomous periodic attractor. The strange attractor has the least amplitude of temperature oscillations of 16.73 K (lower than the autonomous periodic attractor) with an average gasoline yield of 0.438502 (slightly higher than the autonomous periodic attractor). Figure 14 shows different projections for this attractor at Am = 0.532. Computation of the Lyapunov exponent has shown that none of the Lyapunov exponents are positive, with the two largest Lyapunov exponents being very close to zero (L, = -7.65 X 10-4, L2 = -1.7 X 10p4). From Fig. 14 and the analysis of Lyapunov exponents, it is believed that this attractor is a strange non-chaotic attractor, higher-dimensional torus [20,21]. The shrinking size strange non-chaotic attractor resembles the shrinking size chaotic attractor obtained by Holden and Fan [22] in their study of the Rose-Hindmarch model for
(3) I MOO
.41np = 0 2685 12500
13noo
l~sno
I4ono
I Ih
! IX
I.20
I 22
I 24
I 26
I 2x
neuronnl activity. J’hc higher-dimension~ll torus continues to shrink in size and cventuah!; hccomcs a Pl attractor as the forcing amplitude is increased. J‘ahlc I contains a summary 01 the effect of periodic forcing of the autonomous periodic attractor on the amplitude of temperature oscillations and the average gasoline yield at selected forcing amplitude values. It is clear from the table that for the attractor resulting from the forcing of the autonomous periodic attractor the amplitude of temperature oscillations is gcncrally greater than that of the autonomous system (except for the strange non-chaotic attractor. the higher-dimensional torus. which showed lower amplitude of oscillattons). The average gasoline yield is slightly affected by forcing.
4. CONCI,USION In this paper the model developed earlier 1121for gas oil conversion to gasoline in a type 11: industrial lluid catalyttc cracking unit is used to investigate the effect of periodic forcing of air feed temperature on the system performance with special emphasis on gasoline yield and amplitude of temperature oscillations of the forced system. The autonomous case reveals a histability m the region of investigation. The bifurcation behavior of the forced svstem f’or a case in which the center of forcing is moderatelv removed from the homoclinic termination point of the autonomous periodic attractor revealed the rich dynamic behavior ()I’ the system. ,4s the forcing amplitude is increased the attractor resulting from the forcing of the autonomous periodic attractor undergoes period doubling and inverse period doubling bifurcations and eventually loses stability due to a blue-sky catastrophe. Another chaotic attractor has heen found before the first attractor loses stabilitv. A P9 attractor is born by
Chaotic behaviour of forced FCCUs
467
1.29
1.26
(a) Amp = 0.32927
x
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2
1.24
1.191
1.16 C
0.32
0.31
0.33
0.34
0
I
12000
13000
Amp
14000 Time
15000
16000
17000
130000
140000
150000
(d) Amp = 0.32927
0.433: 0.43om 1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.191 100000
110000
120600
Forcing period
YRD 1.28
(e> 1.26
0.442 =:
1.24
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1.22
0.438 0.436 1
1 1.21
1.22
1.23
1.24
Y RD
1.25
1.26
1.:
1.18 1
1.20
1.22
1.24
1.26
1.28
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Fig. 11. (a) Enlargement of Fig. 3 for 0.30 earn 5 0.35. Type 1 intermittency at Am = 0.32927: (b) time trace; (cl phase plane; (d) stroboscopic points histogram showing four laminar channels; (e) two-dimensional stroboscopic map; (f) iterate map showing four distinct approach points.
type 1 intermittency from the previous chaotic attractor and it undergoes period doubling bifurcation leading to banded and fully developed chaos before it regains periodicity by period halving and eventually loses stability. A P13 attractor is also present which undergoes period doubling bifurcations leading to chaos and returns to periodicity (P4) by intermittency. The P4 attractor loses stability and a non-chaotic attractor is born which shrinks in size to give a Pl attractor. The average gasoline yield obtained from the attractor resulting from the forcing of the autonomous periodic attractor is slightly affected by forcing amplitude. As expected, the amplitude of oscillations increases slightly with increase in forcing amplitude (the only exception is the shrinking size torus which oscillates with lower amplitude and higher gasoline yield). It is worth noting that the above analysis shows the robustness of operating in the oscillatory region under significantly high forcing amplitudes on the average gasoline yield,
.A. F.
m&rr,, 0 4x
and S. S. E. f-i. ELNASHAIF
AHASAEED
,
,
,
,
:
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,
, , , 0.56
,
/
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I
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i
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0.60
Forcing amplitude Fig. 12. Enlargement of Fig. 3 for 0.48 5 Am 5 U.64 showing the possibility of having two attractors, a P4 and a shrinkins size stranpc attractor (depending on initial conditions) upon forcing the autonomous periodic attractor.
which is comparable to the optimum gasoline yield (only attainable at extremely high and impractical- values of controller gain) with minimum increase in the amplitude of temperature oscillati6ns.
I.15 1 18 I 20 I 22
1.24
1.26
I.28
0.432 12000
12500
17000
IV00
t4nQn
1
8
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Time Fig. 13 Time trace and phase planes at .4nz = 0.532 ior the two possible resulting attractors from forcing adtonomrrus periodic-attractor: (ah) periodic, P4 and (c.d) strange non-chaotic attractor.
the
Chaotic behaviour of forced FCCUs
‘go
“’
VG ”
‘RD
469
VR -is ‘RD
‘RD
VR “’
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Fig. 14. Projections of the state variables for the shrinking size strange non-chaotic attractor at Am = 0.532 Table 1. Resulting amplitudes and gasoline yield for the attractor resulting from forcing the autonomous periodic attractor at various forcing amplitudes Am Am
0.00000 0.10000 0.15000 0.19874 0.19874 0.20430 0.20920 0.23200 0.26850 0.26850 0.32927 0.53200 0.53200
Amplitude of oscillations (K)
Gasoline yield
Type of attractor
17.65 18.84 20.02 19.76 18.55 18.33 17.71 18.12 17.90 17.76 18.14 20.13 16.73
0.438440 0.438395 0.438403 0.438382 0.438471 0.438478 0.438524 0.438499 0.438491 0.438521 0.438542 0.438412 0.438502
Periodic Pl P2 Pl Chaotic Chaotic P9 Chaotic P18 P13 Chaotic P4 Torus
REFERENCES 1. Iscol, L., The dynamics and stability of a fluid catalytic cracker. In Proceedings of the Automatic Control Conference, Atlanta, GA, 1970, pp. 602-607. 2. Elnashaie, S. S. E. H. and El-Hennawi, I. M., Multiplicity of the steady states in fluidized bed reactors IV. Fluid catalytic cracking. Chem. Eng. Sci., 1979, 34, 1113.
1. f- ABASAEhl)
i-i!
.Ind S. S. F. II. ELNAStlAff-
1 F.dwards. W. M. and Kim. Ii. N.. Mulhple rtcadv stale< in Huidiled bed FC‘(.‘ unit operation. Ch~tn f’rty .Scr.. 1988.37, 161 I. -1. Elshishini. S. S. and Elnashaie, S. S. I:. H.. Digital simulation of mdustrial titud catalytic cracking unltu I. Bifurcation and its implications. Che,n. Eq. Sci., 1990, 45, 553. 5. Elnashaie, S. S. E. H. and Elshishini, S. S.. Digital simulation of industrial flmd catalytic cracking units IV. Dynamic behavior. Chum. Eng. Sci., 1993, 48, 567. 0. Bailey. .I. E.. Periodic operation of chemical reactors: a review. C’hern. f?ng. Cbm,~ut~.. 1973, 1. 111. 7. Al-Taie. A. S. and Kershenbaum. L. S.. Effect of periodic operation cm the selectivity of catalytic rcacticms. /Ir?l. Chrnl. sot.. sv,np. ser.. 1978. (is, 5 12. t(. Sllveslon. P. L., Hudtins, R. R.. Adesina. A. A.. Ross. G. S. and Feimer. J. L.. Activity and selectivitv control through period comp&ion forcing over Fischer-Tropsch catalysts. Gem. Eng. Sci.. 1986. 41. 923. 0. I’lnashaie. S. S. E. H. and Abashar. M. E.. C‘haotic behavior of oeriodicallv forced fluidized-bed catalvtic reactors with consecutive exothermic chr.mical reactions. C’h~m. 6ng.’ Sci., 49, lb93. 2183. II). Can&a, C;. T. and Crossey. M. J.. Chaos in a periodically perturbed CSTR. Uzern. Eng. Cornrnu~., lY87. 51. I. i 1. Abasaeed, A. E. and Elnashaie, S. S. E. H.. The effects of cxtemal disturbances on the performnncc :lnd chaotic behavtour of industrial FCC units. C/XI~S, Solifons & I;ntcruls, lW7, 8, 1941-1955. 1’‘. Elnashaie. S. S. E. H.. Abasaeed, A. E. and Elshishini. S. S.. Digital simulation of industrial fluid catalytic cracking units V. Static and dynamic bifurcation. Chen?. Eng. Sci., 1995, 50, 1615. Bifurcarion md Chuck Hehmuw of CAw Solid I?. Elnashaie, S. S. E. H. and Elshishini, S. S.. Dynnmic, Mndvling. Cafalyfic Reuc~ors. Gordon & Breach. CK, 1996. l-l. Doedel. E. J. and Kernevcz. J. P.. Auto: Sofrwaw fir C’onfinrcar~o~r und iYifurcarro!l !‘rohlems irl Ordlnurj, Differenfiaf Equafions. California Institute of Technology. Pasadena, CA. 1086. i T. Wolf. A., Swift, J. B.. Swinnev. H. L.. and Vastano. J. A.. Determining I.yapunov exponents from a ttmc \crics. Ph?;.kYJ n. 1985. 16, 285. 16. Marek. M. and Schreiber, I.. C.‘haom~ Brhcnlt~~~ rjf I)efermm~.~fic I)r.t.sipnrive .T,.~c*~lr. (‘amhridgc I !ntvcrzit! Press. Cambridge, 1991, p. 70. 17. Jackson, E. A., Perspecrives o/Nonlinear Dynujnics. Cambrtdgc I iniversity Press, Cambridge, 1989, p. I. IK. Pomeau, Y. and Mannevillc, I’.. Intermittent transition to turhulcnce in dissipative dvnamical systems. Conmun.
Math.
Phys.,
1980. 74, 189.
IY. Tanibe. S. S. and Kulkami. B. D.. Intermittency route to chaos 111a periodically forced model reaction system. Chem. Eng Sci.. 1093, 48. 2X I7. XJ. Ding, M., Grebogi, c‘. and Ott, tl.. E vo I u t’ton of attracton in quast-periodically forced systems. from quasi-periodic to strange non-chaotic to chaotic. Pllys. Rev. A. lY8Y. 39, 2593. .! I. Kapitaniak. T.. Ponce, F. and Wojewoda, J.. Route to chao\ via non-chaotic attractor\. 1. t%ys. A: Mafh.. IYYO. 23, 1,387. 22. fiolden. A. V. and Fan, Y.. C‘rlsls-induced chaos in the Rose- Hindmarsh model tor nruronal activity. (.‘hti~. .Solironc