Lo Cholette 1983

MULTIPLICITY OF CONVERSION IN A CASCADE OF IMPERFECTLY STIRRED TANK REACTORSS. N. LOt and A. CHOLElTE* Department of Chemical Engineering, Universite Laval, Qutbec, Canada (Received 9 November 1981; accepted 8 July 1982) Abstract-The multiple steady states of conversion in the model of non-ideal CSTR’s in series have been studied for irreversible, homogeneous exothermic reactions of first order. The iuiluences of the parameters p and n and those of scune kinetic parameters on the occurrence of multiple conversions for a given residence time are presented. The existence of steady-state multiplicity of conversion in each CSTR is shown to be independent of n and depend largely on the values of EIRTo (activation energy), b/T” (heat of reaction), and p. For certain parametric values, more than three steady states can exist in such a model. INTRODUCTION Since Van Heerden [l] published his work in 1953, the study of multiplicity of conversion in reactors has become more and more important to the design and control of chemical reactors. Numerous papers have been published since then on this subject, some of which deal with ideal stirred tank reactors [2-241, controlled cycletank reactor[25,26], and tubular reactors [27-33) in which axial heat and mass transfer was present. Others dealt with tubular reactors with recycle [34,35] and with MT reactor [36]. The particular case of tanksin-series reactor was treated by Gall and Aris [37], Berger and Perlmutter [38], Eguchi and Harada [39], and more recently by other investigators [40,41]. Multiplicity of conversion has also been shown by Lo and Cholette to exist in a non-ideal CSTR 1421, characterized by m (the level of mixing) and (I- n) (the degree of feed channeling), for exothermic reactions. Recently [43], these two authors studied the performance of a model composed of a cascade of p such non-ideal CSTR’s , for simple isothermal and exothermic reactions, and showed how it could be optimized as a funcction of m, n and p and some kinetic parameters. As a result, it was interesting to observe that the existence of multiplicity of conversion in each CSTR was independent of n but depended rather on the values of E/RT,, b/To and p_ A more detailed study on the multiple steady states of conversion in such a model was thus undertaken for adiabatic conditions. Presented hereafter will be the results obtained for exothermic reactions of first order, as an example to show the complexity of the problem of multiplicity of conversion, for the parameters involved. It is believed that, for other exothermic or autocatalytic reactions, there should exist the same kind of complexity as that presented in this paper. For reference, the schematic representation of the model studied is shown again in Fig. I. As assumed previously [43-46], the model is composed of p non ideal CSTR’s in series, the volume of each tank being V; and *Author to whom correspondence should be addressed. ?Present address: Universitt de Quebec B Trois-Rivikres. TroisRivikres, P.Q., Canada, G9A SW. each tank reactor, for any tank in the series, is characterized by the parameters m and n. The definitions of m and n, which can be found elsewhere [4346], are repeated here for convenience. The parameter m, the level of mixing, represents the volume fraction of a tank in which mixing is considered to be perfect, while the parameter n represents the fraction of the feed which enters the zone of perfect mixing. The remaining fraction of the feed, (l-n), is considered to be channeled directly to the tank outlet. As before, the numerical values of the parameters m and n applicable to one tank are also assumed to be applicable to the others in the series. BASIC DIBIGN EQUATIONS FOR STEADY STATE OPERATION Referring to the jth tank shown in Fig. 1, under the condition that the volumetric flow rate q is constant, a material balance around this jth tank leads to, for 1st order irreversible, homogeneous exothermic reactions of the type r’ = KC: I (x,-l - xi) +; K(l - x,‘) = 6 xj=(l-n)xj-,+nx; (2) where x,-, = (C,- C,-,)/C,, x;= (C,- C;)/C, and 6 = mV/q which is the residence time in each tank in the cascade. The heat energy balance for the jth tank is, after substitutions of eon (1) and b = (- AH)CO/gCo into this hiTikJixJi& n L______________, Fig. I. Schematic representation of the model. Variation of _Co/r* with conversion x for first reactions. for a given 0.1) Criterion for multplle steady strifes. as a function of the total residence time &.~} 2(1+ w) (7) = n[%= &i i = 1 and 2 I .cB+exi_. 2..loi= n{( w . curves displaying S shapes such as those illustrated in Fig. (5) IOOO~ / I I of d@ddx(.4 0. the curve showing the variation of Co/r* with x.. The two points L and N in this Fig.. = 0 for Vv)n+n2Z2=0 (6) the roots of which are: [Xc”<l.92 . for all values of n../n. on any of the curves where n < 1.lO. are the values of xc.O& 0. Lo and A. the model can be ascertained. the variation of x with & then yields. vs [e r._....]~. at those two points on the curve for n = 1. is also shown again in Fig.nslJo. MULTIPLE STEADY STATES OF CONVERSION To study the multiplicity of conversion in the model for given values of A and B... The condition for the above eqn (7) to have real roots is: W>4Z(l+Z)orD>4(1+B). for first order exothermic reactions.2. (8) It can be verified further from eqn (7) that [x~.<& is the value of xc..4(1 + w)z?]“. also .~. For reference as well as for convenience.... at point O1 or 0. As seen in Fig.22) r [( w .= where A = EIRT.(1 . xi = x. also by the Newton-Raphson numerical method used in the previous work [43]. 2. Plot of [X.. whereas [x+_~./n) The application of the condition this eqn (5) yields: x&. for different values of A and D. the value of D in eqn (4) must be positive. 3.(22-x. within a certain time interval. K = * e-“RT..... the shape of the plot of Co/r* vs x becomes more and more concave as the values of A and B increase.x+. W = A/B and Z = l/B: 2 e--nwYl(nz+x(n<l))t&... 2.(l+ w)+x. Ij = 0.. As the degree of concavity reaches a certain limit. order X Fig. (a) Multiplicity of conversion when p = 1 (a.. For exothermic reactions..]~. 3. 7 of our previuous paper [43].. 3 represent two stable steady states while the point M the metastable steady state. as shown in Fig. When p = 1...~.368 S. eqns (1).. (3) as follows: (4) The rate constant K in eqn (1) can be written z em. when p = 1. The criterion whereby the existence of multiple conversions in Pi..). N. eqns (1) and (2) together with eqn (4) were solved simultaneously for the outlet conversion x. B = b/T..8 where [x(.& or [xc. CHOLETTE heat equation and simplitication: 7-i = To + bx. has been obtained by forcing the equation of d&ldx = 0 to have two real roots of x which must lie within the interval from 0 to 1.4 e”x. (2) and (4) were combined to give the following equation.... and D = AB. which have already been presented in Fig. after substitutions of x1 = +. 3. where i = 1 and 2.2nZrA . The 4(1+ B). When p > 1.~. bears no relationship with n. The residence time [BT~. The area enclosed [XC by the loop 00. In experimental studies of stability in ideal adiabatic CSTR’s. (a. it can be proved that these two roots satisfy 0 I x. Furthermore. which leads to the following necessary condition imposed upon x*: x.~Jo~ corresponding to point Oi in Fig.n)[(W+2Zn) +(I . 5 1. it would be stable with respect to the curve for n = 0..422) 2 0 must hold.4 where F... its discriminant must be greater than or equal to zero._.]~~ into eqn (5) for xc.hfPQ02 and the two loci 00. the existence of more than three conversions for a given 8~ is possible in certain cases. can be considered as a parameter in the expression for & as a function of x. In that time interval.. a) t0 and F. then the condition ( W .h4PQOz0.. indicating that the existence of multiplicity in the model is independent of feed channeling.-.. would make it possible to determine directly by experiments the conversions on the portion OIMPQ02 of the curve for n = 1...2(1 . in the feed to this tank.1) Criterion for multiple steady states.(x. Since Wr4Z+ 42.(xa)= w+1-XA (11) Fig. and OOz. =0 obtained from eqns (1) and (2). depending on which curve this state belongs to. 3.n)(l . is a region in which a state can be stable or metastable.n)xn] I.5 and metastable when point P belongs to the curve for n = 1. the results reported up to date showed that the metastable steady states defined by the curve 01MPQ02 were obtained indirectly either by feeding and dumping [253 or by temperature perturbation [47]. + (1 .42 . then d&fdx. 4.l~~/le~. As observed. = x~.rA satisfies 0 5 xA 5 1.. where n < 1.): P&A.1. as discussed in the preceding paragraph. n) = (22 . in eqn (lo). implying that the two roots of the equation are real and positive. Letting x.Multiplicity of conversion in a cascade of imperfectly stirred tank reactors inequality of D> satisfies 0 C [xcn< .n)(l -x.w)n . with the K given by eqn (4). An example is shown in Fig..ZZ)nx.Jo~. (b) Multiplicity of conversion when p > 1 (b..6 XP 0._. (14) (9) Obviously . a study on metastable steady states in a CSTR was carried out and the results obtained will be reported in a future publication. which was also obtained by Douglas for n = 1 [4]. the stable states corresponding to the portion 00. the value of xp_. or 02 is a straight line.. can be determined by the substitution of [x~.+ n) 10. for tist order reactions when n =“l..corresponding to each of these two roots was obtained by solving eqns (l).l~~. for j = p yields the following: Since W and Z are positive and ._.loi} = [~~..W[x* + (I.)1xX 369 (12) (13) For eqn (10) to have real roots. In so doing.W must be negative. This again leads to the same inequality of D >4(1+ B) obtained previously for a single CSTR.~(W-42-zz)/(W+4Z+4) 05x*51. The points as 0t4 and P._. composed of the curve 0. 2Z. the existence of multiplicity of conversion in the pth tank depends on the conversion x.2) Determination oj the region in which a state can be stable or metastable. 4 were plotted with the results thus obtained.XJXA . So. the locus of the point 0.Joi = n[~. Taking the state defined at point P as an example. in the interval [BT(~=I)]~ i 0~ < [BT(~=..XA) + ( w . a point such as P or Q is then determined with the conversion obtained at a pair of fixed values of 6. and n. the criterion for the occurrence of multiple conversion at a given f& can be determined also by the preceding method used for p = 1. (2) and (4) with the method of Regula Falsi. indicating that eqn (8) applies to all values of p. 4 for R = 1 when D = 9 and A = 30 and zeTA = 0. Making use tof the method. . An experimental approach is suggested which consists in operating the reactor by varying n with 0. Under the condition of D > 4(1+ B). Effect of p on the variation of the outlet conversion x with the residence time &.(X& n) = n2z2(1. for 05 xA I 1.~.Joi c 1. The value of &. one has F&f.. shown in Fig. on the 0. starting from the origin with a slope of r*{x = . The situation that a state can be either stable or metastable. the variation of pM with these two kinetic parameters is presented in Table 1. _ . p. can exist. while those corresponding to OIPIQl yield such states of Q2R2S2. is seen to increase non-proportionally with D or with B for a constant A. is metastable. For a particular value of p. as when p = 3. it can be concluded that the number of CSTR’s so required in the series as to approach an ideal tubular reactor is quite different. channeling of the feed has an extremely small effect on pM. . 4. corresponding respectively to the points 0. (2) and (4)... * OpM . When p = pM.3 19. also for n = 1. = 1.. . 0.: (xJmax = XA = (W-422)/(W+42+4) (15) in that case eqn (10) has two equal roots. the number of curve segments. 4 and in other figures presented later were the numerical solutions of eqns (1). . 4 shows that the major region where all states are metastable is the loop 0. where i= I. the portion OiPi for which any state is metastable is a function of p when p is smaller than a certain value. As an example. However.for first order exothermic reactions when n = I Kinefic pB=-feTB PM D 6 9 12 15 A 30 30 30 30 B 0. and Pi. Lo and A. any state on the curve O..5 275. Except for the points on the loop QIOz . In other words. Effect of kinetic parameters on the value of pM. (b.4) Influence of n on mukipicity oj conversion of constant p. _p.370 S. 5 and 6. also using A = 30 and D = 6.. 5 and 6. Because of the drastic increase in pM with a small change in B. for first order exothermic reactions when D = 6 and B = 0.2 . PzP. The point OPM is the point at which one stable state becomes possible under the conditions D > 4(1+ I?) and obtained when the value of xA in eqn (14) is at its maximum.3) Influence of p on mukiplicity of conversion al constant n. the portion OiPi converges to point O.5 3. The residence time &oi and 0.. is situated around the neighbourhood of x.. Oz. In other cases. Fig. (b. one of which will be shown later. It has been proved previously from eqn (10) that the existence of multiple conversions in the model is only a function of kinetic parameters: A and D.P. I.6 81.4 Table 2.e. Effect of feed-channeling on the value of p M. . say p. other values on the curves shown in Fig.2) Determination of regions in which a steady state is metasfable... In general. P. (2). compared to A and D. can be determined by means of eqns (l). Table 1. An example of the influence of feed channeling on the performance profile is shown in Fis. N. At this moment no analytical solution concerning these regions has yet been arrived at.. more than one in number. the value at which the existence of multiple conversions is no longer possible and beyond which only one conversion will be obtained at any given residence time &.. by the Newton-Raphson method. Such an effect is exampIified in Table 2 for A = 30 and D = 6.3 0. (b. as shown in Figs 4 and 5.4 0. results obtained from the study for a set of given kinetic parametric values revealed that the value of pm would decrease with n and that the shape of the curve for x. for any value of n. is equal to p when p < pM. This table shows that pM increases with (1 . the value of pw should be a function of A and D. 2. . When n = 1.. . regions of this kind. As seen in Fig. 4. Cno~erre For a particular value of n. such as OZPZ and Q& in Figs.. .2 0. However. there will be no possibility for the model under study to have multiple conversions at any given &when p > pM. as a function of 0~ can be modified considerably by the presence of feed channeling. 4. it is not curve of p = 1 yield the states lying on the part OOzPzQz of the curve for p = 2. the segment QiRi. which presents the various values of pM obtained for several values of n..P. Thus. in Fig. so that there will then be only one conversion corresponding to a given 0. when compared with the results obtained previously for isothermal reactions [43].n) but to a much lesser extent than that shown in Table 1 for the influence of kinetic parameters on pnr.O. . where i = 2. Op. From this table p. (4) and (10). . conversion of reactant in the perfect mixing zone of the jth tank conversion of reactant at the outlet of the XD pth tank E/RT. 6.6 XP 0. K temperature of the feed to the (j + 1)th tank. 5 and 6.Multiplicity of conversion in a cascade of imperfectly stirred tank reactors 371 0. jth tank X. 0. n) function.8 t 0. VT total volume of a tanks-in-series reactor. dimensionless n fraction of the feed entering into a tank reactor. 1. initial reactant concentration in the feed. T K T: temperature in the perfect mixing zone of the jth tank. l. maximum temperature rise for complete conversion of reactant. and n 5 1 conversion of reactant at the outlet of the X.AH)C&iC~). reactant concentration in the perfect mixing c.~l. K B b/To. see eqn (11) F. It can be concluded that in the presence of feed-channeling. see eqn (12) F&A.6 n =0.) R gas constant. cal/(gram-mole) function.4 maximal value of X~ (Xdmax outlet conversion of reactant when p = 1 X(.) Fzh.. zone of the jth tank. gram-mole/l. gram-mole/l. the difference between the two cases can be seen by comparing Figs. gram-mole/l.for first order exothermic reactions when p > 1 and n = 0. dimensionless P number of tanks in series PM the maximal value of p at which the existence of multiplicity of conversion begins to disappear volumetric flow rate. Effect of feed-channeling on the variation of the outlet conversion x. (gram-mole)(K)/cal TO initial temperature of the feed. NOTATION 0. average specific heat of a reaction mixture. with the residence time BT. n) function. A b . sec./sec.5. it would be impossible to attain conversions close to 1 with the model of a cascade of CSTR’s. Effect of feed-channeling on the variation of the outlet conversion x0 with the residence time &.4 Fig. Acknowledgement-The authors express their gratitude to the Natural Sciences and Engineering Research Council of Canada for financial support of this research. X conversion variable conversion parameter X. K V volume of each tank in the tanks-in-series. dimensionless activation energy ( . see eqn (13) heat of reaction.D Eb/RT& dimensionless E activation energy. for tirst order exothermic reactions when p > 1 and n = 1. r(f rate of reaction.5 D= 6 A=30 Fig. caNgram-mole) -(AN) index r: rate of reaction. 1. set-’ m level of mixing. The use of such a model might also introduce additional control problems due to the increase in the value of pM. 5.8 the case when n < 1. dimensionless heat of reaction reactant concentration at the outlet of the G jth tank. call&ram K) .(x. gram-mole/l. . and Engel A. 1974 29 967. Hlav&k Chem. WI Lin K. 1%9 t4 1441. Can. Chem. and Sinkule J. Kagaku Kogaku 1966 4(l) 54 (English version]. and Luss D. 1964 60 1. and Reilly M. L311Gupalo Y. N. F. S. and Cholette A. Van Heerden C. and Amundson N. and Wu L. and Steven W.. Engng Japon 1969 2 95. J. K. Chem. Chem. Douglas I. T.I. Chetn.. 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