The effects of external disturbances on the performance and chaotic behaviour of industrial FCC units

Pergamon Prmted

in Great Britam. All rights rcwrved 0960.07791Y7 $17.00 7 0.M)

PII: SO960-0779(97)00038-6

The Effects of External Disturbances on the Performance and Chaotic Behaviour of Industrial FCC Units A. E. ABASAEED Chemical Engineering

Department, KSU, P.O. Box 800, Riyadh 11421, Saudi Arabia

and S. S. E. H. ELNASHAIEt Chemical and Environmental

Engineering Department, Universiti Putra Malaysia (UPM) 43400 UPM. Serdang, Selangor. Malaysia (Accepted

3 February

1997)

Abstract-The dynamic behaviour of an industrial Type IV fluid catalytic cracking for the production of gasoline unit is investigated for a case where the air feed temperature is periodically forced. The investigation concentrates on the behaviour of the system for a case of bistability for the autonomous system with special emphasis on the effect of forcing on the periodic attractor of the autonomous system. When the centre of forcing is very close to the homoclinical termination point of the autonomous periodic attractor, period-doubling mechanism and Type 1 intermittency have been identified as the routes to chaos for this six-dimensional (6D) system. Chaotic behaviour occurs at very low forcing amplitudes which simulate small disturbances that are unavoidable in the operation of any industrial unit. While in certain ranges of the values of the forcing amplitudes the output amplitudes of the forced system are higher than their counterparts in the autonomous system, other regions show the opposite behaviour. Average gasoline yield in the bistability region for the attractor resulting from the forcing of the autonomous periodic attractor is much higher than that resulting from forcing the autonomous static attractor. This yield is very close to that obtained with the optimum steady state which is unstable and requires prohibitively high values of controller gains to be stabilized. 0 1997 Elsevier Science Ltd

1. NOMENCLATURE A,, A,, A,

= dimensionless group, A, = F,C,sI&&C,,~ F,,C,,I&fW,,, W,sI&dW,a

A, = A, =

= area of bed within clouds for regenerator and reactor, respectively (m’) Am, = amplitude of forced system Am = forcing amplitude = natural amplitude Am,,, C ,111 = concentration of gasoil in input stream (kg/d = specific heat of liquid and vaporized c,,,, c,, gas oil, respectively (kJ/kg) Cm,. Coo = dimensionless group connected with the coke balance in the reactor and regenerator, respectively, C,, = K~IAIRHRC~M, Coo = FcI&A

A,,;,

A,,

tAuthor

= air flow rate (kg/s) FA = catalyst circulation rate (kg/s) F, = gasoil flow rate (kg/s) FG G CR. Cc, = bubble flow rate through any horizontal section for reactor and regenerator. respectively (m’/s) G,,, G,,; = interstitial gas flow rate through any horizontal section for reactor and regenerator, respectively (ma/s) = height of reactor and regenerator, resf&z. Ho pectively (m) HL,, HL,,= heat loss in reactor and regenerator. respectively (so’) HRCB = normalized rate of heat absorption due to endothermic cracking reactions in the reactor (s-l)

for correspondence. 1941

A. E. ABASAEED

1942

= normalized heat of vaporization for gasoil (s-l) = controller gain = coke make rate = dimensionless reactor dense phase temperature = dimensionless regenerator dense phase temperature = dimensionless gasoil concentration = dimensionless gasoline concentration = base value for dimensionless air feed temperature = dimensionless air feed temperature = dimensionless set point reactor temperature = dimensionless reactor temperature = dimensionless gasoil vaporization temperature = catalyst retention in reactor and regenerator, respectively (kg)

H,

KC RC

YKD

Greek C-W.1

and S. S. E. H. ELNASHAIE

letters

= heat of carbon combustion in regenerator (kJ/kg)

a,, a27LYE= dimensionless pre-exponential

PC

factors for the cracking reactions, a1 = K,,,&,,,, (~2= K,,,F. a, = Km&,,, = exothermicity factor for coke combustion reaction, p, = Cm( - AH,.)/ ( r,f G,

Pa)

= dynamrc parameter in heat balance equation for regenerator = dynamic parameter for catalyst adsorp52 tion capacity for gasoil = dynamic parameter for catalyst adsorp53 tion capacity for gasoline = dynamic parameter in coke balance 54 equation for reactor = dynamic parameter in coke balance 55 equation for regenerator & = bed voidage = dimensionless forcing frequency Wf = catalyst activity within reactor +‘R = catalyst activity within regenerator ILO Pa3PI> Pr = density of air, liquid gasoil, vaporized gasoil, respectively (kg/m’) 51

2. INTRODUCTION

Fluid catalytic cracking (FCC) units for the conversion of heavy hydrocarbons such as gasoil to gasoline and light hydrocarbons are important units in the petroleum refining industry. Efficient operation of these units contributes significantly to the overall economics of the refinery. These units consist mainly of a reactor and a regenerator. The two units, reactor and regenerator, are coupled by catalyst circulation between the two vessels. In the reactor, the main cracking reactions take place. The cracking reactions are endothermic and therefore heat must be supplied to the reactor. The carbon formed on the catalyst during cracking is burnt off by air in the regenerator. This is an exothermic reaction and therefore heat must be removed from the regenerator. The catalyst as it circulates between the reactor and the regenerator removes heat from the regenerator and supplies it to the reactor, thus making the overall process mostly autothermic. Previous studies have revealed that a multiplicity of steady state phenomena exist in these units and also that FCC units are usually operated at the middle unstable steady state in order to give high yield of the desired product [l-6]. The reaction scheme used in this investigation involves the conversion of gasoil into gasoline as an intermediate product and the reaction network in its simplest form has the following parallel-consecutive structure:

Gasoil -

k,

Gasoline

k7 5 Coke+Gases

The model developed by Elshishini and Elnashaie for the above reaction scheme was very successful in representing a number of Type IV industrial FCC units and they confirmed the operation of the unit at the middle unstable saddle-type steady state [4]. The above model was extended by Elnashaie et al. to include the dynamics of gasoil as well as gasoline [7]. External disturbances are usually unavoidable in operating industrial units. These disturbances could be very detrimental with respect to the yield of the desired product and/or could lead to ignition or quenching of reactors operated in regions of multiplicity of

Industrial

FCC

units

Therefore, in this preliminary investigation, the effect of external disturbances (represented by periodic forcing of air feed temperature) on the bifurcation behaviour, product yield and temperature oscillations of an industrial Type IV FCC unit is studied. The investigation concentrates on the system behaviour when the centre of forcing is very close to a homoclinical orbit of the autonomous system. Various presentation techniques are used to investigate the periodic and chaotic behaviour of the system.

steady

states.

3. THE MODEL

The model, which was developed earlier [4,7] and successfully tested against a number of Type IV industrial FCC units is used in the present investigation. The details of the model development and its associated assumptions have been given earlier [4,6,7], but the most important features of the model are that (a) it recognizes the two-phase nature of the fluidized beds of the two interconnected vessels, reactor and regenerator, (b) it considers the heat and mass capacitances in the two vessels, (c) it uses the triangular reaction network of Weekman and Nate [I)], and (d) it uses the plant data to obtain the light hydrocarbon fraction as well as the heavy cycle oil fractions. The normalized model equations are given by the following. The reactor dense phase dimensionless temperature is given by

dY,o = B(Yv - YRIJ + A,(Yf - Yv) - Hv + HRCB + A,(Y,;,

-

dr

the regenerator

dense phase dimensionless

~dt

temperature

- YRIl) - HI>,<,

(1)

is given by

= S,(BWY,;, - YG,) + P4CB) + AUK,, - Km) - H&i)>

(2)

the gasoil mass balance in the reactor dense phase is given by

(3) the gasoline mass balance in the reactor dense phase is given by (4)

the catalyst

activity in the reactor dense phase is given by (5)

the catalyst activity in the regenerator

dense phase is given by

dlCIG x = &(CB - G,,(~c;

- (CIR)).

(6)

1944

A. E. ABASAEED

and S. S. E. H. ELNASHAIE

Table 1. Plant data for Type IV industrial FCC unit Fresh feed Recycled HCO Combined feed ratio (CFR) Air flow rate Catalyst circulation rate Combined feed temperature Hydrogen in coke (wt%) HCO/(HCO + CSO + LCO) Coke/(coke + gases) CO,/(CO, + CO) (-AK) Regenerator dimensions Reactor dimensions Catalyst retention in reactor Catalyst retention in regenerator API of raw oil feed Reactor pressure Regenerator pressure Average particle size Pore volume of catalyst Apparent bulk density Catalyst surface area

and the air feed temperature

16.782 kg/s 2.108 kg/s 1.1256 10.670 kg/s 88.605 kg/s 527 “K 4.17 0.286 0.221 0.75 (m’/m’) 31235.6 KJ/kg 5.334 m ID X 14.859 m 3.048 m ID X 12.760 m 17.5 metric tons 50 metric tons 28.7 2254938 kPa 254.8708 kPa 0.00072 m O.ooO31ma/kg 800 kg/m” 215 m’/g

for the forced closed-loop

system is given by

Y,, = Y& + Am sin(w,r) + Kc(Y,, - Y,,).

The parameters section.

and variables

in the above equations

(7)

are defined in the nomenclature

4. RESULTS AND DISCUSSION

AUT086, developed by Doedel and Kernevez [9], is used to construct the bifurcation diagrams using the industrial data given in Table 1. The set point of the proportional controller is the dense phase reactor temperature I’$,, = 1.19314 and the manipulated variable is the air feed temperature Y,,. To ensure the accuracy for the stiff differential equations of this system, the DGEAR subroutine with automatic step size is used for simulation [lo]. Because of the very high numerical sensitivity of the system, the bound on the allowable error was maintained at lo-l2 [6]. 4.1.

Bijkrcation behaviour of the autonomous system

The closed loop autonomous case is represented by eqns (l)-(7) with the value of the forcing amplitude Am in eqn (7) set equal to zero. Figure l(a) and (b) show the bifurcation diagrams for the reactor dense phase temperature Y,, and gasoline yield X,, respectively, with the proportional controller gain KC as the bifurcation parameter. Figure l(a)-1 is an enlargement of the lower branch (the lower left corner of Fig. l(a), showing branches which contain a degenerate Hopf bifurcation point denoted DHB) and Fig. l(a)-2 is an enlargement of the periodic branch emanating from the Hopf Bifurcation point HBl at K, = 2.6863. The main features of the bifurcation diagram, Fig. 1, can be summarized as follows. A region of static bistability extends from K, = 0 to K, = 0.4212, corresponding to a static limit point SLPl, which coincides with DHB as shown in Fig. l(a) and (a)-1. The stable high temperature branch in this region gives low gasoline yield (see Fig. l(b)). The

Industrial

FCC

units

(4

HB2

-

0.0

10.0

20.0 Controller

Fig.

I. Bifurcation (al-1

diagram for reactor Enlargement of lower

gain,

SSB

30.0

Kc

dense phase: (a) temperature; (h) gasoline yield against controller static branches: (a)-2 enlargement of periodic branch at HBl.

gain.

reactor dense phase temperatures for the other two branches (stable and unstable) shown in Fig. l(a)-1 are below the vaporization temperature Y, = 1.078 of gasoil. From Fig. l(b), it is clear that at K, = 0.0 (open loop) the highest gasoline yield XRd = 0.22 is obtained at the middle unstable saddle-type steady state branch. From K, = 0.4212 to K, = 1.099822 (corresponding to SLP2), a unique globally stable steady state branch exists as shown in Fig. l(a) with low gasoline yield (see Fig. l(b)). Another region of static bistability extends from K, = 1.099822 to K, = 2.6863 (corresponding to a Hopf Bifurcation point HBl). The stable periodic branch which emanates from HBl coexists with another static stable branch in the region from K, = 2.6863 to K, = 2.8235 (corresponding to the homoclinic termination point of the periodic branch as shown in Fig. l(a)-2). From K,=2.8235 to K,=3.004 (corresponding to SLP3), the high temperature steady state branch (see Fig. l(a)) which is stable coexists with two unstable static branches. For the region from K, = 3.004 to K, = 25.9423 (corresponding to a Hopf Bifurcation point HB2), there is only a stable periodic branch emanating from the Hopf Bifurcation point HB2 surrounding an unstable

I94h

A.

E. ABASAEED

and

S. S. E. H. ELNASHAIE

steady state branch. At high values of controller gains, greater than 25.9423, a unique globally stable static branch with high gasoline yield exists. The bistability between these attractors for various values of controller gains has been discussed in detail by Abasaeed et nl. [ll]. From the previous bifurcation analysis, it is clear that the highest gasoline yield (about 0.43875) is only attainable at impractically high values of proportional controller gains (K, > 25.9). However, a comparably high gasoline yield (about 0.43754) at practical values of controller gains (2.6863 < K, < 2.8235) can be achieved if the FCC unit is operated on a stable periodic branch. The following bifurcation analysis concentrates on the region in which the periodic branch emanating from the Hopf Bifurcation point HBl at K, = 2.6863 loses its stability ali it collides with the saddle at K, = 2.8235 (see Fig. l(a)-2). Since the air feed temperature Y,., will be used as the external fofcing, the bifurcation diagrams, shown in Fig. 3. arc reconstructed using &;, as the bifurcation parameter at K, = 2.8235. The bifurcation diagrams in this case reveal that a unique static steady state branch (stable branch) exists in the region yI;, < 0.8202 (corresponding to SLPl). The narrow region 0.8202 < I$, < 0.8903 (corresponding to SLP2) contains one stable branch, two unstable steady state branches and a degenerate Hopf Bifurcation point (denoted by DHB in Fig. 2(a)) at Y,:, = 0.882305. This region falls completely in a physically unrealistic region (where the temperature inside the reactor is less than the vaporization temperature of gasoil). An interesting region extends from ylZl = 0.8903 to x:., = 1.35687 (corresponding to SLP3). In this region the bursting phenomenon discussed previously by Elnashaie rf ~1. occurs [7J. In this region only the minimum values of the dense phase reactor temperature for the stable periodic branch are chown in Fig. 2(a). It is postulated that a blue-sky catastrophe has occurred. The term

-C------I

3.00

(a)

!

1.75 2.00

SLP3

3

‘r, SLP4

9 1.50 >

SLPI DHB.SLP2

I 25

Ir

1 --ftt&mg-

(b)

0.0

0.S

IO

1.5 Yfa

Fig. 2. Bifurcation for K, = 2.8235:

2.0

25

1 504

I.506

I 508

I510

Yfa

diagram for reactor dense phase: (a) temperature: (b) gasoline yield against air feed temperature (c) enlargement of periodic branch at HB: (d) variation of period with air feed temperature.

Industrial

FCC

units

6.0

4.0

Fig. 3. Time

trace

t’or dense

phase temperature

at Y,, = 1.25 showing

the bursting

phenomenon

‘blue-sky catastrophe’ describes the discontinuous appearance (birth) or disappearance (death) of Periodic attractors (limit cycles) [16]. Figure 3 shows that (for &, = 1.25) during the bursts, the reactor dense phase temperature reaches prohibitively high values (ignition) and goes down to temperatures lower than the vaporization temperature of gasoil (quenching). The figure also reveals the very high values of the period (about 13,000 s). It is clear from Fig. 2 that in the range from y, = 1.35687 to Y,;, = 1.505187 (corresponding to the homoclinic termination point of the periodic branch emanating from the Hopf Bifurcation point at yl;, = 1.509239, denoted HB in the figure), the high temperature branch is the only stable steady state branch and it coexists with two unstable static branches. This stable branch is characterized by low gasoline yields as shown in Fig. 2(b). The loss of stability of the periodic branch emanating from HB as it collides with the saddle is clear from Fig. 2(c). It is clear from Fig. 2(d) that the period of oscillations of the stable periodic branch increases drastically as the value of yt, approaches 1.505187. In the region from Y,.;,= 1.505187 to Y;, = 1.509239 (corresponding to the Hopf Bifurcation point denoted by HB) there is a bistability between the static high temperature steady state branch and the periodic branch emanating from HB. The region between yl;, = 1.509239 and Y, = 1.924444 (corresponding to SLP4) contains two stable static attractors and one unstable static attractor. The reactor dense phase temperature of the lower stable branch is well above the vaporization temperature of gasoil. The gasoline yield obtained from the lower stable branch is the highest (about 0.43875 close to the HB) while the gasoline yield obtained from the other stable branch (high temperature) is low as shown in Fig. 2(b). At values of air feed temperature higher than 1.924444, the high temperature steady state branch exists as a unique globally stable static branch, with low gasoline yield as shown in Fig. 2(b). 4.2.

B(fiucation

hehaviour

of the forced system

The air feed temperature for this case is given by eqn (7). In this preliminary investigation, the forcing frequency w, is taken as 5 times the frequency of the autonomous system W,,. The centre of forcing Y,;, = 1.505188 for the present case is chosen very close to the homoclinical termination point Y$, = 1.505187 of the periodic branch. For this value Y,, = 1.505188 of air feed temperature, the autonomous system has a bistability between static and periodic attractors. Therefore throughout the range of forcing amplitudes, bistability of the forced

193x

A. E. ABASAEED

and

S. S. E. H.

ELNASHAIE

system should be expected and the attractor obtained after the die-out of initial transients will depend upon the initial conditions of the system. The complete picture of the one parameter stroboscopic Poincare bifurcation diagram is presented in Fig. 4. The figure shows the effect of the forcing amplitude on the reactor dense phase temperature YRD. As mentioned above, since the centre of forcing yl;, = 1.SOS188 is in the bistability region of the autonomous system, it is to be expected that, depending on initial conditions, the forcing will affect either the high temperature static attractor OT the stable periodic attractor of the autonomous system. The effect of forcing on the autonomous high temperature static attractor is shown in Fig. 4(a). The attractor resulting from the forcing of the autonomous periodic attractor is shown in Fig. 4(b). The narrow region containing the five branches ui. h,, c.,5 (I, and ei in Fig. 4(b) is enlarged in Fig. 4(c) for branch c5. Figure 4(a) illustrates the effect of forcing on the bifurcation behaviour of the autonomous high temperature static attractor. Nothing very exciting takes place with regard to this attractor because periodic forcing of this static attractor results in a periodic attractor of period 1 over the entire range of investigation. For values of forcing amplitudes in the ranpc of 0.94 X 1W ’ 0.48 X lo- ‘. all initial conditions tend to take the forced system to the periodic attractor resulting from the forcing of the autonomous high temperature attractor. Figure 4(b) and (c) show the effect of forcing on the bifurcation behaviour of the autonomous periodic attractor. The very low forcing amplitude region which contains branch ci represents very small changes in the air feed temperature, i.e. the temperature changes arc in the range 0.0 to 4.7 X lo-‘“K (i.e. 0.0
Industrial

FCC

units

2.16684

2.16680

4 (b) -. 1.30

::

1.25

a5

-

.

b5

.

c5

-.

d5

_1.15

. . *....

b4

*. . . . . .

. , --a

c4

. . . . . . .

..A

d4

....‘..

**-

-

*

1.20

a4

0.0000

e5 I

,I

1 II

1 I

11 0.0002

II

I

I

I

11

I

II 0.0004

f

11

11

11

11 0.0006

Am 1.24

0.00

0.20

0.40

0.60

0.80

1.00

Am x IO5 Fig. 4. One-parameter stroboscopic Poincare bifurcation diagram for reactor dense phase temperature in the bistability region (strobing every forcing period): (a) the attractor resulting from the forcing of the autonomous static attractor; (b) the attractor resulting from the forcing of the autonomous periodic attractor; (c) enlargement for branch c5 on the left region of (b) ( centre of forcing very close to the homoclinic termination point).

1950

A. E. ABASAEED

and

(b)

S. S. E. H. ELNASHAIE

1.4 $

(4

35,000

Time,

0.42

0.43

0.44

s

X

0.45

0.46

gd

1.24

1.24 cc>

1.23

1,500,000

0

SXP,

1.21

1.22

I 23 Y,,

1.24

(n)

Fig. 5. Chaotic behaviour at Am = 0.9375 X lo-‘: (a) time trace; (b) phase plane. For branch c,: (c) stroboscopic points histogram; (d) first iterate map for reactor dense phase temperature (equivalent to fifth iterate map of the attractor resulting from the forcing of the autonomous periodic attractor).

always converges to the periodic attractor resulting from the forcing of the autonomous static attractor (see Fig. 4(a)). It is postulated here that the loss of stability of the chaotic attractor is due to a boundary crisis at Am = 0.94 X lo--” in favour of the forced high temperature attractor [12]. The term ‘boundary crisis’ is used to describe the loss of stability of chaotic attractors (for certain values of bifurcation parameter) as it collides with an unstable fixed point situated at the boundary of its basin of attraction. This situation continues until Am = 0.245 X lop3 when the attractor resulting from the forcing of the autonomous periodic attractor shows up again with period four P4. It is clear from Fig. 4(b) that at Am = 0.245 X 10-j four branches ad, hS, cq and d, suddenly emerge (P4 attractor) by a possible blue sky catastrophe. The P4 attractor undergoes period-doubling bifurcation leading to chaos similar to that observed above. A two-banded chaotic region terminates at an interior crisis point at about Am = 0.4438 X 10. ‘. Numerical simulation carried out at Am = 0.4375 X lo-’ confirmed the existence of this banded chaos. The results of these simulations are presented in Fig. 6. Figure 6(a) and (b) shows the time trace and phase plane at Am = 0.4375 X 10-j. The two-banded chaos becomes clearer when the stroboscopic points histogram of branch a4 is analysed; the histogram of dense phase reactor temperature against time is shown in Fig. 6(c), where time represents 4 times the forcing period. The appearance of the two bands is very clear from the histogram. The stroboscopic first iterate map for the dense phase temperature (see Fig. 6(d))

Industrial

FCC

units

30,000 Time,

X

s

gd

1.28

-

Cd)

1.28 (cl

+ 5

1.27

1.26 0

I ,ooo,ooo

4XPf

2.000,000

1.26

1.27 YRD

1.28 (n)

Fig. h. Two-banded chaos at Am = 0.4375 X IO -‘: (a) time trace: (h) phase plane. For branch a,: (c) stroboscopic point\ histogram. (d) first itcratc map for the reactor dense phase temperature (equivalent to fourth iterate map of the attractor resulting from the forcing of the autonomous periodic attractor).

for branch CL,clearly shows the two disconnected portions. The two-banded chaos terminates at about Am = 0.4438 X 10-j, and at higher values of forcing amplitude fully developed chaos prevails. At Am = 0.48 X lo-“, a boundary crisis occurs and the chaotic attractor resulting from the forcing of the autonomous periodic attractor loses its stability to the attractor resulting from the forcing of the autonomous high temperature static attractor. According to Pomeau and Manneville [13] chaos may develop through intermittency. Intermittency is characterized by alternate presence of laminar phases (periodic oscillations) and chaotic bursts. Three types of intermittencies (Types 1, 2 and 3) have been proposed, based on the way that the Floquet multiplier exits the unit circle [13]. To analyse the intermittency route to chaos, a narrow region containing periodic windows in Fig. 4(b) is enlarged in Fig. 7(a) for branch a4. On a global scale, Fig. 7(a) shows that, as the forcing amplitude is increased, the attractor resulting from the forcing of the autonomous periodic attractor goes through a chaotic region followed by a narrow region of periodic windows of period 3 (PW3) then a small strip of chaos followed by PW3 and finally period-doubling cascade leading to chaos. The time trace at Am = 0.4674 X 10mm3shown in Fig. 7(b) clearly shows that the periodic oscillations are interrupted by chaotic bursts. It is also clear from the stroboscopic points histogram for branch a4 that three laminar phases sandwiched inside the chaotic regime do exist (see Fig. 7(c)). Of the three types of intermittency, Type 1 (which occurs when the Floquet multiplies passes out of the unit circle through +l) can be

1951

A. E. ABASAEED 1.29

and S. S. E. H. ELNASHAIE

,

1

1.35

1.28

1.27

0.467

0.469

forcing amplitude,

0471

35,000

25,000

lo3

45,000

normalized

55.000

time

1.29

1.29

1.28 1.28

1.27

1.27 1.26

1.26 700,000

lr

1.25 750,000

4XPf

800.000

126

1.25 ‘RD4

cn)

121 ‘RD4

1.28

1.29

cn)

Fig. 7. Intermittency at Am = 0.4674 X 1W ‘: (a) enlargement of the periodic window; (b) time trace showing alternate periodic oscillations and chaotic bursts. For branch a4: (c) stroboscopic points histogram (semi-solid lines indicate iaminar phases): (d) third iterate map confirming Type 1 intermittency.

determined by plotting the ith iterate map [13,14]. For this purpose, the third iterate map for branch uJ is presented in Fig. 7(d). It is clear from the figure that the curve approaches the bisectrix tangentially at three distinct points (shown as points 1, 2 and 3 in Fig. 7(d). This confirms that, for this complex 6D system, Type 1 intermittency does exist. The third iterate map around point 2 (the middle curve tangent to the diagonal) shows that another curve crosses the diagonal. Similar type of behaviour has been observed in a simpler 3D free radical polymerization reaction [15] and is termed ‘leaves’. 4.3.

Effect of forcing on amplitudes of temperuture oscillations

The effect of the forcing amplitude on modulating the reactor dense phase temperature is also investigated. This is a very important aspect in view of our previous investigation on bursting phenomena and its negative effect on gasoline yield [7]. The results are summarized in Table 2. The forcing amplitudes Am given in Table 2 are in degrees Kelvin (K). The dimensionless forcing amplitude can easily be obtained by dividing by 500 (the reference temperature). Column 2 shows the amplitude of temperature oscillations for the attractor

Industrial FCC units

I YS?

Table 2. Effect of forcing on the amplitude of temperature oscillations Am,, and gasoline yield for the attractor resulting from the forcing of the autonomous: (a) static: (b) periodic attractors AM 0.00000 0.00050 0.00100 0.00150

0.00200 0.00250 0.00300 0.00350 0.00400 0.00450 0.0046.5 o.omoo 0.08000 0.1112s 0.12250 0.13500 0.15000 0.17soo 0.20000 0.2 I250 0.21525 0.21625 0.22500 0.23850

Am,

(a)

0.00000

0.00012 0.00012 0.00014 0.00018 0.00020 0.00024 0.00028 0.00030 0.00031 0.0003s 0.00105 0.00287 0.00523 O.OQY40 0.01038 0.01120 0.0131 I 0.01502 0.01563 0.01604 0.01634 0.01681 0.01872

Gasoline yield (a)

Am,, 0~)

Gasoline yield (b)

0.14716 0.14716 0.14716 0.14716 0.14716 0.14716 0.147 16 0.14716 0.14716 0.14716 0.14716 0.14716 0.14716 0.147 17 0.147 I7 0.14717 0.14717 0.14717 0.14717 0.14717 0.14717 0.14717 0.14717 0.147 17

37.467 37.469 37.470 37.470 37.681 37.900 38.010 38.115 39.222 40.591 41.326 unstable unstable unstable 27.461 27.463 27.465 27.46Y 28.532 29.124 29.282 2Y.341 2Y.Y65 31.1 IX

0.43754 0.43754 0.43754 0.43754 0.43754 0.43754 0.43754 0.43753 0.43753 0.43753 0.43753 unstable unstable unstable 0.43802 0.43802 0.43802 0.43803 0.43804 0.43804 0.43805 0.43805 0.43X0$ 0.43805

resulting from the forcing of the autonomous high temperature static attractor. It is clear from this column that the amplitude of oscillations increases with the increase of the forcing amplitude. An increase in the amplitude of the forced static attractor of about 0.01872 K is obtained for an increase of 0.2385 K in forcing amplitude. The results for the effect of forcing on the amplitude of temperature oscillations for the attractor resulting from forcing the autonomous periodic attractor are shown in Column 4 of Table 2. For Yr, = 1.505188, the autonomous periodic attractor (Am = 0) has an amplitude of temperature oscillations of 37.467 K. At very low values of forcing amplitudes (Am < 0.003 K), the forced system amplitudes Am Id are very close to the natural amplitude 37.467 K. However, at Am = 0.003 it is clear that the amplitude of the forced system increases significantly with increasing Am. At a value of forcing amplitude of about 0.00465 K, the attractor resulting from the forcing of the autonomous periodic attractor loses its stability due to a boundary crisis as discussed previously. However, when this attractor shows up again at a forcing amplitude value of about 0.1225 K, its amplitude of temperature oscillations (27.461 K) is clearly less than that of the autonomous periodic attractor (37.467 K). The amplitude of temperature oscillations of the resulting forced attractor increases with the increase of the forcing amplitude but never reaches the amplitude of oscillations of the autonomous periodic attractor. (The highest amplitude of temperature oscillations reached is 31.118 at Am = 0.2385 K.) At a forcing amplitude value of about 0.24 K, the attractor resulting from forcing the autonomous periodic attractor loses completely its stability, as discussed above. It is clear from the above analysis that the amplitude of temperature oscillations for the attractor resulting from the forcing of the autonomous periodic attractor is higher than that of the autonomous periodic attractor for 0.0
A. E. ABASAEED

I953

4.4.

IZfect

o,fforcing

and

S. S. E. H. ELNASHAIE

on the gasoline yield

The results of the effect of forcing on gasoline yield are shown in Column 3 of Table 2 for the attractor resulting from the forcing of the autonomous static attractor and in Column 5 for the attractor resulting from the forcing of the autonomous periodic attractor. It is clear from Column 3 that the gasoline yield obtained with the attractor resulting from the forcing of the static attractor is insignificantly affected by forcing in the low range of forcing amplitude values considered in this preliminary investigation. The gasoline yield obtained with the attractor resulting from the forcing of the autonomous periodic attractor remains about the same for 0.0
In this paper, the mathematical model developed earlier and successfully tested against a number of industrial FCC units is used to investigate the effect of periodic forcing of air feed temperature on these units. The analysis is performed in a region where the autonomous case reveals bistability between a high temperature static attractor and a periodic attractor. It has been established that the autonomous periodic attractor loses its stability at very low values of forcing amplitudes when the centre of forcing is extremely close to the homoclinic termination point. It has also been found that chaotic behaviour prevails in these units when subjected to very small perturbation of the air feed temperature. Period-doubling sequence is found to be the dominant route to chaotic behaviour, but also Type 1 intermittency, blue-sky catastrophe and boundary crisis are identified as the mechanisms for the changes from periodicity to aperiodicity. In certain regions of forcing amplitude values, the amplitude of the forced system has been found to be larger than that of the natural system, whereas in other regions the reverse is true. The attractor resulting from the forcing of the autonomous periodic attractor gives a slightly higher average gasoline yield (0.43805) compared with that obtained with the autonomous periodic attractor (0.43754). REFERENCES I. Iscol.

L.. The dynamics and stability of a fluid catalytic cracker. In Proceerling~ of the Aukmnfir Control Atlanta. Georgia, 1970. pp. 602607. 2. Elnashaie. S. S. E. H. and El-Hennawi. 1. M., Multiplicity of the steady states in Huidized bed reactors IV. Fluid catalytic cracking. C/rem. Engng SC;.. 1979, 34, 1113-1121. 3. Edwards, W. M. and Kim, H. N., Multiple steady states in Ruidized bed FCC unit operation. Chrm. Engng Sd.. 1988. 37, I61 1-1623. Confmwcr.

Industrial

FCC

units

IYSS

4. Elshishini. S. S. and Elnashaie, S. S. E. H.. Digital simulation of industrial fluid catalytic cracking units I. Bifurcation and its implications. Chrm. Engng .Sci.. 1990, 45, 553-559. 5. Arandes. J. M. and De Lasa, H. I., Simulation and multiplicity of the steady states in Ruidized FCCl!s. Chem. Engng SC;.. 1902. 47, 2535-2540. 6. Elnashaie. S. S. E. H. and Elshishini, S. S., Digital simulation of industrial fluid catalytic cracking units IV. Dynamic behaviour. Chem. Engng Sci., I993 48,567-583. 7. 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. Chern. Engng Sci., 1995, 50, 1635-1644. X. Weekman. V. W. and Nate. D. M., Kinetics of catalytic cracking selectivity in fixed, moving and fluid bed reactors. A IChE J.. 1970. 16, 397-404. Y. Doedcl. E. J. and Kernevez. J. P., Auto: Soffware for Conhudon and Bifurcarion Prohlrrns in Ordinaq Diffcvwrtial Eg~utinns. California Institute of Technology, Pasedena, California. lY86. IO. Gear. C. W.. Nutwrical Inirial Value Problems in Ordinary Di,fferential Equations. PrenticeeHall, Ncv, Jersey. 1’)71. Il. Ahasaeed, A. E.. Elnashaic. S. S. E. H.. Elshishini, S. S. and Almutlaq. A.. Static and dynamic bifurcation brhawour of an industrial fluid catalytic cracking unit. In Procerdin~s of‘ the Fowth Saudi Enginrc+n,y ( onfercwcr. Vol. V. Jeddah. Saudi Arabia. 1995, DD. 233-230. 12. Marck. M. and Schreiber, I., Chaotic Brhauiour’b,f Dekwninistic Di.wipntiur Systenzs. Cambridge Universit> Press, Cambridge. 190 I. p. 79. I?. Pomcau, Y. and Manneville, P., Intermittent transition lo turhulencc in dissipative dynamical systems. ( omn~,,n. Marh. Phvs., 1980. 74, 189%197. 14. Tambc. S. S. and Kblkarni, B. D.. Intermittency route to chaos in a periodically forced model reaction system. (‘hem. Engng Sci.. IYY3. 48, 2X17-2821. 15. ‘I cymour, F. A.. The dynamic behavior of free radical solution polymerization reactions in a continuous stirred tank reactor. Ph.D. thesis, University of Wisconsin. Madison, 19x9. 16. Jackson. E. A.. Pcwprc~tiur.\ of Non/&war Dynamics. Vol. I. Cambridge LJniversity Press. Cambridge. 1YXY. p. 2Yh.