Ind. Eng. Chem. Res. 1988, 27, 2218-2224
2218
Lumping Analysis for the Methanol Conversion to Olefins Kinetic Model Octavian M. I o r d a c h e Department of Chemical Engineering, Polytechnic Institute, Bucharest 78126, Romania
Gheorghe C. Maria* a n d Grigore L. Pop ICECHIM-IECB, Chemical and Biochemical Energetics Institute, Spl. Independentei 202, Bucharest 77208, Romania
T h e purpose of this paper is to illustrate the possibility of using stochastic and fuzzy methods to obtain reduced kinetic models of methanol conversion to olefins. Simulations of the reaction system by Markov chains coupled with the calculus of the informational entropy of the lumped chain are presented. A rough description of the kinetics may be supplied by fuzzy relationships. T h e fuzzy approach could also engage practical experience in solving the problem. A fuzzy entropy, considered as the measure of loss of information by lumping, was introduced. 1. Introduction The classical approach in the study of chemical complex systems is to represent the process by means of a deterministic detailed model. Some situations, however, are so complex that it is often impossible to formulate complete mathematical descriptions, let alone a solution of such complex models. A natural trend and a main problem is to simplify the models and to make them adequate for analysis, design, and control. A well-known simplification in the domain of complex kinetics is that of lumping. The complex mixture of chemical species is lumped in classes, and the kinetics of the classes is examined. Wei and Kuo (1969) develop necessary and sufficient criteria for determining whether a system is properly and exactly lumpable. Some recent applications (i-e.,Luss and Golikeri (1975), Hoffmann and Hofman (1977), Aris (1982), Hosten and Froment (1984), Li (1984), Zhu et al. (1985), Frenklach (1985), Coxson and Bischoff (19871, Peereboom (1987) proved that there is interest in this attractive technique. As an observation, it must be pointed out that we are frequently faced with nonlinear reaction systems, and thus the partition of a mixture in classes is imposed by chemical and practical reasons rather than by artificial restrictions such as lumpability conditions (Appendix A). Another route, lumping in the reaction sequence, was developed by Ermakova et al. (1980a,b) using the “ridge regression analysis” of Hoerl and Kennard (1970). In order to reduce a complex kinetic model which describes the methanol conversion to olefins (MTO) process (Mihail et al., 1983, 1987) to a minimum set of equations, Maria and Muntean (1987) use a combinative parameter estimation-reaction sequence reduction strategy. The aim of this paper is to propose and to compare two different methods, namely a stochastic and a fuzzy method, applied for lumping the MTO kinetic model. In the first part of the work, we obtained a lumped scheme using the reaction system simulation by Markov chains (Too et al., 1982) combined with a stochastic method of lumping (Iordache, 1987; Iordache, Corbu, 1987). A study of the informational entropy is used to lump in classes components of comparable entropies and to choose for experimental observations the component or the class corresponding to the maximum entropy. The stochastic approach to solve the problem cannot sufficiently engage the subjective or the practical experience. Otherwise, in many situations it is difficult to estimate the kinetic constants or the probabilities of tran-
sition from one species to another. A fuzzy approach to the problem is proposed in such cases (Negoitii and Ralescu, 1975). Different lumpings could be compared using fuzzy entropy as a criterion of lumping. 2. Kinetic Models for t h e MTO Process Although deeply investigated, the MTO process has an unsufficiently elucidated mechanism. To solve the question of the first C-C bond formation, a carbenium or trialkyloxonium ion intermediate (Derouane et al., 1978; Van den Berg et al., 1980; Perot et al., 1982) or a scheme based on the carbene intermediate (Chang and Silvestri, 1977) was proposed. Among other viewpoints concerning hydrocarbon formation, we notice the olefins electrophilic alkylation with methanol (Anderson et al., 1979), the ethylene formation by transposition of o-ylides to an ethoxy group (Mole and Whiteside, 1982), and the major role of the diffusion phenomena on the possible reaction mechanism (Haag et al., 1982). Derived from the basic scheme, some kinetic models with different orders of complexity were advanced (see, for instance, the review of Liu et al. (1984)). The product reaction mixture contains a great number of chemical compounds. Thus, the proposed models, which include from 1 to 33 chemical reactions (Cobb et al., 1978; Chen and Reagan, 1979; Chang, 1980; Anthony, 1981; Anthony and Singh, 1980; Anthony et al., 1981; Ono and Mori, 1981; Mihail et al., 1983,1987), take into account either scheme with the individual species or use components lumped into a number of groups. Starting from an extended MTO kinetic model with 33 reactions (Mihail et al., 1983) and only 13 linear independent chemical equations, Maria and Muntean (1987) obtained a reduced model, using a combinative reaction sequence reduction strategy. The resulted model (Table I), with 14 reactions, includes 18 compounds. 3. Major Compound Groups for t h e MTO Kinetic Scheme, Identified with a Stochastic Method
Markov chains have been shown to be effective tools for analyzing kinetic models (see for instance Too et al. (1982)). In such an analysis, a state of the Markov chain is associated with every chemical species. The probability that a molecule of type i at time k will be in state j at reaction time k + 1 is denoted by pi,@). Obviously r
Cp,,(k) = 1 /=I
* Author to whom
correspondence should be addressed.
i = I,..., r
!I)
where r denotes the number of chemical species (in the
0888-5885/88/2627-22~8$01.50/0 6 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2219 Table I. MTO Kinetic Model (Maria and Muntean, 1987) and the Entropies ( H Min ) Lumping (I,) no. reaction kinetic parameters" compdsb HM 1 2CH3OH + CH3OCH3 + HZO k, = 14.47 CH2 K1 = 2.71 2 CH30CH3 2CH2 + H 2 0 C2H4 1.5 x 10-7 kz = 14.05 3 CHZ + CHBOCH3 CzH4 + CH3OH C3H6 3.3 x 10-7 k3)nCH2= 7.56 4 CH2 + CHBOCH3 C3H6 + HzO C4H.9 1.1 x 10-7 k,'nCH, = 4.04 5 CH2 + CZH4 C3H6 C5H10 near 0 kS/nCH2= 0.59 6 CHZ + C3H6 C4H8 = 0.93 CH30H 2.2 x k(& = 0.52 CHSOCH3 3.3 x 10-4 7 CHZ + C4H8 C5H10 8 CH2 + Hz CH4 kB)nCH2= 3.14 H2O near 0 9 CzH4 + C4H8 C2H6 + C4H6 kg = 5.85 H2 3.3 x lo+ 10 C3He + C4H.9 C3H8 -I-C4H6 k,, = 26.51 CH4 11 C4H.9 + C4H.9 C4Hlo + C4He kl1 = 92.95 C2H6 12 C5H10 + CSH1O C5H12 + C5H8 kI2 = 259.9 C3H8 13 CHSOH CO + 2Hz k13 = 0.44 C4H10 k14 = 0.01 C5H12 14 CZH4 CH4 + C
- --
+
class no. carbene, !
11
+
-t
-t
c5+
I
light olefins, 1 high olefins, ? oxygenates, 3 water, i hydrogen,
6
paraffins,
B
"Dimensions according to the reaction time (s) and concentrations (mol/mol of feed); values for 370 OC and pure methanol feed. 'CS+ species includes CO, c , C4H6,C5HB.
particular case studied here, r = 15, if the C,+ species includes CO, C, C4H6,C5H8(Table I)). The set of chemical species is denoted by A = (1,2, ..., rj, and the number of moles of type i in the reactor at moment k is denoted by ni(k). The transition probabilities are computed using the kinetic constants k (Table I): (2)
ited to the chemical species indexed by m. Not? thatJhe lumping chain, having only two states, classes 1 and 2, is a non-Markovian chain. Starting with the initial conditions p i = 0, i # m, and pom = 1, we obtained n i ( k ) ,pi,&), and the conditional probabilities pkj (eq A3 in Appendix A). Also by use of eq A4 and A7, the entropies are obtained for every 1, and m (Table I). To obtain accurate results. the reaction time s t e n At. must be sufficiently small to render the transition probability from one state to another to be a very small one. Because the computational time increases if At decreases, the time step was varied according to the strategy used by the Runge-Kutta-Gear integration method for the continuous time kinetic model (Maria and Muntean, 1987). Note that since t = 0.2 s, the entropies are practically constant. Species 1is stationary, while chemical species 10-15 are absorbing. Due to the fact that p i j = 0, the conditional probabilities rest unchanged, Le., pk = S O (eq A3). Consequently, no entropy could be obtained in this case. The results presented in Table I suggest that compounc@ could be lumped in three main classes: light olefins 2, oxygenates (without water) 3, and the paraffins 6. An interesting conclusion according to the obtained entropy is that a good control of the MTO process could be obtained if the measurements are restricted to oxygenates and olefins (great entropies). Observe that l6 or 1, gives more information than other lumpings. Other $asses outlined-in Table I cor_respondto carbene i, water 4, hydrogen 5, and C5H10 7. A remarkable fact is that the components belonging to the same chemical class have comparable entropies. Components 1, 5 , 8, and 10-15, having null (or near null) entropies, could not be discerned by using our stochastic method of analysis. On the basis of the model w$hA14-chemical reactions and by use of the main classes (3, 2, 6) of compounds, with computed entropies > > 0, the lumped kinetic scheme of paraffins (and aromatics) from oxygenates (without water) via olefins (of Chang and Silvestri (1977) (with two chemical reactions) of Chen and Reagan (1979) and Ono and Mori (1981) (with three reactions)) appears as a natural reaction path. The role of t h e carbene intermediate (class 1) and the presence of class 7 (higher olefins) were outlined by Chang (1980) and Anthony (1981) in the following mechanism: paraffins (and aromatics) formation from oxygenates (without water) via carbene-light olefins-higher olefins (four chemical reactions). I
where n, is the number of reactions, ns, is the number of reactant species in reaction u, vju is the stochiometric coefficient, and At is the time increment (Too et al., 1982). For instance, CH30H (species 6 in Table I) has p6,7 = P6,8 = klns(k)At;p6,9 = 2ki& p6,15 = k13At; and P6,6 = 1- b 6 , 7 + P6,8 + P6,9 + P6,15)* The evolution of the number of moles in discrete time is given by
Model 3 is in fact a discretized form of the system of ordinary differential equations that kinetically describe the chemical process. The transition probability matrix of the Markov chain is denoted by P(k) = ( p i j ( k ) )i, and j = 1, ..., r. The elements of P are nonnegative, and the elements in each row sum to unity (see eq 1). The absorbing states of the Markov chain are characterized by p i i = 1 and pji # 0, but pij = 0 (i.e., the nondiagonal elements of the ith row are null). Once a molecule enters such states, it will never return to any of the transient states. If pij = 0 and pp = 0 where i # j (Le., the nondiagonal elements of the zth row and column are null), state i corresponds to an ergodic class and could be studied as a separate chain. The species indexed by i are i? a stationary state. We consider here that the carbene CH2 (i.e., species 1 in Table I) satisfies the so-called quasi-stationarity hypothesis; consequently P11 = 1. The lumping method proposed by Iordache and Corbu (1987) (see Appendix A) is closely followed to lump the states of the Markov chain characterized by matrix P(k). The main difference is that the transition matrix is time dependent (the chain is nonhomogeneous in the case of nonlinear kinetic models). The class of chemical species a t moment k is denoted by ? k . All classes are disjoint s.ts of A . We again consider lumping (1,) in two classes, 1 = {m)and 2 = (1, 2, ..., m-1, m+l, ..., r ) . Such lumping corresponds to practical cases when observations are lim-
2220 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 Table 11. Fuzzy Similarity Matrix, R CHBOH CH30H
1 CHBOCHB
CHSOCH, 0.95
HzO 0.5
1
0.5
HZO
1
species CsHtj CdH,
C2H4 0.5 0.5 0 I
0.3 0.3 0
0.1 0.1 0
0.95
0.5 0.95 1 HO
C3H6
C4Hs
HO”
CH4
Pa
C,+”
Hz
CO
0 0 0 0.2 0.5
0.2 0
0
0.2
0.95 1 CH4
0.1 0.1 1
0 0 0 0.1 0.2 0.4 0.6 0.1 1
0 0 0.1 0.2 0.4 0.6 0 0.2 1
0 0 0 0 0 0 0 0 0.1
0.2 0 0 0 0 0 0 0 0 0 0 1
0 0.1 0.1
P
symmetric
a
c,+
Hz
1
co
HO notes the higher olefins C6HIO;P notes the paraffins C&, C3H8,C4H10;C5+ notes the fraction which includes C5H8,C5H12, C4H6,C.
4. A Fuzzy Analysis for the MTO Process In some studies of complex chemical kinetics, there is no mathematical model of the process but there exists a chemical analysis of the feed and products (or intermediates); the modeling could be based on incomplete or qualitative prior information. Due to the imprecision and nonquantitative factors, the deterministic or stochastic model cannot be used, but the process could be represented in terms of fuzzy relations. Fuzzy relations may be used too as a first rough process representation because the deterministic or stochastic route requires supplementary information and computational effort. We consider a class to consist of a number of similar chemical species gathered together. The primary objective of lumping techniques would be to classify a given set of compounds by assigning them to a reasonable small number of homogeneous classes. The term homogeneous implies that all species in the same class are similar to each other and are not similar to species in other classes. This means that members within each class are sufficiently alike to justify ignoring the individual differences between them. The key problem in fuzzy lumping is to define the subjective similarity. Therefore, the concept of the fuzzy relation would seem to model the similitude between any two species (see NegoitB and Ralescu (1975) and Fan et al. (1985) for excellent tutorial notes). In order to lump in classes ensembles of chemical species described by means of fuzzy sets, fuzzy relations of similitude and fuzzy entropies are defined in the following. The degree of similitude (or similarity index) of species i and j is denoted by a,, E [0,1] where ij E A . In fact, we try to replace the probability of transition, p V , of a stochastic model by the degree of similitude, a,, obtaining a fuzzy model. The relation of similitude is characterized by a fuzzy similarity matrix, R = a, , where ij E A , of the type r X r. Obviously a,, = 1,and a,, = u,~. The selection of good similarity indexes represents a major decision in lumping methodologies (Jardine and Sibson, 1971). For instance, a natural definition of the similarity of two chemical species should be proportional to the number of reaction steps of their closest common ancestor species. Thus, we take a,, = 1 - uk a,, = 1
for i z j
for i = j , k = 1, 2, etc.
(4)
with 0 < a < 1, if there are k steps until species i and j branched off (see Scheme I). In this case, we considered the step of reaction as determinant of the similarity of two species. Considering as another lumping criterion the chemical class to which the component pertains (for in-
Scheme I CH30H
‘ZH6
with 0 < a < 1. The fuzzy analysis method is presented in Appendix B. Let us consider, as a prior information about the MTO process, only the main species (Scheme I) determined from chemical analysis. The carbene and carbenium ions or other possible active intermediates are omitted from the list, not being determinable with common product analysis. In order to determine the reaction mechanism and to construct a kinetic scheme, important experimental efforts are necessary to obtain product distribution under different reaction conditions and special experimental techniques, pointing out the active intermediates and the main reaction path. The question which appears now is how it is possible to construct a lumped kinetic model with major compound classes, using minimum experimental information about the reaction products. The fuzzy technique could be effective if qualitative “historic” information about compound formation is available. With the notations of Appendix B, we may thus construct a fuzzy similarity matrix, R. As an example, in
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2221 Table 111. Elements kij of the Fuzzy Similarity Matrix, R = aij, Where aij = 1 - akb (Equation 4)' species cz' c3/ c4/ Cgl CH3OH CHBOCHQ HzO Hz CH4 Cz C3 C4 C5 3 1 2 1 cz' 3 3 1 1 3 3 3 3 1 1 3 4 4 4 4 1 2 1 c3/ 4 5 1 2 1 1 1 3 4 5 5 c4/ cgl 1 2 1 1 1 3 4 5 6 CH3OH 1 1 1 1 1 1 1 1 CHSOCH, 1 1 1 2 2 2 2 HzO 1 1 1 1 1 1 H2 1 1 1 1 1 C H I 1 1 1 1 c 2 3 3 3 c 3 4 4 symmetric c 4 5
c5
"Notations: C;, C3/, C i , Cgl denote the olefins CzH4,
CO 1 1 1 1 1 1 1 1 1 1 1 1 1
co
C4H6 3 4 5 5 1 2 1 1 1 3 4 5 5 1 C4Hs
C5H,+ C 3 4 5 6 1 2 1 1 1 3 4 5 6 1 5 C5H8 + C
C4H8,C6Hlo;Cz, C3, C4, C5 denote the paraffins CzHe, C3H8,,C4Hlo,C5HI2.
Table 11, the major reaction compounds are listed: CH3OH, CH3OCH3, HzO, CzH4, C3H6, C4H8, CH4, Hz, co;the higher olefiis HO (i.e., C5H10), the paraffins (C2-C4without CH,), and C,+ fraction (including C5H12, C5H8and C4H6, C) are considered as separate groups because of chemical structure reasons and the small amount of products. The similarity matrix elements, aij E [0,1], are as close to unity as species j could be obtained from species i by a direct route. For instance, we take UCH~OH,C~H,= 0.5 > UCH 0H,c3 = 0.3, because C2H4 may be obtained directly from dH30!d easier than C3H6. At the same time, aCH30H,C2a > u C H ~ ~ H , ? O because the olefins formation (with a great amount in product distribution) is considered the main reaction path, while the decomposition of CH30H in CO and Hz is considered a secondary reaction (induced, for instance, by the nature of the reactor wall (Mihail et al., 1987)). As another example, we take aCaQC H > ac &,YO for great MWHSV according to the observadons of Aspmoza (1984), etc. This way to construct the fuzzy matrix, using any previous information on the possible uncoupled reactions, depends on the professional experience of each analyst. The fuzzy analysis (Appendix B) for levels X = 0.8-0.95, applied on matrix R (Table 11),indicates the following main classes: oxygenates (CH30H, CH30CH3),water, olefins (C2H4, C3H6,C4H8,HO), CH4,praffins, C5+, Hz, CO. In this case, H(R) = 72.6, while H(R) = 32.1. The same conclusions can be drawn if aij is perturbed in the range ai, f 0.1. Such a result points out the same major compound classes as the stochastic analysis of section 3 (see also the scheme of Chen and Reagan (1979) and Ono and Mori (1981)) but does not distinguish between the relative importance of each class, as the entropy index for the stochastic analysis. A more quantitative way to construct the fuzzy similarity matrix, R, is to consider, starting from a common ancestor species and using (4),that ai, elements are as close to unity as the number k of reaction steps (in which i and j are not separated) increases (see Scheme I). As an example (Table 111),u C ~ ~ =, 1C- ~a3,~because ~ in the third reaction step ( k = 3) from the common ancestor species, CH30H, the C3H6and C2H6 species are individualized. The approximative reaction path proposed in Scheme I supposes a previous qualitative experience regarding the possible reaction routes to obtain the products and could have an important subjective character. To construct such a reaction path graph, some limitations imposed by the properties of the similarity index given by (4)or (5) must be underlined: (i) a species must appear just once and must be obtained from a single species, avoiding the reversible reaction routes (for instance, reaction 14 in Table
I was neglected in Scheme I); (ii) the graph describing the reaction path should be a tree. The fuzzy analysis of Appendix B applied to matrix R (Table 111)with a = 0.5 indicates for levels X = 0.94496 the following main classes: CH30H, CH30CH3,H20, C2H4, C3H6, C2Hs,C3H8,higher hydrocarbons (C4H8, C4H10, C5H10, C5H12, C4H6, C5H8+ C), HP, CHI, CO. In this case, H(R) = 124, while H(R) = 67.8. The same result is obtained for any a E (O,l), but for another A. Such a conclusions eventually explains the special attention paid in the literature to the methanol to hydrocarbons global kinetics (Anthony et al., 1981; Chang and Silvestri, 1977; Cobb et al., 1978); the CH30H and CH30CH3species are considered at the equilibrium state, and the major products are correlated with the methanol conversion (Anthony et al., 1981). Obviously, in practical situations it is important to choose similar matrices ((4) or (5) or any other), giving a smaller loss of entropy at the same level, A. 5. Conclusions
Throughout the development of the theory of fuzzy sets, it has been emphasized that the concept is basically nonstatistical in nature and the stochastic models are not appropriate for treating the kind of uncertainty which stems from ambiguity of the system. Stochasticity involves uncertainty about the occurrence of an event precisely described. Fuzziness deals with the case of the object itself being intrinsically imprecise. However, it could often be the case that both fuzziness and randomness are present. To find statistical properties of a fuzzy similarity matrix at different levels of lumping seems to be a new problem. If few experimental data are available for a chemical process, but previous qualitative information about possible mechanisms is present, a prior route to obtain lumping models with major compound classes is offered by the fuzzy analysis. A t this stage, the ability to construct the fuzzy similarity matrix between chemical species introduces an important subjective character to the analysis. It should be pointed out that it is difficult to specify the exact numerical values of the similarities, aii. However, the ease of manipulation in the composition and lumping rules is an important factor in the application of the fuzzy approach. If the experimental information about the process is sufficient to propose a mechanism and an extended kinetic model, the lumping stochastic analysis is a worthy alternative to distinguish between major component classes and to reduce the model. This route, using an entropy analysis, indicates the reaction time domain which offers maximum
2222 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
information about the process and points out the relative importance of component classes by comparing the entropies obtained in the whole range of reaction times. Thus, it is possible to indicate several reaction paths for the reduced model. Although the obtained classes in our application leas to the well-known reduced kinetic scheme, the lumping stochastic analysis may underline the possible and nonevident reduced models for other complex kinetic cases. Finally, it should be emphasized that by lumping the species of a kinetic model under the conditions we have stated, one may obtain a new model with a smaller number of species. At the expense of getting less information about the original system, we are faced with a more manageable model.
Nomenclature a = constant ai, = degree of similitude, ij E A A = set of chemical species B, C = matrices defined in Appendix A dij = distance between i and j h = number of classes H(R) = fuzzy entropy HM = informational entropy k = kth step, kth timeincrement; kinetic constant 1, = lumping, 1 = (m), 2 = 11, 2, ..., m-1, m + l , ..., r ) ,m E A M = (nij), i, j E N; transition matrix n i = number of moles of type i n, = number of reactions P k J = probability that the particle is in j E A conditioned on being in the classes (gl, ..., g k ) k E N; probability vector pk = (pk', ..., p i j = probability of transition from i to j, i j E A P = (pij),ij E A ; transition probability matrix P = transition matrix of the lumped chain P(a/b) = probability of a conditioned on b r = number of chemical species R = (aij),i j E A ; fuzzy similarity matrix R, = similarity matrix at the lumping level X S k = (Ski, ..., S k r ) , k E N, possible Value O f Pk At = time increment i k = class at moment k Greek Letters X = lumping level vju = stochiometric coefficient R;, = probability of transition from sito sj nj = element of the stationary vector
Symbols = indicates the class in the lumped system * = composition rule ( = order relation on the set of similarity matrices max = maximum value min = minimum value
in class k , and C is an r X h matrix where the nonnull components of its kth-column are equal to 1 and correspond to the states in k , too, 1 6 k 6 h. The condition of lumpability is
u u
(AI)
CP = PC
(A21
that is, (see Hoffmann and Hofman (1977)).An application of the lumping of a homogeneous Markov chain to aggregation of chemical reaction schemes is due to Wei and Kuo (1969). In this case, the first-order chemical kinetics have been considered. The conditions in wliich the classes of chemical species may be considered independent entities for kinetic purposes were obtained. The lumpability conditions (A2) is not checked in most practical cases. More exactly by Lumping, w,e obtain a non-Markovian chain, defined as A = ( 1 , ..., h]. The method presented in our application consists of replacing the study of the non-Markovian chain of classes by an associated Markov chain that allows an entropy analysis of the problem. The entropies corresponding to different lumpings, l,,, are obtained starting from the conditional probability pkl = PG/Rl, ..., ? k ) , where pkj denotes the probability that the molecule is of the j type at moment k conditioned by the presence of this molecule in classes R1, ..., R k at moments 1, ..., k , respectively. Using Bayes's theorem results in
The vector of conditional probabilities is denoted by pk = ..., pkr). Obviously x & $ k j = 1. T O obtain the entropy of the lumped chain, we considered the sequence of vectors Pk, giving all the possible valu_es of Pk if the marked chemical species start from class 1 (i.e., the molecule is initially of the m type) and then remains in class 2 for any time k 2 1. To avoid confusion, we denote this vector of conditional probabilities by sk. We have s o = (0, ..., 0, 1, 0, ..., 0) where pol = 0, p: = 0 , ...,pom = 1, ..., = 0, s 1 = (PI1, ..., PIm-', 0, P I m f 1 ,..., p { ) , sk = (pkl, ..., pkm-l, 0, pkm+', ..., pkr), etc. The elements of sk are obtained recursively from (A3). The probability of transition from sk to sk+l denoted by T k , k + l is given by
oh1,
..e,
r
A
Appendix A: Lumped Markov Chain It is known that in a Markov chain the impact of the past on the further evolution of the described system is concentrated in the state present at the last moment at which the system was observed. Let us consider a Markov chain with state space A = ( 1 , ..., rtand tran@tion matrix P. Consider a decomposition A = 1 ... h of th_estaie space in pairwise disjoint sets (classes). Denote A = {I, ..., h). A Markov chain is exactly lumpable if A (valued random variable) is again a Markov^chain. The transition matrix P of the lumped chain is P = BPC. Here, B is a h X r matrix where its kth row is a probability vector whose nonnull components are those corresponding to the states
CBPC = PC
rk,k+l
=
r
x
i=lj=l j#m
,
(A41
pk'Pij(k)
In this way, an associated Markov chain, having s k as states and q k + 1 as transition probabilities, results (see Blackwell (1957) and Iordache (1987) for rigorous considerations). The transiton matrix of this chain is 5700
IO
5701
0
0
5712
0 0
7r23 0 0 0 ...........................
5720
:::) ...
Obviously q . 0 = 1- TkTkp+l. The elements of the stationary probability vector of the matrix M, denoted by ?r = (80, nl,r 2 ,...) and calculated from the set z n k a k ; = nj k=O
are
j = 0 , 1, etc.
(A6)
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2223
According to the definition, the entropy of the associated Markov chain is m
HM = -[
1r k ( r k , k + 1 k=O
rk,k+l
+ r k , 0 In rk,O)] (A81
Note also that r k , k + l , M, and H M depend on the time step
k. Appendix B: The Fuzzy Method of Lumping The lumping algorithm (see Negoi$ti and Ralescu (1975)) starts from the sequence of matrices R, R2, ..., Rk,where Rk = Rk-’*R. The max-min composition rule of two fuzzy matrices (written as R*S) is defined by (R*S)ij= maxk [min (rik,skj)]
(B1)
where R = (rij),S = (sij),ij E A . An order relation, (, on the set of similarity matrices is defined by R ( S if and only if rij < sij for all ij E A . Since R has a finite number of elements and the composition rule is max-min, after a finite number of powers, the sequence R” becomes stationary; that is, there exists n so that
R ( R2 ( ... ( R” = R”+’ = etc.
(B2)
The elements of matrix R” are denoted by ai,(”). The partition in classes is established on the basis of the lumping level, X, 0 < X < 1. Two species ij (ij E A ) are assigned to the saqe class if uii(n)> A. In this_way,-we could obtain a new set A = (1,2, ..., fil where 1, 2, ..., h are the classes of the partition ( h < r). The similarity matrix of classes is constructed by using the rule
(R); = min,E;,tEjaSt(n)
(B3)
(RIG= max,E;,tEj
(B4)
or the rule Obviously, matrix R is of the fi X fi type. The similarity matrix obtained by lumping a t the X level is denoted by R,. If X decreases frpm 1 to 0, procedure B3 or B4 leads to lumped matrices, R. The elements of R are smaller than or equal to the original elements of R in procedure B3. To compare different rules of lumping and to obtain a global measure of the degree of similitude of different species characterized by the matrix R = aij, the fuzzy entropy is defined as
H(R) = - [ l x a i j In ai,] - cC(1 - ai$ In (1 - ai,) i J
i
l
(B5)
Observe that the first term in H(R) is formally similar to the entropy of Markov chains. The last term in (B5) could be related to the fact that we take into account the similarities, ai,, as well as the dissimilarities, 1- ai,, in order to lump different species in a class. Note also that R and R = 1- aij have the same fuzzy entropy. R is the so-called dissimilarity matrix. The fuzzy entropy satisfies the following properties: (i) H(R) = 0 if ai, E (0, 11; (ii) H(R) has the maximum value if ai, = 0.5 for all ij, i # j ; (iii) H(Rhl4 H(R) for any 0 < X < 1, and (iv) if XI < X2, then H(R,,) < H(R,,). According to (iii), lumpings lead to a loss of entropy (some elements of the original matrix R disappear). Property (iv) follows frcm the fact that Rh, has more rows and columns than-R,, and that, according to (B3) and (B4), the elements of Rh are the same as the elements of R,, (see also DeLuca ana Termini (1972)). In other words, the level
of imprecision is 0 if and only if aij is the usual characteristic function and is maximum if the degree of similitude is 0.5. The degree of imprecision must decrease when a fuzzy subset is sharpened. It should be noted that the “distance” between two species ij E A , given by dij = 1- ul,, is a nonarchimedean distance in A (see, for details, Jardine and Sibson (1971)). In a nonarchimedean space, A , the usual triangular inequality is replaced by a stronger one: dij < max (dik, dk,}
ij,k
EA
036)
Indeed, taking that into account, according to (B3), ai, =
max [min (a&,ak,)]
(B7)
we have 1 - dij = max [min (l-dlk, 1-dkj)] = max [l - max (dik, dkj)] 2 1 - max (dik, d k j ) (B8) Consequently, inequality B6 results. Moreover, note that due to (B6) all triangles in the space A are either equilateral or isosceles with a small base. All elements of the tree shown in Scheme I hold to inequality B6. Registry No. CH,OH, 67-56-1.
Literature Cited Anderson, J. R.; Foger, K.; Mole, T.; Rajadhyaksha, R. A.; Sanders, J. V. “Reactions on ZSM-5 Type Zeolite Catalysts”. J. Catal. 1979, 58, 114. Anthony, R. G. “A Kinetic Model for Methanol Conversion to Hydrocarbons”. Chem. Eng. Sci. 1981,36, 789. Anthony, R. G.; Singh, B. B. “Kinetic Analysis of Complex Reaction Systems-Methanol Conversion to Low Molecular Weight Olefins”. Chem. Eng. Commun. 1980, 6 , 215. Anthony, R. G.; Singh, B. B.; Vera, E. “Selectivity Data and Empirical Equations for Methanol Conversion to Low Molecular Weight Hydrocarbons”. AIChE Annual Spring Meeting, Houston, 1981. Aris, R. “Residence Time Distribution with Many Reactions and in Several Environments”. In Residence Time Distribution Theory in Chemical Engineering; Petho, A., Noble, R. D., Eds.; Verlag Chemie: Weinheim, 1982. Blackwell, D. “The Entropy of Functions of Finite-State Markov Chains”. In Transactions of the First Prague Conference on Information Theory, Statistics, Decision Functions, Random Processes; Publ. House, Czechoslovak Acad. Sci.: Prague, 1957; pp 13-20. Chang, C. D. “A Kinetic Model for Methanol Conversion to Hydrocarbons”. Chem. Eng. Sci. 1980,35, 619. Chang, C. D.; Silvestri, A. J. “The Conversion of Methanol and Other o-Compounds to Hydrocarbons over Zeolite Catalysts”. J . Catal. 1977, 47, 249. Chen, N. Y.; Reagan, W. J. “Evidence of Autocatalysis in Methanol to Hydrocarbons Reactions over Zeolite Catalysts”. J. Catal. 1979, 59, 123. Cobb, J. T.; Coon, V. T.; Tipnis, P. Final Report DOE, Contract E / W-78-S-02-4691, Pittsburgh University, Sept 1978. Coxson, P. G.; Bischoff, K. B. “Lumping Strategy. 1. Introductory Techniques and Applications of Cluster Analysis”. Znd. Eng. Chem. Res. 1987,26, 1239. ’IeLuca, A.; Termini, S. “A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory”. Inform. Control 1972,20, 301. Derouane, G. E.; Nagy, J. B.; Dejaifve, P.; Van Hooff, J. H.; Spekman, B. P.; VBdrine, C. J.; Naccache, C. “Elucidation of the Mechanism of Conversion of Methanol and Ethanol to Hydrocarbons on a New Type of Synthetic Zeolite”. J . Catal. 1978,53, 40. Ermakova, A.; Valko, P.; Umbetov, A. S. “The Examination Method of Complex Reaction Kinetic Model. I. The Determination of Model Parameters”. Hung. J . Znd. Chem. 1980,8, 197; “11. The Selection of Optimal Route Scheme”. Hung. J . Znd. Chem. 1980, 8, 205. Espinoza, R. L. “Oligomerization vs. Methylation of Propene in the Conversion of DME (or Methanol) to Hydrocarbons”. Ind. Eng. Chem. Product Res. Deu. 1984, 23, 449.
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Fan, L. T.; Shenoi, S.; Gharpuray, M. M.; Lai, F. S. "Chemical and Process System Engineering. Applications of Fuzzy Set Theory: Tutorial Note". Paper presented a t the Summer School on Advances in Chemical Engineering Mathematics, Hannover, West Germany, 1985. Frenklach, M. "Computer Model of Infinite Reaction Sequences: a Chemical Lumping". Chem. Eng. Sci. 1986, 40, 1843. Haag, W. 0.;Lago, R. M.; Rodewald, P. G . "Aromatic Light Olefins and Mechanistic Pathways with ZSM-5 Zeolite Catalyst". J . Molec. Catal. 1982, 17, 161. Hoerl, A. E.; Kennard, R. W. "Ridge Regression: Biased Estimation for Nonorthogonal Problems". Technomet 1970, 12, 55. Hoffmann, U.; Hofman, H. "Reaction Engineering. 8. Kinetics of Multicomponent Multireaction Systems and Simplification of Their Description". Znt. Chem. Eng. 1977, 17, 414. Hosten, L. H.; Froment, G. F. "Kinetic Modelling of Complex Reactions". In Recent Advances in the Engineering Analysis of Chemically Reacting Systems; Doraiswamy, L. K., Ed.; Wiley: New York, 1984. Iordache, 0. "Polystochastic Models in Chemical Engineering"; VNU-Science Press: Utrecht, Netherlands, 1987. Corbu, S. "A Stochastic Model of Lumping". Chem. Iordache, 0.; Eng. Sci. 1987, 42, 125. Jardine, N.; Sibson, K. Mathematical Taxonomy; Wiley: New York, 1971. Li, G. "A Lumping Analysis in Mono- or/and Bimolecular Reaction Systems". Chem. Eng. Sci. 1984, 29, 1261. Liu, L.; Tobias, R. G.; McLaughlin, K.; Anthony, R. G. "Conversion of Methanol to Low-Molecular-Weight Olefins with Heterogeneous Catalysts". In Catal. Conuers. Synth. Gas. Alcohols Chem. Herman, R. G., Ed.; Plenum: New York, 1984. Luss, D.; Golikeri, S. V. "Grouping of Many Species Each Consumed by Two Parallel First-Order Reactions". AZChE J. 1975,21,865. Maria, G.; Muntean, 0. 'Model Reduction and Kinetic Parameters Indentification for the Methanol Conversion to Olefins". Chem. Eng. Sci. 1987, 42, 1451. Mihail, R.; Straja, S.; Maria, G.; Musca, G.; Pop, G. "Kinetic Model
for Methanol Conversion to Olefins". Znd. Eng. Chem. Process Des. Dev. 1983, 22, 532. Mihail, R.; Straja, S.; Maria, G.; Musca, G.; Pop, G . "Reply to Comments on "Kinetic Model for Methanol Conversion to Olefins" with Respect to Methane Formation at Low Conversion". Znd. Eng. Chem. Res. 1987,26,637. Mole, T.; Whiteside, J. A. "Conversion of Methanol to Ethylene over ZSM-5 Zeolite in the Presence of Deuterated Water". J. Catal. 1982, 75, 284. NegoitH, C. V.; Ralescu, D. A. "Applications of Fuzzy Sets t o Systems Analysis"; Birkhauser: Basel, 1975. Ono, Y.; Mori, T. "Mechanism of Methanol Conversion into Hydrocarbons over ZSM-5 Zeolite". J . Chem. SOC., Faraday Trans. 1 1981, 77, 2209. Peereboom, M. "Approximate Lumping Applied to the Isomerization of Methylcyclohexenes". Znd. Eng. Chem. Res. 1987, 26, 1663. Perot, G.; Carmerais, F. X.; Guisnet, M. "Carbon-13 Tracer Study of the Conversion of Dimethyl Ether into Hydrocarbons on Silica-Alumina and HZSM-5 Zeolite". J.Molec. Catal. 1982,17,255. Too, J. R.; Nassar, R.; Fan, L. T. "Simulation of the Performances of a Flow Chemical Reactor by Markov Chains". In Residence Time Distribution Theory in Chemical Engineering; Verlag Chemie: Weinheim, 1982. Van den Berg, J. P.; Wolthuizen, J. P.; Van Hooff, J. H. C. "The Conversion of Dimethyl Ether to Hydrocarbons on Zeolite HZSM-5-The Reaction Mechanism for Formation of Primary Olefins". In Proc. 5th Znt. Conf. on Zeolites; Rees, L. V. C., Ed.; Heyden: London, 1980; pp 649-660. Wei, J.; Kuo, J. C. W. "A Lumping Analysis in Monomolecular Reaction Systems". Znd. Eng. Chem. Fundam. 1969,8, 114. Zhu, K.; Chen, M.; Yan, W. "An Engineering Model of a Network of Multicomponent, Reversible Reactions". Znt. Chem. Eng. 1985, 25, 542. Received for reuiew March 1, 1988 Revised manuscript received July 18, 1988 Accepted July 30, 1988
Inhibition by Product in the Liquid-Phase Hydration of Isobutene to tert -Butyl Alcohol: Kinetics and Equilibrium Studies Enric Velo, Luis Puigjaner, and Francesc Recasens* Department of Chemical Engineering, ETS Enginyers Industrials de Barcelona, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, S p a i n
Intrinsic rates of isobutene hydration t o tert-butyl alcohol on Amberlyst-15 particles were measured to establish a rate equation in a solvent-free, liquid-phase system. The ranges of temperature and concentration are those likely to be found in a multiphase reactor, such as trickle bed or slurry type. Although inhibition by water may also be present, the effect of TBA is far more significant. Thus, the alcohol is found to inhibit the rate more than expected from the values of the equilibrium constant for the hydration reaction, determined in separate experiments. Various rate expressions, including that derived from the accepted hydration mechanism, are tested t o account for product inhibition. Extreme care has been put in ascertaining the effects of internal and external mass-transfer resistances, by use of suitable derived criteria. Introduction tert-Butyl alcohol ( T B A ) is an important oxygenated octane enhancer that is used to replace toxic lead additives in gasoline (O'Sullivan, 1985). For other oxygenates, such as methyl tert-butyl ether, the technology is already well established. For TBA, however, processes have been filed (Franz et al., 1975; Matsuzawa et al., 1973; Moy and Rakow, 1976), but only a few seem to be in commercial operation (O'Sullivan, 1985; Huls, 1983). A high-yield, catalytic route to TBA is based on the direct hydration of isobutene (iB) contained in refinery C4 streams. The synthesis reaction 0888-5885/88/2627-2224$01.50/0
CH3
H3C\ ,C=CHz
4-
H20
e CH3-C-OHI
H3C
(1)
I
CH3
catalyzed by acid, is exothermic and reversible at low temperatures (50-90 "C) and highly selective toward the desired product. Reaction 1 involves a multiphase mixture of a hydrocarbon, an aqueous phase, and a solid catalyst. Successful processes depend on how the problem of the limited miscibility of the components is overcome. Existing pro-
0 1988 American Chemical Society
