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Approach to the analysis of the dynamics of industrial FCC units
J. Proc. Cont. Vol. 8, No. 2, pp. 89±100, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959-1524/98 $19.00 + 0.00
PII: S0959-1524(97)00030-9
Approach to the analysis of the dynamics of industrial FCC units Rafael Maya-Yescas,*y David Bogley{ and Felipe Lo¨pez-Isunza* ¨ de¨ Procesos, Universidad Auto¨noma Metropolitana - Iztapalapa, *Departamento de Ingenieria ¨ Av. Michoaca¨n y Purisima, s/n. Me¨xico 09850 D.F. Me¨xico y Department of Chemical and Biochemical Engineering, University College, London, UK *
Although it is known that FCC units are operated at a controllable steady state, it is commonly assumed that this state is pseudostable. An ignited steady state, theoretically stable, outside the operable domain has also been reported for the same set of operating parameters. The aim of this work is to introduce a theoretical method for the preliminary analysis of the controllability of an industrial FCC unit, at any set of conditions, using a nonlinear model directly. An approach for the analysis of the stability of the system zero dynamics is used to investigate the dynamic behaviour of two common operating policies when the unit is operating either in the standard or in the ignited steady state. It is used to predict some particular characteristics of the process, such as the presence of inverse response after the change in catalyst circulation rate, a common control action. The results derived from the analysis developed are discussed in terms of physical responses and veri®ed by performing open loop dynamic simulations of an industrial FCC unit. The operation of the unit is simulated for both the standard and the ignited steady states. # 1998 Elsevier Science Ltd. All rights reserved Keywords: FCC; mathematical modelling; nonlinear dynamics; open loop simulation; stability; steady state multiplicity; zero dynamics
limitation, there has been a trend to explore the dynamics of nonlinear models. Arbel and co-workers8 present a good review of this subject. In this paper a novel technique to analyse the dynamic behaviour of a nonlinear model for FCC units is given, taking into account some dynamic particularities such as the stability of the zero dynamics, which yields a preliminary study of the controllability of the nonlinear system. One of the earliest nonlinear fundamental models, and also an optimal control scheme, was due to Kurihara9. This model proposes that some FCC units can be controlled by controlling the regenerator only. This assumption is currently represented by considering the reactor (either a riser or a ¯uidised bed) operating under pseudo-steady state conditions. A problem associated with this scheme is that it is not considered as acceptable to the operator. However, it has been pointed out that the same control structure can be reformulated in terms of riser variables4. Hence the assumption of a pseudo-steady riser will be used in this work. An interesting feature of FCC units is that they are adiabatic systems. Due to the moderate endothermic cracking reactions, some heat supply is needed for the riser. This energy is provided by the coke combustion in the regenerator. The vehicle for energy transfer is the solid catalyst. This coupling in energy management
The production of gasoline by ¯uid-bed catalytic cracking (FCC) is one of the most common catalytic processes in the world. The interest in studying its dynamic behaviour is motivated by two main aspects: for academic researchers the complex and highly interacting nature of the reactor±regenerator system, and for industry because of the strong economic incentives associated with the unit. The characterisation of the dynamics of FCC units is dicult. For example, although several authors1±3 have proposed that a FCC unit can exhibit more than one theoretical steady state, there is no fully accepted criterion to con®rm this characteristic3,4. However, there are some available models which consider, in an explicit way, the possibility of multiple steady states arising within the constrained operating zone5. Commercial FCC units are controllable, despite the so called `pseudostability' of the operating steady state3,6. The analysis of the dynamics around this steady state has been explored mainly in the linear sense7 because of the complexity of the unit behaviour, although this approach is limited to a small region around the steady state. Recently, in order to avoid this { To whom correspondence should be addressed. Fax: +44 171 383 2348; e-mail: [email protected]
89
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Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
makes the FCC unit a complex and interesting system. A second important point of view is that the unit is exposed to disturbances in feed composition and temperature, which change the rate of coke deposition. These problems have to be resolved by managing the energy balance changing either the catalyst circulation rate or the ¯ow rate of the air supplied to the regenerator. Theoretically, both actions are able to change the operating steady state10. Moreover, the constraints making up the operable region boundaries also add to the problem of the control of FCC units. Huq et al. suggest some design and operational modi®cations to the model IV FCC unit to improve control performance, in particular by reducing ¯uctuations in the catalyst circulation rate. They report that this FCC unit is ill-conditioned which causes some of the control problems11. Another interesting point is that there is the possibility of catalyst deactivation in both reversible and irreversible ways. In this paper irreversible deactivation was neglected. The expression for reversible deactivation is given as a function of the gasoil mass fraction (hfo ), a critical mass fraction which can cause full catalyst deactivation (!c1 ) and an empirical factor ( ) related to the `coking ability' of the catalyst (Equation (1)). Incorporating this into the model allows the prediction of some dynamic features, such as the possibility of unit quenching if the catalyst becomes fully deactivated. hfo jz ÿ 1 1 jz !c1 ÿ hfo jz The precise kinetic scheme of the cracking reactions is unknown. However, there have been several propositions to represent it. One of the earliest schemes, which is currently used, considers the FCC reactions by lumping them into three dierent groups: gasoil, gasoline and coke12. Although this model has been very useful, it is accepted that it is not accurate8. In this work, a ®ve lump model which considers heavy gasoil, light gasoil, gasoline, volatile gases and coke is used (Figure 1). A comparison of the predicted mass ¯ows at the unit outlet when using this kinetic scheme and industrial data obtained from an industrial unit which processes 40 000 BBL dayÿ1 is shown for three standard operating cases (Figure 2)13. This paper is divided into the following sections. First, the development of the model used is presented; then two steady state solutions are obtained for the model; followed by the development of a novel technique
Figure 2 Comparison of mass ¯ows at the FCC unit outlet
to analyse the zero dynamics of the nonlinear model directly, analysing two important operating policies. The next section is devoted to the open loop dynamic simulations of the FCC unit: two dierent kinds of perturbations are considered and the system responses are brie¯y explained in terms of the physical characteristics of the system. The ®nal two sections concern, ®rst, a discussion of results and the comparison to previous related works followed by the conclusions obtained from this work.
A model for the FCC unit In order to analyse some aspects of the controllability of the FCC unit, a simple nonlinear model is proposed. The riser is eectively a steady state, transported and heterogeneous bed reactor. Some of the riser model equations are reported here; a more complete development can be found in Refs 14 and 15. Dierential mass balances for four of the proposed lumps and energy balances for each phase were developed (Equation (2) and Equation (3)) as well as an algebraic balance for coke, which is the closure condition of the system (no creation of mass). In order to incorporate the possible molar balance, molecular weights of each lump except coke, and stoichiometric coecients are proposed as well as a change in the void fraction and density of the reacting mixture, which are also simulated. This implies that the velocity varies along the riser. It is assumed that both phases are moving in plug ¯ow. Second order kinetics are considered for all the gasoil reactions and ®rst order kinetics for the others16. Because all the reactions that take place in the riser are catalytic, kinetic terms are not included in the balances for the gaseous phase. All these reactions are endothermic. Typical balance for the gaseous phase in the riser: ug @Yg 1 ÿ "1 L @z "1
v Yg
ÿ Ys 0
2
Typical balance for the vicinity and surface of the catalyst in the riser: us @Ys ÿ v Yg ÿ Ys Rr 3 L @z
Figure 1
Kinetic scheme
Due to the nature of the riser reactor, both phases moving in plug ¯ow and only endothermic reactions taking place, phenomena such as bifurcations or steady
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. state multiplicity are not possible8 so this type of dynamic behaviour will occur only in the regenerator model. Instantaneous evaporation of the gasoil supplied to the riser inlet is considered and the catalyst steaming process is neglected. Both reactors are connected by stand pipes which are modelled as time delays. The regenerator is considered as a dynamic CSTR, which consists of bubble and emulsion phases; the former at quasi steady state and the latter exhibiting dynamic behaviour. The model was developed by the classical application of mass and energy balances. The kinetic scheme proposed by Krishna and Parkin17 is used. First order kinetics with respect to the oxygen concentration and constant concentration of coke18 are assumed for the heterogeneous non-catalytic burning of coke. The dynamic mathematical model of the regenerator forms a coupled set of algebraic and dierential equations. The algebraic equations are related to the evaluation of physical properties and stoichiometric coecients as well as mass and energy balances for the bubble phase, which are of the same form as the emulsion phase but with the dynamic term equated to zero. The emulsion phase model, a set of ®rst order dierential equations, consists of mass balances for oxygen (Equation (4)), carbon monoxide (Equation (5)), coke mass fraction (Equation (6)), catalyst hold-up (Equation (7)) and the energy balance (Equation (8)). The contribution of the sensible enthalpy of the gas in the emulsion phase was neglected because the solid fraction terms are larger, by at least two orders of magnitude. In order to have explicit appearance of the time constants, the model is presented in dimensionless form (see nomenclature list for the de®nition of these variables). All the balances are subject to initial conditions given by considering that at time zero the system is operating at the desired steady state. d o2e fe vi o2ei ÿ v o2e d ÿ o2 Dax eÿ x=e o2e
m o2b
ÿ o2e
4
d coe ÿfe v coe m cob ÿ coe d coDax eÿ x=e o2e
5
d!cok Mi !coki ÿ M !cok ÿ e Dax eÿ x=e o2e d
6
d" Mi ÿ M " d
7
de Mi ei ÿ M e h b ÿ e d ÿ x Dax eÿ x=e o2e
8
Now, by de®ning the following vectors: xT o2e ; coe ; !cok ; "; e = regenerator model variables, and uT Mi ; M ; Vi = manipulable input variables, the dynamic part of the model can be transformed into a control ane system (Equation (9)), where G is a rectangular matrix (Equation (10)) which represents the
91
variables directly in¯uenced by the manipulated inputs u, and f is the vector which consists of the transport and reaction terms of the model equations. Control ane systems are a special case of control systems whose main characteristic is that the vector of manipulable variables appears linearly in the dynamic model19. The determination of the dimension of the matrix G is not discussed here, but it is related to the physical system constraints20. This model is used in the analysis of the system dynamics. x_ f x G x u 0
0 B 0 B GB B !coki @ " ei
0 0 ÿ!cok ÿ" ÿe
1 fe o2ei ÿ fe o2e C ÿfe coe C C 0 C A 0 0
9
10
Steady state solutions When the model is solved using the data of an industrial FCC unit, which processes 40 000 BBL dayÿ1 13, it is possible to ®nd three solutions for the energy balance at steady state; two of them are stable21. These steady states are in¯uenced by the riser operation because of the heat removal by the endothermic reactions and the gasoil evaporation3,21. One of the stable states is not of interest because it represents the quenching of the system. The other two steady state regions were calculated by the simulation of the whole unit (Figure 3). The standard operating point was found to be the socalled pseudostable steady state. The existence of the ignited steady state, which exhibits temperatures above the standard operating conditions, has been reported previously1,10,21. It is considered to be stable3,6,21. The temperature when operating at this ignited steady state depends on the catalyst history, as predicted previously10, and interpreted as a bifurcation of the steady state solution. In the next section a method to analyse the system behaviour using the system's zero dynamics in both steady states, standard and ignited, is presented. Two dierent operating policies, commonly applied to industrial FCC units are analysed. Both of them consider the control of catalyst temperature and coke fraction because they aect the riser directly and hence they are usually controlled variables. In order to avoid the afterburning
Figure 3
Steady state multiplicity at regenerator and operating point
92
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
reaction, the other control variable is either carbon monoxide concentration or oxygen concentration.
Analysis of the zero dynamics In order to identify the so called pseudostability of the operating steady state, the zero dynamics of the system must ®rst be characterised. The analysis is performed by using direct, when possible, and indirect Lyapunov methods, in order to predict the possibility of inverse response behaviour. The control ane model of the system is used. The zero dynamics of a system are the dynamics of its inverse22,23. In a linear system, the zero dynamics contain the same information as the zeros of the transfer function. If the model contains Right-Half-Plane (RHP) zeros, the zero dynamics contain RHP poles. These poles aect the system responses in various dierent ways, such as blockage of the response to some exponential inputs or causing inverse response following disturbances to some inputs24. For nonlinear systems it is not possible to de®ne a transfer function, but it is possible to ®nd the zero dynamics. If the system is control ane, the stability features of the zero dynamics can be characterised25 for SISO systems. The idea is to follow the dynamics of an uncontrolled variable when the system is governed by the control of another variable. Because the controlled variable is assumed to remain constant, an ideal vector of control actions can be evaluated, and substituted into the model for the uncontrolled variable. If the evolution of the dynamic behaviour of the uncontrolled variable is not stable, it is possible to conclude that the zero dynamics also are unstable, which is equivalent for a linear system to the presence of a RHP-pole. It is important to note that the evaluation of the control vector is independent of the control algorithm used. Because the internal stability of the closed loop system will depend on the stability of the unforced zero dynamics, a nonlinear stability analysis of these dynamics can be used to determine the spatial region where the dynamics are stable and therefore the closed loop system will be stable26. By following this reasoning, in this work an extension to MIMO systems of the procedure by Trickett25 is presented. It is important to notice that the direct use of a nonlinear model enables the analysis to be applicable to the whole operating region in contrast to a linearised model which is limited to a vicinity around the point used to linearise the model. In order to explore if the dynamic features predicted are a peculiarity of the selected set of control variables, two dierent operating policies are analysed. Analysis of the ®rst operating policy In the ®rst case study, a controller should be able to maintain three controlled variables at their set-point values: oxygen concentration, coke mass fraction and
emulsion temperature. CO concentration and catalyst hold-up dynamics will be controlled within a region around the desired operating state. The objective of this analysis is to anticipate when this surface is de®ned by stable zero dynamics. In order to evaluate the controlling vector necessary to perform this operation, the equation set is divided into two groups (Equation (11)), the ®rst one representing the controlled variables xs o2e , !cok, e) and the second one representing the uncontrolled variables xd coe , "). The vector f and the matrix G are divided according to xs and xd . x_ s f s x Gs x us x_ d f d x Gd x us
11
By allowing the control actions to maintain the time derivatives of the controlled variables at zero it is possible to obtain the steady state system (de®ned by Equation (12)). 0 1 0 1 0 o2e dB C B C @ !cok A @ 0 A d e 0 0 1 ÿ x=es o2e m o2b ÿ o2e ÿ o2 Dax e B C @ ÿe Dax eÿ x=es o2e A 12 0
h b
0 B @ !coki
ÿ e x Dax eÿ x=es o2e 1 0 fe o2ei ÿ fe o2es C ÿ!coks 0 A us
ei
ÿes
0
Since the matrix Gs is not singular, it is possible to calculate its inverse (Equation (13)), and rearranging the steady state expressions it is possible to obtain an expression for the vector of manipulable variables us (Equation (14)) Gÿ1 s 0 B B @
ÿ es !coki!coks ÿei !coks
0
es es !coki ÿei !coks ei es !coki ÿei !coks
ÿ fe o2ei1ÿ o2es
0
0
0
1
C coki C ÿ es !coki!ÿe i !coks A 13
us 0(
1 es !coki ÿei !coks
!coks
h b
ÿ es
)1
C B ÿ x=es B o2es C B ( es e !coks x Dax e )C C B 1 C B es !coki ÿe ! b ÿ e coki h s i !coks C B C B ei e !coki x Dax eÿ x=es o2es C B @ h iA o2b ÿ o2es o2es 1 ÿ x=es m o2ei ÿ o2es ÿ o2 Dax e o2ei ÿ o2es fe 14 Now, by substituting us in the mass balance for CO and catalyst hold-up, it is possible to predict the internal dynamics when operating under this control policy (Equation (15)).
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. d coe D d " 91 08 h i o2b ÿ o2es > > = < m cob ÿ 1 o co e 2ei ÿ o2es C B h i C B B> > co ÿ x=es e ;C C B : Dax e o2ei ÿ o2es o2 co o2es C B C B 9 C B8 " C B > es !coqi ÿe ! ÿ ! b ÿ e coqs coqi h s > i !coqs =C B< C B f es ÿ ei e A @ > > ; : ÿ x=es !coks ÿ !coqi x gDax e o2es 15 In this particular case, the system obtained is a linear one, so it is possible to transform it into canonical form, where yT coe " and cT m cob co Dax eÿ x=es o2es 0. The characteristic matrix A (Equations (16)) is diagonal, so its eigenvalues are its elements. o2b ÿ o2es A11 ÿ m 1 o2ei ÿ o2es 16a o2es o2 Dax eÿ x=es o2ei ÿ o2es A12 0 A21
16b
A22 D2 ="
16c
The evaluation of the eigenvalues of A for both steady states are shown in Table 1. This demonstrates that for the standard operating state the linear zero dynamics contain an RHP-pole26 and hence are not stable. Therefore, if this system is disturbed by using a step function, it may present inverse response24. In the case of the ignited state, the zero dynamics of the system do not contain RHP poles, therefore are stable and inverse response is not expected after a step disturbance. Analysis of the second operating policy By following a similar procedure the second operating policy, when the carbon monoxide concentration is controlled and oxygen concentration is `free', was analysed. In this case, the equation set is divided into a group representing the controlled variables xs coe ; !cok ; e; and another one representing the uncontrolled variables xd o2e ; ". Now, the matrix Gs consists of the second, third and ®fth rows of the original matrix G (Equation (10)), and it is non-singular, so its inverse can be obtained (Equation (17)). 0 1 es coks 0 ÿ es !coki!ÿe es !coki ÿei !coks i !coks B C ei coki ÿ es !coki!ÿe @ 0 Gÿ1 A s es !coki ÿei !coks i !coks 1 ÿ fe co 0 0 es 17 Table 1 Eigenvalues of the internal dynamics
First eigenvalue Second eigenvalue
Standard steady state
Ignited steady state
ÿ44.4327 +1.2417
ÿ35.3030 ÿ1.0770
93
Now, from Equation (11) it is possible to calculate the ideal control vector (Equation (18)): )1 0( 1 es !coki ÿei !coks !coks h b ÿ es C B B es e !coks x Dax eÿ x=es o2e C C B C B( ) C B 1 ! b ÿ e C s coki h us B ! ÿe ! e s coki i coks C B C B ÿ x=e s B ei e !coki x Dax e o2e C C B @ h i A co o 1 ÿ x=es 2e b m coes ÿ 1 co Dax e fe coes 18 By substituting us into the expression for the mass balances for oxygen and the catalyst hold-up, it is possible to obtain the equations which represent the internal dynamics of the system: d o2e d " 8 h i91 0 cob cob < = m o2b ÿ 2 ÿ coes o2e 1 ÿ coe o2ei C B C B: B ÿDax eÿ x=es o2 co o2ei o2e ÿ co o22e ; C C B B 8 9 C " C B > B < es !coki ÿei !coks !coks ÿ !coki h b ÿ es > = C C B A @ f es ÿ ei e > > : ; ÿ x=es !coks ÿ !coki x gDax e o2e 19 In this case, both equations obtained are nonlinear. In the ®rst one, the oxygen concentration is squared and in the second one there is a cross-product between the two analysed states. In order to test the local stability of the internal dynamics (only the operating region is of interest, i.e. 0 o2e1 and 0.4"0.7), it is possible to apply Lyapunov's direct method around each steady state. In order to ®nd a suitable Lyapunov function, Krasovskii's method19 is used here. Because these methods analyse stability around the origin, the model of the zero dynamics is written in terms of deviation variables (Equation (20)) around the steady state (Equation (21)). o2 o2e ÿ o2es
20a
" " ÿ "s 20b ! _ 2 o _ " 9 8 1 0 > > cob > > ÿ 2 ÿ o o o > > m 2b 2 2es > > coes C B > > > > C B > > > > C B = < C B cob ÿ x=e s C B 1 ÿ coe o2ei ÿ Dax e C B > > > > C B > > > > C B > > o o o > > o2 co 2ei 2 2es C B > > > > C B ; : 2 C B o o ÿ co 2 2es C B 9C B8 ""es C B > e ! ÿe ! !coks ÿ !coki h b ÿ es > =C B < s coki i coks C B A @ es ÿ ei e > > ; : ÿ x=es !coks ÿ !coki x Dax e o2 o2es 21
94
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
Following from Krasovskii's theorem, to ensure stability it is sucient to show that the matrix de®ned by ÿF J JT , where J is the Jacobian for the system, is positive de®nite at the origin, and, if this is the case, V o2 , ")T(o2 , ") is a suitable Lyapunov function. For this particular system, F and its determinant were computed (Equations (22) and (23)). ! @f1 @f2 ÿ2 @o @o2 2 22 F @f2 @f2 ÿ @o ÿ2 @" 2
@f1 F 4 @o2
@f2 @f2 ÿ @" @o2
2
23
Now, following Silvester's theorem, to ensure that F is positive de®nite it is sucient to show that its ®rst minor (F11) and its determinant are positive around the origin. In order to evaluate these terms the model derivatives were calculated (Equations (24)±(26)). @f1 cob ÿ m 2ÿ ÿ Dax eÿ x=es @o2 coe 24 o2 co o2ei 2co o2 o2es h @f2 " "s es ÿ ei e @o2 es !coki ÿ ei !coks i !coks ÿ !coki x Dax eÿ x=es @f2 1 @" es !coki ÿ ei !coks n !coks ÿ !coki h b ÿ es
25
26
es ÿ ei e !coks ÿ !coki x o Dax eÿ x=es o2 o2es From the results of the dynamic simulations (shown below), at the ignited state the following results (Equations (27)±(30)) were obtained from Equations (23)±(26): @f1 ÿ4:224 105 ÿ 3:345 105 o2 @o2
27
@f2 1:816 ÿ 4:432 103 " @o2
28
@f2 1:757 10ÿ2 ÿ 2:407 104 o2 @"
29
F 3:221 1010 o2 4:268 1010 o2 ÿ 1:964 107 "2 8:048" 103 2:968 104 30 At the origin of the co-ordinates both F11 and F are positive, therefore the zero dynamics are stable at the ignited steady state.
Now, by evaluating the derivatives and the matrix determinant (Equations (23)±(26)) at the standard steady state the following results are obtained (Equations (31)±(34)): @f1 ÿ3:869 105 ÿ 3:036 105 o2 @o2
31
@f2 ÿ274:395 ÿ 669:583" @o2
32
@f2 1:282 ÿ 685:970o2 @"
33
F 8:332 109 o22 1:060 109 o2 ÿ 4:483 105 "2 ÿ 3:675 105 " ÿ 2:059 106 34 At the origin of the coordinates the minor F11>0 while F < 0. Therefore, it is not possible to use the function obtained to check the stability of that point. This conclusion has been obtained for some SISO systems which present second order zero dynamics25 and the same case is manifested here for a MIMO system. Since Lyapunov's direct method does not provide results, in order to apply Lyapunov's indirect method the equations of the zero dynamics were linearised around the steady state. In order to obtain an accurate result, in the case of a squared expression, as in the oxygen mass balance, it has been proposed that the model be factorised and linearised by using a Taylor series around one of its roots, which represents the operating steady state25. For the ®rst equation of the zero dynamics the proposed procedure leads to the following expression: p! d o2e ÿ 2 ÿ 4 ' o2e ÿ d 2 35 p! 2 ÿ ÿ 4 ' o2e ÿ 2 where co Dax eÿ x=es ; cob ÿ m 2ÿ ÿ o2 co o2ei Dax eÿ x=es ; coes cob o2ei ' m o2b 1 ÿ coes The catalyst hold-up equation is linearised using a Taylor's series expansion around the same steady state (Equation (36)). d" a b o2es " b"s o2e ÿ b"s o2es d where
36
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
a
!coks ÿ !coki h b ÿ es ; es !coki ÿ ei !coks
Table 2 Eigenvalues of the internal dynamics
First eigenvalue Second eigenvalue
es ÿ ei e !coks ÿ !coki x Dax eÿ x=es : b es !coki ÿ ei !coks Now, the system can be written in canonical form (Equation (37)), where A is a lower triangular matrix (Equation (38)). In this way, its eigenvalues are located on the diagonal. dy Ay c d
37
where yT o2e "; cob o cT o 1 ÿ m 2b 2ei coes
ÿ co Dax eÿ x=es o22es ÿ b"s o2es ;
9 08 cob > > ÿ m 2 ÿ co > > = < ei B B ÿo o ÿ 2 o 2 2ei 2es > > AB > B> ; @ : Dax eÿ x=es b"s
95
Standard steady state
Ignited steady state
ÿ864.5953 +1.9717
ÿ352.3270 ÿ1.8781
sidered to be zero time) and then the selected disturbance was applied. Kinetics and deactivation functions were used as proposed in previous sections. The four dynamic cases simulated are listed below (Table 3) which are common control actions. The aim is to explore the behaviour predicted by the model and to compare it with the proposed analysis. Explanations in terms of physical characteristics of the dynamic behaviour are given. Due to the instability of the zero dynamics at the standard steady state, inverse response is expected when the system is operating close to this state. In contrast, inverse responses are not expected when the unit is operating at the ignited steady state because of the stability of the zero dynamics. Changes in catalyst circulation rate
1 0
C C C C A
a b o2es 38
The results obtained when evaluating the eigenvalues of the matrix A for both states are shown in Table 2. The linearised internal dynamics of the simulated system have a RHP pole at the standard operating steady state, therefore the zero dynamics are not stable at this state. For the ignited state there are no RHP poles in the linearised internal dynamics, therefore the zero dynamics are stable at the ignited steady state. The results obtained by using the approach proposed yielded that the system zero dynamics are not stable at the standard operating state. In contrast, at the ignited steady state it was found that the zero dynamics are stable. In this way, it is expected that if the unit is disturbed by a step change when operating around the standard steady state it may exhibit inverse response. However, if the unit is operating at the ignited steady state, no inverse response behaviour is expected for the same situation. The next section is devoted to presenting and explaining the results obtained for open loop dynamic simulations of the FCC unit.
When a FCC unit is disturbed by changing its catalyst circulation rate, the net eect is to alter the energy balance in both reactors, riser and regenerator. This situation arises because the solid is the vehicle for the transfer of energy between the reactors. A reduction in the catalyst circulation rate increases the energy hold-up in the regenerator. In this way, there is a larger amount of energy, supplied by the coke burning at the catalyst surface, available to enhance any chemical reaction. So, if the coke production is large enough, the regenerator warms up after the perturbation. Simultaneously, for the riser, a reduction in catalyst circulation is manifested as a smaller amount of energy supplied. However, the situation changes when the catalyst starts to become hotter due to the change in the temperature of the regenerator, compensating the ®rst eect. In this way, some minutes after the disturbance has been applied, there is a change in the global behaviour, a situation that could give rise to inverse response. The response of coke production when a decrease in catalyst circulation rate is applied shows inverse response when the disturbance is introduced (Figure 4). This situation arises because of the enhancement of coke burning in the regenerator followed by an increase in coke production in the riser. Its eect on the coke ¯ow response is not so evident because of the fast increase in the regenerator temperature, and consequently, the coke is completely depleted and the riser Table 3 Operating case studies
Dynamic simulations of the FCC unit In order to verify the results found in the analysis of the zero dynamics, responses for open loop mode dynamics were simulated after the unit had been exposed to different kinds of disturbances. In all the cases, the system was simulated until the steady state was reached (con-
1. A 3% step decrease of catalyst circulation rate and re-establishment of the original steady state conditions after 120 min 2. A 3% step increase of catalyst circulation rate and re-establishment of the original steady state conditions after 120 min15 3. A step increase of 5% in supplied air for 60 min and re-establishment of the original steady state conditions 4. A step decrease of 5% in supplied air for 60 min and re-establishment of the original steady state conditions
96
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
temperature is also increased. The coke elimination is re¯ected also by a fall in the coke present at the riser outlet which is due to a decrease in coke recirculation rather than a drop in its production. When the disturbance is removed after 120 min, as expected inverse response behaviour is not observed because the regenerator has been shifted to the ignited steady state, which has stable zero dynamics. The existence of the ignited steady state and the possibility to reach it after a decrease and re-establishment in catalyst circulation rate has been reported previously10. Similarly, Arbel et al.8,21 analysed the same problem and also found inverse response after a step change in catalyst circulation rate, although their data seems to correspond to temperatures and conversions of the ignited steady state. In the case of an increase in catalyst circulation rate, there is an increase in the amount of energy ¯owing between reactor and regenerator, but the energy holdup in the regenerator is decreased proportionally to the catalyst residence time. As a result there is a smaller speci®c amount of energy available to promote the regeneration reactions. At the same time, in the riser there is a larger ¯ow of energy, a situation that improves gasoil evaporation and reacting conditions, which results in a cooling of the catalyst. However, coke production is also enhanced accelerating the catalyst deactivation and stopping the reactions. Due to these opposing eects, the behaviour after an increment in catalyst ¯ow between the reactors is not obvious: even if the coke production is higher, its burning rate could drop, decreasing the source of energy in the unit. This larger amount of deposited coke can either heat the unit for `good' reaction conditions or take the unit to a quenching state, which is known as `snowball' quenching3, and is another reason to expect a non-intuitive response to this action. Although the system is operating near the standard steady state, when an increase in catalyst circulation rate is introduced (Figure 5) inverse response behaviour is not visible. This situation can arise if dynamic phe-
nomena are combined, similar in the linear sense to a pole-zero cancellation25. Due to the availability of more energy, which is supplied by the hot catalyst coming from the regenerator, the riser temperature is increased. As can be seen, yield to gasoline follows the same trend of the riser temperature and the yield of coke is also enhanced. Because the catalyst activity behaviour is opposed to the coke accumulation, eventually the global conversion of gasoil falls as coke on the catalyst increases. The regenerator temperature is increased because of the enhancement in coke combustion. In this way, the coke burning rate also increases resulting in a slightly falling trend for the coke hold-up at the regenerator. In the riser, coke production continues stopping the favourable situation of possible high gasoil conversion because of the higher catalyst deactivation. When original operating conditions are re-established, after 120 min, the riser temperature experiences an instantaneous drop which, for the values considered in the simulation, is suciently large to produce the quenching of the unit. This is evidence of the lack of stability of the operating steady state. It can also be seen that at the same time inverse response behaviour is shown for coke ¯ow and gasoil conversion, as predicted by the analysis, because the unit continues operating around the standard state. The regenerator temperature falls quickly following the trend established by the riser. Due to this situation, coke burning in the regenerator is insucient and a higher amount of coke on the catalyst surface is recirculated to the riser, stopping the process. This phenomenon ®nally takes the process to the coldest steady state. In both previously simulated cases two important characteristics of this system have been established. First, the dynamics of the FCC unit are strongly dependent on catalyst hold-up and, consequently, on coke hold-up. These variables have a strong in¯uence on the regenerator energy balance, which dictates the energy hold-up for the whole unit. Because of this situation, the riser follows the conditions determined by the regenerator. Second, the eect of the manipulation
Figure 4
Figure 5 System response to an increase in catalyst circulation rate
System response to a decrease in catalyst circulation rate
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. of the catalyst circulation ¯ow as a control action is not trivial because the system can present dierent dynamic responses depending on its coke content and on the residence time of the catalyst in the regenerator. The fact that coke is one of the most important variables of the dynamic system, and is not an independent one, was noted a long time ago1,27. The mathematical analysis for the model predicted that the standard steady state is subject to inverse response, in contrast to the ignited steady state. The simulations are in agreement with these observations. It was shown previously that the standard state has unstable zero dynamics. However, no conclusions about the state stability have been given. Arbel and co-workers21 proposed that the slope condition of the heat removed and heat produced functions, typically applied to analyse the energy balance, could be insucient to assess stability in the case of heterogeneous systems, such as the regenerator analysed. In the case of the model of a heterogeneous system, it is possible that the presence of another phase, which was considered pseudostationary, gives stability to this point. Perhaps a different conclusion about stability could be obtained if the short-time dynamics of the bubble phase were to be considered for short intervals after the disturbances are applied. The authors do not know related works in this area. A deeper analysis of this situation is beyond the scope this work. Changes in air supply Another control action which is commonly applied to FCC units is the change in air supplied to the regenerator. Because this air ¯ow is colder than the prevailing regenerator temperature, the energy balance is aected. The heat capacity for the gases is lower than for the solids, so the change in temperature should not be large. On the other hand, the availability of more oxygen, one of the reactants in the coke burning, works in favour of the reaction rate. In this way, the production of energy is enhanced by a higher oxygen concentration. Due to this opposing behaviour, a change in air supply could be followed by inverse response. The next two simulation studies illustrate the unit behaviour after this perturbation. The parameters and kinetics are the same as in previous cases. The response of the system to an increase in air supplied to the regenerator shows initially a cooling of the bed (Figure 6). All variables show inverse response behaviour some minutes after the disturbance is applied. Immediately, the coke burning rate decreases for some minutes and the ¯ow of this entity to the riser is increased. The combined eect of this cooling and catalyst which is partially deactivated by a larger amount of coke deposited provokes a fast drop in gasoil conversion. About 35 min after the disturbance was applied, before the re-establishment of the original operating conditions, the regenerator recovers. This is due to the availability of higher oxygen concentration, increasing the rate of coke burning, causing the regen-
97
erator temperature to rise. The riser temperature, and consequently yield and conversion, increase following the trend established by the regenerator. After 60 min the disturbance is removed and the system dynamics continue following the trend initiated by the recovery of the regenerator. Again, even when the system is operating near the standard steady state, inverse response behaviour is not visible. At the end of the simulation time, the system trend is to operate again around the standard steady state. In this way, it is possible to notice that the steady state attracts the solution and therefore is stable. So, the pseudostability of the whole unit referred previously can be related exclusively to the instability of the regenerator zero dynamics. When the FCC unit is disturbed by a decrease in the ¯ow of supplied air there are two opposing eects. First, the eect of one of the cooling streams is reduced. So, the ®rst response of the regenerator temperature could be an increase. However, at the same time, the main source of energy, the burning reaction, lacks one of the most important reactants, the oxygen. In this way, the coke could accumulate cooling the catalyst. In all the simulated cases, as in industrial practice, a small excess of air (about 6%) above that necessary to burn o all the coke in the ignited state was considered, so this eect could be ameliorated. Second, the riser temperature will be aected following the regenerator temperature trend. Meanwhile, due to a lesser activity of the deactivated catalyst, the endothermic cracking reactions become slower, the gasoil conversion will drop and, consequently, the use of energy in the riser will be smaller. In this way, the regenerator will be able to recover from the disturbance. Due to this sequence of events, inverse response is expected after this action. Immediately after the system is disturbed by a decrease in supplied air all the variables show inverse response (Figure 7), as in the previous case. Because the colder air was decreased, the next response of the regenerator's temperature is to rise. This situation provokes an increase in the riser temperature as well.
Figure 6
System response to an increase in air supply
98
Figure 7
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
System response to a decrease in air supply
Because the unit was originally working under the in¯uence of an excess of air, the regenerator is hot enough to burn o the coke produced, so its temperature continues increasing and coke ¯ow falls. In this way, the trend presented by this variable now shows inverse response, which causes a similar eect in the gasoil conversion. The riser continues warming-up because of the arrival of this hotter catalyst, and therefore, the global conversion of gasoil increases. In this way, the system has been shifted to the ignited steady state. When the original air supply level is recovered, after 60 min, the regenerator temperature does not change any more because of the stability of the new state. Moreover, there is no inverse response after this action because the zero dynamics are also stable at the ignited steady state. The responses obtained after the disturbance of air supply showed a behaviour similar to the response after a perturbation in catalyst circulation rate. The analysis predicted inverse response phenomena when the unit is disturbed around the standard steady state. This phenomenon was possibly hidden when the action provoked an increase in the regenerator temperature as the ®rst eect. If the unit is disturbed when operating at the ignited steady state, inverse response is not detected in any of the simulated cases.
Discussion In the cases presented above the inverse response observed in the simulations was predicted by the analysis, although in two cases it was predicted but not observed. The two exceptions were the response to an increase in catalyst circulation rate and the recovery of the system after an increase in air supply. In both cases, the applied action increases the regenerator temperature. Because all the phenomena which warm the regenerator up could be faster than those which cool it down, it is not unexpected that the inverse response
behaviour is hidden. At the ignited steady state inverse response behaviour was not observed at all after a step change because the zero dynamics are stable. Multiplicity of the operating steady states was predicted for a single set of parameters. Two steady states which are very close, denoted the standard and the ignited states, and another far from the operating zone, known as the quenched state, were predicted. These states have been also predicted by Arbel et al.21 and described qualitatively by Edwards and Kim3. The three models mentioned have in common that they emphasise the consideration of the energy used in the endothermic cracking reactions, and the energy necessary to evaporate the gasoil. Another point under discussion is whether the unit operates at the ignited steady state or at an intermediate steady state, i.e. in between the quenched and the ignited ones. The analysis presented here identi®es that the zero dynamics of the ignited steady state are stable, therefore, the presence of inverse response implies that operation at the standard (intermediate) steady state should be preferred. A feature that makes the ignited steady state dicult to identify is that it has been proposed that this state is characterised by the total burning of the carbon monoxide which can be generated by the coke combustion3,6. These authors characterise the CO burning as a cause of the possible shifting of steady state. However, Elshishini and Elnashaie10 proposed that it is possible to predict that the ignited steady state is reachable with or without the total combustion of CO. Arbel et al.21 found that it is possible to predict some ignited steady states which exhibit CO, although their ignited temperatures correspond to the intermediate, called standard, steady state. In this work CO was not found at the ignited steady state. Nevertheless, this phenomenon is seen as a consequence of the rise in temperature which also accelerates the homogeneous burning of CO, and hence this reaction contributes to maintain the unit operating at the ignited state. The stability of the operating point is another unresolved topic. When the unit is considered to operate at the ignited steady state, because of the considerations of the energy balance commonly used to assess stability, it is assumed that the operating point is stable. It was found that the standard operating point corresponds to the intermediate steady state, which in fact attracts the solution. This state has unstable zero dynamics. Due to the linearisation of the model, the proof of stability is only valid in the vicinity of this operating point. The ignited state was predicted to be stable when analysing the full range of the nonlinear model.
Conclusions A method to analyse the stability and some features of the dynamic responses of a control ane nonlinear MIMO system has been proposed. A comprehensive nonlinear model for an FCC unit has been used to test
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. this method. A full nonlinear model was used to develop the method and to identify inverse response behaviour by analysing the stability properties of the zero dynamics. The analysis was based on the proposition that the standard operating steady state of an FCC unit is controllable, but possibly pseudostable. The stability of the zero dynamics of the case study were analysed using Lyapunov's methods. The analysis of the standard steady state predicted unstable zero dynamics. The often quoted pseudostability of the standard operating point was shown to be identi®ed by the lack of stability of the zero dynamics of the proposed system model. The zero dynamics of the theoretical ignited state were predicted to be stable. In order to test the method, dynamic simulations were performed using the full nonlinear FCC unit model. Inverse response to disturbances when the unit was operating around the standard steady state and the possibility of a shift of steady state were found. These two phenomena were explained on the basis of the expected behaviour of an industrial system. In order to explore whether the dynamic characteristics predicted were a peculiarity of the selected set of control variables, two dierent operating policies were analysed. The same results were obtained, even though the zero dynamics are linear in one case and nonlinear in the other. While many of the phenomena such as multiplicity and pseudostability were identi®ed many years ago, a detailed analysis and approach to predict a priori such phenomena have only very recently become available. Such an approach will help to clarify the causes of complex phenomena that arise in the design and control of chemical processes.
8.
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21. 22. 23. 24.
Acknowledgements The authors thank CONACYT (MeÂxico) and The British Council (UK) for ®nancial support and the Instituto Mexicano del PetroÂleo and PEMEX (MeÂxico) for some data and useful discussions.
25. 26. 27.
99
Between Process Design and Process Control, ed. J. D. Perkins. London, 1992, pp. 127±132. Arbel, A., Huang, Z., Rinard, I. H., Shinnar, R. and Sapre, A. V., Dynamics and control of ¯uidized catalytic crackers. 1. Modeling the current generation of FCC's. Ind. Eng. Chem. Res., 1995, 34, 1228±1243. Gould, L. A., Evans, L. B. and Kurihara, H., Optimal control of ¯uid catalytic cracking processes. Automatica, 1970, 6, 695±703. Elshishini, S. S. and Elnashaie, S. S. E. H., Digital simulation of industrial ¯uid catalytic cracking units: bifurcation and its implications. Chem. Eng. Sci., 1990, 45, 553±559. Huq, I., Morari, M. and Sorensen, R. C., Modi®cations to model IV ¯uid catalytic cracking units to improve dynamic performance. AIChE J, 1995, 41, 1481±1496. Weekman Jr V. W. and Nace, D. M., Kinetic of catalytic cracking selectivity in ®xed, moving and ¯uid bed reactors. AIChE J., 1970, 16, 397±404. LoÂpez-Isunza, F., Dynamic modelling of an industrial ¯uid catalytic cracking unit. Comp. Chem. Eng., 1992, 16, S139. Maya-Yescas, R. and LoÂpez-Isunza, F., Un Esquema CineÂtico para el Craqueo CatalõÂtico de GasoÂleos en un Reactor de Lecho Transportado. Avances Ing. QuõÂm, 1993, 14, 71±78. Maya-Yescas, R., Bogle, D. and LoÂpez-Isunza, F., Steady state and dynamic simulation of FCC reactor units. Pre-prints of The 4th IFAC Symposium on Dynamics and Control of Chemical Reactors and Distillation columns: DYCORD+95. ed. J. Rawlings. Helsingùr, Denmark, pp. 153±158. Decrooq, D., Catalytic Cracking of Heavy Petroleum Fractions. Institute FrancËais du PeÂtrole. Paris, 1984. Krishna, A. and Parkin, E., Modeling the regenerator in commercial ¯uid catalytic cracking units. Chem. Eng. Progress., 1985, 57±61 Levenspiel, O., Chemical Reactions Engineering. 2nd edn. John Wiley & Sons, New York, 1972. Slotine, J. J. E. and Li, W., Applied Nonlinear Control. Prentice Hall Inc, Englewood Clis, NJ, 1991. Newell, R. B. and Lee, P. L., Applied Process Control: A Case Study, (Appendix D). Prentice-Hall, Englewood Clis, NJ, 1989. Arbel, A., Rinard, I. H., Shinnar, R. and Sapre, A. V., Dynamics and control of ¯uidized catalytic crackers. 2. Multiple steady states and instabilities. Ind. Eng. Chem. Res., 1995, 34, 3014±3026. Isidori, A., Nonlinear Control Systems. 3rd edn. Springer-Verlag, London, 1995. Kravaris, C. and Kantor, J. C., Geometric methods for nonlinear process control. 1. Background. Ind. Eng. Chem. Res., 1990, 29, 2295±2310. Holt, B. R. and Morari, M., Design of resilient processing plantsÐVI. The eect of right-half-plane zeros on dynamic resilience. Chem. Eng. Sci., 1985, 40, 59±74. Trickett, K. J., Quanti®cation of inverse response for controllability assessment of non linear processes. Ph.D. thesis. University of London, 1994. Daoutidis, P. and Kravaris, C., Inversion and zero dynamics in nonlinear multivariable control. AIChE J, 1991, 37, 527±538. Pohlenz, J. B., How operating variables aect ¯uid catalytic cracking. Oil & Gas J., 1963, 124±143.
Nomenclature References 1. Iscol, L., The dynamics and stability of a ¯uid catalytic cracker. Proceedings of the 1970 JACC Meeting, Atlanta, GA, 1970, PP. 602±607. 2. Elnashaie, S. S. E. H. and El-Hennawi, I. M., Multiplicity of the steady state in ¯uidized bed reactors-IV. Fluid catalytic cracking (FCC). Chem. Eng. Sci., 1979, 34, 1113±1121. 3. Edwards, W. M. and Kim, H. N., Multiple steady states in FCC unit operation. Chem. Eng. Sci., 1988, 43, 1825±1830. 4. Denn, M. M., Process Modeling. Longman Scienti®c & Technical and John Wiley & Sons. New York, 1986. 5. McFarlane, R. C., Reineman, R. C., Bartee, J. F. and Georgakis, C., Dynamic simulator for a model IV ¯uid catalytic cracking unit. Comp. Chem. Eng., 1993, 17, 275±300. 6. Lee, W. and Kugelman, A. M., Number of steady-state operating points and local stability of open-loop ¯uid catalytic cracker. Ind. Eng. Chem. PDD., 1973, 12, 197±204. 7. Wol, E. A., Skogestad, S., Hovd, M., and Mathisen, K. W., A Procedure for Controllability Analysis. Pre-prints of Interactions
av = volumetric speci®c surface of the catalyst (6.02105 mÿ1) Dax = kxcoketref = dimensionless characteristic time of the chemical reaction (2.70106) fe = volume fraction of air fed directly to the emulsion at the regenerator inlet (0.19) kg = regenerator mass transfer coecient (9.2610ÿ4 m/s) kx = coke burning reaction rate constant (kmol.m3.kgÿ1.sÿ1) L = riser length (30 m) qair = volumetric air supply rate (28.14 m3 sÿ1) tref = characteristic residence time of gasoil and products in the riser (3.9 s) Tref = reference temperature (930.0 K)
100
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
u = velocity (m/s) Vreg = total regenerator volume (2407.45 m3) Y = concentration (kmol/m3) or temperature (K) in the model for the riser z = axial co-ordinate in the riser
coke = = m =
mcat tref Mcat
v
=
qair tref 1ÿ"e Vreg
c
=
Cc Co2ref
=
kg av tref
Greek letters
=
x
=
ÿHx Co2ref cat Cpcat Tref
x
=
Ex Rg Tref
"
=
p =
=
e
=
Tp Tref
heat of reaction (Equation (2)). In the mass balance it is 1.0 dimensionless heat of reaction of the coke burning reaction (2.9610ÿ3) dimensionless activation energy of the coke burning reaction (2.44) volumetric fraction occupied by emulsion phase (0.41) dimensionless temperature of the phase p stoichiometric coecient in the coke burning reaction stoichiometric coecient for the coke
m
!cok =
coke density (kg/m3) (unknown) dimensionless time dimensionless characteristic residence time for the solids in the emulsion dimensionless characteristic residence time for the gases in the bubble dimensionless concentration of the compound C dimensionless characteristic time for interphase mass transfer mass fraction, related to the gasoil feed amount, of coke adsorbed by the catalyst.
Indices *
b d e i s
Ð Ð Ð Ð Ð Ð
refers to the partitioned equation system; refers to the bubble phase; dynamic conditions; refers to the emulsion phase; conditions at regenerator inlet; conditions at the steady state.
PII: S0959-1524(97)00030-9
Approach to the analysis of the dynamics of industrial FCC units Rafael Maya-Yescas,*y David Bogley{ and Felipe Lo¨pez-Isunza* ¨ de¨ Procesos, Universidad Auto¨noma Metropolitana - Iztapalapa, *Departamento de Ingenieria ¨ Av. Michoaca¨n y Purisima, s/n. Me¨xico 09850 D.F. Me¨xico y Department of Chemical and Biochemical Engineering, University College, London, UK *
Although it is known that FCC units are operated at a controllable steady state, it is commonly assumed that this state is pseudostable. An ignited steady state, theoretically stable, outside the operable domain has also been reported for the same set of operating parameters. The aim of this work is to introduce a theoretical method for the preliminary analysis of the controllability of an industrial FCC unit, at any set of conditions, using a nonlinear model directly. An approach for the analysis of the stability of the system zero dynamics is used to investigate the dynamic behaviour of two common operating policies when the unit is operating either in the standard or in the ignited steady state. It is used to predict some particular characteristics of the process, such as the presence of inverse response after the change in catalyst circulation rate, a common control action. The results derived from the analysis developed are discussed in terms of physical responses and veri®ed by performing open loop dynamic simulations of an industrial FCC unit. The operation of the unit is simulated for both the standard and the ignited steady states. # 1998 Elsevier Science Ltd. All rights reserved Keywords: FCC; mathematical modelling; nonlinear dynamics; open loop simulation; stability; steady state multiplicity; zero dynamics
limitation, there has been a trend to explore the dynamics of nonlinear models. Arbel and co-workers8 present a good review of this subject. In this paper a novel technique to analyse the dynamic behaviour of a nonlinear model for FCC units is given, taking into account some dynamic particularities such as the stability of the zero dynamics, which yields a preliminary study of the controllability of the nonlinear system. One of the earliest nonlinear fundamental models, and also an optimal control scheme, was due to Kurihara9. This model proposes that some FCC units can be controlled by controlling the regenerator only. This assumption is currently represented by considering the reactor (either a riser or a ¯uidised bed) operating under pseudo-steady state conditions. A problem associated with this scheme is that it is not considered as acceptable to the operator. However, it has been pointed out that the same control structure can be reformulated in terms of riser variables4. Hence the assumption of a pseudo-steady riser will be used in this work. An interesting feature of FCC units is that they are adiabatic systems. Due to the moderate endothermic cracking reactions, some heat supply is needed for the riser. This energy is provided by the coke combustion in the regenerator. The vehicle for energy transfer is the solid catalyst. This coupling in energy management
The production of gasoline by ¯uid-bed catalytic cracking (FCC) is one of the most common catalytic processes in the world. The interest in studying its dynamic behaviour is motivated by two main aspects: for academic researchers the complex and highly interacting nature of the reactor±regenerator system, and for industry because of the strong economic incentives associated with the unit. The characterisation of the dynamics of FCC units is dicult. For example, although several authors1±3 have proposed that a FCC unit can exhibit more than one theoretical steady state, there is no fully accepted criterion to con®rm this characteristic3,4. However, there are some available models which consider, in an explicit way, the possibility of multiple steady states arising within the constrained operating zone5. Commercial FCC units are controllable, despite the so called `pseudostability' of the operating steady state3,6. The analysis of the dynamics around this steady state has been explored mainly in the linear sense7 because of the complexity of the unit behaviour, although this approach is limited to a small region around the steady state. Recently, in order to avoid this { To whom correspondence should be addressed. Fax: +44 171 383 2348; e-mail: [email protected]
89
90
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
makes the FCC unit a complex and interesting system. A second important point of view is that the unit is exposed to disturbances in feed composition and temperature, which change the rate of coke deposition. These problems have to be resolved by managing the energy balance changing either the catalyst circulation rate or the ¯ow rate of the air supplied to the regenerator. Theoretically, both actions are able to change the operating steady state10. Moreover, the constraints making up the operable region boundaries also add to the problem of the control of FCC units. Huq et al. suggest some design and operational modi®cations to the model IV FCC unit to improve control performance, in particular by reducing ¯uctuations in the catalyst circulation rate. They report that this FCC unit is ill-conditioned which causes some of the control problems11. Another interesting point is that there is the possibility of catalyst deactivation in both reversible and irreversible ways. In this paper irreversible deactivation was neglected. The expression for reversible deactivation is given as a function of the gasoil mass fraction (hfo ), a critical mass fraction which can cause full catalyst deactivation (!c1 ) and an empirical factor ( ) related to the `coking ability' of the catalyst (Equation (1)). Incorporating this into the model allows the prediction of some dynamic features, such as the possibility of unit quenching if the catalyst becomes fully deactivated. hfo jz ÿ 1 1 jz !c1 ÿ hfo jz The precise kinetic scheme of the cracking reactions is unknown. However, there have been several propositions to represent it. One of the earliest schemes, which is currently used, considers the FCC reactions by lumping them into three dierent groups: gasoil, gasoline and coke12. Although this model has been very useful, it is accepted that it is not accurate8. In this work, a ®ve lump model which considers heavy gasoil, light gasoil, gasoline, volatile gases and coke is used (Figure 1). A comparison of the predicted mass ¯ows at the unit outlet when using this kinetic scheme and industrial data obtained from an industrial unit which processes 40 000 BBL dayÿ1 is shown for three standard operating cases (Figure 2)13. This paper is divided into the following sections. First, the development of the model used is presented; then two steady state solutions are obtained for the model; followed by the development of a novel technique
Figure 2 Comparison of mass ¯ows at the FCC unit outlet
to analyse the zero dynamics of the nonlinear model directly, analysing two important operating policies. The next section is devoted to the open loop dynamic simulations of the FCC unit: two dierent kinds of perturbations are considered and the system responses are brie¯y explained in terms of the physical characteristics of the system. The ®nal two sections concern, ®rst, a discussion of results and the comparison to previous related works followed by the conclusions obtained from this work.
A model for the FCC unit In order to analyse some aspects of the controllability of the FCC unit, a simple nonlinear model is proposed. The riser is eectively a steady state, transported and heterogeneous bed reactor. Some of the riser model equations are reported here; a more complete development can be found in Refs 14 and 15. Dierential mass balances for four of the proposed lumps and energy balances for each phase were developed (Equation (2) and Equation (3)) as well as an algebraic balance for coke, which is the closure condition of the system (no creation of mass). In order to incorporate the possible molar balance, molecular weights of each lump except coke, and stoichiometric coecients are proposed as well as a change in the void fraction and density of the reacting mixture, which are also simulated. This implies that the velocity varies along the riser. It is assumed that both phases are moving in plug ¯ow. Second order kinetics are considered for all the gasoil reactions and ®rst order kinetics for the others16. Because all the reactions that take place in the riser are catalytic, kinetic terms are not included in the balances for the gaseous phase. All these reactions are endothermic. Typical balance for the gaseous phase in the riser: ug @Yg 1 ÿ "1 L @z "1
v Yg
ÿ Ys 0
2
Typical balance for the vicinity and surface of the catalyst in the riser: us @Ys ÿ v Yg ÿ Ys Rr 3 L @z
Figure 1
Kinetic scheme
Due to the nature of the riser reactor, both phases moving in plug ¯ow and only endothermic reactions taking place, phenomena such as bifurcations or steady
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. state multiplicity are not possible8 so this type of dynamic behaviour will occur only in the regenerator model. Instantaneous evaporation of the gasoil supplied to the riser inlet is considered and the catalyst steaming process is neglected. Both reactors are connected by stand pipes which are modelled as time delays. The regenerator is considered as a dynamic CSTR, which consists of bubble and emulsion phases; the former at quasi steady state and the latter exhibiting dynamic behaviour. The model was developed by the classical application of mass and energy balances. The kinetic scheme proposed by Krishna and Parkin17 is used. First order kinetics with respect to the oxygen concentration and constant concentration of coke18 are assumed for the heterogeneous non-catalytic burning of coke. The dynamic mathematical model of the regenerator forms a coupled set of algebraic and dierential equations. The algebraic equations are related to the evaluation of physical properties and stoichiometric coecients as well as mass and energy balances for the bubble phase, which are of the same form as the emulsion phase but with the dynamic term equated to zero. The emulsion phase model, a set of ®rst order dierential equations, consists of mass balances for oxygen (Equation (4)), carbon monoxide (Equation (5)), coke mass fraction (Equation (6)), catalyst hold-up (Equation (7)) and the energy balance (Equation (8)). The contribution of the sensible enthalpy of the gas in the emulsion phase was neglected because the solid fraction terms are larger, by at least two orders of magnitude. In order to have explicit appearance of the time constants, the model is presented in dimensionless form (see nomenclature list for the de®nition of these variables). All the balances are subject to initial conditions given by considering that at time zero the system is operating at the desired steady state. d o2e fe vi o2ei ÿ v o2e d ÿ o2 Dax eÿ x=e o2e
m o2b
ÿ o2e
4
d coe ÿfe v coe m cob ÿ coe d coDax eÿ x=e o2e
5
d!cok Mi !coki ÿ M !cok ÿ e Dax eÿ x=e o2e d
6
d" Mi ÿ M " d
7
de Mi ei ÿ M e h b ÿ e d ÿ x Dax eÿ x=e o2e
8
Now, by de®ning the following vectors: xT o2e ; coe ; !cok ; "; e = regenerator model variables, and uT Mi ; M ; Vi = manipulable input variables, the dynamic part of the model can be transformed into a control ane system (Equation (9)), where G is a rectangular matrix (Equation (10)) which represents the
91
variables directly in¯uenced by the manipulated inputs u, and f is the vector which consists of the transport and reaction terms of the model equations. Control ane systems are a special case of control systems whose main characteristic is that the vector of manipulable variables appears linearly in the dynamic model19. The determination of the dimension of the matrix G is not discussed here, but it is related to the physical system constraints20. This model is used in the analysis of the system dynamics. x_ f x G x u 0
0 B 0 B GB B !coki @ " ei
0 0 ÿ!cok ÿ" ÿe
1 fe o2ei ÿ fe o2e C ÿfe coe C C 0 C A 0 0
9
10
Steady state solutions When the model is solved using the data of an industrial FCC unit, which processes 40 000 BBL dayÿ1 13, it is possible to ®nd three solutions for the energy balance at steady state; two of them are stable21. These steady states are in¯uenced by the riser operation because of the heat removal by the endothermic reactions and the gasoil evaporation3,21. One of the stable states is not of interest because it represents the quenching of the system. The other two steady state regions were calculated by the simulation of the whole unit (Figure 3). The standard operating point was found to be the socalled pseudostable steady state. The existence of the ignited steady state, which exhibits temperatures above the standard operating conditions, has been reported previously1,10,21. It is considered to be stable3,6,21. The temperature when operating at this ignited steady state depends on the catalyst history, as predicted previously10, and interpreted as a bifurcation of the steady state solution. In the next section a method to analyse the system behaviour using the system's zero dynamics in both steady states, standard and ignited, is presented. Two dierent operating policies, commonly applied to industrial FCC units are analysed. Both of them consider the control of catalyst temperature and coke fraction because they aect the riser directly and hence they are usually controlled variables. In order to avoid the afterburning
Figure 3
Steady state multiplicity at regenerator and operating point
92
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
reaction, the other control variable is either carbon monoxide concentration or oxygen concentration.
Analysis of the zero dynamics In order to identify the so called pseudostability of the operating steady state, the zero dynamics of the system must ®rst be characterised. The analysis is performed by using direct, when possible, and indirect Lyapunov methods, in order to predict the possibility of inverse response behaviour. The control ane model of the system is used. The zero dynamics of a system are the dynamics of its inverse22,23. In a linear system, the zero dynamics contain the same information as the zeros of the transfer function. If the model contains Right-Half-Plane (RHP) zeros, the zero dynamics contain RHP poles. These poles aect the system responses in various dierent ways, such as blockage of the response to some exponential inputs or causing inverse response following disturbances to some inputs24. For nonlinear systems it is not possible to de®ne a transfer function, but it is possible to ®nd the zero dynamics. If the system is control ane, the stability features of the zero dynamics can be characterised25 for SISO systems. The idea is to follow the dynamics of an uncontrolled variable when the system is governed by the control of another variable. Because the controlled variable is assumed to remain constant, an ideal vector of control actions can be evaluated, and substituted into the model for the uncontrolled variable. If the evolution of the dynamic behaviour of the uncontrolled variable is not stable, it is possible to conclude that the zero dynamics also are unstable, which is equivalent for a linear system to the presence of a RHP-pole. It is important to note that the evaluation of the control vector is independent of the control algorithm used. Because the internal stability of the closed loop system will depend on the stability of the unforced zero dynamics, a nonlinear stability analysis of these dynamics can be used to determine the spatial region where the dynamics are stable and therefore the closed loop system will be stable26. By following this reasoning, in this work an extension to MIMO systems of the procedure by Trickett25 is presented. It is important to notice that the direct use of a nonlinear model enables the analysis to be applicable to the whole operating region in contrast to a linearised model which is limited to a vicinity around the point used to linearise the model. In order to explore if the dynamic features predicted are a peculiarity of the selected set of control variables, two dierent operating policies are analysed. Analysis of the ®rst operating policy In the ®rst case study, a controller should be able to maintain three controlled variables at their set-point values: oxygen concentration, coke mass fraction and
emulsion temperature. CO concentration and catalyst hold-up dynamics will be controlled within a region around the desired operating state. The objective of this analysis is to anticipate when this surface is de®ned by stable zero dynamics. In order to evaluate the controlling vector necessary to perform this operation, the equation set is divided into two groups (Equation (11)), the ®rst one representing the controlled variables xs o2e , !cok, e) and the second one representing the uncontrolled variables xd coe , "). The vector f and the matrix G are divided according to xs and xd . x_ s f s x Gs x us x_ d f d x Gd x us
11
By allowing the control actions to maintain the time derivatives of the controlled variables at zero it is possible to obtain the steady state system (de®ned by Equation (12)). 0 1 0 1 0 o2e dB C B C @ !cok A @ 0 A d e 0 0 1 ÿ x=es o2e m o2b ÿ o2e ÿ o2 Dax e B C @ ÿe Dax eÿ x=es o2e A 12 0
h b
0 B @ !coki
ÿ e x Dax eÿ x=es o2e 1 0 fe o2ei ÿ fe o2es C ÿ!coks 0 A us
ei
ÿes
0
Since the matrix Gs is not singular, it is possible to calculate its inverse (Equation (13)), and rearranging the steady state expressions it is possible to obtain an expression for the vector of manipulable variables us (Equation (14)) Gÿ1 s 0 B B @
ÿ es !coki!coks ÿei !coks
0
es es !coki ÿei !coks ei es !coki ÿei !coks
ÿ fe o2ei1ÿ o2es
0
0
0
1
C coki C ÿ es !coki!ÿe i !coks A 13
us 0(
1 es !coki ÿei !coks
!coks
h b
ÿ es
)1
C B ÿ x=es B o2es C B ( es e !coks x Dax e )C C B 1 C B es !coki ÿe ! b ÿ e coki h s i !coks C B C B ei e !coki x Dax eÿ x=es o2es C B @ h iA o2b ÿ o2es o2es 1 ÿ x=es m o2ei ÿ o2es ÿ o2 Dax e o2ei ÿ o2es fe 14 Now, by substituting us in the mass balance for CO and catalyst hold-up, it is possible to predict the internal dynamics when operating under this control policy (Equation (15)).
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. d coe D d " 91 08 h i o2b ÿ o2es > > = < m cob ÿ 1 o co e 2ei ÿ o2es C B h i C B B> > co ÿ x=es e ;C C B : Dax e o2ei ÿ o2es o2 co o2es C B C B 9 C B8 " C B > es !coqi ÿe ! ÿ ! b ÿ e coqs coqi h s > i !coqs =C B< C B f es ÿ ei e A @ > > ; : ÿ x=es !coks ÿ !coqi x gDax e o2es 15 In this particular case, the system obtained is a linear one, so it is possible to transform it into canonical form, where yT coe " and cT m cob co Dax eÿ x=es o2es 0. The characteristic matrix A (Equations (16)) is diagonal, so its eigenvalues are its elements. o2b ÿ o2es A11 ÿ m 1 o2ei ÿ o2es 16a o2es o2 Dax eÿ x=es o2ei ÿ o2es A12 0 A21
16b
A22 D2 ="
16c
The evaluation of the eigenvalues of A for both steady states are shown in Table 1. This demonstrates that for the standard operating state the linear zero dynamics contain an RHP-pole26 and hence are not stable. Therefore, if this system is disturbed by using a step function, it may present inverse response24. In the case of the ignited state, the zero dynamics of the system do not contain RHP poles, therefore are stable and inverse response is not expected after a step disturbance. Analysis of the second operating policy By following a similar procedure the second operating policy, when the carbon monoxide concentration is controlled and oxygen concentration is `free', was analysed. In this case, the equation set is divided into a group representing the controlled variables xs coe ; !cok ; e; and another one representing the uncontrolled variables xd o2e ; ". Now, the matrix Gs consists of the second, third and ®fth rows of the original matrix G (Equation (10)), and it is non-singular, so its inverse can be obtained (Equation (17)). 0 1 es coks 0 ÿ es !coki!ÿe es !coki ÿei !coks i !coks B C ei coki ÿ es !coki!ÿe @ 0 Gÿ1 A s es !coki ÿei !coks i !coks 1 ÿ fe co 0 0 es 17 Table 1 Eigenvalues of the internal dynamics
First eigenvalue Second eigenvalue
Standard steady state
Ignited steady state
ÿ44.4327 +1.2417
ÿ35.3030 ÿ1.0770
93
Now, from Equation (11) it is possible to calculate the ideal control vector (Equation (18)): )1 0( 1 es !coki ÿei !coks !coks h b ÿ es C B B es e !coks x Dax eÿ x=es o2e C C B C B( ) C B 1 ! b ÿ e C s coki h us B ! ÿe ! e s coki i coks C B C B ÿ x=e s B ei e !coki x Dax e o2e C C B @ h i A co o 1 ÿ x=es 2e b m coes ÿ 1 co Dax e fe coes 18 By substituting us into the expression for the mass balances for oxygen and the catalyst hold-up, it is possible to obtain the equations which represent the internal dynamics of the system: d o2e d " 8 h i91 0 cob cob < = m o2b ÿ 2 ÿ coes o2e 1 ÿ coe o2ei C B C B: B ÿDax eÿ x=es o2 co o2ei o2e ÿ co o22e ; C C B B 8 9 C " C B > B < es !coki ÿei !coks !coks ÿ !coki h b ÿ es > = C C B A @ f es ÿ ei e > > : ; ÿ x=es !coks ÿ !coki x gDax e o2e 19 In this case, both equations obtained are nonlinear. In the ®rst one, the oxygen concentration is squared and in the second one there is a cross-product between the two analysed states. In order to test the local stability of the internal dynamics (only the operating region is of interest, i.e. 0 o2e1 and 0.4"0.7), it is possible to apply Lyapunov's direct method around each steady state. In order to ®nd a suitable Lyapunov function, Krasovskii's method19 is used here. Because these methods analyse stability around the origin, the model of the zero dynamics is written in terms of deviation variables (Equation (20)) around the steady state (Equation (21)). o2 o2e ÿ o2es
20a
" " ÿ "s 20b ! _ 2 o _ " 9 8 1 0 > > cob > > ÿ 2 ÿ o o o > > m 2b 2 2es > > coes C B > > > > C B > > > > C B = < C B cob ÿ x=e s C B 1 ÿ coe o2ei ÿ Dax e C B > > > > C B > > > > C B > > o o o > > o2 co 2ei 2 2es C B > > > > C B ; : 2 C B o o ÿ co 2 2es C B 9C B8 ""es C B > e ! ÿe ! !coks ÿ !coki h b ÿ es > =C B < s coki i coks C B A @ es ÿ ei e > > ; : ÿ x=es !coks ÿ !coki x Dax e o2 o2es 21
94
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
Following from Krasovskii's theorem, to ensure stability it is sucient to show that the matrix de®ned by ÿF J JT , where J is the Jacobian for the system, is positive de®nite at the origin, and, if this is the case, V o2 , ")T(o2 , ") is a suitable Lyapunov function. For this particular system, F and its determinant were computed (Equations (22) and (23)). ! @f1 @f2 ÿ2 @o @o2 2 22 F @f2 @f2 ÿ @o ÿ2 @" 2
@f1 F 4 @o2
@f2 @f2 ÿ @" @o2
2
23
Now, following Silvester's theorem, to ensure that F is positive de®nite it is sucient to show that its ®rst minor (F11) and its determinant are positive around the origin. In order to evaluate these terms the model derivatives were calculated (Equations (24)±(26)). @f1 cob ÿ m 2ÿ ÿ Dax eÿ x=es @o2 coe 24 o2 co o2ei 2co o2 o2es h @f2 " "s es ÿ ei e @o2 es !coki ÿ ei !coks i !coks ÿ !coki x Dax eÿ x=es @f2 1 @" es !coki ÿ ei !coks n !coks ÿ !coki h b ÿ es
25
26
es ÿ ei e !coks ÿ !coki x o Dax eÿ x=es o2 o2es From the results of the dynamic simulations (shown below), at the ignited state the following results (Equations (27)±(30)) were obtained from Equations (23)±(26): @f1 ÿ4:224 105 ÿ 3:345 105 o2 @o2
27
@f2 1:816 ÿ 4:432 103 " @o2
28
@f2 1:757 10ÿ2 ÿ 2:407 104 o2 @"
29
F 3:221 1010 o2 4:268 1010 o2 ÿ 1:964 107 "2 8:048" 103 2:968 104 30 At the origin of the co-ordinates both F11 and F are positive, therefore the zero dynamics are stable at the ignited steady state.
Now, by evaluating the derivatives and the matrix determinant (Equations (23)±(26)) at the standard steady state the following results are obtained (Equations (31)±(34)): @f1 ÿ3:869 105 ÿ 3:036 105 o2 @o2
31
@f2 ÿ274:395 ÿ 669:583" @o2
32
@f2 1:282 ÿ 685:970o2 @"
33
F 8:332 109 o22 1:060 109 o2 ÿ 4:483 105 "2 ÿ 3:675 105 " ÿ 2:059 106 34 At the origin of the coordinates the minor F11>0 while F < 0. Therefore, it is not possible to use the function obtained to check the stability of that point. This conclusion has been obtained for some SISO systems which present second order zero dynamics25 and the same case is manifested here for a MIMO system. Since Lyapunov's direct method does not provide results, in order to apply Lyapunov's indirect method the equations of the zero dynamics were linearised around the steady state. In order to obtain an accurate result, in the case of a squared expression, as in the oxygen mass balance, it has been proposed that the model be factorised and linearised by using a Taylor series around one of its roots, which represents the operating steady state25. For the ®rst equation of the zero dynamics the proposed procedure leads to the following expression: p! d o2e ÿ 2 ÿ 4 ' o2e ÿ d 2 35 p! 2 ÿ ÿ 4 ' o2e ÿ 2 where co Dax eÿ x=es ; cob ÿ m 2ÿ ÿ o2 co o2ei Dax eÿ x=es ; coes cob o2ei ' m o2b 1 ÿ coes The catalyst hold-up equation is linearised using a Taylor's series expansion around the same steady state (Equation (36)). d" a b o2es " b"s o2e ÿ b"s o2es d where
36
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
a
!coks ÿ !coki h b ÿ es ; es !coki ÿ ei !coks
Table 2 Eigenvalues of the internal dynamics
First eigenvalue Second eigenvalue
es ÿ ei e !coks ÿ !coki x Dax eÿ x=es : b es !coki ÿ ei !coks Now, the system can be written in canonical form (Equation (37)), where A is a lower triangular matrix (Equation (38)). In this way, its eigenvalues are located on the diagonal. dy Ay c d
37
where yT o2e "; cob o cT o 1 ÿ m 2b 2ei coes
ÿ co Dax eÿ x=es o22es ÿ b"s o2es ;
9 08 cob > > ÿ m 2 ÿ co > > = < ei B B ÿo o ÿ 2 o 2 2ei 2es > > AB > B> ; @ : Dax eÿ x=es b"s
95
Standard steady state
Ignited steady state
ÿ864.5953 +1.9717
ÿ352.3270 ÿ1.8781
sidered to be zero time) and then the selected disturbance was applied. Kinetics and deactivation functions were used as proposed in previous sections. The four dynamic cases simulated are listed below (Table 3) which are common control actions. The aim is to explore the behaviour predicted by the model and to compare it with the proposed analysis. Explanations in terms of physical characteristics of the dynamic behaviour are given. Due to the instability of the zero dynamics at the standard steady state, inverse response is expected when the system is operating close to this state. In contrast, inverse responses are not expected when the unit is operating at the ignited steady state because of the stability of the zero dynamics. Changes in catalyst circulation rate
1 0
C C C C A
a b o2es 38
The results obtained when evaluating the eigenvalues of the matrix A for both states are shown in Table 2. The linearised internal dynamics of the simulated system have a RHP pole at the standard operating steady state, therefore the zero dynamics are not stable at this state. For the ignited state there are no RHP poles in the linearised internal dynamics, therefore the zero dynamics are stable at the ignited steady state. The results obtained by using the approach proposed yielded that the system zero dynamics are not stable at the standard operating state. In contrast, at the ignited steady state it was found that the zero dynamics are stable. In this way, it is expected that if the unit is disturbed by a step change when operating around the standard steady state it may exhibit inverse response. However, if the unit is operating at the ignited steady state, no inverse response behaviour is expected for the same situation. The next section is devoted to presenting and explaining the results obtained for open loop dynamic simulations of the FCC unit.
When a FCC unit is disturbed by changing its catalyst circulation rate, the net eect is to alter the energy balance in both reactors, riser and regenerator. This situation arises because the solid is the vehicle for the transfer of energy between the reactors. A reduction in the catalyst circulation rate increases the energy hold-up in the regenerator. In this way, there is a larger amount of energy, supplied by the coke burning at the catalyst surface, available to enhance any chemical reaction. So, if the coke production is large enough, the regenerator warms up after the perturbation. Simultaneously, for the riser, a reduction in catalyst circulation is manifested as a smaller amount of energy supplied. However, the situation changes when the catalyst starts to become hotter due to the change in the temperature of the regenerator, compensating the ®rst eect. In this way, some minutes after the disturbance has been applied, there is a change in the global behaviour, a situation that could give rise to inverse response. The response of coke production when a decrease in catalyst circulation rate is applied shows inverse response when the disturbance is introduced (Figure 4). This situation arises because of the enhancement of coke burning in the regenerator followed by an increase in coke production in the riser. Its eect on the coke ¯ow response is not so evident because of the fast increase in the regenerator temperature, and consequently, the coke is completely depleted and the riser Table 3 Operating case studies
Dynamic simulations of the FCC unit In order to verify the results found in the analysis of the zero dynamics, responses for open loop mode dynamics were simulated after the unit had been exposed to different kinds of disturbances. In all the cases, the system was simulated until the steady state was reached (con-
1. A 3% step decrease of catalyst circulation rate and re-establishment of the original steady state conditions after 120 min 2. A 3% step increase of catalyst circulation rate and re-establishment of the original steady state conditions after 120 min15 3. A step increase of 5% in supplied air for 60 min and re-establishment of the original steady state conditions 4. A step decrease of 5% in supplied air for 60 min and re-establishment of the original steady state conditions
96
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
temperature is also increased. The coke elimination is re¯ected also by a fall in the coke present at the riser outlet which is due to a decrease in coke recirculation rather than a drop in its production. When the disturbance is removed after 120 min, as expected inverse response behaviour is not observed because the regenerator has been shifted to the ignited steady state, which has stable zero dynamics. The existence of the ignited steady state and the possibility to reach it after a decrease and re-establishment in catalyst circulation rate has been reported previously10. Similarly, Arbel et al.8,21 analysed the same problem and also found inverse response after a step change in catalyst circulation rate, although their data seems to correspond to temperatures and conversions of the ignited steady state. In the case of an increase in catalyst circulation rate, there is an increase in the amount of energy ¯owing between reactor and regenerator, but the energy holdup in the regenerator is decreased proportionally to the catalyst residence time. As a result there is a smaller speci®c amount of energy available to promote the regeneration reactions. At the same time, in the riser there is a larger ¯ow of energy, a situation that improves gasoil evaporation and reacting conditions, which results in a cooling of the catalyst. However, coke production is also enhanced accelerating the catalyst deactivation and stopping the reactions. Due to these opposing eects, the behaviour after an increment in catalyst ¯ow between the reactors is not obvious: even if the coke production is higher, its burning rate could drop, decreasing the source of energy in the unit. This larger amount of deposited coke can either heat the unit for `good' reaction conditions or take the unit to a quenching state, which is known as `snowball' quenching3, and is another reason to expect a non-intuitive response to this action. Although the system is operating near the standard steady state, when an increase in catalyst circulation rate is introduced (Figure 5) inverse response behaviour is not visible. This situation can arise if dynamic phe-
nomena are combined, similar in the linear sense to a pole-zero cancellation25. Due to the availability of more energy, which is supplied by the hot catalyst coming from the regenerator, the riser temperature is increased. As can be seen, yield to gasoline follows the same trend of the riser temperature and the yield of coke is also enhanced. Because the catalyst activity behaviour is opposed to the coke accumulation, eventually the global conversion of gasoil falls as coke on the catalyst increases. The regenerator temperature is increased because of the enhancement in coke combustion. In this way, the coke burning rate also increases resulting in a slightly falling trend for the coke hold-up at the regenerator. In the riser, coke production continues stopping the favourable situation of possible high gasoil conversion because of the higher catalyst deactivation. When original operating conditions are re-established, after 120 min, the riser temperature experiences an instantaneous drop which, for the values considered in the simulation, is suciently large to produce the quenching of the unit. This is evidence of the lack of stability of the operating steady state. It can also be seen that at the same time inverse response behaviour is shown for coke ¯ow and gasoil conversion, as predicted by the analysis, because the unit continues operating around the standard state. The regenerator temperature falls quickly following the trend established by the riser. Due to this situation, coke burning in the regenerator is insucient and a higher amount of coke on the catalyst surface is recirculated to the riser, stopping the process. This phenomenon ®nally takes the process to the coldest steady state. In both previously simulated cases two important characteristics of this system have been established. First, the dynamics of the FCC unit are strongly dependent on catalyst hold-up and, consequently, on coke hold-up. These variables have a strong in¯uence on the regenerator energy balance, which dictates the energy hold-up for the whole unit. Because of this situation, the riser follows the conditions determined by the regenerator. Second, the eect of the manipulation
Figure 4
Figure 5 System response to an increase in catalyst circulation rate
System response to a decrease in catalyst circulation rate
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. of the catalyst circulation ¯ow as a control action is not trivial because the system can present dierent dynamic responses depending on its coke content and on the residence time of the catalyst in the regenerator. The fact that coke is one of the most important variables of the dynamic system, and is not an independent one, was noted a long time ago1,27. The mathematical analysis for the model predicted that the standard steady state is subject to inverse response, in contrast to the ignited steady state. The simulations are in agreement with these observations. It was shown previously that the standard state has unstable zero dynamics. However, no conclusions about the state stability have been given. Arbel and co-workers21 proposed that the slope condition of the heat removed and heat produced functions, typically applied to analyse the energy balance, could be insucient to assess stability in the case of heterogeneous systems, such as the regenerator analysed. In the case of the model of a heterogeneous system, it is possible that the presence of another phase, which was considered pseudostationary, gives stability to this point. Perhaps a different conclusion about stability could be obtained if the short-time dynamics of the bubble phase were to be considered for short intervals after the disturbances are applied. The authors do not know related works in this area. A deeper analysis of this situation is beyond the scope this work. Changes in air supply Another control action which is commonly applied to FCC units is the change in air supplied to the regenerator. Because this air ¯ow is colder than the prevailing regenerator temperature, the energy balance is aected. The heat capacity for the gases is lower than for the solids, so the change in temperature should not be large. On the other hand, the availability of more oxygen, one of the reactants in the coke burning, works in favour of the reaction rate. In this way, the production of energy is enhanced by a higher oxygen concentration. Due to this opposing behaviour, a change in air supply could be followed by inverse response. The next two simulation studies illustrate the unit behaviour after this perturbation. The parameters and kinetics are the same as in previous cases. The response of the system to an increase in air supplied to the regenerator shows initially a cooling of the bed (Figure 6). All variables show inverse response behaviour some minutes after the disturbance is applied. Immediately, the coke burning rate decreases for some minutes and the ¯ow of this entity to the riser is increased. The combined eect of this cooling and catalyst which is partially deactivated by a larger amount of coke deposited provokes a fast drop in gasoil conversion. About 35 min after the disturbance was applied, before the re-establishment of the original operating conditions, the regenerator recovers. This is due to the availability of higher oxygen concentration, increasing the rate of coke burning, causing the regen-
97
erator temperature to rise. The riser temperature, and consequently yield and conversion, increase following the trend established by the regenerator. After 60 min the disturbance is removed and the system dynamics continue following the trend initiated by the recovery of the regenerator. Again, even when the system is operating near the standard steady state, inverse response behaviour is not visible. At the end of the simulation time, the system trend is to operate again around the standard steady state. In this way, it is possible to notice that the steady state attracts the solution and therefore is stable. So, the pseudostability of the whole unit referred previously can be related exclusively to the instability of the regenerator zero dynamics. When the FCC unit is disturbed by a decrease in the ¯ow of supplied air there are two opposing eects. First, the eect of one of the cooling streams is reduced. So, the ®rst response of the regenerator temperature could be an increase. However, at the same time, the main source of energy, the burning reaction, lacks one of the most important reactants, the oxygen. In this way, the coke could accumulate cooling the catalyst. In all the simulated cases, as in industrial practice, a small excess of air (about 6%) above that necessary to burn o all the coke in the ignited state was considered, so this eect could be ameliorated. Second, the riser temperature will be aected following the regenerator temperature trend. Meanwhile, due to a lesser activity of the deactivated catalyst, the endothermic cracking reactions become slower, the gasoil conversion will drop and, consequently, the use of energy in the riser will be smaller. In this way, the regenerator will be able to recover from the disturbance. Due to this sequence of events, inverse response is expected after this action. Immediately after the system is disturbed by a decrease in supplied air all the variables show inverse response (Figure 7), as in the previous case. Because the colder air was decreased, the next response of the regenerator's temperature is to rise. This situation provokes an increase in the riser temperature as well.
Figure 6
System response to an increase in air supply
98
Figure 7
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
System response to a decrease in air supply
Because the unit was originally working under the in¯uence of an excess of air, the regenerator is hot enough to burn o the coke produced, so its temperature continues increasing and coke ¯ow falls. In this way, the trend presented by this variable now shows inverse response, which causes a similar eect in the gasoil conversion. The riser continues warming-up because of the arrival of this hotter catalyst, and therefore, the global conversion of gasoil increases. In this way, the system has been shifted to the ignited steady state. When the original air supply level is recovered, after 60 min, the regenerator temperature does not change any more because of the stability of the new state. Moreover, there is no inverse response after this action because the zero dynamics are also stable at the ignited steady state. The responses obtained after the disturbance of air supply showed a behaviour similar to the response after a perturbation in catalyst circulation rate. The analysis predicted inverse response phenomena when the unit is disturbed around the standard steady state. This phenomenon was possibly hidden when the action provoked an increase in the regenerator temperature as the ®rst eect. If the unit is disturbed when operating at the ignited steady state, inverse response is not detected in any of the simulated cases.
Discussion In the cases presented above the inverse response observed in the simulations was predicted by the analysis, although in two cases it was predicted but not observed. The two exceptions were the response to an increase in catalyst circulation rate and the recovery of the system after an increase in air supply. In both cases, the applied action increases the regenerator temperature. Because all the phenomena which warm the regenerator up could be faster than those which cool it down, it is not unexpected that the inverse response
behaviour is hidden. At the ignited steady state inverse response behaviour was not observed at all after a step change because the zero dynamics are stable. Multiplicity of the operating steady states was predicted for a single set of parameters. Two steady states which are very close, denoted the standard and the ignited states, and another far from the operating zone, known as the quenched state, were predicted. These states have been also predicted by Arbel et al.21 and described qualitatively by Edwards and Kim3. The three models mentioned have in common that they emphasise the consideration of the energy used in the endothermic cracking reactions, and the energy necessary to evaporate the gasoil. Another point under discussion is whether the unit operates at the ignited steady state or at an intermediate steady state, i.e. in between the quenched and the ignited ones. The analysis presented here identi®es that the zero dynamics of the ignited steady state are stable, therefore, the presence of inverse response implies that operation at the standard (intermediate) steady state should be preferred. A feature that makes the ignited steady state dicult to identify is that it has been proposed that this state is characterised by the total burning of the carbon monoxide which can be generated by the coke combustion3,6. These authors characterise the CO burning as a cause of the possible shifting of steady state. However, Elshishini and Elnashaie10 proposed that it is possible to predict that the ignited steady state is reachable with or without the total combustion of CO. Arbel et al.21 found that it is possible to predict some ignited steady states which exhibit CO, although their ignited temperatures correspond to the intermediate, called standard, steady state. In this work CO was not found at the ignited steady state. Nevertheless, this phenomenon is seen as a consequence of the rise in temperature which also accelerates the homogeneous burning of CO, and hence this reaction contributes to maintain the unit operating at the ignited state. The stability of the operating point is another unresolved topic. When the unit is considered to operate at the ignited steady state, because of the considerations of the energy balance commonly used to assess stability, it is assumed that the operating point is stable. It was found that the standard operating point corresponds to the intermediate steady state, which in fact attracts the solution. This state has unstable zero dynamics. Due to the linearisation of the model, the proof of stability is only valid in the vicinity of this operating point. The ignited state was predicted to be stable when analysing the full range of the nonlinear model.
Conclusions A method to analyse the stability and some features of the dynamic responses of a control ane nonlinear MIMO system has been proposed. A comprehensive nonlinear model for an FCC unit has been used to test
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al. this method. A full nonlinear model was used to develop the method and to identify inverse response behaviour by analysing the stability properties of the zero dynamics. The analysis was based on the proposition that the standard operating steady state of an FCC unit is controllable, but possibly pseudostable. The stability of the zero dynamics of the case study were analysed using Lyapunov's methods. The analysis of the standard steady state predicted unstable zero dynamics. The often quoted pseudostability of the standard operating point was shown to be identi®ed by the lack of stability of the zero dynamics of the proposed system model. The zero dynamics of the theoretical ignited state were predicted to be stable. In order to test the method, dynamic simulations were performed using the full nonlinear FCC unit model. Inverse response to disturbances when the unit was operating around the standard steady state and the possibility of a shift of steady state were found. These two phenomena were explained on the basis of the expected behaviour of an industrial system. In order to explore whether the dynamic characteristics predicted were a peculiarity of the selected set of control variables, two dierent operating policies were analysed. The same results were obtained, even though the zero dynamics are linear in one case and nonlinear in the other. While many of the phenomena such as multiplicity and pseudostability were identi®ed many years ago, a detailed analysis and approach to predict a priori such phenomena have only very recently become available. Such an approach will help to clarify the causes of complex phenomena that arise in the design and control of chemical processes.
8.
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21. 22. 23. 24.
Acknowledgements The authors thank CONACYT (MeÂxico) and The British Council (UK) for ®nancial support and the Instituto Mexicano del PetroÂleo and PEMEX (MeÂxico) for some data and useful discussions.
25. 26. 27.
99
Between Process Design and Process Control, ed. J. D. Perkins. London, 1992, pp. 127±132. Arbel, A., Huang, Z., Rinard, I. H., Shinnar, R. and Sapre, A. V., Dynamics and control of ¯uidized catalytic crackers. 1. Modeling the current generation of FCC's. Ind. Eng. Chem. Res., 1995, 34, 1228±1243. Gould, L. A., Evans, L. B. and Kurihara, H., Optimal control of ¯uid catalytic cracking processes. Automatica, 1970, 6, 695±703. Elshishini, S. S. and Elnashaie, S. S. E. H., Digital simulation of industrial ¯uid catalytic cracking units: bifurcation and its implications. Chem. Eng. Sci., 1990, 45, 553±559. Huq, I., Morari, M. and Sorensen, R. C., Modi®cations to model IV ¯uid catalytic cracking units to improve dynamic performance. AIChE J, 1995, 41, 1481±1496. Weekman Jr V. W. and Nace, D. M., Kinetic of catalytic cracking selectivity in ®xed, moving and ¯uid bed reactors. AIChE J., 1970, 16, 397±404. LoÂpez-Isunza, F., Dynamic modelling of an industrial ¯uid catalytic cracking unit. Comp. Chem. Eng., 1992, 16, S139. Maya-Yescas, R. and LoÂpez-Isunza, F., Un Esquema CineÂtico para el Craqueo CatalõÂtico de GasoÂleos en un Reactor de Lecho Transportado. Avances Ing. QuõÂm, 1993, 14, 71±78. Maya-Yescas, R., Bogle, D. and LoÂpez-Isunza, F., Steady state and dynamic simulation of FCC reactor units. Pre-prints of The 4th IFAC Symposium on Dynamics and Control of Chemical Reactors and Distillation columns: DYCORD+95. ed. J. Rawlings. Helsingùr, Denmark, pp. 153±158. Decrooq, D., Catalytic Cracking of Heavy Petroleum Fractions. Institute FrancËais du PeÂtrole. Paris, 1984. Krishna, A. and Parkin, E., Modeling the regenerator in commercial ¯uid catalytic cracking units. Chem. Eng. Progress., 1985, 57±61 Levenspiel, O., Chemical Reactions Engineering. 2nd edn. John Wiley & Sons, New York, 1972. Slotine, J. J. E. and Li, W., Applied Nonlinear Control. Prentice Hall Inc, Englewood Clis, NJ, 1991. Newell, R. B. and Lee, P. L., Applied Process Control: A Case Study, (Appendix D). Prentice-Hall, Englewood Clis, NJ, 1989. Arbel, A., Rinard, I. H., Shinnar, R. and Sapre, A. V., Dynamics and control of ¯uidized catalytic crackers. 2. Multiple steady states and instabilities. Ind. Eng. Chem. Res., 1995, 34, 3014±3026. Isidori, A., Nonlinear Control Systems. 3rd edn. Springer-Verlag, London, 1995. Kravaris, C. and Kantor, J. C., Geometric methods for nonlinear process control. 1. Background. Ind. Eng. Chem. Res., 1990, 29, 2295±2310. Holt, B. R. and Morari, M., Design of resilient processing plantsÐVI. The eect of right-half-plane zeros on dynamic resilience. Chem. Eng. Sci., 1985, 40, 59±74. Trickett, K. J., Quanti®cation of inverse response for controllability assessment of non linear processes. Ph.D. thesis. University of London, 1994. Daoutidis, P. and Kravaris, C., Inversion and zero dynamics in nonlinear multivariable control. AIChE J, 1991, 37, 527±538. Pohlenz, J. B., How operating variables aect ¯uid catalytic cracking. Oil & Gas J., 1963, 124±143.
Nomenclature References 1. Iscol, L., The dynamics and stability of a ¯uid catalytic cracker. Proceedings of the 1970 JACC Meeting, Atlanta, GA, 1970, PP. 602±607. 2. Elnashaie, S. S. E. H. and El-Hennawi, I. M., Multiplicity of the steady state in ¯uidized bed reactors-IV. Fluid catalytic cracking (FCC). Chem. Eng. Sci., 1979, 34, 1113±1121. 3. Edwards, W. M. and Kim, H. N., Multiple steady states in FCC unit operation. Chem. Eng. Sci., 1988, 43, 1825±1830. 4. Denn, M. M., Process Modeling. Longman Scienti®c & Technical and John Wiley & Sons. New York, 1986. 5. McFarlane, R. C., Reineman, R. C., Bartee, J. F. and Georgakis, C., Dynamic simulator for a model IV ¯uid catalytic cracking unit. Comp. Chem. Eng., 1993, 17, 275±300. 6. Lee, W. and Kugelman, A. M., Number of steady-state operating points and local stability of open-loop ¯uid catalytic cracker. Ind. Eng. Chem. PDD., 1973, 12, 197±204. 7. Wol, E. A., Skogestad, S., Hovd, M., and Mathisen, K. W., A Procedure for Controllability Analysis. Pre-prints of Interactions
av = volumetric speci®c surface of the catalyst (6.02105 mÿ1) Dax = kxcoketref = dimensionless characteristic time of the chemical reaction (2.70106) fe = volume fraction of air fed directly to the emulsion at the regenerator inlet (0.19) kg = regenerator mass transfer coecient (9.2610ÿ4 m/s) kx = coke burning reaction rate constant (kmol.m3.kgÿ1.sÿ1) L = riser length (30 m) qair = volumetric air supply rate (28.14 m3 sÿ1) tref = characteristic residence time of gasoil and products in the riser (3.9 s) Tref = reference temperature (930.0 K)
100
Approach to the analysis of the dynamics of industrial FCC units: R. Maya-Yescas et al.
u = velocity (m/s) Vreg = total regenerator volume (2407.45 m3) Y = concentration (kmol/m3) or temperature (K) in the model for the riser z = axial co-ordinate in the riser
coke = = m =
mcat tref Mcat
v
=
qair tref 1ÿ"e Vreg
c
=
Cc Co2ref
=
kg av tref
Greek letters
=
x
=
ÿHx Co2ref cat Cpcat Tref
x
=
Ex Rg Tref
"
=
p =
=
e
=
Tp Tref
heat of reaction (Equation (2)). In the mass balance it is 1.0 dimensionless heat of reaction of the coke burning reaction (2.9610ÿ3) dimensionless activation energy of the coke burning reaction (2.44) volumetric fraction occupied by emulsion phase (0.41) dimensionless temperature of the phase p stoichiometric coecient in the coke burning reaction stoichiometric coecient for the coke
m
!cok =
coke density (kg/m3) (unknown) dimensionless time dimensionless characteristic residence time for the solids in the emulsion dimensionless characteristic residence time for the gases in the bubble dimensionless concentration of the compound C dimensionless characteristic time for interphase mass transfer mass fraction, related to the gasoil feed amount, of coke adsorbed by the catalyst.
Indices *
b d e i s
Ð Ð Ð Ð Ð Ð
refers to the partitioned equation system; refers to the bubble phase; dynamic conditions; refers to the emulsion phase; conditions at regenerator inlet; conditions at the steady state.