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Mediating Operation of Catalytic CSTR: A Novel Possibility for Stabilizing Intermediate Steady State
| Content Provider | Semantic Scholar |
|---|---|
| Author | Gol'dshtein, Vladimir Pan, Vadim |
| Copyright Year | 1996 |
| Abstract | A novel approach for stabilizing intermediate steady state of CSTR is proposed by using a special type of periodic forced operation, socalled mediating operation. Mediating operation enables to produce new additional steady states for which one of new intermediate steady states is stable. Thus proposed approach employs a periodic forcing for stabilization of a selected steady state by control of a steady state multiplicity. Feed ow rate is considered as a manipulated variable. Changing of CSTR multiplicity is investigated for two{step control inputs. It is shown, analytically, that under two{step control CSTR can exhibit at most ve steady states having stable intermediate steady state. A constructive procedure is proposed for nding control parameters corresponded to maximal multiplicity, i.e. for stabilizing intermediate steady state by two{step control. INTRODUCTION A novel approach for achievement of stable intermediate steady state of a catalytic Continuous Stirred Tank Reactor (CSTR) is considered by using a special type of periodic forcing, so-called mediating operation. Mediating operation as a new periodic operation approach for catalytic CSTR with widely separated time scales has been proposed by Gol'dshtein et al. (1996). Mediating operation is introduced as a periodic intermediate mode operation with respect to di erent reactor time scales. The main goal of operating method is to control two system responses by manipulating a single control input. The study aims to demonstrate that mediating operation by feed ow rate inputs can produce additional steady states of CSTR for which one of new intermediate steady states is stable. Thus reactor behavior for intermediate temperature area can be stabilized. The proposed novel approach can be formulated as an application of mediating operation to control a steady state multiplicity for the achievement of a stable steady state in the selected area. The stabilization of chemical reactor operation can be viewed as a potential bene t for forced periodic operation approach. Unforced classical CSTR with single reaction may have single stable steady state as a minimum or three steady states as a maximum. For maximal three multiplicity the lower and upper steady states are stable and intermediate one is unstable (see, for example, Uppal et al., 1974; 1976). Traditionally, stabilizing reactor operation in intermediate domain is considered as moving stable steady state 1 (lower or upper or single) of periodically forced CSTR in an unstable operation domain of the unforced reactor. In this sense, vibrational control by feed ow rate manipulation can change multi{stability to a single stable steady state (Bellman et al., 1983) or can move an upper steady state in some intermediate operation domain (Cinar et al., 1987a,b). Mediating operation by feed ow rate manipulation can result also to a single stability or can move a stable lower steady state in the vicinity of unstable intermediate steady state of the unforced reactor (Gol'dshtein et al., 1996). A novel possibility for stabilizing intermediate steady state arises due to ability of mediating operation to control a steady state multiplicity. Gol'dshtein et al. (1990) have shown that two{step periodic forcing of intermediate mode may lead to ve steady states for which one of intermediate steady states is stable. Hence mediating operation enables to achieve a stable steady state for intermediate temperature area by creating additional multi{stability of CSTR. For this purpose two{step mediating operation is examined by using singular perturbation and averaging methods. Multiplicity features of averaged steady state system are studied by using catastrophe theory (see, for example, Br ocker and Lander, 1975; Poston and Stewart, 1978; Golubitsky and Schae er, 1985). In general chemically reacting systems may exhibit complicated steady state multiplicity. Steady state multiplicity and possible types of bifurcation diagrams for CSTR as a multi{reaction system have been researched in detail 2 by Balakotaiah and Luss (1982, 1983, 1984, 1988). Dynamic features of systems with widely separated time scales have been studied by Sheintuch and Luss (1987) for limiting case. Multiplicity problem of CSTR-s coupled in series has been considered by Dangenlmayr and Stewart (1985), Retzlo et al. (1992). This communication addresses a new multiplicityproblem of CSTR caused by periodic manipulation of external input as a feed ow rate. It is shown, analytically, that for two{step control multi{parameter CSTR problem is described by standard butter y singularity. As a result CSTR can exhibit at most ve steady states having stable intermediate steady state. The systematic numerical procedure is presented for nding amplitude parameters of two{step control which provide ve steady states and enable stabilizing intermediate steady state by such a way. In particular, shown, that only stroboscopical control can lead to desired stabilization. It is shown also that correlation between a coolant temperature and a feed ow temperature is essential for realization of maximal multiplicity. 1 REACTOR MODEL AND MEDIATING OPERATION Chemical reactor is considered as a two{phase well{stirred tank reactor with a gas mixture reacting on the surface of a solid catalyst. A thermal equilibrium is assumed between di erent phases in the reactor at all times. For a single rst{order reaction the heat and mass balance equations of the 3 gas{solid CSTR have the form (V cp +mcs)dT dt0 = cpF (T Tf) hA(T Tc) +V ( H)Ck0 exp( E RT ) V dC dt0 = F (C Cf) V Ck0 exp( E RT ) (1) For reducing to dimensionless form it would be desirable that F as a control variable should be a linear parameter of a dimensionless system as it was in the initial system (1). So following to Gol'dshtein et al. (1996) the system (1) has the following dimensionless form: d dt = u ( c) +B(1 )k( ) d dt = u + (1 )k( ) (2) The ratio of a gas heat capacity to the total reactor heat capacity, is considered as a small parameter, because the solid density is 103 times greater than the gas density. Thus system (2) is a singularly perturbed system with a small parameter 1. Further following asymptotic analysis scheme (see, for example, Gol'dshtein and Sobolev 1988,1992) zero{approximation, = 0, of system (2) is considered, d dt = u ( c) +B(1 )k( ) 0 = u + (1 )k( ) =) = k( ) k( )+u (3) Gol'dshtein et al. (1996) have shown that heat processes are approximately 1= times slower than concentration processes in gas{solid CSTR, conc: temp:. Mediating operation is introduced as an intermediate mode 4 operation with respect to slow and fast reactor processes, conc: temp:. Taking into account the considered type of operation, traditional averaging method (Bogoliubov and Mitropolsky, 1961) can be used with period of averaging de ned by control variable period, . Note that temperature as a slow variable can be regarded as a constant during the interval of averaging. The averaged equations of (3) become d dt = û ( c) +B(1 u0011̂)k( ) ̂= 1 R 0 k( ) k( )+u dt (4) Steady states of the averaged system (4) are de ned by the relation d dt = 0 which leads 0 = û ( c) +B(1 ̂)k( ) ̂= 1 R 0 k( ) k( )+u dt (5) 2 MAXIMAL MULTIPLICITY OF TWO{STEP MEDIATING OPERATION Introducing two{step periodic input of intermediate mode can lead to essential change of CSTR dynamic behavior. In Gol'dshtein et al. (1990) was demonstrated existence of ve steady states for two{step mediating operation. In this part of paper we aim to show analytically that multiplicity features of system (5) are described by so-called butter y singularity (see, for example, Poston and Stewart, 1978). Hence for two{step mediating operation CSTR may exhibit ve steady states as a maximum. Let us underline that 5 for maximal ( ve) multiplicity of butter y singularity at least one of three intermediate solutions is a stable one. Thus stabilization of an intermediate steady state is achieved by realization of maximal multiplicity. Bifurcation set of butter y singularity is a four{dimension space described by four bifurcation parameters. So, we also determine here four critical (bifurcation) parameters to which the reactor steady state system (5) characterized by a large number of parameters may be reduced. For investigating an in uence of two{step mediating operation on dynamic behavior of CSTR it would be desirable to separate two{step control parameters from other bifurcation parameters. In particular the separation enables to nd out two{step control values for which ve steady states with stable intermediate one can be achieved (see next part). Let us consider zero{average oscillations of feed ow rate, û= u0 in form of two{step periodic input (see Fig.1) ~ u(t) = ( u1 for n t (n+ ) u2 for (n+ ) < t (n+ 1) (6) where is duty fraction and n = 0, 1, 2, ... . Note that the selected forcing function gives the same average output as widely used nonsymmetric rectangular waveform (see, for example, Meerkov, 1980). Equations describing steady states of averaged system (5) under two{step periodic control are 0 = u0 ( c) +B(1 u0011̂)k( ) ̂= k( )(k( ) + u2 + u1 u0)=[(k( ) + u1)(k( ) + u2)] (7) 6 Equations of system (7) can be combined to give a single equation (u0 + ) c = Bk( ) k( )u0 + k( )u1u2 (k( ) + u1)(k( ) + u2) (8) For reducing (8) de ne new variable z, new reactor parameters z = 1+ c u0+ + c r = c u0+ + c q = B u0 u0+ + c s = 1 c u0+ + c = 1 (1 r) (9) and new forcing parameters n1 = u1=u0 n2 = u2=u0 (10) Denominator of equation (8) is positive, so the equation can be reduced [(1 + q)z sq]e2(z+r)+ u0[(n1 + n2 + n1n2q)z n1n2sq]ez+r + u20n1n2z = 0 (11) Note that feasible region of described above parameters can be determined as follows. Reactor parameters q, s, u0 must be positive, parameter r can be negative or positive but close to zero. Control parameters satisfy, for example, the condition, 0 n1 1 n2, because relative amplitudes of control pulses are symmetric variables for averaged r |
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| Resource Type | Article |
